HABILITATIONSSCHRIFT

Gravity in 2D and 3DA romance of lower dimensions

Presented to the Faculty of PhysicsVienna University of Technology

By

Daniel Grumiller

Institute for Theoretical Physics

E-mail: grumil@hep.itp.tuwien.ac.at

Vienna, February 24, 2010

Abstract

Gravity in lower dimensions provides a useful expedient for testing ideasabout quantum gravity in higher dimensions. Technical simplifications inlower dimensions often lead to exact results, and this helps to address someof the conceptual problems posed by classical and quantum gravity. At thesame time, these simplifications could remove some of the features that makegravity interesting. Striking a balance between models that are tractable andmodels that seem relevant is an art in its own right.

Two is the lowest dimension that admits gravity models containing blackhole solutions. In particular, 2D dilaton gravity exhibits a wide range of in-teresting phenomena, for instance a rich structure of thermodynamic observ-ables like black hole entropy or free energy. Conceptual issues of holographicrenormalization with arbitrary spacetime asymptotics can be resolved com-paratively easily. 2D gravity thus provides valuable insights for correspondinghigher-dimensional theories of gravity.

Three is the lowest dimension that admits gravity models containing blackhole solutions and gravitons. In particular, Cosmological Topologically Mas-sive Gravity exhibits a wide range of interesting features: it contains differentblack hole solutions, asymptotically Anti-deSitter solutions as well as solu-tions with different asymptotics and massive graviton excitations. In the pastthree years 3D gravity has been studied vigorously, and my recent researchresults have contributed significantly to this field. One of the main goals isto obtain a useful and soluble model of quantum gravity. Another pertinentgoal is to apply the gauge/gravity duality to 3D gravity in order to describecertain condensed matter systems. Both is work in progress.

In this Habilitationsschrift I collect a selection of my papers on 2D and3D gravity and provide a brief guideline to these papers as well as to currentresearch of my START group at the Vienna University of Technology.

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Acknowledgments

I am grateful to my collaborators, colleagues and students for scores ofhours of discussions. I thank in particular my co-authors on the papersthat constitute this Habilitationsschrift: Wolfgang Kummer for numerousvaluable advises during the early stages of my career and for being such agood person; Dima Vassilevich for an enjoyable long-term collaborationon 2D gravity and the hospitality in Leipzig; Luzi Bergamin for being sogood-natured, reliable and fun to work with; Rene Meyer for being suchan excellent, agreeable and successful student; Roman Jackiw for morethan two fruitful and pleasurable years of collaboration at the MassachusettsInstitute of Technology; Robert McNees for his excellent introduction toholographic renormalization and his hospitality at the Perimeter Institute;Niklas Johansson for being extremely brilliant and pleasant at the sametime, and for being my first post-doc; Peter van Nieuwenhuizen for in-venting and explaining supergravity, as well as for all the wonderful stories;Ivo Sachs for his permanently good spirit in the face of difficult tasks; andOlaf Hohm for inventing and explaining new massive gravity. Moreover,I thank Toni Rebhan, head of the Fundamental Interactions group at theInstitute for Theoretical Physics of the Vienna University of Technology, fordoing the right things at the right time in the right way. Finally, I thankLaurenz Widhalm, not just for splendid collaboration on outreach issuesat teilchen.at and in schools, but also for his friendship and for introducingus to the world of geocaching.

Privately I would like to thank so many people — family and friends— that this acknowledgment would grow without bound. Thus, I’ll restrictmyself to three highlights in print and let everyone else know my gratefulnessin private. Wiltraud. What can I say? Life without you wouldn’t be my life,and I am happy that you took all the moving within Europe and betweenEurope and the US with good humour. Moving back to Vienna was thehardest part, but we did it! Laurin and Armin. You are the best boys afather can wish for: curious, cheeky, playful and reliable when it counts — itwas phantastic to live with you in Leipzig, in Boston and it is phantastic tolive with you in Vienna. In addition I am grateful for each and every privatevisit during the past six years, so to all our family members and friends whovisited us in Leipzig or Boston, who kept contact with us, who helped usmoving, who brought us poppy seeds or Topfen, who gave me an excellentexcuse for serious cooking, who made trips with us within Leipzig or to NewYork or to the Blue Hills Reservoir or in Burgenland or in Zeutschach, whohelped us survive the first 1.5 years back in Vienna: thank you. Thanks.

This Habilitationsschrift was supported by the START project Y435-N16of the Austrian Science Foundation (FWF).

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ToThe Inhabitants of SPACE IN GENERAL

And H. C. IN PARTICULARThis Work is Dedicated

By a Humble Native of FlatlandIn the Hope that

Even as he was Initiated into the MysteriesOf THREE Dimensions

Having been previously conversantWith ONLY TWO

So the Citizens of that Celestial RegionMay aspire yet higher and higher

To the Secrets of FOUR FIVE OR EVEN SIX DimensionsThereby contributing

To the Enlargement of THE IMAGINATIONAnd the possible Development

Of that most rare and excellent Gift of MODESTYAmong the Superior RacesOf SOLID HUMANITY

Edwin A. Abbot alias A. Square, “Flatland — A romance of manydimensions” (1884)

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Contents

I Guide to the Papers 1

1 Gravity in two dimensions (2002-2007) 3

2 Gravity in three dimensions (2008-2010) 7

Bibliography 11

II Papers on 2D Gravity 13

1 Dilaton gravity in two dimensions (2002) 15

2 The classical solutions of the dimensionally reduced gravita-tional Chern-Simons theory (2003) 159

3 Supersymmetric black holes are extremal and bald in 2Ddilaton supergravity (2004) 171

4 Virtual black holes and the S-matrix (2004) 201

5 An action for the exact string black hole (2005) 229

6 Ramifications of lineland (2006) 271

7 Duality in 2-dimensional dilaton gravity (2006) 317

8 Thermodynamics of black holes in two (and higher) dimen-sions (2007) 331

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III Papers on 3D Gravity 395

1 Instability in cosmological topologically massive gravity atthe chiral point (2008) 397

2 Canonical analysis of cosmological topologically massive grav-ity at the chiral point (2008) 417

3 Consistent boundary conditions for cosmological topologi-cally massive gravity at the chiral point (2008) 429

4 Holographic counterterms from local supersymmetry with-out boundary conditions (2009) 435

5 AdS3/LCFT2 – Correlators in Cosmological Topologically Mas-sive Gravity (2009) 445

6 AdS3/LCFT2 – Correlators in New Massive Gravity (2009) 509

7 Gravity duals for logarithmic conformal field theories (2010)519

CV 535

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Part I

Guide to the Papers

1

Chapter 1

Gravity in two dimensions(2002-2007)

Two’s company; three’s a crowd

Proverb

Part II of this Habilitationsschrift contains eight papers on 2D dilatongravity [1–8]. The theory considered in most papers is 2D dilaton gravity,the action of which is given by

S2D = −

∫

d2x√

|g|[

XR−U(X)(∇X)2 −2V (X)]

+boundary terms (1.1)

where g is a 2D metric, X the dilaton field and U, V arbitrary functionsthereof. Here is a brief description and list of abstracts of these papers.

Ref. [1] is an invited review article on 2D dilaton gravity. It sum-marizes my earliest research achievements until 2002 and provides a compre-hensive overview on 2D dilaton gravity. Here is its abstract:The study of general two dimensional models of gravity allows to tackle ba-sic questions of quantum gravity, bypassing important technical complicationswhich make the treatment in higher dimensions difficult. As the physicallyimportant examples of spherically symmetric Black Holes, together with stringinspired models, belong to this class, valuable knowledge can also be gained forthese systems in the quantum case. In the last decade new insights regardingthe exact quantization of the geometric part of such theories have been ob-tained. They allow a systematic quantum field theoretical treatment, also ininteractions with matter, without explicit introduction of a specific classicalbackground geometry. The present review tries to assemble these results in a

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coherent manner, putting them at the same time into the perspective of thequite large literature on this subject.

Ref. [2] constructs all classical solutions globally of a specific 3Dtheory reduced to 2D. This paper led to several invitations and to closercontact with the Massachusetts Institute of Technology, in particular withAlfredo Iorio, Roman Jackiw and Carlos Nunez. Here is its abstract:The Kaluza-Klein reduction of the 3d gravitational Chern-Simons term to a2d theory is equivalent to a Poisson-sigma model with fourdimensional targetspace and degenerate Poisson tensor of rank 2. Thus two constants of motion(Casimir functions) exist, namely charge and energy. The application ofwell-known methods developed in the framework of first order gravity allowsto construct all classical solutions straightforwardly and to discuss their globalstructure. For a certain fine tuning of the values of the constants of motionthe solutions of hep-th/0305117 are reproduced. Possible generalizations arepointed out.

Ref. [3] discusses all supersymmetric solutions of 2D dilaton su-pergravity. This paper considerably extends the discussion on 2D dilatonsupergravity as compared to the review [1]. Here is its abstract:We present a systematic discussion of supersymmetric solutions of 2D dilatonsupergravity. In particular those solutions which retain at least half of the su-persymmetries are ground states with respect to the bosonic Casimir function(essentially the ADM mass). Nevertheless, by tuning the prepotential appro-priately, black hole solutions may emerge with an arbitrary number of Killinghorizons. The absence of dilatino and gravitino hair is proven. Moreover, theimpossibility of supersymmetric dS ground states and of nonextremal blackholes is confirmed, even in the presence of a dilaton. In these derivations theknowledge of the general analytic solution of 2D dilaton supergravity plays animportant role. The latter result is addressed in the more general context ofgPSMs which have no supergravity interpretation. Finally it is demonstratedthat the inclusion of non-minimally coupled matter, a step which is alreadynontrivial by itself, does not change these features in an essential way.

Ref. [4] is an invited review article on virtual black holes. Here isits abstract:A brief review on virtual black holes is presented, with special emphasis onphenomenologically relevant issues like their influence on scattering or onthe specific heat of (real) black holes. Regarding theoretical topics results im-portant for (avoidance of) information loss are summarized. After recalling

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Hawking’s Euklidean notion of virtual black holes and a Minkowskian notionwhich emerged in studies of 2D models, the importance of virtual black holesfor scattering experiments is addressed. Among the key features is that vir-tual black holes tend to regularize divergences of quantum field theory and thata unitary S-matrix may be constructed. Also the thermodynamical behaviorof real evaporating black holes may be ameliorated by interactions with vir-tual black holes. Open experimental and theoretical challenges are mentionedbriefly.

Ref. [5] constructed for the first time a local action for a specificfamily of string theoretic black holes. This paper circumvented anearlier no-go result by relaxing one of its premises and led to numerous in-vitations as well as to the collaboration with Robert McNees. Here is itsabstract:A local action is constructed describing the exact string black hole discoveredby Dijkgraaf, Verlinde and Verlinde in 1992. It turns out to be a special 2DMaxwell-dilaton gravity theory, linear in curvature and field strength. Twoconstants of motion exist: mass M ≥ 1, determined by the level k, and U(1)-charge Q ≥ 0, determined by the value of the dilaton at the origin. ADMmass, Hawking temperature TH ∝

√

1 − 1/M and Bekenstein–Hawking en-tropy are derived and studied in detail. Winding/momentum mode dualityimplies the existence of a similar action, arising from a branch ambiguity,which describes the exact string naked singularity. In the strong couplinglimit the solution dual to AdS2 is found to be the 5D Schwarzschild blackhole. Some applications to black hole thermodynamics and 2D string theoryare discussed and generalizations — supersymmetric extension, coupling tomatter and critical collapse, quantization — are pointed out.

Ref. [6] is another invited review on 2D dilaton gravity. Its focusis particularly on the more recent developments between 2002-2006. Thispaper was written with my undergraduate student Rene Meyer in Leipzig.Here is its abstract:A non-technical overview on gravity in two dimensions is provided. Appli-cations discussed in this work comprise 2D type 0A/0B string theory, BlackHole evaporation/thermodynamics, toy models for quantum gravity, for nu-merical General Relativity in the context of critical collapse and for solidstate analogues of Black Holes. Mathematical relations to integrable models,non-linear gauge theories, Poisson-sigma models, KdV surfaces and non-commutative geometry are presented.

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Ref. [7] discovers a new duality in 2D dilaton gravity that hadbeen overlooked so far. This paper emerged as an unexpected spin-offof a longer work with Roman Jackiw. Here is its abstract:We descry and discuss a duality in 2-dimensional dilaton gravity.

Ref. [8] provides a first principle derivation of all boundary termsin 2D dilaton gravity. In particular, the holographic counterterms wereconstructed and exploited to derive the free energy of a large class of blackholes, as well as other thermodynamic properties. Here is its abstract:A comprehensive treatment of black hole thermodynamics in two-dimensionaldilaton gravity is presented. We derive an improved action for these theoriesand construct the Euclidean path integral. An essentially unique boundarycounterterm renders the improved action finite on-shell, and its variationalproperties guarantee that the path integral has a well-defined semi-classicallimit. We give a detailed discussion of the canonical ensemble described bythe Euclidean partition function, and examine various issues related to sta-bility. Numerous examples are provided, including black hole backgroundsthat appear in two dimensional solutions of string theory. We show that theExact String Black Hole is one of the rare cases that admits a consistentthermodynamics without the need for an external thermal reservoir. Our ap-proach can also be applied to certain higher-dimensional black holes, such asSchwarzschild-AdS, Reissner-Nordstrom, and BTZ.

Outlook on 2D gravity While currently my research focus is mostly on3D gravity or higher-dimensional gravity, I still keep working in this fieldwhere I am one of the leading experts world-wide. I pursue this topic incollaboration with students at the Vienna University of Technology, andwith my collaborators Luzi Bergamin, Roman Jackiw, Robert McNees, ReneMeyer, Dimitri Vassilevich and others.

Main goal of this line of research: thorough understanding of classi-cal, semi-classical and quantum gravitational effects and black holeproperties. Since many higher-dimensional black holes can be describedby 2D dilaton gravity (Schwarzschild, Reissner-Nordstrom, BTZ, etc.) resultsobtained in two dimensions can provide valuable insights also for higher di-mensions, where the “holy grail” — a consistent and comprehensive quantumtheory of gravity — is still out of reach at present.

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Chapter 2

Gravity in three dimensions(2008-2010)

Professor, you should be commendedOn your theory so geniusly splendid.But some say it’s luck,And you really just suck,’Cause your theory’s not what you intended!

Physics Limericks (Harvard University),http://www.physics.harvard.edu/academics/undergrad/limericks.html

Part III of this Habilitationsschrift contains seven papers on 3D gravity [9–15]. The theory considered in most papers below is cosmological topologicallymassive gravity, whose action (up to boundary terms) is given by

S3D =1

16πG

∫

d3x√

|g|[

R+2

ℓ2+

1

2µεαβγΓρ

ασ

(

∂βΓσγρ+

2

3Γσ

βτΓτγρ

)

]

(2.1)

where G is Newton’s constant, g is a 3D metric, ℓ is the AdS radius and µthe Chern–Simons coupling. Here is a brief description and list of abstractsof these papers.

Ref. [9] discovered graviton excitations in a specific 3D gravitytheory that were thought to be absent by others. Moreover, it con-jectured a logarithmic conformal field theory as gauge theory dual. Thispaper is my best-known recent work, and most of the papers below are basedon it. It was written with the graduate student Niklas Johansson, who isnow my post-doc in Vienna. Here is its abstract:We consider cosmological topologically massive gravity at the chiral point with

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positive sign of the Einstein–Hilbert term. We demonstrate the presence ofa negative energy bulk mode that grows linearly in time. Unless there arephysical reasons to discard this mode, this theory is unstable. To addressthis issue we prove that the mode is not pure gauge and that its negativeenergy is time-independent and finite. The isometry generators L0 and L0

have non-unitary matrix representations like in logarithmic CFT. While thenew mode obeys boundary conditions that are slightly weaker than the onesby Brown and Henneaux, its fall-off behavior is compatible with spacetimebeing asymptotically AdS3. We employ holographic renormalization to showthat the variational principle is well-defined. The corresponding Brown–Yorkstress tensor is finite, traceless and conserved. Finally we address possibilitiesto eliminate the instability and prospects for chiral gravity.

Ref. [10] showed that the graviton mode discovered above persistsnon-perturbatively. In addition we provided a novel reformulation of cos-mological topologically massive gravity in this work. Here is its abstract:Wolfgang Kummer was a pioneer of two-dimensional gravity and a strongadvocate of the first order formulation in terms of Cartan variables. In thepresent work we apply Wolfgang Kummer’s philosophy, the “Vienna Schoolapproach”, to a specific three-dimensional model of gravity, cosmological topo-logically massive gravity at the chiral point. Exploiting a new Chern–Simonsrepresentation we perform a canonical analysis. The dimension of the phys-ical phase space is two per point, and thus the theory exhibits a local physicaldegree of freedom, the topologically massive graviton.

Ref. [11] showed that there are boundary conditions consistent withasymptotic AdS behavior that encompass the graviton excitationsdiscovered above. These boundary conditions relax the famous Brown–Henneaux boundary conditions, but the asymptotic symmetry algebra con-sists of two copies of the Virasoro algebra, just like in 3D Einstein gravitywith Brown–Henneaux boundary conditions. Here is its abstract:We show that cosmological topologically massive gravity at the chiral point al-lows not only Brown–Henneaux boundary conditions as consistent boundaryconditions, but slightly more general ones which encompass the logarithmicprimary found in 0805.2610 as well as all its descendants.

Ref. [12] provided a novel way to derive holographic counterterms,namely from supersymmetry. This paper was written together withthe co-inventor of supergravity and provides the basis of current collabora-tion with Anton Rebhan and Peter van Nieuwenhuizen. Here is its abstract:

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We show in some lower-dimensional supergravity models that the holographiccounterterms which are needed in the AdS/CFT correspondence to make thetheory finite, coincide with the counterterms that are needed to make the ac-tion supersymmetric without imposing any boundary conditions on the fields.

Ref. [13] corroborates the logarithmic CFT conjecture made aboveby calculating 2- and 3-point correlators. Besides these slightly lengthycalculations the paper contains also a lot of material that will be useful for fu-ture purposes, like the construction of all normalizable and non-normalizablemodes in cosmological topologically massive gravity on an AdS background.Here is its abstract:For cosmological topologically massive gravity at the chiral point we calcu-late momentum space 2- and 3-point correlators of operators in the postu-lated dual CFT on the cylinder. These operators are sourced by the bulk andboundary gravitons. Our correlators are fully consistent with the proposalthat cosmological topologically massive gravity at the chiral point is dual toa logarithmic CFT. In the process we give a complete classification of nor-malizable and non-normalizeable left, right and logarithmic solutions to thelinearized equations of motion in global AdS3.

Ref. [14] shows that also a related 3D gravity theory, New MassiveGravity, can be dual to a logarithmic CFT for a certain tuningof parameters. This paper was written with one of the inventors of NewMassive Gravity, Olaf Hohm. Here is its abstract:We calculate 2-point correlators for New Massive Gravity at the chiral pointand find that they behave precisely as those of a logarithmic conformal fieldtheory, which is characterized in addition to the central charges cL = cR = 0by “new anomalies” bL = bR = −σ 12ℓ

GN

, where σ is the sign of the Einstein–Hilbert term, ℓ the AdS radius and GN Newton’s constant.

Ref. [15] is an invited proceedings contribution. It summarizes therecent evidence that led to the proposals for specific gravity duals of logarith-mic CFTs and provides an outlook towards condensed matter applications.Here is its abstract:Logarithmic conformal field theories with vanishing central charge describesystems with quenched disorder, percolation or dilute self-avoiding polymers.In these theories the energy momentum tensor acquires a logarithmic part-ner. In this talk we address the construction of possible gravity duals forthese logarithmic conformal field theories and present two viable candidatesfor such duals, namely theories of massive gravity in three dimensions at achiral point.

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Outlook on 3D gravity Currently I am working very focussed on 3D grav-ity in collaboration with undergraduate students, my PhD student SabineErtl, my post-doc Niklas Johansson, my visitors Branislav Cvetkovic, OlafHohm, Roman Jackiw, Ivo Sachs and Dima Vassilevich, as well as with ad-ditional international collaborators. In addition to support from my STARTproject Y435-N16 my research group on 3D gravity is supported by the FWFproject P21927-N16.

Main goal of this line of research: constructing a consistent 3Dtheory of quantum gravity and applying the AdS/CFT correspon-dence to provide novel gravity duals for certain 2D conformal fieldtheories. The elusive theory of quantum gravity is often called “the holygrail of theoretical physics”. Of all the attempts to quantize gravity, stringtheory is the best developed theory, but it is still poorly understood andpoorly supported by experiment. Thus, it is prudent to consider simplerapproaches to quantum gravity, where results can be expected on a rea-sonable time-scale. Lower-dimensional models of gravity provide such anapproach. In the past 20 years studies of 2-dimensional gravity led to nu-merous exciting results for classical and quantum black holes. However, thesimplicity of these black hole models eliminates two important features ofhigher-dimensional gravity: they do not contain gravitons, and they lack agood analog for the horizon area of black holes. Both these deficiencies canbe remedied by considering a suitable model of gravity in three dimensions,like cosmological topologically massive gravity. Even if this ambitious goalmight be out of reach, the study of 3D gravity can lead to useful applicationsin unexpected fields, such as condensed matter physics: the AdS/CFT cor-respondence relates gravity theories on AdS to conformal field theories thatlive on the boundary of AdS. Currently we continue to unravel novel featuresof purported CFT duals to cosmological topologically massive gravity as wellas possible condensed matter physics applications, e.g. in the description ofsystems with quenched disorder.

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Bibliography

Selected literature on 2D gravity

[1] D. Grumiller, W. Kummer, and D. V. Vassilevich, “Dilaton gravity intwo dimensions,” Phys. Rept. 369 (2002) 327–429, hep-th/0204253;invited review article. p. 15 ff.

[2] D. Grumiller and W. Kummer, “The classical solutions of thedimensionally reduced gravitational Chern-Simons theory,” AnnalsPhys. 308 (2003) 211 hep-th/0306036. p. 159 ff.

[3] L. Bergamin, D. Grumiller and W. Kummer, “Supersymmetric blackholes are extremal and bald in 2-D dilaton supergravity,” J. Phys. A37(2004) 3881 hep-th/0310006. p. 171 ff.

[4] D. Grumiller, “Virtual black holes and the S-matrix,” Int. J. Mod.Phys. D13 (2004) 1973 hep-th/0409231; invited review article. p. 201ff.

[5] D. Grumiller, “An action for the exact string black hole,” JHEP 0505(2005) 028 hep-th/0501208. p. 229 ff.

[6] D. Grumiller and R. Meyer, “Ramifications of lineland,” Turk. J. Phys.30 (2006) 349–378, hep-th/0604049; invited review article. p. 271 ff.

[7] D. Grumiller and R. Jackiw, “Duality in 2-dimensional dilatongravity,” Phys. Lett. B642 (2006) 530–534, hep-th/0609197. p. 317 ff.

[8] D. Grumiller and R. McNees, “Thermodynamics of black holes in two(and higher) dimensions,” JHEP 04 (2007) 074, hep-th/0703230.p. 331 ff.

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Selected literature on 3D gravity

[9] D. Grumiller and N. Johansson, “Instability in cosmologicaltopologically massive gravity at the chiral point,” JHEP 07 (2008)134; arXiv:0805.2610 [hep-th]. p. 397 ff.

[10] D. Grumiller, R. Jackiw and N. Johansson, “Canonical analysis ofcosmological topologically massive gravity at the chiral point,”contribution to Wolfgang Kummer Memorial Volume, World Scientific;arXiv:0806.4185 [hep-th]. p. 417 ff.

[11] D. Grumiller and N. Johansson, “Consistent boundary conditions forcosmological topologically massive gravity at the chiral point,” Int. J.Mod. Phys. D17 (2009) 2367; arXiv:0808.2575 [hep-th]. p. 429 ff.

[12] D. Grumiller and P. van Nieuwenhuizen, “Holographic countertermsfrom local supersymmetry without boundary conditions,” Phys. Lett.B682 (2010) 462; arXiv:0908.3486 [hep-th]. p. 435 ff.

[13] D. Grumiller and I. Sachs, “AdS3/LCFT2 – Correlators inCosmological Topologically Massive Gravity,” arXiv:0910.5241

[hep-th], to be published in JHEP (2010). p. 445 ff.

[14] D. Grumiller and O. Hohm, “AdS3/LCFT2 – Correlators in NewMassive Gravity,” arXiv:0911.4274 [hep-th], submitted to Phys.Lett. B (2010). p. 509 ff.

[15] D. Grumiller and N. Johansson, “Gravity duals for logarithmicconformal field theories,” arXiv:1001.0002 [hep-th], submitted toJ. Phys. Conf. Ser. (2010). p. 519 ff.

Comments on the papers constituting this Habilitationsschrift:

• In all publications above I have contributed as main author.

• Papers [1–5,7–9,11–12] are published in peer reviewed internationaljournals. Paper [6] is published as a peer reviewed proceedingscontribution. Paper [10] published as a book contribution. Paper [13]is accepted for publication in JHEP. Paper [14] was submitted forpublication to PLB. Paper [15] is an invited proceedings contribution.

• Page numbers at the end of each reference direct to thestarting page of the corresponding paper in thisHabilitationsschrift.

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Part II

Papers on 2D Gravity

Dilaton Gravity in Two Dimensions

D. Grumiller a,W. Kummer a,D.V. Vassilevich b,c

aInstitut fur Theoretische Physik, TU Wien, Wiedner Hauptstr. 8–10, A-1040Wien, Austria

bInstitut fur Theoretische Physik, Universitat Leipzig, Augustusplatz 10, D-04109Leipzig, Germany

cV.A. Fock Insitute of Physics, St. Petersburg University, 198904 St. Petersburg,Russia

Abstract

The study of general two dimensional models of gravity allows to tackle basic ques-tions of quantum gravity, bypassing important technical complications which makethe treatment in higher dimensions difficult. As the physically important examplesof spherically symmetric Black Holes, together with string inspired models, belongto this class, valuable knowledge can also be gained for these systems in the quan-tum case. In the last decade new insights regarding the exact quantization of thegeometric part of such theories have been obtained. They allow a systematic quan-tum field theoretical treatment, also in interactions with matter, without explicitintroduction of a specific classical background geometry. The present review triesto assemble these results in a coherent manner, putting them at the same time intothe perspective of the quite large literature on this subject.

Key words: dilaton gravity, quantum gravity, black holes, two dimensional modelsPACS: 04.60.-w, 04.60.Ds, 04.60.Gw, 04.60.Kz, 04.70.-s, 04.70.Bw, 04.70.Dy,11.10.Lm, 97.60.Lf

Email addresses: grumil@hep.itp.tuwien.ac.at (D. Grumiller),wkummer@tph.tuwien.ac.at (W. Kummer), vassil@itp.uni-leipzig.de (D.V.Vassilevich).

Preprint submitted to Elsevier Science 1 February 2010

Contents

1 Introduction 18

1.1 Structure of this review . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2 Differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.2.1 Short primer for general dimensions . . . . . . . . . . . . . . 24

1.2.2 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Models in 1 + 1 Dimensions 33

2.1 Generalized Dilaton Theories . . . . . . . . . . . . . . . . . . . . . . 33

2.1.1 Spherically reduced gravity . . . . . . . . . . . . . . . . . . . 33

2.1.2 Dilaton gravity from strings . . . . . . . . . . . . . . . . . . . 34

2.1.3 Generalized dilaton theories – the action . . . . . . . . . . . . 35

2.1.4 Conformally related theories . . . . . . . . . . . . . . . . . . 37

2.2 Equivalence to first-order formalism . . . . . . . . . . . . . . . . . . 39

2.3 Relation to Poisson-Sigma models . . . . . . . . . . . . . . . . . . . 42

3 General classical treatment 46

3.1 All classical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Global structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2.1 Schwarzschild metric . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.2 More general cases . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Black hole in Minkowski, Rindler or de Sitter space . . . . . . . . . . 59

4 Additional fields 63

4.1 Dilaton-Yang-Mills Theory . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Dilaton Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3 Dilaton gravity with matter . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.1 Scalar and fermionic matter, quintessence . . . . . . . . . . . 71

4.3.2 Exact solutions – conservation law for geometry and matter . 72

5 Energy considerations 76

5.1 ADM mass and quasilocal energy . . . . . . . . . . . . . . . . . . . . 76

5.2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6 Hawking radiation 84

6.1 Minimally coupled scalars . . . . . . . . . . . . . . . . . . . . . . . . 84

6.2 Non-minimally coupled scalars . . . . . . . . . . . . . . . . . . . . . 89

7 Nonperturbative path integral quantization 94

7.1 Constraint algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.2 Path integral quantization . . . . . . . . . . . . . . . . . . . . . . . . 99

7.3 Path integral without matter . . . . . . . . . . . . . . . . . . . . . . 101

7.4 Path integral with matter . . . . . . . . . . . . . . . . . . . . . . . . 105

7.4.1 General formalism . . . . . . . . . . . . . . . . . . . . . . . . 105

7.4.2 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 105

7.4.3 Exact path integral with matter . . . . . . . . . . . . . . . . 107

16

8 Virtual black hole and S-Matrix 109

8.1 Non-minimal coupling, spherically reduced gravity . . . . . . . . . . 1108.2 Effective line element . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128.3 Virtual black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138.4 Non-local φ4 vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . 1148.5 Scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 1158.6 Implications for the information paradox . . . . . . . . . . . . . . . . 117

9 Canonical quantization 119

10 Conclusions and discussion 122

Acknowledgement 125

A Spherical reduction of the curvature 2-form 126

B Heat kernel expansion 127

References 131

List of Figures

2.1 A selection of dilaton theories . . . . . . . . . . . . . . . . . . 423.1 Killing norm for Schwarzschild metric . . . . . . . . . . . . . . 553.2 Derivative of the second null direction . . . . . . . . . . . . . . 553.3 Second null direction . . . . . . . . . . . . . . . . . . . . . . . 553.4 Conformal coordinates with “compression factor” . . . . . . . 563.5 Reorientation of Fig. 3.4: patch A . . . . . . . . . . . . . . . . 563.6 Mirror image of Fig. 3.5: patch B . . . . . . . . . . . . . . . . 573.7 Further flips: patches C and D . . . . . . . . . . . . . . . . . . 573.8 CP diagram for the Schwarzschild solution . . . . . . . . . . . 573.9 Basic patch of Reissner-Nordstrom metric . . . . . . . . . . . 583.10 Penrose diagram for Reissner-Nordstrom metric . . . . . . . . 583.11 A possible RN-kink . . . . . . . . . . . . . . . . . . . . . . . . 593.12 “Phase” diagram of CP diagrams . . . . . . . . . . . . . . . . 618.1 CP diagram of the VBH . . . . . . . . . . . . . . . . . . . . . 1138.2 Total V (4)-vertex with outer legs . . . . . . . . . . . . . . . . . 1148.3 Kinematic plot of s-wave cross-section dσ/dα . . . . . . . . . 116

17

1 INTRODUCTION

1 Introduction

The fundamental difficulties encountered in the numerous attempts tomerge quantum theory with General Relativity by now are well-known evenfar outside the narrow circle of specialists in these fields. Despite many valiantefforts and new approaches like loop quantum gravity [371] or string theory 1

a final solution is not in sight. However, even many special questions searchan answer 2 .

Of course, at energies which will be accessible experimentally in the fore-seeable future, due to the smallness of Newton’s constant, respectively thelarge value of the Planck mass, an effective quantum theory of gravity can beconstructed [129] in a standard way which in its infrared asymptotical regimeas an effective quantum theory may well describe our low energy world. Itsextremely small corrections to classical General Relativity (GR) are in fullagreement with experimental limits [436]. However, the fact that Newton’sconstant carries a dimension, inevitably makes perturbative quantum gravityinconsistent at energies of the order of the Planck mass.

In a more technical language, starting from a fixed classical background,already a long time ago perturbation theory has shown that although puregravity is one-loop renormalizable [404] this renormalizability breaks downat two loops [188], but already at one-loop when matter interactions aretaken into account. Supergravity was only able to push the onset of non-renormalizability to higher loop order (cf. e.g. [224,38,119]). It is often arguedthat a full treatment of the metric, including non-perturbative effects from thebackreaction of matter, may solve the problem but to this day this remains aconjecture 3 . A basic conceptual problem of a theory like gravity is the doublerole of geometric variables which are not only fields but also determine the(dynamical) background upon which the physical variables live. This is e.g. ofspecial importance for the uncertainty relation at energies above the Planckscale leading to Wheeler’s notion of “space-time-foam” [434].

Another question which has baffled theorists is the problem of time. Inordinary quantum mechanics the time variable is set apart from the “observ-ables”, whereas in the straightforward quantum formulation of gravity (theso-called Wheeler-deWitt equation [435,121]) a variable like time must be in-troduced more or less by hand through “time-slicing”, a multi-fingered timeetc. [232]. Already at the classical level of GR “time” and “space” changetheir roles when passing through a horizon which leads again to considerablecomplications in a Hamiltonian approach [10, 272].

Measuring the “observables” of usual quantum mechanics one realizesthat the genuine measurement process is related always to a determination of

1 The recent book [360] can be recommended.2 A brief history of quantum gravity can be found in ref. [371].3 For a recent argument in favor of this conjecture using Weinberg’s argument of“asymptotic safety” cf. e.g. [296].

18

1 INTRODUCTION

the matrix element of some scattering operator with asymptotically definedingoing and outgoing states. For a gauge theory like gravity, existing proofs ofgauge-independence for the S-matrix [279] may be applicable for asymptoti-cally flat quantum gravity systems. But the problem of other experimentallyaccessible (gauge independent!) genuine observables is open, when the dynam-ics of the geometry comes into play in a nontrivial manner, affecting e.g. thenotion what is meant by asymptotics.

The quantum properties of black holes (BH) still pose many questions.Because of the emission of Hawking radiation [211,412], a semi-classical effect,a BH should successively lose energy. If there is no remnant of its previousexistence at the end of its lifetime, the information of pure states swallowed byit will have only turned into the mixed state of Hawking radiation, violatingbasic notions of quantum mechanics. Thus, of special interest (and outside therange of methods based upon the fixed background of a large BH) are the laststages of BH evaporation.

Other open problems – related to BH physics and more generally to quan-tum gravity – have been the virtual BH appearing as an intermediate stage inscattering processes, the (non-)existence of a well-defined S-matrix and CPT(non-)invariance. When the metric of the BH is quantized its fluctuations mayinclude “negative” volumes. Should those fluctuations be allowed or excluded?The intuitive notion of “space-time foam” seems to suggest quantum gravityinduced topology fluctuations. Is it possible to extract such processes from amodel without ad hoc assumptions? From experience of quantum field theoryin Minkowski space one may hope that a classical singularity like the one inthe Schwarzschild BH may be eliminated by quantum effects – possibly at theprice of a necessary renormalization procedure. Of course, the latter may justreflect the fact that interactions with further fields (e.g. other modes in stringtheory) are not taken into account properly. Can this hope be fulfilled?

In attempts to find answers to these questions it seems very reasonableto always try to proceed as far as possible with the known laws of quantummechanics applied to GR. This is extremely difficult 4 in D = 4. Therefore, formany years a rich literature developed on lower dimensional models of gravity.The 2D Einstein-Hilbert action is just the Gauss-Bonnet term. Therefore,intrinsically 2D models are locally trivial and a further structure is introduced.This is provided by the dilaton field which naturally arises in all sorts ofcompactifications from higher dimensions. Such models, the most prominentbeing the one of Jackiw and Teitelboim (JT), were thoroughly investigatedduring the 1980-s [22, 123, 405, 122, 124, 238, 250, 251, 312, 388]. An excellentsummary (containing also a more comprehensive list of references on literaturebefore 1988) is contained in the textbook of Brown [59]. Among those modelsspherically reduced gravity (SRG), the truncation of D = 4 gravity to itss-wave part, possesses perhaps the most direct physical motivation. One caneither treat this system directly in D = 4 and impose spherical symmetry in

4 A recent survey of the present situation is the one of Carlip [79].

19

1 INTRODUCTION

the equations of motion (e.o.m.-s) [276] or impose spherical symmetry alreadyin the action [36,412,33,409,205,324,407,244,276,295,195], thus obtaining adilaton theory 5 . Classically, both approaches are equivalent.

The rekindled interest in generalized dilaton theories in D = 2 (hence-forth GDTs) started in the early 1990-s, triggered by the string inspired[310,137,443,127,316,233,117,254] dilaton black hole model 6 , studied in theinfluential paper of Callan, Giddings, Harvey and Strominger (CGHS) [71].At approximately the same time it was realized that 2D dilaton gravity canbe treated as a non-linear gauge-theory [426,230].

As already suggested by earlier work, all GDTs considered so far couldbe extracted from the dilaton action [373,349]

L(dil) =∫d2x

√−g[XR

2− U(X)

2(∇X)2 + V (X)

]+ L(m) , (1.1)

where R is the Ricci-scalar, X the dilaton, U(X) and V (X) arbitrary functionsthereof, g is the determinant of the metric gµν , and L(m) contains eventualmatter fields.

When U(X) = 0 the e.o.m. for the dilaton from (1.1) is algebraic. Forinvertible V ′(X) the dilaton field can be eliminated altogether, and the La-grangian density is given by an arbitrary function of the Ricci-scalar. A recentreview on the classical solution of such models is ref. [381]. In comparison withthat, the literature on such models generalized to depend also 7 on torsion T a

is relatively scarce. It mainly consists of elaborations based upon a theory pro-posed by Katanaev and Volovich (KV) which is quadratic in curvature andtorsion [250,251], also known as “Poincare gauge gravity” [322].

A common feature of these classical treatments of models with and with-out torsion is the almost exclusive use 8 of the gauge-fixing for the D = 2metric familiar from string theory, namely the conformal gauge. Then thee.o.m.-s become complicated partial differential equations. The determinationof the solutions, which turns out to be always possible in the matterless case(L(m) = 0 in (1.1)), for nontrivial dilaton field dependence usually requiresconsiderable mathematical effort. The same had been true for the first paperson theories with torsion [250, 251]. However, in that context it was realizedsoon that gauge-fixing is not necessary, because the invariant quantities R andT aTa themselves may be taken as variables in the KV-model [390,389,391,392].This approach has been extended to general theories with torsion 9 .

5 The dilaton appears due to the “warped product” structure of the metric. Fordetails of the spherical reduction procedure we refer to appendix A.6 A textbook-like discussion of this model can be found in refs. [183,399].7 For the definition of the Lorentz scalar formed by torsion and of the curvaturescalar, both expressed in terms of Cartan variables zweibeine eaµ and spin connection

ωabµ we refer to sect. 1.2 below.8 A notable exception is Polyakov [362].9 A recent review of this approach is provided by Obukhov and Hehl [348].

20

1 INTRODUCTION

As a matter of fact, in GR many other gauge-fixings for the metric havebeen well-known for a long time: the Eddington-Finkelstein (EF) gauge, thePainleve-Gullstrand gauge, the Lemaıtre gauge etc. . As compared to the “di-agonal” gauges like the conformal and the Schwarzschild type gauge, theypossess the advantage that coordinate singularities can be avoided, i.e. thesingularities in those metrics are essentially related to the “physical” ones inthe curvature. It was shown for the first time in [291] that the use of a temporalgauge for the Cartan variables (cf. eq. (3.3) below) in the (matterless) KV-model made the solution extremely simple. This gauge corresponds to the EFgauge for the metric. Soon afterwards it was realized that the solution could beobtained even without previous gauge-fixing, either by guessing the Darbouxcoordinates [377] or by direct solution of the e.o.m.-s [290] (cf. sect. 3.1). Thenthe temporal gauge of [291] merely represents the most natural gauge fixingwithin this gauge-independent setting. The basis of these results had been afirst order formulation of D = 2 covariant theories by means of a covariantHamiltonian action in terms of the Cartan variables and further auxiliary fieldsXa which (beside the dilaton field X) take the role of canonical momenta (cf.eq. (2.17) below). They cover a very general class of theories comprising notonly the KV-model, but also more general theories with torsion 10 . The mostattractive feature of theories of type (2.17) is that an important subclass ofthem is in a one-to-one correspondence with the GDT-s (1.1). This dynamicalequivalence, including the essential feature that also the global properties areexactly identical, seems to have been noticed first in [248] and used extensivelyin studies of the corresponding quantum theory [281,285,284].

Generalizing the formulation (2.17) to the much more comprehensive classof “Poisson-Sigma models” [379, 396] on the one hand helped to explain thedeeper reasons of the advantages from the use of the first oder version, on theother hand led to very interesting applications in other fields [3], includingespecially also string theory [382, 387]. Recently this approach was shown torepresent a very direct route to 2D dilaton supergravity [140] without auxiliaryfields.

Apart from the dilaton BH [71] where an exact (classical) solution ispossible also when matter is included, general solutions for generic D = 2gravity theories with matter cannot be obtained. This has been possible onlyin restricted cases, namely when fermionic matter is chiral 11 [278] or whenthe interaction with (anti)selfdual scalar matter is considered [356].

Semi-classical treatments of GDT-s take the one loop correction frommatter into account when the classical e.o.m.-s are solved. They have beenused mainly in the CGHS-model and its generalizations [41, 117, 374, 44, 115,147, 256, 446, 445, 209, 210, 423]. In our present report we concentrate onlyupon Hawking radiation as a quantum effect of matter on a fixed (classical)

10 In that case there is the restriction that it must be possible to eliminate allauxiliary fields Xa and X (see sect. 2.1.3).11 This solution was rediscovered in ref. [393].

21

1 INTRODUCTION

geometrical background, because just during the last years interesting insighthas been obtained there, although by no means all problems have been settled.

Finally we turn to the full quantization of GDTs. It was believed byseveral authors (cf. e.g. [373, 349, 242, 139, 138]) that even in the absence ofinteractions with matter nontrivial quantum corrections exist and can be com-puted by a perturbative path integral on some fixed background. Again theevaluation in the temporal gauge [291], at first for the KV-model showed thatthe use of other gauges just obscures a very simple mechanism. Actually alldivergent counter-terms can be absorbed into one compact expression. Aftersubtracting that in the absence of matter the solution of the classical theoryrepresents an exact “quantum” result. Later this perturbative argument hasbeen reformulated as an exact path integral, first again for the KV-model [204]and then for general theories of gravity in D = 2 [281,285,284,196,157,199].

In our present review we concentrate on the path integral approach, withDirac quantization only referred to for sake of comparison. In any case, thecommon starting point is the Hamiltonian analysis which in a theory for-mulated in terms of Cartan variables in D = 2 possesses substantial techni-cal advantages. The constraints, even in the presence of matter interactions,form an algebra with momentum-dependent structure constants. Despite thatnonlinearity the simplest version of the Batalin-Vilkovisky procedure [27] suf-fices, namely the one also applicable to ordinary nonabelian gauge theoriesin Minkowski space. With a temporal gauge fixing for the Cartan variablesalso used in the quantized theory, the geometric part of the action yields theexact path integral. Possible background geometries appear naturally as ho-mogeneous solutions of differential equations which coincide with the classicalones, reflecting “local quantum triviality” of 2D gravity theories in the ab-sence of matter, a property which had been observed as well before in theDirac quantization of the KV-model [377].

These features are very difficult to locate in the GDT-formulation (1.1),but become evident in the equivalent first order version with a “Hamiltonian”action.

Of course, non-renormalizability persists in the perturbation expansionwhen the matter fields are integrated out. But as an effective theory in caseslike spherically reduced gravity, specific processes can be calculated, relyingon the (gauge-independent) concept of S-matrix elements. With this method,scattering of s-waves in spherically reduced gravity has provided a very di-rect way to create a “virtual” BH as an intermediate state without furtherassumptions [157].

The structure of our present report is determined essentially by the ap-proach described in the last paragraphs. One reason is the fact that a very com-prehensive overview of very general classical and quantum theories in D = 2is made possible in this manner. Also a presentation seems to be overdue inwhich results, scattered now among many different original papers can be inte-grated into a coherent picture. Parallel developments and differences to otherapproaches will be included in the appropriate places.

22

1 INTRODUCTION

1.1 Structure of this review

This review is organized as follows:• Section 1 in its remaining part contains a short primer on differential geom-

etry (with special emphasis on D = 2). En passant most of our notationsare fixed in that subsection.

• Section 2 motivates the study of GDTs and introduces its action in thethree most frequently used forms (dilaton action, first order action, andPoisson-Sigma action) and describes the relations between them.

• Section 3 gives all classical solutions of GDTs in the absence of matter. Theglobal structure of such theories is discussed using Schwarzschild space-time as a simple example. As a further illustration we consider a family ofdilaton models describing a single black hole in Minkowski, Rindler or deSitter space-time.

• Section 4 extends the discussion to additional gauge-fields, supergravity and(bosonic or fermionic) matter fields.

• Section 5 considers the role of energy in GDTs. In particular, the ADMmass, quasilocal energy, an absolute conservation law and its correspondingNother symmetry are discussed.

• Section 6 leaves the classical realm providing a concise treatment of (semi-classical) Hawking radiation for minimally and non-minimally coupled mat-ter.

• Section 7 is devoted to non-perturbative path integral quantization of thegeometric sector of GDTs with (scalar) matter, giving rise to a non-localand non-polynomial effective action depending solely on the matter fieldsand external sources. The matter sector is treated perturbatively.

• Section 8 shows some consequences of the previously developed perturbationtheory: the virtual black hole phenomenon, the appearance of non-localvertices, and S-matrix elements for s-wave gravitational scattering.

• Section 9 describes the status of Dirac quantization for a typical exampleof that approach.

• Section 10 concludes with a brief summary and an outlook regarding openquestions.

• Appendix A recalls the spherical reduction procedure in the Cartan formal-ism.

• Appendix B collects some basic properties of the heat kernel expansionneeded in Section 6.

Several topics are closely related to the subject of this review, but are not

included:(1) Various calculations and explanations of the BH entropy [169, 355] be-

came a large and rather independent field of research which shows, how-ever, overlaps [165,171] with the general treatment of the dilaton theoriespresented in this review. We do not cover approaches which imply fur-ther physical assumptions which transgress the orthodox application ofquantum theory to gravity [34, 35, 43, 24, 31].

23

1 INTRODUCTION

(2) The ideas of the holographic principle [403,400] and of the AdS/CFT cor-respondence [309,200,444] are now being actively applied to BH physics(see, e.g. [375] and references therein).

(3) There exist different approaches to integrability of gravity models in twodimensions [338,269,268,339]. In particular, a rather sophisticated tech-nique has been applied to solve the effective 2D models emerging aftertoroidal reduction (instead of the spherical reduction considered in this re-view) of the four-dimensional Einstein equations [39,417]. Recently againinteresting developments should be noted in Liouville gravity [151, 406].Some relations between 2D dilaton gravity and the theory of solitonswere discussed in [70, 336].

Each of these topics deserves a separate review, and in some cases such reviewsexist. Therefore, we have restricted ourselves in those fields to just a few(somewhat randomly selected) references which hopefully will permit furtherorientation.

1.2 Differential geometry

1.2.1 Short primer for general dimensions

In the comprehensive approach advocated for D = 2 gravity the useof Cartan variables (zweibeine, spin-connection) plays a pivotal role. As anintroduction and in order to fix our notations we shall review briefly thisformalism. For details we refer to the mathematical literature (cf. e.g. [334]).

On a manifold with D dimensions in each point one introduces viel-beine eµa(x), where Greek indices refer to the (holonomic) coordinates xµ =(x0, x1, . . . , xD−1) and Latin indices denote the ones related to a (local) Lorentzframe with metric η = diag (1,−1, . . . ,−1). The dual vector space is spannedby the inverse vielbeine 12 eaµ(x):

eµaebµ = ηab (1.2)

SO(1, D− 1) matrices Lab(x) of the (local) Lorentz transformations obey

Lac Lbc = δab . (1.3)

A Lorentz vector V a = eaµVµ transforms under local Lorentz transformations

asV ′a(x) = Lab(x)V

b(x) (1.4)

This implies a covariant derivative

(Dµ)ab = δab ∂µ + ωµ

ab , (1.5)

12 For simplicity we shall use indiscriminately the term “vielbein” for the vielbein,the inverse vielbein and the dual basis of 1-forms (the components of which aregiven by the inverse vielbein) whenever the meaning is clear either from the contextor from the position of indices.

24

1 INTRODUCTION

if the spin-connection ωµab is introduced as the appropriate gauge field with

transformationω′µab = −Lbd (∂µL

ad) + Lac ωµ

cd Lb

d . (1.6)

The infinitesimal version of (1.6) follows from Lab = δab + lab + O(l2) wherelab = −lba .

Formally also diffeomorphisms

xµ(x) = xµ − ξµ(x) + O(ξ2) (1.7)

can be interpreted, at least locally, as gauge transformations, when the Lievariation is employed which implies a transformation referring to the samepoint. In

∂xµ

∂xν= δµν − ξµ, ν ,

∂xν

∂xµ= δνµ + ξν, µ (1.8)

partial derivatives with respect to xν have been abbreviated by the index aftera comma.

For instance, for the Lie variation of a tensor of first order Vµ(x) =∂xν

∂xµ Vν(x) one obtains

δξVµ(x) = Vµ(x) − Vµ(x) = ξν , µ Vν + ξνVµ, ν . (1.9)

For the dual to the tangential space, e.g. V µ∂µ = V µ(∂µxν)∂ν = V µ∂µ one

derives the analogous transformation

δξVµ = V µ(x) − V µ(x) = −ξµ, νV ν + ξνV µ, ν . (1.10)

The metric gµν in the line element is a quadratic expression of the viel-beine

(ds)2 = gµν dxµdxν = eaµe

bν ηab dx

µdxν , (1.11)

and, therefore, a less elementary variable. Also the reparametrization invariantvolume element√

(−)D−1g dD x =√

(−)D−1 det gµν dD x =

=√

(−)D−1(det eaµ)2 det η dD x = | det eaµ| dDx = |e| dDx (1.12)

is of polynomial form if expressed in vielbein components.The advantage of the form calculus [334] is that diffeomorphism invariance

is automatically implied, when the Cartan variables are converted into oneforms

eaµ → ea = eaµdxµ, ωµ

ab → ωab = ωµ

ab dx

µ (1.13)

which are special cases of p-forms

Ωp =1

p!Ωµ1 ... µp

dxµ1 ∧ dxµ2 ∧ · · · ∧ dxµp . (1.14)

Due to the antisymmetry of the wedge product dxµ∧dxν = dxµ⊗dxν−dxν⊗dxµ = −dxν ∧ dxµ all totally antisymmetric tensors Ωµ1...µp

are described in

25

1 INTRODUCTION

this way. Clearly Ωp = 0 for p > D. The action of the (p+ q)-form Ωq ∧Ωq onp+ q vectors is defined by

Ωp ∧ Ξq(V1, . . . , Vp+q) =1

p!q!

∑

π

δπΩ(Vπ(1), . . . , Vπ(p))Ξ(Vπ(p+1), . . . , Vπ(p+q)),

(1.15)where the sum is taken over all permutations π of 1, . . . , p + q and δπ is +1for an even number of transpositions and −1 for an odd number of transpo-sitions. It is convenient at this point to introduce the condensed notation for(anti)symmetrization:

α[µ1...µp] :=1

p!

∑

π

δπαaπ(1)...aπ(p), σ(µ1...µp) :=

1

p!

∑

π

σaπ(1)...aπ(p), (1.16)

where the sum is taken over all permutations π of 1, . . . , p and δπ is definedas before. In the volume form

Ωp=D =1

D!a[µ1...µD ] ǫ

µ1 ... µD dDx =1

D!a[µ1 ... µD] |e|ǫµ1 ... µDdDx (1.17)

the product of differentials must be proportional to the totally antisymmetricLevi-Civita symbol ǫ01...(D−1) = −1 or, alternatively, to the tensor ǫ = |e|−1ǫ(cf. (1.12)). The integral of the volume form

∫MD

ΩD on the manifold MD con-tains the scalar a = a[µ1 ... µD ] ǫ

µ1 ... µD which is the starting point to constructdiffeomorphism invariant Lagrangians.

By means of the metric (1.11) a mixed ǫ-tensor

ǫµ1 ... µp

µp+1 ... µp+q = gµ1ν1 gµ2ν2 . . . gµpνpǫν1 ... νpµp+1...µp+q (1.18)

can be defined which allows the introduction of the Hodge dual of Ωp as aD − p form

∗ Ωp = Ω′D−p =

1

p!(D − p)!ǫµ1 ... µD−p

ν1 ... νp Ων1 ... νpdxµ1 ∧ . . . ∧ dxµD−p .

(1.19)In D = even and for Lorentzian signature we obtain for a p-form

∗ ∗ Ωp = (−1)p+1Ωp. (1.20)

The exterior differential one form d = dxµ∂µ with d2 = 0 increases the formdegree by one:

dΩp =1

p!∂µΩµ1...µp

dxµ ∧ dxµ1 ∧ · · · ∧ dxµp (1.21)

Onto a product of forms d acts as

d (Ωp ∧ Ωq) = dΩp ∧ Ωq + (−1)pΩp ∧ dΩq . (1.22)

26

1 INTRODUCTION

We shall need little else from the form calculus [334] except the PoincareLemma which says that for a closed form, obeying dΩp = 0, in a certain(“star-shaped”) neighborhood of a point xµ on a manifold M, Ωp is exact, i.e.can be written as Ωp = dΩ′

p−1.In order to simplify our notation we shall drop the ∧ symbol whenever

the meaning is clear from the context.The Cartan variables expressed as one forms (1.13) in view of their

Lorentz-tensor properties are examples of algebra valued forms. This is alsothe case for the covariant derivative (1.5), now written as

Dab = δabd+ ωab , (1.23)

when it acts on a Lorentz vector.From (1.13) and (1.23) the two natural quantities to be defined on a

manifold are the torsion two-form

T a = Dab e

b (1.24)

(“First Cartan’s structure equation”) and the curvature two-form

Rab = Da

c ωcb (1.25)

(“Second Cartan’s structure equation”). From (1.23) immediately follows

(D2)ab = DacD

cb = Ra

b , (1.26)

Bianchi’s first identity. Using (1.26) D3 can be written in two equivalent ways,

DabR

bc −Ra

bDbc = 0, (1.27)

corresponding to Bianchi’s second identity

(dRab) + ωacRcb + ωbcR

ac =: (DR)ab = 0 . (1.28)

The l.h.s. defines the action of the covariant derivative (1.23) on Rab, a Lorentztensor with two indices. The brackets indicate that those derivatives only actupon the quantity R and not further to the right. The structure equationstogether with the Bianchi identities show that the covariant action for anygravity action in D dimensions depending on ea, ωab can be constructed as avolume form depending solely on Rab, T a and ea. The most prominent exampleis Einstein gravity in D = 4 [136, 135] which in the Palatini formulationreads [352]

LHEP ∝∫

M4

Rabeced ǫabcd , (1.29)

having used the definition ǫabcd = ǫµνστ eµaeνbeσc eτd. The condition of vanishing

torsion T a = 0 for this special case already follows from varying ωab indepen-dently in (1.29).

27

1 INTRODUCTION

In the usual textbook formulations of Einstein gravity, in terms of themetric, the affine connection Γµν

ρ appears as the only variable in the covariantderivative, e.g. for a contravariant vector Xν

Xν;µ := ∇µX

ν = (∇µ)νρX

ρ = (∂µδνρ + Γµρ

ν)Xρ. (1.30)

In the vielbein basis eaµ we relate Xb = ebρXρ and let (1.5) act onto that Xb.

Multiplying by the inverse vielbein (1.2) and comparing with (1.30) yields

Γµνρ = ea

ρ[(Dµ)

ab e

bν

]. (1.31)

The same identification follows, of course, from the covariant derivative of acovariant vector:

Xν;µ := ∂µXν − Γµνρ Xρ (1.32)

Covariant derivatives may be constructed easily also for tensors withmixed space-time and local Lorentz indices. For instance, that derivative act-ing upon the vielbeine eρc

(Dµ e)νa = [(Dµ)

νρ ]a

c eρc := (∇µ)νρ e

ρa + (ωµ)a

c eνc = 0 (1.33)

is seen to vanish. By (1.2) this implies the same result for analogously definedvielbeine eaρ

(Dµe)aρ = 0 . (1.34)

From (1.34) and the antisymmetry of ωab = −ωba (one version of metricity)corresponding to its property as a Lorentz generator of SO(1, D − 1) imme-diately

∇µgρσ = 0 (1.35)

can be derived, the version of the metricity usually employed in torsionlesstheories.

Comparing the antisymmetrized part of the affine connection Γ[µν]ρ =

12(Γµν

ρ − Γνµρ) of (1.31) with the components of the torsion (1.24), multiplied

by the inverse vielbein, shows that the expressions are identical:

eρa Taµν = Γ[µν]

ρ . (1.36)

This allows to express the full affine connection

Γµνρ = Γ(µν)

ρ + Tµνρ (1.37)

in terms of Christoffel symbols µ, ν, ρ and the contorsion K

Γ(µν)ρ = gρσ Γ(µν)σ = µ, ν, ρ + K(µν)ρ (1.38)

by the standard trick of considering (1.35) in the form

gνρ,µ = Γµνλ gλρ + Γµρ

λ gλν (1.39)

28

1 INTRODUCTION

with (1.38) and by taking the linear combination of the identity (1.39) minusthe one for gµν,ρ plus the one for gρµ,ν . In this way the Christoffel symbol

µ, ν, ρ =1

2(gνρ,µ + gµρ,ν − gµν,ρ) , (1.40)

but also the additional contorsion contribution K from the nonvanishing tor-sion in (1.38)

K(µν)ρ = T[ρµ]ν + T[ρν]µ (1.41)

can be found. Nonvanishing torsion and thus also a nonvanishing contorsionare important for the determination of the global properties of a certain solu-tion of a generic theory of gravity.

In contrast to ordinary Minkowski space field theories, the variables ofgravity – in the most general case the independent Cartan variables e andω – in the dynamical evolution also determine the non-Minkowski dynamicalbackground upon which the theory lives. Thus, for the investigation of thatbackground a device must be found which acts like a test charge in an elec-tromagnetic field. The simplest possibility in gravity is to add the Lagrangianof a point particle with path xµ = xµ(τ) to the original action ( ˙xµ = dxµ/dτwith the affine parameter τ),

L(p) = −mτ2∫

τ1

ds = −mτ2∫

τ1

√gµν(x) ˙xµ ˙xν dτ , (1.42)

with a mass m, small enough to be of negligible gravitational influence. Vari-ation of L(p) with respect to xµ leads to the usual geodesic equation

¨xµ + Γ(ρσ)µ ˙xρ ˙xσ = 0 , (1.43)

where, by construction from (1.42), Γ(ρσ)µ = gµα ρ, σ, α only “feels” the

Christoffel part (1.40) of the affine connection and not the contorsion (1.41).Alternatively, also the full affine connection Γ may be considered in (1.43)(“autoparallels”) [217,218]. For that modified geodesic equation for xα(τ) alsoa (non-local) action replacing (1.42) can be found in the literature [160, 262].In order to explore the local and topological properties of a certain manifoldwhich corresponds to a solution of a generic gravity theory all points mustbe connected which can be reached by a device like the geodesic (1.43) bymeans of a time-like, but also space-like or light-like path. The classificationof possible extensions of a certain patch uses the notion of “geodesic” incom-pleteness: a geodesic which has only a finite range of affine parameter, butwhich is inextendible 13 in at least one direction is called incomplete. A space-time with at least one incomplete (time/space/light-like) geodesic is called(time/space/light-like) geodesically incomplete. The notion of incompleteness

13 This means the corresponding geodesic must have (at least) one endpoint. Fordetails we refer to [216,428].

29

1 INTRODUCTION

also yields the most satisfactory classification of (geometric) singularities. Forexample, a singularity like the one in the Schwarzschild metric [386] can bereached by at least one (time- or light-like) geodesic with finite affine param-eter (i.e. with finite proper time for massive test particles).

For D = 2 theories a complete discussion of “geodesic topology” forany generic theory can be carried out (cf. sect. 3.2.) [430, 264]. Here we justwant to emphasize the importance of the type of device to be used for thedetermination of the “effective” topology of the manifold which, in principle,may be different for geodesics, autoparallels, spinning particles etc. .

1.2.2 Two dimensions

In D = 2 the Lorentz transformations (1.3),(1.4) simply reduce to a boostwith velocity v

Lab =

cosh v sinh v

sinh v cosh v

a

b

= δab + ǫab v + O(v2) , (1.44)

where in local Lorentz indices with metric ηab = ηab (η = diag(+1,−1)) theLevi-Civita symbol ǫab = ηac ǫcb (ǫ01 = −ǫ01 = +1) coincides with the tensor.It is related to the tensor ǫµν in holonomic coordinates (cf. (1.17)) by (explicitvalues of Lorentz indices in (1.47) are underlined)

ǫ = −1

2ǫabe

a ∧ eb (1.45)

ǫµν = eaµebνǫab = |e|ǫµν = |e|−1gµρgνσ ǫ

ρσ , (1.46)

|e| = det eaµ = e00 e

11 − e

10 e

01 . (1.47)

It should be noted that in (1.45) we choose the sign, which differs from (1.14),in order to be consistent with some original literature. As there is only onegenerator εab in SO(1, 1) (cf. (1.44)) the spin connection one-form simplifiesto a single term ωab = ω ǫab and hence the one quadratic in ω of Rab (1.25)vanishes:

Rab = ǫab dω . (1.48)

From now on for simplicity we shall refer to the 1-form ω as the “spin connec-tion”.

This shows that the curvature in D = 2 only possesses one independentcomponent which we take to be the Ricci-scalar 14 :

R = 2 ∗ dω = 2|e|−1ǫρσ∂ρωσ . (1.49)

14 Our convention corresponds to the contraction Rµννµ = Rµν abe

aνebν where Rµν abare the tensor components of Rab. Rµν

ρσ then coincides with the usual textbookdefinition [428].

30

1 INTRODUCTION

It is clear from this expression that the Hilbert-Einstein action in two di-mensions is a total divergence. In (compact) Euclidean space (

√−g → √g)

without boundaries it becomes the Euler characteristic of a 2D Riemannianspace with genus γ ∫

Mγ

d2x√g R = 8π(1 − γ) . (1.50)

Also the torsion simplifies to a volume form

T a =1

2Tµν

adxµ ∧ dxν , Tµνa = (Dµeν)

a − (Dνeµ)a , (1.51)

with(Dµ)

ab = ∂µδ

ab + ωµǫ

ab . (1.52)

The Hodge dual of T a here is a diffeomorphism scalar:

τa := ∗ T a = |e|−1 ǫµν (Dµeaν) (1.53)

In D = 2 the inverse of the zweibeine from (1.2) obeys the simple relation

eµa = −|e|−1ǫµν ǫab ebν . (1.54)

The formula for the change of the Ricci scalar under a conformal trans-formation of the metric gµν = e2ρ gµν is most easily derived from a transfor-mation eaµ = eρ eaµ of the zweibeine for vanishing torsion T a = 0, i.e. withω = ω = ea ∗ dea in the Ricci scalar (1.49)

|e|R = 2|e| ∗ d(ea ∗ dea) = 2 ǫτσ∂τ

(eσa ǫµν

|e| ∂µeaν). (1.55)

Remembering e = |e|e2ρ and using (1.54) for eµaeνbη

ab = gµν yields ((e) =√−g)

an important identity:√−g R =

√−gR− 2∂τ (√−ggτσ ∂σρ) . (1.56)

Light-cone Lorentz vectors are especially useful in D = 2,

X± :=1√2

(X0 ±X1) , (1.57)

yielding X2 = XaXa = X aXa = ηabXaX b = 2X+X− with metric

ηab =

0 1

1 0

(1.58)

and the corresponding Lorentz ǫ-tensor ǫab = ηa c ǫc b with ε±± = ±1. The lightcone components of the torsion (1.51) become

T± = (d± ω) e±. (1.59)

31

1 INTRODUCTION

Since we are going to discuss fermionic matter (as well as supergravity)we have to fix our spinor notation. The γa-matrices are defined in a localLorentz frame

γa, γb

= 2ηab (1.60)

γ0 =

0 1

1 0

, γ1 =

0 1

−1 0

γ∗ := −γ0γ1 =

1 0

0 −1

= −1

2

[γ0, γ1

].

(1.61)

In light cone components we obtain a representation in terms of nilpotentmatrices

γ+ =√

2

0 1

0 0

, γ− =

√2

0 0

1 0

(1.62)

The covariant derivative acting on two-dimensional Dirac fermions

Dµ = ∂µ −1

2γ∗ ωµ (1.63)

is determined by the Lorentz generator for spinors [γ0, γ1]/4 = −γ∗/2.

32

2 MODELS IN 1 + 1 DIMENSIONS

2 Models in 1 + 1 Dimensions

There are (at least) four different motivations to study generalized dilatontheories (GDT) in D = 2:• Starting from Einstein gravity in D ≥ 4 and imposing spherical symmetry

one reproduces a certain GDT• A certain limit of (super-)string theory yields a particular GDT as effective

action• GDTs can be viewed as toy models for quantization of gravity and as a

laboratory for studying BH evaporation• In a first order formulation the underlying Poisson structure reveals relations

to non-commutative geometry and deformation quantization. Again, GDTsare a convenient laboratory to elucidate these new concepts and techniques.

Moreover, a result obtained along one route is of course also valid for all otherapproaches after having translated the jargon from one field to the others. Inthis sense, GDTs may even serve as a link between general relativity (GR),string theory, BH physics and non-commutative geometry.

We base our discussion on the first (somewhat more phenomenological)route and show the links to the other fields in this section.

2.1 Generalized Dilaton Theories

2.1.1 Spherically reduced gravity

The introduction of dilaton fields allows the treatment of the dynamicsfor a generic higher dimensional (D > 2) theory of gravity in an effectivetheory at lower dimension D1 < D, which is still diffeomorphism invariant. Incertain special cases the isometry group of the D-dimensional metric is suchthat it allows for a reduction to D1 = 2. Important examples for D = 4 aretoroidal reduction [273, 189, 178, 225, 56] and spherical reduction [36, 412, 33,409,205,324,407,244,276,295,195]. The latter is of special importance, becauseit covers the Schwarzschild BH. Therefore, we concentrate on that example.

Splitting locally the D-dimensional manifold MD into a direct productM2 × SD−2 the line element becomes

(ds)2(D) = gµν(x)dx

µdxν − λ−2X2

D−2 (dΩ)2SD−2 (2.1)

where (dΩ)2SD−2 is the surface element of the (D−2)-dimensional sphere, xµ =

x0, x1 are the coordinates in M2, and λ is a parameter of mass dimensionone. A straightforward calculation (cf. e.g. [197]; explicit formulae for thecurvature 2-form, the ensuing Ricci-scalar and the Euler- and Pontryagin-classcan be found in appendix A) for the D-dimensional Hilbert-Einstein action

33

2 MODELS IN 1 + 1 DIMENSIONS

LHE =∫dDx

√−g(D)R(D) yields ((∇X)2 = gµν∂µX∂νX)

L(SRG) =OD−2

λD−216πGN

∫d2x

√−g[XR+

D − 3

D − 2

(∇X)2

X− λ2(D − 2)(D − 3)X

D−4D−2

]. (2.2)

In the prefactor, which will be dropped consistently in the following, OD−2

denotes the surface of the unit sphere SD−2. Fixing the 2D diffeomorphisms(partially) as X = (λr)D−2 (the radius r representing one of the coordinatesand λ > 0) eq. (2.1) yields the usual spherically symmetric line element inwhich r > 0 is required.

Another way to obtain a 2D theory from a higher dimensional one is tosuppose that the D-dimensional manifold is a direct product MD = M2 ⊗TD−2, where TD−2 is a torus, and that all fields are independent of the D −2 extra coordinates. This procedure is called dimensional reduction. It alsoproduces a dilaton theory in 2D if the higher dimensional theory alreadycontains the dilaton [180].

2.1.2 Dilaton gravity from strings

Developments in string theory contributed much to the increase of interestin dilaton gravity in the 1990s. The simplest way to obtain it from strings isto consider the conditions for world-sheet conformal invariance [72].

The starting point is the non-linear sigma model action for the closedbosonic string,

L(σ) =1

4πα′

∫d2ξ

√−h

[gµνh

ij∂iXµ∂jX

ν + α′ΦR], (2.3)

where ξ is a coordinate on the string world-sheet, hij is a metric 15 there, Rrepresents the corresponding scalar curvature. The other symbols denote: thetarget space coordinates (Xµ), the target space metric (gµν), and the dilatonfield (Φ). As usual, α′ is the inverse string tension. The antisymmetric B-fieldis set to zero.

It is essential for string consistency that, as a quantum field theory, thesigma model be locally scale invariant. This is equivalent to the requirementthat the trace of the 2D world-sheet energy-momentum tensor vanishes. Itsgeneral structure is

2πT ii = βΦR + βgµνhij∂iX

µ∂jXν , (2.4)

where the “beta functions” βΦ and βgµν are local functionals of the couplingsgµν and Φ, usually calculated in the form of a power series in α′. Note that thefirst term in L(σ) is conformally invariant and contributes to the β-functions

15 This metric should not be confused with gµν restricted to D = 2 in (2.8).

34

2 MODELS IN 1 + 1 DIMENSIONS

at the quantum level only through the conformal anomaly. It corresponds toO(α′)0. The second term in (2.3) breaks local scale invariance already at theclassical level. Due to the factor α′ its contributions to the trace (2.4) also startwith the zeroth power of α′. The leading terms in βΦ and βgµν were calculatedin ref. [72]. With our sign conventions they read:

βΦ

α′ = − λ2

4π2− 1

16π2

(4(∇Φ)2 − 4∇µ∇µΦ − R

), (2.5)

βgµν = Rµν + 2∇µ∇νΦ , (2.6)

where ∇µ is the covariant derivative in target space, R is the scalar curvatureof the target space manifold. The constant λ depends on the central charge.For the bosonic string it is

λ2 =26 −D

12α′ . (2.7)

This constant vanishes for critical strings.The key observation regarding the beta functions (2.5) and (2.6) is that

the conditions of conformal invariance βΦ = 0 and βgµν = 0 are equivalent tothe e.o.m.-s to be derived from the dilaton gravity action

L(dil) =∫dDX

√−ge−2Φ[R + 4(∇Φ)2 − 4λ2

]. (2.8)

In particular, the dilaton e.o.m. is equivalent to βΦ = 0. The Einstein equationsare given by a combination of the two beta functions, βgµν − 8π2gµνβ

Φ/α′ = 0.For D = 2 the action (2.8) describes the geometric part of the “string

inspired” dilaton (CGHS) model [71] which has been studied since the early1990-s [326,77,32,330,277,416]. It is intimately related to the SO(2, 1)/U(1)-WZW exact conformal field theory 16 [310, 137, 443,127].

An amusing feature of (2.8) with D = 2 is that after the identificationX = e−2Φ it can be obtained from (2.2) by taking there the limit D → ∞keeping λ2(D − 2)(D − 3) → const. = 4λ2 . This corresponds to the classicallimit α′ → ∞.

2.1.3 Generalized dilaton theories – the action

A result like (2.2) or (2.8) suggests the consideration of GDTs

L(dil) =∫d2x

√−g[R

2X − U(X)

2(∇X)2 + V (X)

], (2.9)

where the overall factor has been chosen for later convenience. Clearly aneven more general action could contain still another arbitrary function Z(X),replacing X in the first term of the square bracket [19, 349]. However, we

16 The non-compact form is SO(2, 1)/SO(1, 1). An early review on 2D gravity and2D string theory from the stringy point of view is ref. [186].

35

2 MODELS IN 1 + 1 DIMENSIONS

assume that Z(X) is invertible for the range of X to be considered 17 . Thisallows the inversion X = Z−1 (X) and the reduction to the form (2.9). Indeedthe “physical” applications seem to be always of that type. The BH singularityof SRG reveals itself in the singular factor U of the dynamical term for thedilaton field. This is the first hint to the fact that the “strength” of thatsingularity in the solution of (2.2) is not fixed by the action; it will actuallyturn out to be a “constant of motion” which for the BH coincides with theADM mass (cf. sect. 5).

An alternative representation is suggested by (2.8):

L(dil) =1

2

∫d2x

√−ge−2Φ[R− U(Φ)(∇Φ)2 + 2V (Φ)

], (2.10)

with U(Φ) = 4 exp (−2Φ)U(exp (−2Φ)) and V (Φ) = exp (2Φ)V (exp (−2Φ)).Eqs. (2.9) and (2.10) are related by the redefinition of the dilaton field

X = e−2Φ, (2.11)

explicitly taking into account positivity ofX which is required in many models.Among the GDTs (2.9) with U(X) = 0 the simplest nontrivial choice of

refs. [22, 123, 405,122,124,239]

VJT = ΛX, UJT = 0 , (2.12)

the Jackiw-Teitelboim (JT) model, has played a decisive role for the under-standing of 2D (lineal) gravity [238]. Depending on the sign of Λ it describesa 2D (anti-) de Sitter manifold with constant positive or negative curvature.The symmetry properties of the model are related to the Lie algebra SO(1, 2).It has been explored in detail in the quoted references. Below this algebra willturn out to represent the special linear case of some, in general, nonlinear (fi-nite W -) algebra [118] associated with a generic dilaton theory (2.9) (cf. Sect.2.3).

More complicated models with U(X) = 0, but V (X) exhibiting a singu-larity in X, among others may also involve solutions with space-time structureof a BH or its generalizations. E.g. the choice 18

VRN = −2M

X2+

Q2

4X3(2.13)

produces a line element like the one for the Reissner-Nordstrom BH withcharge Q and mass M [364, 346]. Evidently in this case the singularities are

17 To the best of our knowledge there is no literature on nontrivial models whereZ(X) is not invertible (cf. also [397]). By a suitable redefinition a different simpli-fication with U(X) = 1, Z(X) 6= 1 was proposed in ref. [373].18 Solving the general theory in Sect. 3.1 we shall find that the potentials U andV as in (2.9) determining a dilaton action can even be ‘designed’, starting from agiven line-element.

36

2 MODELS IN 1 + 1 DIMENSIONS

kept fixed by parameters of the action. They cannot be related to the conser-vation law referred to already above for a “dynamical” model with singularnonvanishing U(X) and regular V (X). A final remark for the case U = 0 con-cerns the possibility to eliminate the dilaton field altogether by means of thealgebraic equations of motion produced by varyingX in (2.9), V ′ (X) = −R/2.If this equation can be inverted, the dilaton Lagrangian for U = 0 turns intoa Lagrangian depending on the function of R alone [380, 168,381,397]:

L =∫d2x

√−g f(R) (2.14)

As compared to such theories (2.14), the literature on models generalized soas to depend also on torsion (cf. (1.53))

L =∫d2x

√−g h (R, τaτa) (2.15)

is relatively scarce. It mainly consists of elaborations based upon the model ofKatanaev and Volovich [250, 251] where the function h in (2.15) is quadraticin R and linear in τaτa, also known as “Poincare gauge gravity” [390,389,391,392,348,322].

Models with U(X) 6= 0 and different assumptions for that function andV (X) have been studied extensively (cf. e.g. [312,19,349,350,374,373,41,116,311, 173, 304, 303, 298]). For their solution throughout these works the con-formal or the Schwarzschild gauge have been used, leading to complicatede.o.m.-s, the solution of which often requires considerable mathematical ef-fort. Because we shall avoid this complication altogether (sect. 3) no explicitexamples of this approach will be given here.

2.1.4 Conformally related theories

Sometimes, it is convenient [306, 337, 255, 114, 293, 88, 112, 84, 89] to usea conformal transformation (1.56) with ρ(X) = −1/2

∫X U(y) dy in (2.9) tosimplify the dynamics by the transition to a new theory with U = 0 andV (X) = V (X) exp(−2ρ). One has to keep in mind, however, that the twotheories need not be equivalent physically. To interpret the results one mustalways return to the original theory. This subtlety was sometimes ignored. Onesource of this misunderstanding seems to be that in field theory the transfor-mation of field variables in a fixed flat Minkowski background is allowed, aslong as such a transformation is regular. For a GDT (2.9) with singular Uthis has to fail for two reasons. The first one is that such a conformal trans-formation must be singular in order to compensate for a singularity in U(X).Still one could argue that locally such a transformation should be permissible.However, and this is the second crucial reason, in gravity the field theory in itsvariables at the same time determines the (dynamical!) manifold upon whichit lives. For a singular conformal transformation the new manifold can possesscompletely different topological properties.

37

2 MODELS IN 1 + 1 DIMENSIONS

An extreme example is the CGHS model (2.8) [71] which from a Schwarz-schild-like topology may be transformed into flat (Minkowski) space. The rea-son can be seen most easily in the transformation behavior of geodesics: onlynull geodesics are mapped onto (in general non-affinely parameterized) nullgeodesics and their corresponding affine parameters are related by [428]:

dτ

dτ∝ e2ρ (2.16)

If ρ approaches infinity at a certain point, by such a singular conformaltransformation geometric properties like geodesic (in)completeness can be al-tered 19 .

In fact, this misunderstanding had been clarified already half a centuryago [153] in connection with the Jordan-Brans-Dicke theory in D = 4 [240,241, 51]. There already in D = 4 a “Jordan-field” X in a D = 4 action like(2.9) with U(X) = const. is introduced. The D = 4 version of identity (1.56),together with an appropriate transformation of X may be used to transformthat action so that the term involving R is reduced to the Hilbert-Einsteinform. At that time a controversy arose whether the latter (the “Einstein-frame”) or the original one (the “Jordan frame”) was the “correct” one. Asargued by Fierz [153] the answer to that questions depends on the definition ofgeodesics, to be used for the determination of the global topology (cf. sect. 1.2).A geodesic depending on the metric g in the Jordan frame is quite differentfrom the one which feels the metric of the conformally transformed g in theEinstein-frame. Of course, for a (globally) regular conformal transformationΩ2, gµν = Ω2 gµν it would be perfectly correct to simultaneously transform ginto the Jordan frame. But then the equation of the geodesic, when expressedin terms of g acquires an additional dependence on Ω(X), i.e. the test particlewould feel a non-geodesic external force exerted by the Jordan-field X.

The confusion in D = 2 probably also originated from the by now veryfamiliar situation in string theory [191,360]. Its conformal invariance does notcarry over automatically to the world-sheet, where it is achieved by imposingthe e.o.m.’s in target space (cf. sect. 2.1.2). String theory yields dilaton gravityas its low energy limit also in higher dimensions. In that context the Jordanframe usually now is called the string frame and the old discussion referred toin the previous paragraph has been resurrected in modern language [78, 126,83, 152, 5].

A simple example of a singular conformal transformation leading to achange of (timelike) geodesic (in)completeness can be found in fig. 9.1 of [428].Another obvious case is provided by the Schwarzschild metric, eq. (3.37) be-low. A (singular) conformal transformation with Ω2 = ξ−1 = (1 − 2M/r)−1

19 Since the usual conformal transformation involved in this context is proportionalto the integral of U(X) and the latter has a singularity in practically all physicallyinteresting models there will be at least one such singular point in addition to the(asymptotic) singularity at X → ∞ [193].

38

2 MODELS IN 1 + 1 DIMENSIONS

and a (singular) coordinate transformation r =∫ r dy/ξ(y) leads to Minkowski

spacetime. This is, of course, a rather trivial consequence of (patchwise) con-formal flatness of any 2D metric. It will be discussed below why ADM mass(sect. 5.1) and Hawking radiation (sect. 6) are, in general, different in confor-mally related theories.

2.2 Equivalence to first-order formalism

Cartan variables have been introduced in sect. 1.2 in order to formulate avery general class of D = 2 first order gravity (FOG) theories by the covariantHamiltonian action

L(FOG) =∫

M2

[Xa(De)a +Xdω + ǫV(XaXa, X)] , (2.17)

which seems to have been introduced first for the special case (V = 0) instring theory [426], then considered for a special model in ref. [230] and finallygeneralized to the in D = 2 most general form (2.17) for a theory of puregravity in refs. [396, 379]. It depends on auxiliary fields Xa and X so that itis sufficient to include only the first derivatives of the zweibeine (torsion) andof the spin connection (curvature). The whole dynamical content is encodedin a (Lorentz-invariant) potential V multiplied by the volume form (1.45). Inthe following very often light-cone coordinates (1.57) and (1.59) will be used:

Xa(De)a = X+(d− ω) e− +X−(d+ ω) e+ (2.18)

We also recall (1.49), the relation 2 ∗ dω = R between spin-connection andcurvature scalar.

The component version of (2.17) with (1.49) follows from the identifica-tion (cf. (1.53),(1.55)) implying the Hodge-duals,

(d± ω)e± ⇒ ǫµνd2x1

2T±µν = (e) τ±d2x ,

dω ⇒ ǫµν∂µωνd2x = (e)

R

2d2x ,

ǫ⇒ −1

2ǫabe

aµebν ǫµνd2x = (e) d2x ,

(2.19)

as

L(FOG) =∫d2x

ǫµν [X+(∂µ − ωµ)e

−ν +X−(∂µ + ωµ)e

+ν

+X ∂µων ] + (e)V(2X+X−, X). (2.20)

The original intention of the formulation (2.17) had been to express ageneral 2D Lagrangian involving the only independent geometric quantities

39

2 MODELS IN 1 + 1 DIMENSIONS

(Ricci scalar R, and torsion scalar T 2 = τaτa = 2τ+τ−, cf. (1.53),(2.15))

L(R,τ2) =∫d2x

√−g h(R, τ 2) (2.21)

in a simpler fashion. The variables Xa and X can be eliminated by the alge-braic e.o.m.-s from variation δXa, δX in (2.17) or (2.20),

τa +∂V∂Xa

= 0 ,R

2+∂V∂X

= 0 , (2.22)

provided the Hessian | ∂2V/∂XA∂XB | does not vanish (XA = X,Xa). Evi-dently this is not always possible, but also, inversely, not every action L(R, τ 2)permits a reformulation as L(FOG) in (2.17) 20 .

Fortunately the relation of (2.17) to GDT (2.9) and especially to modelswith a physical motivation (e.g. SRG) is more immediate and subjected toweaker conditions. Then, instead, only Xa and the torsion-dependent part ofthe spin connection are eliminated by e.o.m.-s which are linear and algebraicand thus may be reinserted into the action 21 . From the definition for ∗T a(1.53) with ω = ωaea in the local Lorentz basis ea, the identities ea∧eb = −ǫab·ǫand ∗ ǫ = 1, one gets

∗T a = ∗ dea − ωa (2.23)

orω = ωaea = ea ∗ dea − ea ∗ T a =: ω − ∗T , (2.24)

where ω represents the torsion free part of the spin connection.The e.o.m. from variation of Xa in (2.17)

dea + ǫab ω ∧ eb + ǫ∂V∂Xa

= 0 , (2.25)

after taking the Hodge dual, multiplication with ea and comparison with theidentity (2.24) yields the relation between ∗T and V

∗T = −ea∂V∂Xa

. (2.26)

Reinserting this algebraic eq. into (2.17) produces

L(FOG)1 =

∫

M2

[Xaǫab

∂V∂ Xc

ec ∧ eb +Xdω − dX ∧ ec ∂V∂Xc

+ ǫV], (2.27)

where the torsion dependent part of ω now has been eliminated, but thedependence on Xa is retained. For potentials ∂V/∂Xa = 0 eq. (2.27) already

20 For a mathematically more precise discussion of this point we refer to ref. [397].21 This equivalence has been published first in ref. [248] for the KV-model [250,251].The proof below follows the formulation used in ref. [141] for the even more generalcase of 2D dilaton supergravity (cf. also sect. 4.2).

40

2 MODELS IN 1 + 1 DIMENSIONS

by the second and third eq. (2.19) can be identified directly as GDT (2.9) withU = 0,V(X) = V (X). When ∂V/∂Xa 6= 0 the e.o.m. from δXa in (2.27) mustbe used,

(dX ∧ ec +Xcǫ)∂2V

∂Xc ∂Xa= 0 , (2.28)

which for nonvanishing Hessian of V, now with respect to the Xa alone, leadsto 22

Xa = ∗ (ea ∧ dX) =ǫµν

(e)eaµ (∂νX) . (2.29)

For easy comparison with the GDT (2.9), before using (2.29) and (2.27),the latter action is rewritten in component form. After cancellation of twoterms with ∂V/∂Xa the final result is very simple

L(dil) =∫d2x (e)

[XR

2+ V(−(∇X)2, X)

], (2.30)

where, according to (2.29), the argument XaXa in V has been replaced by asecond derivative term of X, to be identified also here with the same dilatonfield as in (2.9). The curvature scalar R = ∗2dω refers to the torsionless partof the spin connection in (2.24). Thus it may be expressed equally well directlyby the 2D metric gµν .

For potentials quadratic in the torsion-related variable Xa

V = U(X)XaXa

2+ V (X) (2.31)

the action (2.30) exactly coincides with (2.9) in which torsion had been zerofrom the beginning. As we have used the algebraic (and even only linear)e.o.m.-s to reduce the configuration space of L(FOG) to the one of L(dil), the twoactions lead to the same dynamics. This equivalence can be verified easily aswell by the study of the explicit analytic solution (cf. sect. 3.1). We anticipatealso that at the quantum level the steps above can be simply reinterpreted as“integrating out” the torsion dependent part of ω and Xa [281], cf. footnote61 on page 99.

Apart from covering torsionless dilaton theories (2.9), the first order for-mulation (2.17) also permits the inclusion of 2D theories with nonvanishingtorsion. The choice

VKV =α

2XaXa +

β

2X2 − Λ , (2.32)

after elimination of Xa and X according to (2.22) produces the KV model[250, 251] which is quadratic in curvature and torsion and thus of the type of“Poincare gauge” theory [217, 218]. By our equivalence relation it could alsohave been written as the corresponding dilaton theory (2.9), of course.

22 For potentials V of the form (2.31) eq. (2.29) does not hold necessarily at pointsX0 where U(X0) = 0.

41

2 MODELS IN 1 + 1 DIMENSIONS

Model V (X) U(X) Reference

SRG (D > 3) −λ2

2(D − 2)(D − 3)X

(D−4)(D−2) − (D−3)

(D−2)Xsect. 2.1.1

CGHS −2λ2X − 1X

sect. 2.1.2

JT ΛX 0 sect. 2.1.3

KV β2X2 − Λ α [250, 251]

Fig. 2.1. A selection of dilaton theories

A final remark concerns the overall normalization of our action. By com-paring the term ∝ R in (2.9) and in SRG (2.2), the factor OD−2/(16πGNλ

D−2)is replaced by 1/2 in (2.9). We shall find it more convenient to stick to thelatter normalization so that e.g. for SRG (2.2)

USRG = − (D − 3)

(D − 2)X, VSRG = −λ

2

2(D − 2)(D − 3)X(D−4)/(D−2) . (2.33)

Of course, when introducing matter by spherical reduction (cf. sect. 4.1) thesame overall normalization must be chosen.

The potentials for the most frequently used dilaton models are summa-rized in table 2.1.

2.3 Relation to Poisson-Sigma models

Possibly the most important by-product of the approach to 2D-gravitytheories as presented in this report has been the realization that all models oftype (2.17) are a special case of the more comprehensive concept of Poisson-Sigma models (PSM) [229,379,376,378] with the action

L(PSM) =∫

M2

[dXI ∧AI +

1

2PIJAJ ∧ AI

], (2.34)

defined on a 2D base manifold M2 with target space N with coordinatesXI . Those coordinates as well as the gauge fields AI are functions of thecoordinates xµ on the base manifold (XI(x), AI(x)). The same symbols areused to denote the mapping of M2 to N . The dXI stand for the pullback ofthe target space differential dXI = dxµ∂µX

I and AI are one-forms on M2

with values in the cotangent space of N .The nontrivial (topological) content of a certain PSM is encoded in the

Poisson tensor PIJ = −PJI which only depends on the target space coordi-

42

2 MODELS IN 1 + 1 DIMENSIONS

nates. This tensor may be related to the Schouten-Nijenhuis bracket [383,340]

XI , XJ = PIJ (2.35)

which is assumed to obey a vanishing bracket of P with itself, i.e. nothingelse than a Jacobi identity which expresses the vanishing of the Nijenhuistensor [340]

PIL ∂PJK

∂XL+ cycl (I, J,K) = 0 . (2.36)

Only for PIJ linear in XI (in gravity theories the Jackiw-Teitelboim model[22, 123, 405, 122, 124, 238]), eq. (2.36) reduces to the Jacobi identity for thestructure constants of a Lie algebra and becomes independent of X. In generalthe algebra (2.35) with (2.36) covers a class of finite W-algebras [118]. Earlyversions of this nonlinear algebras from 2D gravity were discussed as constraintalgebra of the Hamiltonian in the context of the KV-model in [192], and withscalar and fermionic matter in [278]. The interpretation as a nonlinear gaugetheory in a related approach goes back to [230, 229].

Although we are dealing with bosonic fields in the present section ournotation anticipates already the graded PSM (gPSM) of supergravity in sect.4.3. Thus the index summation in (2.34) is in agreement with the conventionused in supersymmetry and (just here and in sect. 4.3) we shall also defineinstead of (1.22) the exterior differentiation to act from the right:

d(Ωp ∧ Ωq) = Ωp ∧ dΩq + (−1)qdΩp ∧ Ωq (2.37)

In the bosonic PSM for 2D gravity the action (2.34) reduces to (2.17) withthe identification

XI → (X,Xa) , AI → (ω, ea) . (2.38)

The component PaX of PIJ is determined by local Lorentz transformation forwhich (cf. (2.43) below)

PaX = Xbǫba (2.39)

is the generator. The components

Pab = V ǫab (2.40)

contain the potential V(Y,X) which determines the specific model (Y =XaXa/2).

With the present convention (2.37), the e.o.m.-s from (2.34) become

dXI + PIJAJ = 0 (2.41)

dAI +1

2

(∂PJK

∂XI

)AK ∧AJ = 0 . (2.42)

The identities (2.36) are the essential ingredient to show the validity of the

43

2 MODELS IN 1 + 1 DIMENSIONS

symmetries 23

δXI = PIJ ǫJ , (2.43)

δAI = −d ǫI −(∂PJK

∂XI

)ǫK AJ (2.44)

in terms of the local infinitesimal parameters ǫI(x). Eq. (2.44) reveals the gaugefield property of AI . Whereas for 2D gravity with (2.38), (2.39), (2.40) localLorentz-transformations (ǫI → ǫX) can be extracted easily from (2.43) and(2.44), diffeomorphisms (1.9) are obtained by considering ǫI = ξµAµI [396].Evidently (2.34) is invariant under target space diffeomorphisms too. Onlywhen those transformations are diffeomorphisms also globally, the topology ofM2 remains unchanged. It should be noted that conformal transformationsof the world sheet metric can be expressed as target space diffeomorphisms.Otherwise the problems discussed in sect. 2.1.4 are relevant also at the present,more general, level. Singular target space reparametrization (analogous to theconformal transformations discussed there) could eliminate singularities of themanifold M2 if the identification (2.38) of the PSM variables is retained.Of course, an appropriate simultaneous (singular) redefinition in the relationbetween AI and the Cartan variables could formally keep the topology ofM2 in terms of the new variables intact, at the price of those singularitiesappearing in the relation between AI and (ea, ω).

In 2D gravity the Poisson tensor PIJ is not of full rank, because thenumber of target space coordinates is odd. This also may happen for generalPSM-s and it implies the existence of “Casimir functions”, whose commutatorwith XI in the sense of (2.35) vanishes. In 2D gravity there is only one suchfunction 24

XI , C = PIJ ∂C∂XJ

= 0 . (2.45)

The conservation of C with respect to both coordinates of the manifold

dC = dXI ∂C∂XI

= −PIJ AJ∂C∂XI

= 0 (2.46)

follows from (2.45) and the use of (2.41) in (2.46). Lorentz-invariance requiresC = C(Y,X) with Y = XaXa/2 and thus according to (2.45) C must obey (cf.(2.39) and (2.40))

∂C∂X

− V(Y,X)∂C∂Y

= 0 . (2.47)

23 Applying (2.43) and (2.44) to the commutator of infinitesimal transformationsthe resulting one is again a symmetry only if the e.o.m.-s (2.41) are used, or if PIJ

is linear in X [397].24 For more details regarding generic PSM-s with more Casimir functions C we referto ref. [397].

44

2 MODELS IN 1 + 1 DIMENSIONS

This partial differential equation has a simple analytic solution for the physi-cally most interesting potentials of type (2.31). It will be discussed in connec-tion with the solution in closed form in sect. 3.1.

The rank of the Poisson tensor is not constant in general but may changeat special points in the target space or corresponding points on the world-sheet.A noteable example is a Killing-horizon. Thus, the introduction of so-calledCasimir-Darboux coordinates in which the Poisson tensor

PIJCD =

0 0 0

0 0 1

0 −1 0

(2.48)

is constant only works patchwise. Such singular points may be modelled by“Casimir-non-Darboux” coordinates ZI

PIJCnD =

0 0 0

0 0 Z1

0 −Z1 0

. (2.49)

This allows the extension of patches over a point which is singular in Casimir-Darboux coordinates since for Z1 = 0 the rank changes from 2 to 0. In ad-dition, however, such a coordinate system still may change the singularitystructure of the original theory: e.g. a singularity like the one at X = 0 inSRG is not visible in (2.49); thus the transformation between these coordi-nate systems must be singular at X = 0.

A different route to simplify the target space structure is symplectic ex-tension [142]. By adding an auxiliary target space coordinate one can elevatethe Poisson structure to a symplectic structure. Again, this works only patch-wise in general since the determinant of the Poisson-tensor can be singular.At a physical level, the symplectic extension resembles Kuchar’s geometrody-namics of the Schwarzschild BHs [276]: one introduces a canonically conjugatevariable for the conserved quantity (in Kuchar’s scenario on the world-sheetboundary, in the symplectic extension in the bulk of target space).

For the application to gravity and supergravity theories in D = 2 weshall not need to know more about the PSM formulation. However, the fieldof PSM-theories recently has attracted substantial interest in string theory[382,387] and, quite generally, in mathematical physics in connection with theKontsevich formula for the non-commutative star product 25 [85, 86].

The quantization of general PSM-s [220, 221] essentially follows the ap-proach which will be presented in sect. 7 for the special case of dilaton gravity.

25 For the definition and physical applications of deformation quantization the sem-inal papers [28,29] may be consulted.

45

3 GENERAL CLASSICAL TREATMENT

3 General classical treatment

Simple counting of degrees of freedom shows that dilaton gravity withoutmatter fields in 2D has no propagating modes. Therefore, in terms of suitablevariables, the dynamics may be made essentially trivial. This suggests (but inno way guarantees!) that all classical solutions can be found in a closed form.

As pointed out already above, the fact that the solutions for dilaton the-ories of type (2.9) can be obtained in analytic form had been tested in manyspecific cases [168, 312,173,391,304,174], always using the conformal gauge

ds2 = 2e2ρdx+dx− (3.1)

or the Schwarzschild gauge (cf. (3.34) below). With (3.1) even for a theory assimple as (2.12) the solution of a Liouville equation

ρ = −Λeρ (3.2)

had been necessary.The advantages of the light cone gauge for Lorentz indices combined with

a temporal gauge for the Cartan variables

e+0 = 0, e−0 = 1, ω0 = 0 (3.3)

was realized first [291] in connection with the classical solution of the KV-model [250]. In this gauge the line element for any 2D gravity theory becomes

(ds)2 = 2e+1 dx1(dx0 + e−1 dx

1) (3.4)

which in GR represents an Eddington-Finkelstein (EF) gauge [134, 156]. Interms of the Killing field kα = (0, 1), the existence of which is a property of thesolutions (∂gµν/∂x

1 = 0) and redefining the x0-coordinate by dx0 = e+1 (x0)dx0,the line element (3.4) may be rewritten as

(ds)2 = dx1(2dx0 + k2dx1) (3.5)

with the Killing norm k2(x0) = kαkα containing all the information of the sys-tem (like ρ(x) in the conformal gauge (3.1)). The key advantage of the ingoing(outgoing) EF gauge as compared to the conformal or the Schwarzschild gaugeis that it is free from coordinate singularities on an ingoing (outgoing) horizon.The only singularities of k2(x0) correspond to singularities of the curvature;zeros k2(x0) = 0 describe horizons. This gauge will turn out to be intimatelyrelated to the natural solution of the e.o.m.-s for all models in the first orderformulation.

In the first subsection 3.1 all classical solutions of GDTs without matterare determined in a very simple way, maintaining gauge invariance. Amongthe specific gauge choices the EF gauge emerges as the most natural one, alsofor the analysis of the global structure of these solutions. The most important

46

3 GENERAL CLASSICAL TREATMENT

dilaton gravity models (cf. fig. 2.1) belong to a two parameter sub-family ofall possible theories. This family of models is considered in more detail in thelast subsection.

3.1 All classical solutions

In anticipation of what we shall need in sect. 4.3 we derive the e.o.m.sfrom an action (2.17) supplemented as L = L(FOG) + L(m) by an, as yet,unspecified matter part L(m). The quantities

W± := δL(m)/δe∓, W := δL(m)/δX (3.6)

contain the couplings to matter. A dependence of L(m) on the spin connec-tion or the auxiliary fields X± will be discarded (cf. sect. 4.3). Variation ofδω, δe∓, δX, and δX∓, respectively, yields the e.o.m.-s

dX +X−e+ −X+e− = 0 , (3.7)

(d± ω)X± ∓ Ve± +W± = 0 , (3.8)

dω + ǫ∂V∂X

+W = 0 , (3.9)

(d± ω)e± + ǫ∂V∂X∓ = 0 . (3.10)

The first equation (3.7) can be used to eliminate the auxiliary fields X± interms of e± and dX. The second pair (3.8) is contained in the set of higher-dimensional Einstein equations for the special case of dimensionally reducedgravity. Eq. (3.9) yields the dilaton current W which is proportional to thetrace of the higher-dimensional energy momentum tensor for dimensionallyreduced gravity, and (3.10) entails the torsion condition. If the potential Vis independent of the auxiliary fields X± the condition for vanishing torsion(1.59) is obtained. Of course, in addition to (3.7)-(3.10) the e.o.m.-s for matterδL(m)/δφA = 0 for generic matter fields φA must be taken into account as well.

In the present section we are interested only in the direct solution of (3.7)-(3.10) without fixing any gauge [290], in the absence of matter (W = W± = 0).Linear combination of the two equations (3.8), multiplied, respectively, by X−

and X+ and using (3.7) leads to (Y = XaXa/2 = X+X−)

d(X+X−) + V(Y,X)dX = 0 . (3.11)

This indicates the existence of a conservation law for a function C(Y,X) =C0 = const. which is nothing else than the Casimir function of the Poisson-Sigma model of sect. 2.3. In the application to physically motivated 2D models,potentials of the form (2.31) were found to be the most important ones. Wetherefore concentrate on those. Multiplying (3.11) by the integrating factorexpQ with

Q =∫ X

U(y)dy, (3.12)

47

3 GENERAL CLASSICAL TREATMENT

one obtains the conservation law

d C = 0 (3.13)

for the Casimir functionC = eQ Y + w (3.14)

with

w(X) =

X∫eQ(y) V (y)dy . (3.15)

Of course, any function of C is also absolutely conserved. Therefore, for somespecific model, among others, a suitable convention must be used to fix thelower limit of integration in Q (influencing an overall factor of C) and thelower limit in (3.15) (yielding an additive overall contribution).

We assume X+ 6= 0 which will be realized (cf. (3.8)) if V (X) = 0 doesnot possess a nontrivial solution for X. This is true in SRG, but e.g. in theKV-model [250] such a “point-solution” may appear for certain values of theparameters [377]. If X+ 6= 0 the first component of eq. (3.8) with a new oneform Z := e+/X+

ω = −dX+

X++ ZV (3.16)

determines the spin connection in terms of the other variables. In a similarway eq. (3.7) may be taken to define e−:

e− =dX

X++X−Z (3.17)

From (3.16) and (3.17) and eq. (3.10) with the upper sign, recalling that now

ǫ = −e−e+ = −dX Z (3.18)

for the potential (2.31), the short relation

dZ − dXZU = 0 (3.19)

follows. The ansatz Z = Z expQ, with the same integrating factor (3.12) as theone introduced above for C, reduces eq. (3.19) to dZ = 0. Now by application ofthe Poincare Lemma (cf. sect. 1.2) Z = df is the only “integration” necessaryfor the full solution 26 :

e+ = X+eQdf (3.20)

e− =dX

X++X−eQdf (3.21)

ω = −dX+

X++ V eQ df (3.22)

C = eQX+X− + w(X) = C0 = const. (3.23)

26 This type of solution has been given first in ref. [377], starting from the Darbouxcoordinates for the KV-model [250].

48

3 GENERAL CLASSICAL TREATMENT

Indeed, all the other equations are easily checked to be fulfilled identically. Eq.(3.23) can be used to express X− in terms X and X+, so that in addition to fbeside the constant C0 we have the free functions X and X+. Eqs. (3.7-3.10)are symmetric in the light cone coordinates. Therefore, the whole derivationcould have started as well from the assumption X− 6= 0.

It is straightforward, although eventually tedious in detail, to generalizethe solution (3.20)-(3.23) to dilaton theories where in (2.9) the factor of theRicci scalar is a more general (non-invertible) function Z(X).

Comparing the number of arbitrary functions (f , X, X+) in the solution(3.20)-(3.23), with the three continuous gauge degrees of freedom, the the-ory is a topological one 27 , albeit of a different type from other topologicaltheories like the Chern-Simons theory [384,385,440,441,442]: there is no dis-crete topological charge like the winding number associated to the solutions.In agreement with sect. 2.3 the only variable which determines the differentsolutions for a given action is the constant C0 ∈ R.

The key role of C is exhibited by the line element

(ds)2 = 2e+ ⊗ e− = eQ(X)df ⊗ [2dX + 2(C0 − w(X)) df ] (3.24)

after elimination of X− by (3.23). Whenever a redefinition of X by

dX = dX expQ (3.25)

is possible 28 eq. (3.24) becomes

(ds)2 = 2df ⊗ dX + ξ(X)df ⊗ df , (3.26)

ξ(X) = 2eQ(C0 − w)X=X(X) ; (3.27)

i.e. the EF gauge is obtained when f and X are taken directly as the co-ordinates. Then ξ(X) coincides with the Killing norm k2 (cf. (3.5)). As weshall see in the next subsection the “topological” properties of ξ(X), i.e. thesequence of singularities and zeros (horizons), and the behavior at the bound-aries of the range for X, completely determines the global structure of a so-lution. Eqs. (3.24-3.27), together with the definitions (3.12), (3.15) representthe main result of this section. They are exact expressions for the geometricvariables and thus also for the line element (ds)2 valid for (almost) arbitrarydilaton gravity models without matter. It is now easy to specify other gaugesby taking (3.24) as the point of departure, that is after having solved thee.o.m.-s (3.7)-(3.10) in the simple manner demonstrated above, namely in-serting (x0 = t, x1 = r, F ′ = ∂F/∂r, F = ∂F/∂t)

dX = X ′dr +˙Xdt , df = f ′dr + fdt , (3.28)

27 In the sense that no continuous physical degrees of freedom are present [42].28 If the redefinition is not possible for all X the computation of geodesics shouldstart from (3.24).

49

3 GENERAL CLASSICAL TREATMENT

into (3.26) with (3.27) with appropriate choices for X and f . Of particularinterest are diagonal gauges, a class of gauges to which prominent choices(Schwarzschild and conformal gauge) belong. The absence of mixed terms drdtin the metric can be guaranteed in a certain patch by the gauge conditions

X = X(r) , (3.29)

X ′ + ξf ′ = 0 , (3.30)

and f 6= 0. The solution 29 for f from (3.30)

f = −r∫dx

1

ξ (X(x))· dX(x)

dx+ f(t) =

= −K(X)

2+ f(t) , (3.31)

contains the integral K(X) defined by

K(r) = 2

r∫

r0

dyξ−1(y). (3.32)

The diagonal line element

(ds)2 = ξ[(fdt)2 − (f ′dr)2

], (3.33)

for f = f ′ = 1 attains the form of the conformal gauge. Requiring furthermoredet g = −1 as in Schwarzschild type gauges yields

(ds)2 = ξ(dt)2 − ξ−1(dr)2. (3.34)

As a concrete example we take SRG (2.33) with D = 4, where QSRG =∫X USRG(y)dy = −1/2 lnX with a natural choice y = 1 for the lower limitof integration and wSRG = −2λ2

√X with the lower limit X = 0 (cf.(3.12),

(3.15)). The conserved quantity (3.14) becomes

CSRG =X+X−√X

− 2λ2√X = C0 , (3.35)

and the Killing norm (3.27) reads

k2SRG = ξSRG =

2C0√X

+ 4λ2 . (3.36)

In terms of the new variable X = 2√X the EF gauge (3.26) follows with

ξ(X) = 4C0/X + 4λ2. Further trivial redefinitions (r = X/(2λ), u = 2λf ,

29 f(t) is arbitrary except ˙f 6= 0.

50

3 GENERAL CLASSICAL TREATMENT

dt = du+ dr/ξ) yield the Schwarzschild metric [386]

(ds)2sch =

(1 − 2M

r

)(dt)2 −

(1 − 2M

r

)−1

(dr)2 , (3.37)

where, as expected, C0 is related to the mass M

M = − C0

4λ3. (3.38)

From the steps leading to the line element (3.24) or to one of its subsequentversions it is obvious that these steps can be retraced backwards as easily, sayfrom a Killing norm ξ(X) in (3.27) towards an action. This procedure is notunique, because one function ξ(X) is to be related to two other functions (Uand V ).

For an associated model (2.9) with U = 0, from (3.15) and (3.27) the po-tential V (X) simply results by differentiation dw/dX. However, as emphasizedalready above, these models essentially encode their topology in the parame-ters of the action determined by V (X). In the Killing norm ξ = 2(C0 −w) thevalue of C0 by which the solutions differ in those models only influences theposition of the zeros of ξ, the horizons. For instance, the Reissner-Nordstromsolution of eq. (2.13) follows from the potential anticipated in that equation.As will be seen in sect. 4, when interactions with matter are turned on, C ofthe present chapter is part of the conservation law involving matter contribu-tions. Thus e.g. the influx of matter only changes C and hence in models withU = 0 the position of the horizons, but does not change the “strength” of thesingularities which is fixed here by the mass and the charge Q.

Actually SRG belongs to the interesting class of models in which a givenξ(X) is related to both functions U and V in a very specific way, namely, suchthat C0 = 0 corresponds to a flat (Minkowski) manifold. According to (3.24)this can be simply achieved by the condition

eQw = α = const. , (3.39)

because then the only dependence on X resides in the term ∝ C0 [248]. Thusall models of this class (“Minkowski ground state dilaton theories”) are char-acterized by the relation between U and V in (2.9) following from (3.39):

V (X) =α

2

d

dXexp

−2

X∫U(y)dy

(3.40)

A last related remark concerns the conformal transformation (1.56) of adilaton theory represented by the first order action (2.17). It is always possible,starting from a theory with V (X) and vanishing U(X) to arrive at a model

51

3 GENERAL CLASSICAL TREATMENT

with nonvanishing U(X) by the transformation

ea = ea eQ/2

Xa = Xa e−Q/2

ω = ω +U

2Xa ea

V = e−Q V (X) +XaXa

2U(X) , (3.41)

where Q is defined as in (3.12). In the language of the PSM formulation (sect.2.3) this is an explicit example of a target space diffeomorphism 30 . However, aspointed out already several times, this mathematical transformation connectstwo models with solutions of, in general, completely different topology and/orproperties regarding the role of the conserved quantity C0.

3.2 Global structure

As emphasized in sect. 1.2 the global properties of a solution for thegeometric variables are obtained by following the path of some device onthe manifold. The most important example is the geodesic (1.43) which maypenetrate horizons, but ends when singularities are encountered at finite affineparameter. When no geodesic can reach a boundary of the space-time for finitevalues of the affine parameter the space-time is called geodesically complete,otherwise geodesically incomplete. It should be emphasized that the procedurepresented below does not require the explicit or implicit knowledge of Kruskal-like global coordinates.

For the analysis of the global structure it is convenient to use outgoing oringoing EF coordinates. In a simplified notation [264] the line element (3.26)is written for the outgoing case (f ∝ u, X ∝ r, k2(r) = ξ(r), ξ∞ := +1)

(ds)2out = du(2dr + ξ(r)du) . (3.42)

The ingoing EF gauge (still with ξ∞ = +1)

(ds)2in = dv(2dr− ξ(r)dv) , (3.43)

which will be used here in order to construct patch A of the conformal diagramfor Schwarzschild space-time (see below), could have been obtained if one usesZ := e−/X− for X− 6= 0 in (3.16-3.23). Eq. (3.43) is the most suitable startingpoint for our subsequent arguments in the present example.

For the ingoing metric (xα = v, r)

gµν =

−ξ(r) 1

1 0

(3.44)

30 Cf. the discussion after (2.44).

52

3 GENERAL CLASSICAL TREATMENT

the geodesics (v = v(τ), r = r(τ)) obey

v +v2

2ξ′ = 0 , (3.45)

r − rvξ′ +v2

2ξ′ξ = 0 . (3.46)

These are the e.o.m.-s of (1.42) if the affine parameter τ is identified with s.The Killing field ∂/∂v implies a constant of motion (kα = (1, 0))

gαβkαxβ = r − ξ(r)v =

√|A| = const. (3.47)

which could have been derived as well from (3.46) and (3.45) by taking a properlinear combination. From (3.47) the affine parameter τ can be identified withparameters describing the line-element (3.43)

±(dτ)2 =1

|A|(dr − ξdv)2 = (ds)2 , (3.48)

where the two signs correspond to time-like, resp. space-like geodesics. Thefirst order differential equation

dv

dr=

1

ξ(r)

1 ∓

(1 +

ξ

A

)−1/2 , (3.49)

for the two signs in (3.49) describes two types of geodesics v(1)(r) and v(2)(r)at each point v, r. To avoid confusion with the association of signs, in thenew constant A = ±|A| the two signs from (3.48) have been absorbed so thatnow A > 0 and A < 0 correspond to time-like, resp. space-like geodesics.Inserting (3.49) into (3.48) provides the relation

s(r) = ± 1√|A|

r∫

r0

dy1√

1 + ξ(y)/A. (3.50)

“Special” geodesics A = 0 with

dv

dr= ξ−1(r) , (3.51)

s =

r∫

r0

dy |ξ(y)|−1/2 , (3.52)

and “degenerate” ones obeying A = ξ(r) = const. must be considered sepa-rately.

The advantage of the EF-gauge is visible e.g. in (3.49). The geodesic withthe upper sign in the square bracket clearly passes continuously (C∞) through

53

3 GENERAL CLASSICAL TREATMENT

a horizon where ξ(rh) = 0 which, therefore, does not represent a boundary ofthe patch for the solution (provided A 6= 0).

For a first orientation of the global properties of a manifold it is sufficientto study the behavior of null-directions. Light-like directions are immediatelyread off from (ds)2 = 0 in (3.43):

(dv = 0) → v(1) = const. := v(1)0 (3.53)

(dv 6= 0) → v(2) = K(r) + const. := K(r) + v(2)0 (3.54)

with K defined in (3.32). In terms of the new variables

v = v(1) , u = v(2) −K(r) (3.55)

those null-directions become the straight lines v = const. u = const.. The lineelement in these (conformal) coordinates, of course, exhibits (coordinate) sin-gularities at the horizons. It should be stressed that conformal coordinates arebeing used here only in order to be in agreement with standard diagrammaticrepresentations.

Furthermore, it is convenient to map (ds)2 by a conformal transformationonto a finite region by considering e.g. [428]

(ds)2 =2dudv ξ(r(u))

(1 + u2)(1 + v2)(3.56)

where the powers of the two factors in the denominator are chosen appropri-ately. Light-like geodesics are mapped onto light-like geodesics, i.e. the causalstructure is not changed by this transformation. The conformal diagrams ob-tained in this way have been introduced by Carter and Penrose [81, 357].

The trivial example is Minkowski space with ξ = 1. From (3.56) with(3.32) and (3.57) both light-cone variables

u = v − 2(r − r0) , v = v (3.57)

lie in −∞ ≤ u, v ≤ +∞. By the compactification (3.56), in the line element

(ds)2 = 2dUdV (3.58)

the new variables U = arctan u, V = arctan v are restricted to the finiteinterval −π/2 ≤ U, V ≤ +π/2.

3.2.1 Schwarzschild metric

As a typical nontrivial example for the general procedure in curved space[264] we take the Schwarzschild BH with

ξ(r) = 1 − 2M

r. (3.59)

54

3 GENERAL CLASSICAL TREATMENT

The second light-like coordinate, solving (3.54) with (3.32)

v(2) = v(2)0 + 2r∗ = v

(2)0 + 2r + 4M ln

∣∣∣∣1 − r

2M

∣∣∣∣ , (3.60)

1 2 3 4 5 6

-3

-2

-1

1

A

ξ(r)

B

r

Fig. 3.1.

1 2 3 4 5 6

-6

-4

-2

2

4

6

8

A

dv(2)/dr

B r

Fig. 3.2.

1 2 3 4 5 6

-5

5

10

15

A

v(2)

B r

Fig. 3.3.

is intimately related to the “Regge-Wheeler tortoise coordinate” r∗, but, aswe see below, the actual integration neednot even be performed. It is sufficient tojust regard the general features of thecurves.

The steps from Fig. 3.1 to Fig. 3.3are obvious by inspection. The change toconformal (null) coordinates in Fig. 3.4implies the introduction of u as the hori-zontal axis. Thus the curves v(2) = const.in Fig. 3.3 are to be “straightened” intovertical lines. Above the line u = 0 thispushes the lines r = const. in the regionsA and B of Fig. 3.1 to 3.3 together sothat they all terminate in the point (a) in3.4. For negative u those lines are pushedapart to end in the corners (b) and (c).The value r = 2M corresponds to thelines (b)-(c) and (a)-(e) with the excep-tion of the endpoints (a), (b) and (c).Similarly, the value r = ∞ correspondsto the lines (a)-(d) and (c)-(d), except forthe endpoints (a) and (c). The integration

constant v(2)0 , the endpoint of those curves

for r = 0 in Fig. 3.3, always terminatesat some finite value which is smaller thanall v(1)|r=0 > v

(2)0 . Therefore, the left-hand

boundary in Fig. 3.4 for r = 0 experiencesa “cut off”, described by the line from (a) to (b) 31 . In the language of generalrelativity the nomenclature for the points (a), (c), (d) and (e) is, respectively,i+, i−, i0 and the bifurcation-two-sphere. The lines (a)-(b), (a)-(d), (d)-(c) and(a)-(e) are, respectively, the singularity, I +, I − and the Killing horizon.

We emphasize again that in the EF gauge the whole patch of Fig. 3.4is connected by continuous geodesics. A treatment in the conformal gauge[245], although using simpler geodesics, suffers from the drawback that the

31 Whether this is really a straight line as drawn in Fig. 3.4 depends among otherson the compression factor. The same is also true for the other boundaries at r = ∞,u = −∞. However, the shape of those curves is irrelevant, as far as the topologicalproperties are concerned which are determined by their mutual arrangement only.

55

3 GENERAL CLASSICAL TREATMENT

u=

v=

~u=vA B

r=0

(a)

~

~

v=v=v~

(c)(b)

(0)(1)

(0)(2)

(e)

(d)

8

8+

Fig. 3.4. Conformal coordinates with “compression factor”

A

B

Fig. 3.5. Reorientation of Fig. 3.4: patch A

connection between the regions A andB must be made by explicit continuationthrough the coordinate singularity at the horizon r = 2M . We now turn Fig.3.4 by 45 (Fig. 3.5) and call it patch A. Clearly r → ∞ is complete in thesense of sect. 1.2.1 because there the space becomes asymptotically flat (cf.(3.59)). The singularity at r = 0 can be reached for finite affine parameter.At the edge v = −∞ incompleteness is observed, and (in conformal gauge) acoordinate singularity. Therefore, an extension must be possible.

Indeed, introducing coordinates vB, rB in patch B by

rB = r

vB = K(r) − v(3.61)

with

dr = drB

dv = −dvB + 2ξ−1(rB) drB(3.62)

again transforms the line element (3.43) into itself, but with the replacementr → rB, v → vB. Moreover, we obtain the same differential equations as theones in the patch A except for the change of sign v → −v (cf. (3.61)). As aconsequence patch B is given by Fig. 3.6, the mirror image of Fig. 3.5.

Further patch solutions C and D can be obtained by simply changing bothsigns on the right-hand side of (3.55) resp. (3.61), yielding the patches of Fig.3.7. Now the key observation is that the lines r = const. correspond to thesame variable in the regions A of A and A′ of B. The same is true in B′ and B′′

for B and C, and for A′′ and A′′′ for C and D. Superimposing those regions wearrive at the well-known Carter-Penrose (CP) diagram for the Schwarzschildsolution (Fig. 3.8).

56

3 GENERAL CLASSICAL TREATMENT

A’

B’

Fig. 3.6. Mirror image of Fig. 3.5: patch B

B’’A’’ A’’’

B’’’

Fig. 3.7. Further flips: patches C and D

Fig. 3.8. CP diagram for the Schwarzschild solution

3.2.2 More general cases

We have glossed over several delicate points in this procedure [263, 264].As pointed out at the beginning of this section in a more complicated case acareful analysis of geodesics is necessary at external boundaries and, especially,at the corners of a diagram like Fig. 3.8. One may encounter “completeness”in this way in some corners, but also in the middle of a diagram, resemblingFig. 3.8. Still, in all those cases the analysis does not need the full solutionof the geodesic equations (3.45), (3.46). It suffices to check their properties inthe appropriate limits only.

Also the diagram alone may not be sufficient to read off some important“physical” properties. The line of reasoning, passing through the Figs. 3.1 - 3.5shows that obviously all Killing norms ξ(r) with one singularity, one (single)zero and ξ∞ = 1 will lead to the same diagram Fig. 3.8. However, e.g. theincomplete boundary at the singularity may behave differently. For the CGHSmodel [71] in which the power r−1 (or r−(D−3) for SRG from D > 4) is replacedby an exponential e−r, only time-like geodesics are incomplete at r = 0. Thismeans that light signals take “infinitely long time” to reach the singularity(null completeness) , whereas massive objects do not.

Another important point to be checked is whether by superimposingpatches around some center as the bifurcation 2-sphere in Fig. 3.8 one re-ally arrives at B = B′′′ (uniqueness), or whether this can be obtained only byimposing certain further conditions. Otherwise not a planar picture like Fig.3.8, but an infinite continuation in the form of a “spiral staircase” extending

57

3 GENERAL CLASSICAL TREATMENT

Fig. 3.9. Basic patch of Reissner-Nordstrom metric

Fig. 3.10. Penrose diagram for Reissner-Nordstrom metric

above (and below) the drawing plane may emerge.When ξ exhibits two zeros as for the Reissner-Nordstrom metric

ξRN = 1 − 2M

r+Q2

r2(3.63)

the basic patch of Fig. 3.9 with the superposition method described above,leads to the well-known [234,265] one-dimensional infinite periodic extensionof Fig. 3.10.

When even three zeros are present in the Killing norm the global diagrambecomes periodic in two directions (cf. e.g. Fig. 2.1-2.3 in [197]), i.e. coversthe whole plane 32 .

Such one, two or more dimensional lattices exhibit discrete symmetries,

32 Here we have even discarded the “uniqueness”-problem, referred to above.

58

3 GENERAL CLASSICAL TREATMENT

identify

Fig. 3.11. A possible RN-kink (cf. [267])

which, in turn, may be used to compactify manifolds by identifying certaincurves. If this identification occurs in a nontrivial manner, “solitonic” mani-folds are produced [267], as illustrated by the example Fig. 3.11.

3.3 Black hole in Minkowski, Rindler or de Sitter space

As a further illustration for the application of the methods described inthis section a family of dilaton gravities [249] is considered which includes thephysically most interesting models describing a single BH in Minkowski (cf.(2.33)), Rindler or de Sitter space. The potentials U and V are assumed to beof a simple monomial form,

U(X) = − a

X, V (X) = −B

2Xa+b. (3.64)

Among the constants a, b and B only a and b distinguish between physicallyinequivalent models. B plays the same role as λ2 in (2.33), defining an overallscale factor.

In the line element (3.24) the functions Q and w read (cf. (3.12) and(3.15))

eQ(X) = X−a , w(X) = − B

2(b+ 1)Xb+1 , (3.65)

59

3 GENERAL CLASSICAL TREATMENT

so that

(ds)2 = X−adf ⊗[2dX + 2

(C0 +

B

2(b+ 1)Xb+1

)df

]. (3.66)

The equation w = C0 has at most one solution on the positive semi-axis. Hencethe metric (3.66) exhibits at most one horizon.

The most interesting models correspond to positive a for which the func-tion X−a diverges at X = 0. For X > 0 in terms of the alternative definitionof the dilaton field Φ (2.11) the dilaton action (2.10) with the potentials (3.64)becomes

L(dil) =1

2

∫d2x

√−ge−2Φ[R + 4a(∇Φ)2 −Be2(1−a−b)Φ

], (3.67)

which may be more familiar to a string audience.It is also instructive to calculate the scalar curvature:

R = 2aC0Xa−2 +

Bb

b+ 1(b+ 1 − a)Xa+b−1 . (3.68)

In what follows only models with b 6= −1 will be considered (although b = −1can be analyzed too [249]). For the “ground state” solutions C0 = 0 only thesecond term in (3.68) survives. If a = b + 1 or b = 0 the scalar curvature ofthe ground state is zero. A more detailed analysis shows that the first case(a = b+1) corresponds to Minkowski space, and the second (b = 0) to Rindlerspace. The condition a = b + 1 for the Minkowski ground state models alsofollows from (3.40). If a = 1 − b the ground state has constant curvature andcorresponds to (anti-) de Sitter space.

For the general solutions (3.66) with C0 6= 0 it follows from (3.68) that,depending on the values of a and b, they may show a curvature singularity atX = 0, at X = ∞, or at both values. In the special cases considered abovethere could be only one singularity. Therefore, these models describe (in asomewhat generalized sense) a single BH immersed in Minkowski, or Rindler,or de Sitter space.

Many interesting and important models belong to the two-parameter fam-ily of this section. The SRG models (2.2) for general dimension D lie on theline a = b + 1 between a = 1/2 (this point corresponds to D = 4) and a = 1.As D grows, these models approach the point a = 1, corresponding to theCGHS model [71]. The point a = 0, b = 1 describes the Jackiw-Teitelboimmodel [22,405,238]. Lemos and Sa [298] gave the global solutions for b = 1−a,Mignemi [323] considered a = 1 and all values of b. The models of [148] cor-respond to b = 0 and a ≤ 1. The general solution for the whole plane wasobtained in ref. [249] and is summarized in Fig. 3.12.

In order to calculate quantities like the ADM mass and the Hawking fluxit is essential to re-write the line element (3.66) for asymptotically Minkowski,de Sitter and Rindler models in such a form that it becomes the standard one

60

3 GENERAL CLASSICAL TREATMENT

b

10 a2

1

at singularity

incomplete

null

SBH

b=a-1

Fig. 3.12. “Phase” diagram showing the CP diagrams related to the (a- andb-dependent) action (3.67). Bold lines in those diagrams denote geodesically in-complete boundaries. Spherically reduced models lie on the half-line b = a − 1,b ≤ 0, the endpoint of which corresponds to CGHS. The special case of SRG fromD = 4 is depicted by the point labelled by SBH.

in the asymptotic region. Here we give an explicit expression for the asymptot-ically Minkowski solutions only (other cases can be found in ref. [299]) wheresuch a representation is possible for a ∈ (0, 2). We repeat the steps which ledbefore from the metric (3.24) to the Schwarzschild black hole (3.37). Namely,we first introduce the coordinates X = r, f = u (cf. (3.26))

u =

√B

af sign(1 − a) , r =

√a

B

1

|1 − a|X1−a (3.69)

and write the metric in EF form (3.27) where ξ now reads

ξ (r) = 2C0 |1 − a| aa−1 r

aa−1

(B

a

) 2−a2(a−1)

+ 1 (3.70)

Following the steps after (3.28) one arrives at the generalized Schwarz-schild metric (3.34) with ξ as in (3.70).

For a ∈ (0, 1) the asymptotic region corresponds to r → ∞, while for

61

3 GENERAL CLASSICAL TREATMENT

a ∈ (1, 2) it is reached with the limit r → 0. In both cases ξ(r) → 1, and themetric assumes the standard Minkowski form gµν → diag(1,−1).

For the asymptotically Rindler and de Sitter solution with a belonging tothe same interval, a ∈ (0, 2), the quantity ξ becomes

ξ(r) = rB122 −Mr

aa−1 (3.71)

for the Rindler solutions, and

ξ(r) = r2B2 −Mra

a−1 (3.72)

for the de Sitter solutions. Explicit expressions for the constants M and B2

and for the variables t and r can be found in ref. [299]. The presence of typicallinear (Rindler) and quadratic (de Sitter) terms in (3.71) and (3.72) shouldbe noted.

Although the CGHS model a = 1, b = 0 belongs to the family of theasymptotically Minkowski models considered above, the equations (3.69) aresingular at a = 1. Appropriate coordinates for this case are

u = −√Bf , r = − 1√

BlnX . (3.73)

The line element (3.27) now contains

ξ(r) = 1 +2C0e

√Br

B. (3.74)

A somewhat non-standard feature of (3.74) is that the asymptotic region issituated at r → −∞.

62

4 ADDITIONAL FIELDS

4 Additional fields

In 2D there are neither gravitons nor photons, i.e. no propagating physicalmodes exist [42]. This feature makes the inclusion of Yang-Mills fields in 2Ddilaton gravity or an extension to supergravity straightforward. Indeed, bothgeneralizations can be treated again as a PSM (2.34) with generalized AI andXI . More locally conserved quantities (Casimir functions) may emerge andthe integrability concept is extended.

Beside gauge fields also scalar and spinor fields may be added. If the latterare derived from higher dimensions through spherical reduction they are non-minimally coupled to the dilaton. The introduction of those fields in generaldestroys the integrability. Only in special cases exact solutions still can beobtained. An example are chiral fermions [278] or (anti-)selfdual scalars [356].

4.1 Dilaton-Yang-Mills Theory

The interaction with additional one-form Yang-Mills fields Aa related tolocal gauge transformations belonging to a compact Lie group G is simply in-cluded in (2.17) by introducing further auxiliary variables Za (additional tar-get space coordinates in the PSM language) in the dilaton-Yang-Mills (DYM)action

L(DYM) =∫

M2

[XaDea +Xdω + Za(DA)a

+ ǫV(XaXa, X, c1(Z), c2(Z), . . . cn−1(Z)) ] . (4.1)

The gauge covariant derivative

(DA)a = dAa + gfabcAbAc (4.2)

contains the structure constants fabc and the gauge coupling g. The potentialV, invariant under local Lorentz transformations and transformations G nowalso may depend on the Casimir invariants ci of the group G. For instancein G = SU(N) there are N − 1 independent invariant polynomials of degree2, 3, . . . , N − 1 in terms of the components Za.

The abelian case (f = 0 in (4.2)) is especially simple. There V onlydepends on the single variable Z. Variation of (4.1) with respect to A directlyyields dZ = 0, i.e. Z = Z0 = const is conserved, an additional Casimir functionin the PSM-interpretation of (4.1). Because Z = Z0 is the result of solving adifferential equation it cannot be simply reinserted into (4.1). Variation of Zyields

dA = −ǫ ∂V∂Z

∣∣∣∣∣Z=Z0

. (4.3)

The remaining e.o.m-s can be solved as for (2.17) by (3.20-3.23) with just anadditional dependence on the constant Z = Z0 in the potential V in (3.22)

63

4 ADDITIONAL FIELDS

and in w(X,Z) of (3.23) [289, 305].For a nonabelian gauge group G the coupling between the gauge fields

Aa and the geometric variables is somewhat more complicated, but as a PSMit can be treated again along the lines of sect. 3.1. For a potential of thetype V(X+X−, X) + α(X)ZaZa the solution of the geometric sector can beobtained like the one from of an ordinary GDT, because ZaZa is constanton-shell [263, 264, 397]. Such a potential correctly reproduces the e.o.m.-s forordinary D=2 dilaton-Yang-Mills theory:

(DA)a = −ǫα(X)Za , d(ZaZa) = 0 (4.4)

Some explicit solutions for the dilaton-Maxwell-Scalar system can be found inref. [353].

The action (4.1) does not contain the special case which emerges fromspherical reduction of Einstein-Yang-Mills (EYM) in D = 4. The reasonfor this is obvious: While the Killing vectors ξs associated with sphericalsymmetry act trivially on the metric, δξsgµν = 0 with δξs being the Lie-derivative with respect to ξs, a corresponding transformation of the gauge fieldAµ → Aµ − εsδξsAµ can be compensated by a suitable gauge transformation,Aµ → Aµ + εsDµWs, such that δξsAµ = DµWs [162]. Already for SU(2) themost general solution compatible with spherical symmetry, sometimes called“Witten’s ansatz” [438], yields terms which are not described by (4.1), namelyan additional (minimally coupled) charged scalar field with dilaton-dependentmass term and quartic self-interaction 33 .

4.2 Dilaton Supergravity

Already a long time ago the superfield approach has been applied insupersymmetric extensions of GDTs [223]. Superfields are expressed in termsof supercoordinates zM = (xm, θµ), where the bosonic coordinates xm aresupplemented 34 by anticommuting (Grassmann) coordinates θµ.

We assume the latter to be Majorana spinors, i.e. we restrict ourselves toN = 1 superspace inD = 2. Beside the Z2-grading property for coordinates 35 ,

zMzN = (−1)MNzNzM , (4.5)

33 Cf. [427] for a comprehensive review on non-trivial EYM solutions.34 Throughout this subsection we employ the generally accepted supersymmetrynotation with Latin indices from the middle of the alphabet for the holonomicbosonic coordinates xn (denoted “xµ” in the rest of this Report). Greek indices arereserved for the fermionic coordinates θµ. A similar notation is used in tangentialspace for Lorentz vectors XA = (Xa,Xα) with indices from the beginning of thealphabet.35 In the exponent of (−1), M or N is zero for a bosonic component, for a fermionicone M resp. N are 1.

64

4 ADDITIONAL FIELDS

the derivatives with respect to z are defined to act to the right→

∂M =→

∂/∂zM .Only those derivatives will appear in the following, so the arrow will bedropped.

Any vector-field in superspace V = V M∂M is invariant under non-degen-erate coordinate changes zM → zM (z)

V M∂M = V M ∂zL

∂zM∂zN

∂zL∂

∂zN, (4.6)

which shows the advantage of the conventional summation of indices “ten tofour” in supersymmetry, already introduced in the section on PSM-s (sect.2.3).

Now any formula of differential geometry in ordinary space of the PSMcan be copied to superspace notation. The p-forms eq. (1.14), of sect. 1.2.1turn into the same expressions written in terms of dzM instead of dxµ, adheringstrictly to the summation of supersymmetry:

Φ =1

p!dzMp ∧ · · · ∧ dzM1ΦM1···Mp

(4.7)

Also the external differential is defined in agreement with sect. 2.3 as

dΦ =1

p!dzMp ∧ · · · ∧ dzM1 ∧ dzN∂NΦM1···Mp

(4.8)

which implies the Leibniz rule (2.37) for superforms, already introduced in thatsection in anticipation of the graded PSM (gPSM) approach below. Clearlythe (anti-)symmetry properties of the tensor ΩM1 ...Mp

now depend on thegraded commutation properties (4.5) of the indices. Instead of (1.13), (1.14)the one-forms of superzweibein and superconnection are EM

A and

ΩMAB = ΩMLA

B , (4.9)

where in (4.9) the simplification for two dimensions with

LAB =

ǫab 0

0 −12(γ∗)α

β

(4.10)

already has been taken into account. The fermionic part in (4.10), the gener-ator of Lorentz transformations in spinor space, agrees with (1.61). Covariantderivatives, defined by analogy to (1.23)

∇MVA = ∂MV

A + ΩMVBLB

A ,

∇MVA = ∂MVA − ΩMLABVB .

(4.11)

lead to the expressions for the components of supercurvature and supertorsion

RMNAB =

(∂MΩN − ∂NΩM(−1)MN

)LA

B =: FMNLAB, (4.12)

65

4 ADDITIONAL FIELDS

TMNA = ∂MEN

A + ΩMENBΩMLB

B − (M ↔ N)(−1)MN (4.13)

Again Bianchi identities, direct generalizations of (1.26) and (1.28), must hold(cf. [142], eqs. (1.58), (2.20)). They restrict the component fields, contained inEM

A,ΩM when these expressions are expanded in terms of (a finite numberof) ordinary fields, appearing as coefficients of powers Θµ and Θ2 = ΘµΘµ.It turns out that to deal with supertorsion it is more convenient to use itsanholonomic components:

TABC = (−1)A(B+N) EB

NEAMTMN

C (4.14)

The literature on 2D supergravity (cf. e.g. [223,60,315,370,368,297]) is stronglyinfluenced by its close relation to string theory, where the bosonic torsion van-ishes, Tab

c = 0. It uses the further constraints on TABC

Tαβc = 2i (γc)αβ, Tαβ

γ = 0 , (4.15)

the first of which is dictated by the requirement that in the limit of globaltransformations ordinary supersymmetry should be restored. The second oneturns out to be a convenient choice, because then the Bianchi identities in aWess-Zumino type gauge are fulfilled identically [141].

In the application to 2D gravity including bosonic torsion it seems nat-ural to retain (4.15), but to simply drop the zero bosonic torsion condition.However, as a consequence of the Bianchi identities it turned out [141] that thesuperfield components, obtained in this manner, did not permit the construc-tion of 2D supergravity Lagrangians with nonvanishing bosonic torsion, afterall. Only after replacing (4.15) by the weaker set (Fαβ = Eα

M EβN FNM(−1)M ,

cf. (4.12))

(γa)βα Tαβ

c = −4iδac, Tαβ

γ = 0, (γa)βαFαβ = 0 (4.16)

a solution can be found [142]. However, the mathematical complexity of thisapproach becomes considerable.

Instead, we turn to the generalization of the PSM, adding fermionic targetspace coordinates χα and corresponding Rarita-Schwinger 1-form fields ψα tothe degrees of freedom in (2.34), (2.38) as 36

XI = (X,Xa, χα) ,

AI = (ω, ea, ψα) .(4.17)

Apart from that, the PSM action retains the form (eq. (2.34)).

36 Under the title “free differential algebras” this has been proposed for simplemodels in ref. [237], cf. also refs. [228,229,398].

66

4 ADDITIONAL FIELDS

Both χα and ψα denote Majorana fields, when, as in what follows, N = 1supergravity is considered 37 . The graded Poisson tensor PIJ = (−1)IJ+1PJI

instead of (eq. (2.36)) is now assumed to fulfill a graded Jacobi identity

PIL→

∂LPJK + gcycl(IJK) = 0 . (4.18)

Except for the range of fields to be summed over, the e.o.m-s are again (2.41),(2.42). The symmetries (2.43), (2.44) depend on infinitesimal local parametersǫI = (ǫφ, ǫa, ǫα).

The mixed components PαX are constructed by analogy to PaX in (2.39)with the appropriate generator (−γ∗/2) of Lorentz transformations in 2Dspinor space (cf. eq. (4.10)). Then dǫα in the second set of eq. (2.44) acquiresan additional term casting it into the covariant combination (Dǫ)α, with co-variant derivative (1.63). This is precisely the form required for the (dilatondeformed) supergravity transformation of the “gravitino” ψα. As the Poissontensor PIJ also here is not of full rank, Casimir functions C(Y, φ, χ2) existwhich, following the same line of argument as in sect. 2.3 obey dC = 0. FromLorentz invariance a bosonic C in supergravity is of the form

C = c+ 12χ2c2 , (4.19)

where c coincides with the quantity denoted by C in the pure bosonic case(2.46,3.23). However, also fermionic Casimir functions may appear (see below).

The determination of all possible minimally extended supergravities re-duces to the solution of the Jacobi identities (4.18). In the general ansatz forPIJ

Pab = V ǫab , (4.20)

Pbφ = Xa ǫab , (4.21)

Pαφ = −1

2χβ(γ∗)β

α , (4.22)

Pαb = χβ(F b)βα , (4.23)

Pαβ = vαβ +χ2

2vαβ2 , (4.24)

the function

V = V(X, Y ) +χ2

2v2(X, Y ) (4.25)

contains the original bosonic potential V. As explained above, eqs. (4.21) and(4.22) are fixed completely by Lorentz invariance. This invariance also impliesthat the (symmetric) spinor-tensor V αβ in (4.24) can be expanded furtherinto three scalar functions of X and Y , multiplying the symmetric matrices

37 In higher N supergravity Majorana fields(1)

χα · · ·(N)

χα and corresponding(i)

ψα areneeded with an additional SO(N) symmetry.

67

4 ADDITIONAL FIELDS

(γ∗)αβ, γαβa , Xa(γ∗γa)

αβ:

V αβ = Uγ∗αβ + i UXaγaαβ + i UXaǫa

bγbαβ =

= vαβ +χ2

2vαβ2 .

(4.26)

Each function U again has a pure bosonic part and, as indicated in the secondline of (4.26), a term proportional to χ2. In a similar manner the spinor F b

in (4.23) can be expressed in terms proportional to δβα, (γa)β

α(γ∗)βα which

finally requires eight scalar functions of X and Y , multiplied by appropriatefactors constructed with the help of Xa and ǫab [140]. In (4.23) the multiplyingfactor χ precludes terms with factor χ2 in F . Thanks to (4.21) and (4.22) theJacobi identities with one index referring to X are fulfilled automatically. Theremaining ones can be solved algebraically, provided a quite specific sequenceof steps is followed (for details see [140]). Three main cases are determined bythe rank of the 2 × 2 spinor matrix vαβ in (4.26).

For full rank (det vαβ 6= 0) the solution is found to depend on five scalarfunctions of X, Y and the derivatives thereof, if the bosonic potential V in(4.25) is assumed to be given.

If the fermionic rank is reduced, beside the bosonic Casimir function(4.19) one or two fermionic ones exists. They are of the generic form

C(±) = χ±∣∣∣∣∣X−−

X++

∣∣∣∣∣

±1/4

c(±) (X, Y ) (4.27)

and owe their Lorentz invariance to the interplay of the abelian boost trans-formations exp(±β) of the light cone coordinates X±± related to Xa, andexp(±β/2) of the chiral spinor components χ± of χα. For fermionic rank 1 thegeneral solution contains four arbitrary functions beside V and one additionalCasimir function of type (4.27). For rank zero of the fermionic extension inPIJ beside (4.19) both fermionic Casimir functions (4.27) are conserved andthree functions remain arbitrary for a given bosonic potential.

In order to avoid solving differential equations by imposing the Jacobiidentities (4.18) also for reduced fermionic rank, it is important to make in-tensive use of the information on the Casimir functions.

The arbitrariness of the solution of the Jacobi identities can be under-stood as well by studying reparametrizations of the target space, spanned bythe XI in the gPSM. Those reparametrizations may generate new models.Therefore, they can be useful to create a more general gPSM from a simplerone, although this approach is difficult to handle if V in (4.25) is assumed tobe the given starting point. However, the subset of those reparametrizationsmay be analyzed which leaves a given bosonic theory unchanged. Again thesame number of arbitrary functions emerges for the different cases describedin the paragraphs above.

As an illustration we quote eq. (4.252) from [140] which represents one (ofmany) supergravity actions which possess a bosonic potential (2.31) quadratic

68

4 ADDITIONAL FIELDS

in torsion 38 :

L(QBT) =∫

MXdω +XaDea + χαDψα + ǫ

(V + 1

2XaXaU + 1

2χ2v2

)

+U

4Xa(χγ3γaγ

bebψ) − i V

2u(χγaeaψ)

− i

2Xa(ψγaψ) − 1

2

(u+

U

8χ2)

(ψγ3ψ),

v2 = − 1

2u

(V U + V ′ +

V 2

u2

).

(4.28)

In this formula U(X) is the quantity defined in (2.31). The prepotential u isrelated to U(X) and V (X) by

u2(X) = −2 e−Q(X) w(X) , (4.29)

where Q and w have been defined in (3.12) and (3.15).The supergravity transformations of ea and ψα with small fermionic pa-

rameter ǫ for this action (4.28)(cf. eqs. (4.255) - (4.259) of [140] and footnote38) are of the form

δ ea = i (ǫγaψ) + · · · , δ ψα = −(D ǫ)α + · · · . (4.30)

Thus, they contain the essential terms, but, not shown here, also others, be-cause of the deformation by the dilaton field. SRG is the special case (2.33)for U and V , but as a supergravity extension (4.28) is not unique.

A generic property of the fermionic extensions obtained in this analysis isthe appearance of obstructions, which is a typical feature of supersymmetrictheories (cf. e.g. [133]). The first type of those consists in singular functions ofthe bosonic variables X and Y , multiplying the fermionic parts of a supergrav-ity action, when no such singularities are present in the bosonic part. But evenin the absence of such additional singularities, a relation like (4.29) betweenthe original potential and some prepotential u, dictated by the correspondingsupergravity theory, either leads to a restriction of the range of X and/or Yas given by the original bosonic one, or even altogether prevents any exten-sion of the latter. Remarkably, a known 2D supergravity model like the oneof Howe [223] which originally had been constructed with the full machineryof the superfield technique, is one example which escapes such obstructions.There, in our language, the PSM potential V = −2λ2X3 permits an expansionin terms of the prepotential u(X) through V = −du2/dX because Q = 0. Anexample where obstructions seem to be inevitable is the KV-model [250] withquadratic bosonic torsion.

On the other hand, the supergravity extension of SRG following from theaction (4.28) is free from such problems. However, it is not the only possible

38 We comply with our present notation by the replacements φ→ X,Z → U(X) inref. [140]. Furthermore an arbitrary constant is now fixed as u0 = 1.

69

4 ADDITIONAL FIELDS

extension of the bosonic theory. Indeed, the hope that a link could be foundbetween the possibility of reducing the arbitrariness of extensions referred toabove, and of the absence of such obstructions, did not materialize. Severalcounter examples could be given including different singular and nonsingularextensions of SRG.

Another very important point concerns the “triviality” of supergravityextensions, proved earlier by Strobl [398]. It was based upon the observationthat locally a formulation of the dynamics in terms of Darboux coordinatesallows to elevate the infinitesimal transformations (2.43,2.44) (on-shell) to fi-nite ones. Then the latter may be used to gauge the fermionic fields in 2Dsupergravity to zero. Providing now the explicit form of those Darboux coor-dinates in the explicit solution of a generic model, additional support has beengiven to the original argument of ref. [398]. However, the appearance of theobstructions and the ensuing singular factors in the transition to the Darbouxcoordinates may introduce a new aspect. When those new singularities ap-pear at isolated points without restriction of the range for the original bosonicfield variable, they may be interpreted and discarded much like coordinatesingularities. Another way to circumvent this problem in the presence of re-strictions to the range and thus to retain triviality is to allow a continuationof our (real) theory to complex variables. This triviality disappears anyhow,when interactions with additional matter fields are introduced, obeying thesame symmetry as given by the gPSM-theory. A proposal in this direction canbe found in ref. [237].

In order to eliminate the arbitrariness of superdilaton extensions the onlyviable argument seems to start from a supergravity theory in higher dimen-sions (e.g. D = 4) and to reduce it (spherically or toroidally) to a D = 2effective theory. It turns out that the Killing spinors needed in that case mustbe Dirac spinors, requiring the generalization of the work of ref. [140] to (atleast) N = 2, where, however, the same technique of gPSM-s can be applied.

As in the bosonic case an action like (4.28) or, better, directly its gPSMform (2.34) can be converted into a dilaton theory by elimination of the torsiondependent part of the spin connection and of Xa. Also for supergravity theequation for Xa is independent of the potential V, eq. (2.29) being replacedby

Xa = −eamǫmn[(∂nX) +1

2(χγ∗ψn)] . (4.31)

The corresponding dilaton action in the gPSM notation becomes

L =∫

M

[Xdω + χα(Dψ)α +1

2PAB |Xa eB eA ] , (4.32)

where ω is the torsionless part of the curvature as in (2.27) and |Xa meansthat the components of the Poisson tensor (in the anholonomic basis) are tobe taken with Xa given by (4.31). When (4.32) is written in components (cf.eq. (4.246) of [140]) with the bosonic potential (2.31) for the Howe model

70

4 ADDITIONAL FIELDS

(V = −12X3, U = 0), it can be checked that the superdilaton theory, obtained

in this way differs from the direct superextension of the bosonic theory [354].The reason is that in our approach X is directly promoted to be the bosoniccomponent of the superfield, whereas in ref. [354] the dilaton represents thebosonic part of yet another scalar superfield.

4.3 Dilaton gravity with matter

4.3.1 Scalar and fermionic matter, quintessence

For the inclusion of scalar matter as in sect. 2.1 we start with the ex-ample of spherical reduction of Einstein theory. When massless scalar fieldsare coupled minimally in D dimensions ((∇(D)Φ)2 = gMN∂MΦ ∂NΦ ; M,N =0, 1, . . . D − 1)

L(m,D)(φ) =

1

2

∫dDx

√−g(D) (∇(D)Φ)2 , (4.33)

for scalar fields φ = φ(x0, x1) the 2D reduced action in terms of the compo-nents of gMN as derived from the line-element (2.1), becomes

L(m)(φ) =

OD−2

λD−2

∫d2x

√−g F (X) (∇φ)2 , (4.34)

FSRG(X) =X

2. (4.35)

Such an interaction is an example of nonminimal coupling (F (X) 6= const.) inthe reduced case. Admitting a general function F (X), a dilaton field depen-dence different from SRG (eq. (4.35)) can be covered as well. An especiallysimple theory follows for F = const., minimal coupling at the D = 2 level 39 .Below we shall absorb the relative factor 1/2 between the dilaton action (2.2)and the first order action (2.17) – which of course must also be adjusted prop-erly in the matter action – into the coupling function F (X).

Dropping as in our convention for the geometrical part of the FOG actionin (2.17) the prefactor 2OD−2/λ

D−2, the action (4.34) can be written also as

L(m)(φ) =

1

2

∫

M2

F dφ ∗ dφ =1

2

∫F (dφea) ∗ (dφeb) ηab . (4.36)

In order to avoid the delicate subject of Killing spinors, necessary forthe (spherical) reduction of fermions 40 (cf. e.g. [358] for D = 4) we shallonly deal with fermions introduced directly in D = 2. The diffeomorphism

39 This is not to be confused with F ∝ X, corresponding to minimal coupling in theoriginal dimension in the case of SRG. We will refer to that case as non-minimalcoupling.40 Such a reduction yields a dilaton dependent “mass” term and coupling of spinorsto the auxiliary fields X±.

71

4 ADDITIONAL FIELDS

covariant generalization (cf. (1.63)) of the Dirac action

L(m)(ψ) =

i

4

∫d2x

√−g eµa (ψγa↔D

(ψ)

µ ψ) (4.37)

as in Minkowski space must contain a two-sided derivative a↔Db = a(Db) −

(Da)b in order to yield a real action. The cancellation of ω in that derivative

is a peculiar feature of D = 2, i.e. the simplification↔Dµ =

↔∂µ is possible

there. Thus both interactions (4.36) and (4.37) do not depend on the spinconnection, as anticipated already in deriving the (classical) e.o.m.-s withmatter, eqs. (3.6)-(3.10).

An additional geometrical degree of freedom may also appear already atthe D = 4 level. Recently in connection with the observation of supernovaeat high values of the redshift [367, 365, 366, 359] the validity of the Hilbert-Einstein theory has been put into doubt [12]. The simplest theoretical descrip-tion requires the introduction of a (still very small) cosmological constant Λ(de Sitter theory). Therefore, extensions of the Einstein theory towards theold Jordan-Brans-Dicke (JBD) theory [153,241,51], have been revived, wherealready at D = 4 an additional scalar field Φ (Jordan field, “quintessence”) isassumed to exist [433,431,80,447,363]. Then, already in D = 4 an action like(2.9) is postulated with X → Φ and appropriate assumptions for functionsU(4)(Φ), V(4)(Φ) so that the “effective cosmological constant” is driven to itspresent (small) value 41

L(Q) =∫d4x

√−g(4)

[R(4) Φ + U(4)(Φ) (∇(4)Φ)2 + V(4)(Φ)

]. (4.38)

After spherical reduction of (4.38), even without including a genuine matterinteraction a 2D theory emerges where the (in D = 4 geometric) variable Φturns into something like an additional scalar field (beside the genuine D = 2dilaton field X). These “two-dilaton theories” have been studied in more detailin [194]. The most interesting feature of such theories is that one dilaton fieldplays the role of a “geometric” dilaton field inD = 2 and the other one behaveslike matter, providing continuous physical degrees of freedom. We shall discussthe classical and quantum properties of dilaton gravity with (4.34), (4.35) insect. 7.

4.3.2 Exact solutions – conservation law for geometry and matter

In the presence of interactions with additional fields which – in contrastto gauge fields (cf. subsection 4.1) or supergravity (subsection 4.2) – cannot beincorporated into the PSM approach, the possibility to find analytic solutions

41 Recently dilaton gravity in 4D has been discussed in a framework where even thedilaton can be understood in geometrical terms [190].

72

4 ADDITIONAL FIELDS

is restricted 42 . Nevertheless, the interest in such solutions had been raised, es-pecially by the work on the CGHS model (cf. sect. 2.1.2). It possesses a globalstructure very much like the one of the genuine Schwarzschild BH (Fig. 3.8).Also application of the (singular) dilaton field dependent conformal transfor-mation (1.56) with dilaton dependent conformal factor

ρCGHS = −1

2

∫ X

UCGHS(X′)dX ′ =

1

2lnX (4.39)

is found to cancel the torsion term (cf. sect. 2.1.4). The transformed potentialsread

U(X) = 0, V (X) = VCGHS(X)eQCGHS(X) = const. (4.40)

Then also the action for the scalar field (4.36) becomes the one in a flatbackground (gµν → ηµν). In this model F is taken to be constant (minimalcoupling). After (trivially) solving for the (free) scalar or fermionic fields, theinverse conformal transformation is applied. This method has also been ex-tended to include one-loop quantum effects in the semi-classical approach, bydescribing that lowest order quantum effect through the Polyakov effectiveaction [361]. Adding to this action another piece, adapted suitably so that theexact solubility is maintained, more semi-classical solutions have been stud-ied [374, 44, 115, 147, 256, 446, 445]. An approximate analysis of the solutionsis possible, of course, for a larger class of models. For instance, it has beendemonstrated [252] that adding the Polyakov term to SRG shifts and attenu-ates the BH singularity.

In the exact solution without matter, the important step has been to useone of the eqs. (3.8) to express ω as in (3.16). If the theory only dependson one type of chiral fields either W− or W+ in (3.8) vanishes. Then thesame elimination of ω can be used. Separating e.g. Dirac fermions into chiralcomponents as

Ψ =4√

2

χR

χL

, Ψ =

4√

2(χ†R, χ

†L

), (4.41)

the interaction (4.37) may be written as

L(m)(Ψ) = −

∫

M2

(e+J− + e−J+) , (4.42)

J−,+ = JR,L = i[χ†R,L (dχR,L) − (dχ†

R,L)χR,L]. (4.43)

Also “amplitudes” k and “phases” ϕ can be introduced,

χR,L =1√2kR,L e

iϕR,L , (4.44)

42 It is possible, however, to adjust the dilaton potentials in such a way that someexact solutions with matter can be obtained [154,155].

73

4 ADDITIONAL FIELDS

in terms of which the chiral currents (4.43) become

J+,− = −k2L,R dϕL,R . (4.45)

Since the theory only depends upon one type of chiral fields, either J+ orJ− in (4.42) is zero [278, 356]. Still an equation like (3.16) without mattercontribution holds, and the further steps of the solution for the geometricvariables are exactly as in the matterless situation, except for the mattercontribution in the conservation law. The general case from (3.8) and (3.7)may be derived from

d(X+X−) + V(X, Y ) dX +X−W+ +X+W− = 0 . (4.46)

For chiral fermions only one of the two last terms remains. As the consequencesof (4.46) are of more general importance we will come back to that relationwhen the situation without restrictions on matter will be discussed below.

The matter equation for chiral fermions kL = 0 in (4.44) by variation ofϕR and k2

R

e+d ϕR = 0 , (4.47)

d (k2R e

+) = 0 (4.48)

are solved easily, because e+ is the previous solution (3.20). Eq. (4.47) impliesthat ϕR = ϕR (f) where f is the arbitrary function introduced in (3.20).Inserting the latter into (4.48) determines the amplitude:

k2R =

e−Q(X)

X+g(f) (4.49)

An analogous procedure works for chiral scalars. Variation with respect to e∓

of the action (4.36) for F = const. in light-like coordinates (φ± = ∗(dφe±))yields

W± = −F[φ± dφ∓ e±φ+φ−

], (4.50)

so that for (anti-)selfdual scalars with either φ+ = 0 (selfdual: ∗dφ = dφ) orφ− = 0 (anti-selfdual: ∗dφ = −dφ) again one of the eqs. (3.8) is independentof the matter contribution. The subsequent steps to obtain the exact solutionproceed as for chiral fermions.

The e.o.m. for minimally coupled scalars can be written as

d ∗ dφ = d(φ−e+) = d(φ+e−) = 0 (4.51)

where the last two forms are to be used, respectively, in the (anti-)selfdualcases. Thus (anti-)selfdual matter identically solves (4.51). Furthermore, e.g.for selfdual φ the condition e+dφ = 0 makes the lines of φ = const. light-like.With the same solution (3.20) for e+ as without matter, the latter relationleads to dφdf = 0, i.e. φ = φ(f), or vice versa. As f represented a null coor-dinate in the EF line element, the peculiar light-like nature of φ is confirmed.

74

4 ADDITIONAL FIELDS

The similarity between (anti-)selfdual scalars and chiral fermions is not sur-prising in view of the well-known close relation between scalars and fermionsin D = 2 [439].

Other nontrivial examples of exactly soluble systems are static solu-tions 43 [159] or continuously self-similar solutions [369] of SRG with a massless(non-minimally coupled) scalar field.

Minimally coupled scalars with SRG for the geometric sector can be solvedin a perturbative manner [329] or in the static limit [177, 176].

With the exception of aforementioned special cases, the CGHS model,the models of refs. [154,155] or some other simple potentials like e.g. constantV (i.e. Rindler metric) and of teleparallelism (pure torsion, V = 0) [410] noexact solution with general (scalar or fermionic) matter seems to be known 44 .Indeed, one of the main open problems in classical dilaton theory with matteris an analytic (as opposed to numerical) description of non-trivial systemsshowing the feature of critical collapse 45 .

43 An extensive discussion of these solutions can be found in refs. [247,197].44 Recently cosmological solutions in the JT model have been obtained [68,271].45 Here ref. [103] for the seminal work of Choptuik should be quoted. It had beentriggered by previous analytic studies of Christodoulou [107, 108, 110, 109]. Recentreviews are refs. [201,202].

75

5 ENERGY CONSIDERATIONS

5 Energy considerations

It is important to clarify the relation of certain quantities appearing ingeneric D = 2 gravity theories (like the absolutely conserved quantity C(g))with respect to corresponding concepts well-known from D = 4 Einstein grav-ity.

5.1 ADM mass and quasilocal energy

Because of diffeomorphism invariance the Hamiltonian density vanisheson the surface of the constraints in all gravity theories including all 2D dilatonmodels 46 . However, a boundary term must be included in the Hamiltonianwhich can be used to define a global or “quasilocal” energy. This is the essenceof the Arnowitt–Deser–Misner procedure [7]. As discussed in detail by Fad-deev [149] in the context of 4D Einstein gravity, this boundary is the one at(spatial) infinity, and the value of the gravitational energy is very sensitive tothe choice of asymptotic conditions and it is not invariant under the changeof coordinates at infinity. This reflects the fact that the energy is related toan observer connected with the asymptotic coordinate system who measuresthis energy. In generic 2D dilaton gravity the situation becomes even morecomplicated since a natural asymptotic coordinate system does not exist forsome of the models 47 . This lack of asymptotic diffeomorphism invariance (andof conformal invariance) has led to much confusion in the literature.

To facilitate comparison with other works on the ADM approach we con-sider here the second order dilaton action (3.67) with the exponential param-eterization X = e−2Φ for the dilaton field. Since only the first term e−2ΦRis essential for the calculation of the ADM mass, this does not imply anyrestrictions on the potentials U and V .

The diagonal gauge for the metric (N =√ξf ,Λ =

√ξf ′ in (3.33))

(ds)2 = N2(dt)2 − Λ2(dr)2 . (5.1)

is sufficient for our purposes since the lapse N is a Lagrange multiplier forthe Hamiltonian constraint in generally covariant theories. A more completecanonical analysis can be found in refs. [276, 307,295,280,299].

The next step is to supplement the volume action (3.67) by a suitableboundary term. The role of the latter is to convert second order derivativesto first order ones. More exactly, such a term must provide standard variationequations for the boundary data, the induced metric on the boundary. Inparticular, this means that the first variation of the total action must notcontain boundary contributions with normal derivative for the variation of thelapseN . Since only the curvature term contains second derivatives, it also must

46 Cf. also the general Hamiltonian analysis in sect. 7 below.47 In the absence of asymptotic flatness as in the KV-model [250, 251] other possi-bilities than the ADM-mass were discussed in ref. [290].

76

5 ENERGY CONSIDERATIONS

not depend on the potentials U and V in (2.9). Therefore, we may adopt forall models the expression for the SRG model. It can be obtained by sphericalreduction of the standard extrinsic curvature term in four dimensions. Thisyields

Lb =∫

∂M

Ndt e−2ΦK , (5.2)

where we assume that ∂M corresponds to a constant value of r. N dt is thesurface element on the boundary, e−2Φ is produced by the spherical reduction.K is the extrinsic curvature,

K = − 1

N∂nN = ∓ 1

Λ∂r(logN) , (5.3)

where ∂n denotes the derivative with respect to an outward pointing unitnormal. The upper (lower) sign in (5.3) should be taken on the “right” (“left”)component of the boundary.

For linear variations around a static background in the diagonal gauge(3.33) the curvature follows from the simplified formula

√−gR = 2∂r1

Λ∂rN . (5.4)

Now the lapse N is varied in the total action

δ(Ldil + Lb) =∫

M

(δN)[e.o.m]d2x+∫

∂M

dt(δN)2e−2Φ∂nΦ . (5.5)

The second integral generates the so-called quasilocal energy associatedwith the observer at the boundary:

Eql = 2e−2Φ∂nΦ = −∂nX (5.6)

To obtain the Hamiltonian (or the ADM mass) (5.6) must be multiplied byN :

H = EADM = −N∂nX (5.7)

Evidently, solutions with constant dilaton (see, e.g., [259]) will lead to zeroADM mass.

If one moves the boundary to the asymptotic region for a generic D = 2dilaton theory, the right hand side of (5.7) diverges. In order to arrive at a finitevalue of a “generalized” ADM mass a procedure like the Gibbons-Hawkingsubtraction [179,58] is needed. A rather natural idea [1,215] is to subtract fromthe total action an action functional calculated for some reference space-timewith the same induced metric on the spatial boundary (Φ or X for SRG). Notethat normal derivatives of the boundary data and the normal metric (Λ) maybe different for the reference space-time. Effectively, this means to subtractfrom the quasi-local “physical” energy the one of some reference (“empty

77

5 ENERGY CONSIDERATIONS

space”) configuration, denoted by a subscript 0:

Eregql = −∂nX + [∂nX]0 . (5.8)

To obtain the ADM mass measured in a physical space-time (5.8) should bemultiplied by the lapse function N corresponding also to the physical space-time:

M regADM = N (−∂nX + [∂nX]0) . (5.9)

Eq. (5.9) may be evaluated directly for dilaton theories admitting solu-tions which are flat everywhere, i.e. where U and V are related by (3.40). SRGis a special case of this class, where it is natural to take the reference frameto be the Minkowski space solution with C0 = 0. Instead of studying the mostgeneral situation we concentrate on the subclass of models (3.67). Identifyingvalues of the dilaton field on the boundary for physical and reference space-times is equivalent to identifying the coordinate r of the boundary. This yieldsthe total ADM mass

M regADM = ± lim

r→I

√ξ(r)

(−√ξ(r) +

√ξ0(r)

)∂rX , (5.10)

where the upper sign (+) should be taken if the asymptotic region I cor-responds to r → ∞, and the lower one (−) if I corresponds to r → 0. Bysubstituting e.g. the expressions (3.70) and (3.69) for ξ(r) and r(X) and takinginto account the position of the asymptotic region one obtains

M regADM = −C0

√a

B. (5.11)

It is instructive to check whether the mass of the Schwarzschild black hole fitscorrectly into this procedure. For D = 4 the values a = 1

2and B = 2λ2 follow

from (2.33) and (3.64). Multiplying (5.11) by the coefficient which has beenomitted while passing from (2.2) to (2.9) indeed yields

M regADM = − C0

4GNλ3. (5.12)

This value exactly coincides with (3.38) where units with GN = 1 have beenused.

For the CGHS model (a = 1 in (5.11)) the different coordinate system(3.73) has to be used. Substituting also (3.74) in (5.10) and taking the lowersign there - since the asymptotic region corresponds to the lower limit of r -the ADM mass becomes

M regADM = −C0B

− 12 , (5.13)

which somewhat surprisingly coincides with the naive limit a → 1 in (5.11).This value is also consistent with the calculations existing in the literature.For example, Witten’s result [443] is recovered if we take B = 8/k′, replace C0

by a constant shift of the dilaton Φ and take into account the overall factorof 1/2 assumed in our action.

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5 ENERGY CONSIDERATIONS

It should be noted that in all examples considered above positive massBHs correspond to negative values of C0 whereas positive values of the latterdescribe naked singularities.

For Minkowski ground state theories (cf. (3.40)) of which SRG is a specialcase, that subtraction procedure appears very natural. For more complicatedmodels it might be preferable to subtract the total energy of a reference spacefrom the total energy of the physical space [299]:

M regADM = −[N∂nX] + [N∂nX]0 . (5.14)

This formula, of course, cannot reproduce the correct mass e.g. in the case ofthe Schwarzschild BH, but may be useful is a different context. There are con-siderable variations in the details of such a subtraction being used by differentauthors and in different models. Sometimes, it appeared more appropriate tosubtract an extremal BH solution instead of the Minkowski space [37]. Asnoted in [253] the subtraction procedures of [443, 182, 335, 311, 299], appliedto the so-called exact string BH [127], lead to different results. Treating thismodel is especially tricky since the corresponding action is not known.

It should always be kept in mind that a strong dependence on the asymp-totic conditions and on the subtraction procedure has a clear physical origin:energy depends on the observer who measures it. This is true for both referenceand physical space-times.

Definitions of the ADM mass for higher dimensional dilaton gravities havebeen considered in refs. [101, 231]. For asymptotically Rindler and de Sittermodels the ADM mass has been calculated in ref. [299] (see also [258]). Thisconcept can be also introduced in the presence of radiation [257] and of ashock wave of matter fields [113]. An extension of the Hamiltonian analysis tothe case of charged BH-s is possible too [319].

Obviously according to this procedure the ADM mass is not conformallyinvariant. This means that it will change, in general, if a conformal transfor-mation, for example, removes the kinetic term for the dilaton. The reason wasclearly stated in ref. [101,100]. Even though it is possible to make the unregu-larized energy conformally invariant for a selected class of models, the subtrac-tion term will inevitably destroy that invariance since an “empty” referencespace is mapped into a non-trivial configuration. In other words, physical andreference observers are being transformed differently.

A final remark concerns approaches where instead of the ADM mass theconserved quantity C(g) has been related directly to a quasilocal energy ex-pression. The notion of quasilocal energy has been investigated thoroughly inthe context of General Relativity by Brown and York [58, 57] and in the con-text of 2D dilaton gravity by Kummer and Lau [280]. As shown in ref. [290]a relation to C(g) is possible following the arguments leading to Wald’s energydensity [429,236]. Approaches which require no explicit subtraction have beensuggested as well [290, 311].

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5 ENERGY CONSIDERATIONS

5.2 Conservation laws

Even in the most general dilaton theory including matter interactions,when no exact solution is known, the conservation law which may be derivedfrom (4.46) contains important information. Again potentials with quadratictorsion (2.31) only are considered because there the integrating factor expQcan be determined easily. Attaching that factor as in (3.14), (3.15) yields

dC(g) +W (m) = 0 , (5.15)

where C(g) is the quantity defined in (3.14) for the geometric variable and (cf.(3.8), (3.6))

W (m) = eQ (X+W− +X−W+) . (5.16)

Clearly W (m) from (5.15) must obey the integrability condition dW (m) = 0,a relation which in turn must be expressible in terms of the e.o.m.-s. WithW (m) = dC(m) eq. (5.15) simply becomes 48

dC(tot) = d(C(g) + C(m)) = 0 (5.17)

and C(tot) = C0 = const. is an absolutely (i.e. in both coordinates) conservedquantity [290]. Obviously in the presence of further gauge fields the presentargument can be generalized easily following the steps of sect. 4.1.

Using the integrability condition for the components Wµ in W (m) =Wµ dx

µ, eq. (5.15) can be integrated in two equivalent ways (x0 = t, x1 = r,C0 is an integration constant).

C(g)(t, r; t0, r0) = −t∫

t0

dt′ W0(t′, r) −

r∫

r0

dr′W1(t0, r′) + C0 =

= −r∫

r0

dr′W1(t, r′) −

t∫

t0

dt′W0(t′, r0) + C0 (5.18)

for any first order gravity action (2.17) or its equivalent dilaton form (2.9).SRG may serve as a concrete example [195]. In the “diagonal” gauge

widely used in spherical BH simulations [103, 201, 202] with g00 = α2(t, r),g11 = −a2 = −(1 − 2m/r)−1, g01 = g10 = 0, the zweibein must be fixedas e+0 = e−0 = α/

√2, e+1 = −e−1 = a/

√2 and the dilaton field by X =

λ2r2/4. Then the quantity m(t, r) (sometimes called mass aspect function) isproportional to C(g) with the proportionality factor for SRG given by (3.38).

48 An early version of a conservation law of this type [311] was not general enoughto cover interacting scalars and fermions as introduced in the present chapter.

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5 ENERGY CONSIDERATIONS

As shown in [195] e.g. the first version of (5.18) turns into

m(t, r; t0, r0) =

t∫

t0

dt′4πr2

a2(∂′0φ)(∂1φ)+

+

r∫

r0

dr′ 2πr′2

((∂0φ)2

α2+

(∂′1φ)

a2

)

t=t0

+m0 . (5.19)

Defining the ADM-mass as mADM = m(t0,∞, t0, r0) in the limit r → ∞in (5.19) and using asymptotically free ingoing and outgoing spherical scalarwaves φ ∼ [f+(t − r) + f−(t + r)]/r

√4π yields the effective time-dependent

mass of the (eventual) BH

m(eff)BH (t) = m(t,∞; t0, 0) = mADM +

t∫

t0

dt′ [(f ′−)2 − (f ′

+)2] . (5.20)

The matter contribution has the intuitive interpretation as the total incoming,resp. outgoing flux to infinity of matter at a certain time t, starting fromt = t0. In fact, when such fluxes exist, it is necessary to use m

(eff)BH as a measure

of BH formation rather than mADM alone [201] 49 . It is remarkable that, incontrast to the situation in D = 4, in D = 2 something like a standard energyconservation law can be formulated in this manner.

Clearly, the importance of that conservation (5.17) and (5.18) is not re-stricted to SRG where the ADM-mass can be defined from an asymptoticallyflat region. For generic 2D dilaton theories (2.9) or (2.17) there is a close re-lation of C(tot) to the concept of “quasilocal energy” [58, 290, 299, 280] whichhas been dealt with also in the previous subsection.

5.3 Symmetries

The final topic of this subsection is the question of symmetries, to beattached to (5.17). It should be emphasized that these symmetries are quitedifferent from the (gauge-like) ones incorporated automatically in the PSM ap-proach, because the latter is valid for the geometric part of the action alone.When matter is absent the Nother symmetry of C(tot) = C(g) is realized by atranslation in the Killing direction [289]. In the presence of matter the integra-bility condition dW (m) = 0 for (5.15) can be interpreted as a conservation lawof another one form current W (m) which is related to a symmetry transforma-tion with another type of parameters. Both ingredients are necessary for the

49 As pointed out in ref. [195] the first order formulation also seems to be muchmore convenient in gauges of the Sachs-Bondi type, where no coordinate singularityis created at the horizon. Then the introduction of the extrinsic curvature as anadditional variable [314] can be avoided altogether.

81

5 ENERGY CONSIDERATIONS

peculiar “two-stage” Nother symmetry which has been encountered here [286].It seems to be yet another special feature of a generic D = 2 theory.

In order to simplify the discussion of this unusual symmetry, the mecha-nism is explained in the frame of a toy-model [289] which, nevertheless, con-tains all essential features:

L =∫

M2

(Xdw +Kwdφ) . (5.21)

The first (“geometric”) term in (5.21) can be considered as a simplificationof the Lagrangian (2.17) whereas the second (“matter”) term resembles thefermion interaction, as written in (4.42) with (4.43), and a current expressedin terms of amplitude (1-form w), phase (0-form φ), and Lagrange multipliers(0-forms X,K). In the e.o.m-s to be derived from (5.21)

dX +Kdφ = 0 , (5.22)

dw = 0 , (5.23)

wdφ = 0 , (5.24)

d(Kw) = 0 , (5.25)

eq. (5.22) represents the analogue of the conservation law in the form (4.46)withW (m) = Kdφ. The integrability condition dW (m) = 0 becomes dKdφ = 0.This implies K = K(φ) so that Kdφ = d (

∫ φy0K(y) dy), and

dC = d

X +

φ∫

y0

K(y) dy

= 0 (5.26)

is the counterpart of (5.17). Thus C = C0 = const. characterizes the solutionsof this theory. From (5.23) and (5.24) similarly w = w(φ) can be concluded,so that (5.25) is fulfilled identically.

In the “matterless” case (K = 0) the “geometric” symmetry transforma-tions are constant translations δw = δγ = const. The integrability conditiond(Kdφ) = 0 allows an expansion in terms of e.o.m.-s (5.23)-(5.25) which cor-respond, respectively, to the variations δL/δX, δL/δK, δL/δφ:

d (Kdφ) =

(δL

δφ−K

δL

δX

)∂0φ

w0+δL

δK

∂0K

w0(5.27)

The apparent dependence on the specific coordinate x0 is spurious (e.g. ∂0φ/w0 =∂1φ/w1 from (5.24)). Thus (5.27) permits the introduction of a “matter” sym-metry with global parameter δρ

δφ =∂0φ

w0δρ, δX = −K∂0φ

w0δρ, δK =

∂0K

w0δρ (5.28)

or an equivalent one with ∂0 → ∂1, w0 → w1. It can be checked that the

82

5 ENERGY CONSIDERATIONS

Lagrangian L in L =∫ L of (5.21) indeed transforms as a total divergence:

δL = d

(Kdφ−K

∂0φ

w0w

)δρ (5.29)

The related conserved Nother one form current becomes J = Kdφ, or ∗Jµ =ǫµνK∂νφ in components for the Hodge dual of J . Hence the conservation lawfor the complete expression (5.26) is related to a simultaneous transformationof the action L with respect to both the symmetry parameters δγ and δρ thesecond of whom belongs to a different (one-form) current W (m) = Kdφ.

It is straightforward to apply the procedure, as outlined in this simpleexample, to a general theory with matter interactions in D = 2. The resultingformulas are quite lengthy (cf. [286]) and, therefore, will not be reproducedhere.

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6 HAWKING RADIATION

6 Hawking radiation

One of the main motivations for studying low dimensional gravity theoriesis the hope to get insight into the dynamics of a BH, its quantum radiation andeventual evaporation [211]. Therefore, it is important to make sure that espe-cially the effect of Hawking radiation still exists in two-dimensional theoriesand to study its basic properties like the temperature-mass relation.

It should be kept in mind, though, that this effect, discovered more thana quarter of a century ago, is a fixed background phenomenon. No quantumgravity is involved; only the matter field action is taken into account in theone-loop approximation. The vacuum polarization is described by the energymomentum tensor, induced by this quantum effect,

Tµν =2√−g

δW

δgµν, (6.1)

where W is the one-loop effective action for the matter fields on a classicalbackground manifold with metric gµν . For minimal coupling of scalars in 2DW in (6.1) is the famous Polyakov action [361]. In a suitable coordinate systemthe Hawking flux is given by the light-cone component T−− calculated in theasymptotic region.

To this end various methods have been developed [170]. Most of them canbe applied in 2D. Variation of W as in (6.1) allows the direct determinationof Tµν . Alternatively, the thermal particle distribution may be reproduced bycomparing different vacuum states from the Bogoliubov coefficients [105,413].

In this review we follow the approach of Christensen and Fulling [104]based upon the conformal anomaly. Like the comparison of thermal distribu-tions it should not be sensitive to the dimensionality of space-time. Here thecomputation for minimally coupled scalars is very simple, and a closed expres-sion for the energy momentum tensor may be given for any dilaton gravitymodel. For non-minimal coupling, the situation is much more complicated.Several problems still remain unsolved, although the result for the flux fromD = 4 can be reproduced correctly. A detailed and elementary discussion ofthe non-minimal case can be found in ref. [288], where it was shown that theuse of the fully integrated effective action could be avoided altogether.

6.1 Minimally coupled scalars

The simplest example is a minimally coupled scalar field with action (4.34)and FOD−2/λ

D−2 = 1/2:

Lmin(φ) =1

2

∫d2x

√−ggµν(∂µφ)(∂νφ) . (6.2)

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6 HAWKING RADIATION

If φ is taken to be an on-shell classical field the energy-momentum tensorsatisfies the usual conservation equation

∇µTµν = 0 . (6.3)

The same relation holds for 1-loop quantum corrections with a trivial back-ground field φ = 0 where in (6.3) the effective action W by (6.1) appears. Eq.(6.3) is most conveniently analyzed in the conformal gauge (3.1). We changevariables dz = dr/ξ(r) in the generalized Schwarzschild gauge (3.34) to obtain(3.33) in the form

(ds)2 = ξ(r)((dt)2 − (dz)2) . (6.4)

In light cone coordinates x± = (t ± z)/√

2 the line element (6.4) will beexpressed as

(ds)2 = 2e2ρdx+dx− , ρ =1

2log(ξ) . (6.5)

For the asymptotically Minkowski models 0 < a < 1 considered in (3.70)it is convenient to write ξ as

ξ(r) = 1 −(rhr

) a1−a

, (6.6)

where

rh = (−2C0)1−a

a1

|1 − a|(B

a

)a−22a

(6.7)

is the value of r at the horizon. The explicit form (6.7) will not be neededuntil the very end of this calculation.

There are only two non-zero components of the Levi-Civita connection:Γ++

+ = 2∂+ρ and Γ−−− = 2∂−ρ. The minus component of ν in (6.3) yields

∂+T−− + ∂−T+− − 2(∂−ρ)T+− = 0 . (6.8)

On static backgrounds, which depend on the variable r alone, the relations

∂+ = −∂− =1√2∂z =

1√2ξ(r)∂r , (6.9)

between partial derivatives hold. Therefore, (6.8) becomes a simple first orderordinary differential equation

(∂z − 2(∂zρ))T+− = ∂zT−− . (6.10)

The flux component T−− can be found easily from the trace [104]

T µµ = 2e−2ρT+− . (6.11)

As the classical trace of Tµν for a massless field is zero in D = 2, the wholecontribution to T µµ arises from the conformal (or Weyl) anomaly (cf. [132] for

85

6 HAWKING RADIATION

a historical review). With minimally coupled scalars in 2D the calculationsare especially simple, but they permit to illustrate several important points.As a first step, in the action (6.2) an integration by parts is performed and itis continued to the Euclidean domain,

LE =1

2

∫d2x

√gφAφ , (6.12)

where A = −∆ = −gµν∇µ∇ν is the Laplace operator on the curved back-ground.

The path integral measure is defined by the relation

1 =∫

(Dφ) exp(−∫d2x

√gφ2

), (6.13)

so that the procedure maintains diffeomorphism invariance and thus preservesthe conservation equation (6.3). It is also possible to trade part of the diffeo-morphism invariance for Weyl invariance [243,239,6,294,115], but this optionwill not be considered here.

The partition function for the field φ reads

Z =∫

(Dφ) exp(−∫d2x

√gφAφ

)= (detA)−

12 . (6.14)

where the determinant is divergent. The zeta function regularization [131,212]

W = − lnZ = −1

2ζ ′A(0), ζA(s) = Tr(A−s) , (6.15)

is very convenient in the present context. Prime denotes differentiation withrespect to s. Strictly speaking, to keep the argument of the zeta functionin (6.15) dimensionless, one has to multiply it by µ2s where µ is a param-eter with mass dimension one. Then the effective action W will be shiftedby −1

2ζA(0) lnµ2 which represents the usual renormalization ambiguity. This

term, however, does not contribute to the anomaly.The following analysis will be valid for an arbitrary conformally covariant

operator which means that under an infinitesimal conformal transformationδgµν = 2gµνδρ(x) of the metric (6.5) the operator A changes as

δA = −2(δρ(x))A . (6.16)

Because of this property, the variation of the zeta function is simply

δζA(s) = −sTr((δA)A−1−s) = 2sTr((δρ)A−s) , (6.17)

i.e. the operator A−s is restored with its original power. The correspondingchange of the effective action is expressed in terms of a generalized (“smeared”)zeta function:

δW = −ζ(0|δρ, A) , ζ(s|δρ, A) := Tr((δρ)A−s) . (6.18)

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6 HAWKING RADIATION

At vanishing argument s→ 0 eq. (6.18) can be evaluated easily by heat kernelmethods. For the operator A = −∆ the result is 50

ζ(0|δρ,−∆) =1

24π

∫d2x

√gRδρ . (6.19)

On the other hand, the definition of the energy-momentum tensor (6.1) yields

δW =1

2

∫d2x

√gTµνδg

µν = −∫d2x

√gT µµ δρ . (6.20)

Comparing (6.18) with (6.20) and (6.19) the well-known expression for thetrace anomaly

T µµ =1

24πR (6.21)

follows, which remains unchanged after continuation back to Minkowski sig-nature.

In conformal gauge the Ricci scalar becomes 51

R = 2e−2ρ∂2zρ . (6.22)

In light cone coordinates (6.11) yields

T+− =1

24π∂2zρ . (6.23)

With this input, the conservation equation (6.10) is solved easily,

T−− =1

24π

[∂2zρ− (∂zρ)

2]+ t− , (6.24)

where t− is the integration constant.Different choices of t− correspond to different “quantum vacua” [45, 412,

235,181]. There is nothing specific for 2D models in this respect. We assumethat the Killing horizon is non-degenerate, i.e. ξ(r) has a simple zero at r = rhas for ξ(r) in (6.6). To ensure regularity of the energy-momentum tensor at thehorizon in global (Kruskal) coordinates one has to require that T−− exhibits asecond order zero at r = rh. There is only one integration constant t− available.Therefore, fixing it by the requirement

T−−|h = 0 , (6.25)

it must be checked later on whether (6.25) indeed produces a second orderzero.

50 See Appendix B for the details.51 This expression may be obtained most easily from the identity (1.56) with gµνfrom the line element (6.5), R = 0, ∂r = ∂z.

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6 HAWKING RADIATION

In terms of the function ξ the energy momentum tensor (6.24) can beexpressed as (cf. (6.5) and (6.9))

T−− =1

96π

[2ξξ′′ − (ξ′)2

]+ t− . (6.26)

With t− determined from (6.25) it is an easy exercise to show that for theasymptotically Minkowski models (6.6) the Hawking flux in the asymptoticregion becomes

T−−|as =a2

96π (a− 1)2r2h

. (6.27)

This flux defines the Hawking temperature TH of the BH. In 2D the Stefan-Boltzmann law contains T 2

H :

T−−|as =π

6T 2H . (6.28)

Comparing (6.27) and (6.28) the value of TH agrees with the one derived fromsurface gravity TH = 1

4πξ′|rh. These equations together with (6.7) and (5.11)

fix the dependence of the Hawking temperature on the ADM mass for thisclass of models:

TH ∝ (MADM)a−1

a . (6.29)

The well known inverse mass law for the Schwarzschild BH (a = 1/2) isreproduced. Eq. (6.29) reveals an intriguing property [299] of the class of 2Dmodels discussed in sect. 3.3: depending on the parameter a the Hawkingtemperature may be proportional to a negative, but also a positive power ofthe BH mass.

It is easy to check that near the horizon indeed

T−−|r→rh∼ (r − rh)

2 (6.30)

for all values of a, i.e. the requirement of a continuous flux in Kruskal coordi-nates is fulfilled.

Again, the CGHS model must be considered separately. By substituting(3.74) in (6.26) the Hawking flux

T−−|as =B

96π(6.31)

is obtained, consistent with the earlier calculation [71]. It is important to notethat in the CGHS model Hawking radiation does not depend on the ADMmass.

Hawking radiation can be studied as well for asymptotically Rindlerand de Sitter models. Explicit expressions can be found in ref. [299]. T−−for “exotic” configurations with constant dilaton has been calculated in refs.[259, 258]. It is very sensitive with respect to asymptotic conditions on themetric. Physically this means that one has to fix length and time scales used

88

6 HAWKING RADIATION

in its measurement. Clearly, different scales yield different results, as may beseen by comparing refs. [148] and [299] where asymptotically Rindler spaceswere studied. By choosing an accelerated reference system Hawking radiationmay be converted into Unruh radiation [67].

It should be stressed that Hawking radiation behaves quite differently inconformally related models as witnessed by the results of ref. [69] vs. ref. [299].Conformal transformations change, in general, also the asymptotic behaviorof the metric and of the path integral measure. Indeed the very existence ofthe conformal anomaly means conformal non-invariance of the theory.

The case of minimally coupled spinor fields interacting (again minimally)with an abelian gauge field can be also analyzed along the same lines. One hasto add a contribution of the chiral anomaly to the Polyakov action [341,351] 52 .Another generalization [106] consists in considering the Casimir force due to aminimally coupled scalar field between two surfaces on a CGHS background.

6.2 Non-minimally coupled scalars

The scalar field action (4.34),(4.35) contains a non-minimal coupling tothe dilaton from spherical reduction. On dimensional and symmetry groundsfor a GDT in the path integral measure also a general function Ψ of the dilatonΦ may be introduced,

1 =∫

(Dφ) exp(−∫d2x

√ge−2Ψφ2

), (6.32)

instead of the standard mode normalization condition following fromD dimen-sional spherical reduction (Ψ = Φ), using the exponential parameterization ofthe dilaton (2.11). Then the rescaled field ϕ = e−Ψφ still possesses the stan-dard dilaton independent path integral measure in 2D (6.13). In terms of thisnew field the action (4.34) reads

L(nm) =1

2

∫d2x

√gϕA(nm)ϕ , (6.33)

A(nm) = −e2(Ψ−Φ)gµν(∇µ∇ν + 2(Ψ,µ − Φ,µ)∂ν+Ψ,µν + Ψ,µΨ,ν − 2Ψ,µΦ,ν), (6.34)

where an integration by parts has been performed and an irrelevant overallfactor in the action has been dropped (Ψ,µ = ∇µΨ).

The first calculation of the conformal anomaly for non-minimally coupledscalar fields with the spherically reduced path integral measure (Ψ = Φ) hasbeen presented by Mukhanov, Wipf and Zelnikov [333] who were also the firstto address the problem of Hawking radiation for spherically reduced matter.

52 The case of neutral matter on the background of a charged BH is even simpler.One has to modify only the metric in the Polyakov action [125]. The expression(6.26) still holds in terms of a different ξ.

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6 HAWKING RADIATION

Their result was confirmed later [102,226] and extended to arbitrary measure[282] 53 .

As in the minimally coupled case the zeta function regularization (6.15)may be employed. The operator A(nm) being conformally covariant, the equa-tions (6.16) – (6.18) and (6.20) (after the replacement A → A(nm)) are stillvalid. The conformal anomaly is derived from ζ(0|δρ, A(nm)). Again the gen-eral formulae from Appendix B can be used, because A(nm) may be expressedin the standard form (B.1)

A(nm) = −(gµν∇µ∇ν + E) , (6.35)

with the “effective” metric gµν = e2(Ψ−Φ)gµν and the covariant derivative

∇µ = ∂µ + Γµ + ωµ , ωµ = Ψ,µ − Φ,µ , (6.36)

where Γ is the Christoffel connection for the metric g. Here the potential Ereads

E = gµν(−Φ,µΦ,ν + Φ,µν) . (6.37)

According to Appendix B, (eq. (B.5) with (B.9) and (B.13)), after returningto Minkowski space one obtains for the smeared ζ-function (6.18)

ζ(0|δρ, A(nm)) =1

24π

∫d2x

√−g(R + 6E)δρ , (6.38)

where R is the scalar curvature determined from g, so that (6.37) and (6.38)yield the trace anomaly [282]

T µµ =1

24π(R− 6(∇Φ)2 + 4∇2Φ + 2∇2Ψ) . (6.39)

For the spherically reduced measure Ψ = Φ this expression agrees withrefs. [333,102,226]. Different expressions for the conformal anomaly with var-ious choices of the measure were reported too [48, 342, 325, 343]. It is oftenimportant to keep track of total derivatives (or zero modes in the compactcase) in computations of T µµ . A careful analysis of this type has been per-formed by Dowker [130] (cf. also [283, 288]) who confirmed the result (6.39)for SRG.

When scalars are coupled nonminimally to a dilaton field the conservationlaw for the one-loop energy-momentum tensor has to be modified,

∇µTµν = −(∂νΦ)1√−g

δW

δΦ, (6.40)

as can be seen by applying the usual assumption of diffeomorphism invarianceto a φ dependent matter action. In the absence of classical matter fluxes the

53 The literature on this subject is quite large (cf. e.g. [48,342,325,344,227, 345]).

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6 HAWKING RADIATION

matter action can be replaced again by the one-loop effective action of non-minimally coupled scalars [222].

In contrast to conformal transformations, a shift of the dilaton field Φ →Φ + δΦ does not act on the operator A(nm) in a covariant way 54 . Therefore,

the variation of(A(nm)

)−sin the zeta function does not yield a power of A(nm)

anymore. As a consequence the variation of the effective action cannot beexpressed in terms of known heat kernel coefficients.

A way to overcome this difficulty has been suggested in [287]. By keepingthe same classical action, but by changing the hermiticity requirements ofrelevant operators, A(nm) has been transformed into a product of two operatorsof Dirac type each of which transforms homogeneously under the shifts ofthe dilaton field. This allowed us to calculate the energy-momentum tensorand the effective action in a closed analytical form. Although this procedurechanges the original spectral problem, the results exhibit several attractivefeatures which have to be present in spherically reduced theories. For example,the Hawking temperature coincides with its geometrical expression throughsurface gravity. Thus, in hindsight one may conjecture [288] that this proceduresomehow takes into account the “dimensional reduction anomaly” (see below).Another model where the energy-momentum tensor can be calculated exactlyhas been proposed recently [167].

In this connection it should be remarked that in 2D a generic differentialoperator can be represented in “dilaton” form 55

A = −(eΨ∇µe−Φ

) (e−Φ∇µe

Ψ)

:= LµL†µ . (6.41)

This parameterization proved very convenient in resumming the perturbativeexpansion of the effective action [203]. It also allows to prove some symmetryrelations between functional determinants even in higher dimensions or if Ψand Φ are matrix-valued fields [421]. Roughly speaking, these symmetry rela-tions allow to interchange Lµ and L†

µ inside the determinant which is a rathernon-trivial operation because of the summation over µ in (6.41).

In order to study the quantum back reaction upon the classical BH, solu-tions of the field equations obtained from an action containing both classicaland one-loop parts are needed. It is not possible to solve such equations ingeneral, even if the quantum effects are represented by the simplest Polyakovaction. However, for particular dilaton theories exact solutions can be ob-tained [374, 372, 44, 115, 147, 256, 446, 445], although in some of these papersthe “quantum” part was rather introduced by hand than derived.

54 From now on we assume that the function Ψ in the measure is fixed. The mostrelevant choice (SRG) is Ψ = Φ.55 In 2D locally any operator of Laplace type can be represented as a product ofan operator of Dirac type and its conjugate [420]. In contrast to the present case,however, these Dirac operators will not necessarily transform homogeneously underthe shift of the dilaton.

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6 HAWKING RADIATION

An effective action for non-minimally coupled fields was presented in [287].Admittedly, its derivation lacked complete rigor. Many authors [49,50,47,65,62, 61, 63, 320, 321,317,318,21] employ the “conformal” action

W (conf) =1

96π

∫d2x

√−g[R

1

2R − 12(∇Φ)2 1

2R + 12RΦ

](6.42)

which correctly reproduces the conformal anomaly (6.39) for Ψ = Φ but ne-glects (an undetermined) conformally invariant part 56 . The first term underthe integral in (6.42) yields the Polyakov action. It is interesting to note thatalthough (6.42) differs from the full effective action obtained in [287], manyphysical predictions are identical.

Even if it is assumed thatW (conf) provides a correct description of one-loopeffects for non-minimally coupled matter, many problems remain open. Thefirst one is how to deal with the non-local terms in (6.42). Direct variation ofthis equation with respect to the metric leads to very complicated expressions[302]. It was proposed in ref. [64] to convert (6.42) into a local action byintroducing two auxiliary fields, f1 = (1/2)R and f2 = (1/2)(∇Φ)2. Variousversions of this method were frequently used since (e.g. [65,62,61,63,14,15]).As the new action in terms of f1 and f2 is local it is quite straightforward tovary it with respect to the metric in order to arrive at the energy momentumtensor. However, since f1 and f2 are to be found from R and (∇Φ)2 by solvingsecond order differential equations, the energy momentum tensor obtainedin this way will in general depend on four integration constants. This is anindication that such an extended action does not necessarily yield the samephysics as the original one. For the latter a single first order equation (6.40)must be solved in the conformal gauge. Indeed, many physical predictions, as,e.g., the BH “anti-evaporation” [49] may depend on the way the action (6.42)is treated.

It has been noted [13, 14, 15] that T−− for non-minimal coupling at thehorizon behaves as (r−rh)2 ln(r−rh) instead of (6.30) for the minimal coupling.This means that in Kruskal coordinates the energy momentum tensor exhibitsa singularity, although a rather weak (logarithmic, integrable) one [288]. Inref. [16] this singularity has been attributed to a breakdown of the WKBapproximation.

In the case of non-minimally coupled scalars derived from SRG, action andpath integral measure (the mode normalization condition) coincide with theones for the s-wave parts of the corresponding quantities in four dimensions.Does this guarantee that the 2D Hawking flux will be just the s-wave part ofthe Hawking flux in four dimensions? The answer is negative, because renor-malization and dimensional reduction do not commute. Indeed, even if eachindividual angular momentum contribution to the energy-momentum tensoror to the effective action were finite, the sum over the angular momenta will,

56 Some physical motivations why the conformally non-invariant part may dominateat a certain energy scale can be found in ref. [347].

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6 HAWKING RADIATION

in general, diverge. In fact, the effective action W can be written in the zetafunction regularization as

W reg = −1

2Γ(s)

∑

l

∑

nl

λ−sl,nl=∑

l

W regl , (6.43)

where the “partial wave” effective action is

W regl = −1

2Γ(s)

∑

nl

λ−sl,nl. (6.44)

Here λl,nlare eigenvalues of the kinetic operator in four dimensions correspond-

ing to the angular momentum l. To remove divergences in W regl as s→ 0 one

must subtract the pole terms:

δWl =1

2sζl(0) + . . . , ζl(s) =

∑

nl

λ−sl,nl. (6.45)

Here dots denote finite renormalization terms. After that one obtains thefamiliar expression (6.15) for each l with an appropriate operator A. However,the sum

W =∑

l

(W regl + δWl) (6.46)

will diverge. Thus, a subtraction term which is needed to make (6.43) finitehas nothing to do with the sum over l of the individual pole terms (6.45).This latter sum simply does not exist! This means that the four-dimensionaltheory requires more counterterms and counter terms of a different type thanthe spherically reduced one. This problem was noted long ago [437] in calcu-lations of tunnel determinants. In the context of SRG the non-commutativityof renormalization and dimensional reduction has been called “dimensionalreduction anomaly” [166]; it has been the subject of extensive studies overrecent years [401, 16, 111].

We conclude this section by noting that for massive matter fields the situ-ation is simpler than for massless ones. One can apply e.g. the high frequencyapproximation [17] to estimate the energy-momentum tensor. The zero masslimit in such calculations is, of course, singular. The massive case is also lessinteresting because the Hawking flux is suppressed by the mass.

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

7 Nonperturbative path integral quantization

As pointed out in the previous section, dilaton gravity is a convenientlaboratory for studying semi-classical effects like Hawking radiation. For quan-tum effects the simplicity of 2D theories becomes even more important, sincequantum gravity is beset with well-known conceptual problems, which are es-sentially independent of the considered dimension (cf. e.g. [79]). Thus, a studyin a framework were the purely technical challenges are not as demanding asin higher-dimensional theories is desirable.

GDT in 2D with matter as a theoretical laboratory for quantum gravityhas several advantages as compared to other models:• As outlined in the introduction, it encompasses many different theories pro-

posed in the literature, including models with strong physical motivation,like SRG.

• It exhibits continuous physical degrees of freedom and thus provides physicalscattering processes with a non-trivial S-matrix, as opposed to pure SRGor GDT without matter.

• It is still simple enough to allow a non-perturbative treatment in the geo-metric sector.

The main advantage of the first point is that the same techniques can be useduniformly for a large class of theories. The second point will be elaborated indetail in the next section, where the S-matrix for s-wave gravitational scat-tering will be calculated. The third point is conceptually and technically veryimportant: the split of geometric variables into background plus fluctuationsin perturbation theory is something which can be avoided here. From theviewpoint of GR this is very attractive.

After integrating out geometry exactly a non-local and non-polynomialaction is obtained, depending solely on the matter fields and external sources.When perturbation theory is introduced at this point geometry can be re-constructed self-consistently to each given order. In particular, the properback-reaction from matter is included automatically.

From a technical point of view the use of Cartan variables in a first orderformulation has been crucial. The ensuing constraint algebra also with mattershares the essential features with the one in the PSM model (cf. sect. 2.3)which governs the matterless case: It becomes a finite W-algebra for minimallycoupled matter and a Lie-algebra for the JT model [22,123,405,122,124,238].Moreover it still closes with δ-functions rather than derivatives of them. Withinthe BRST quantization procedure the “temporal” gauge has turned out to beextremely useful. It will lead to an effective metric in Sachs-Bondi form. Asseen above (cf. sect. 3.1) that gauge appeared to be already the most naturalone in the absence of matter interactions. Even when the latter are presentthe classical action in that gauge remains linear in the canonical coordinatesof the geometrical sector. Consequently, by integration three functional deltafunctions are generated which are used to perform an exact path integrationover the corresponding canonical momenta. If no matter fields are present in

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

this way an exact generating functional for Green functions is obtained. Theeffective field theory with non-local interactions for the case with matter fieldswill be considered in detail in sect. 8.

7.1 Constraint algebra

A prerequisite for the proper formulation of a path integral is the Hamil-tonian analysis. The key advantage of the formulation (2.17) for the geometricpart of the action is its “Hamiltonian” form. The component version (2.20), to-gether with the one for scalar fields, nonminimally coupled by F (X) 6= const.to the dilaton field (cf. (4.36)),

L(m) = (e)F (X)[1

2ηab (ǫ

µνeaν∂µφ)(ǫκλebκ∂λφ

)− f(φ)

](7.1)

in terms of the canonical coordinates φ and

qi = (ω1, e−1 , e

+1 ), qi = (ω0, e

−0 , e

+0 ) (7.2)

allow the identification of the respective canonical momenta 57 from the totalLagrangian L = L(g) + L(m) (L =

∫d2xL, ∂0qi = qi etc.)

pi =∂L∂qi

= (X,X+, X−) , (7.3)

π =∂L∂φ

, (7.4)

pi =∂L∂ ˙

iq= 0 . (7.5)

Eqs. (7.5) are three primary constraints. The canonical Hamiltonian density

Hc = piqi + πφ− L, (7.6)

after elimination of φ and qi becomes

Hc = −qiGi, (7.7)

with the secondary first class constraints

Gi(q, p, φ, π) := G(g)i (q, p) +G

(m)i (q, p, φ, π), (7.8)

57 Strictly speaking the relation between pi and the variables X,X± yields primarysecond class constraints. However, the canonical procedure using Dirac bracketsin the present case justifies the shortcut implied by (7.3). We use the standardnomenclature of Hamiltonian analysis (cf. e.g. [187, 219, 128]). Note that LFOG =−∫d2xL(g), the minus sign being a consequence of our notation which relates the

first order action (2.17) to minus the second order action (1.1).

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

the geometric part of which is given by (V = V(p2p3, p1) as defined in (2.31))

G(g)1 = ∂1p1 + p3q3 − p2q2, (7.9)

G(g)2 = ∂1p2 + q1p2 − q3V, (7.10)

G(g)3 = ∂1p3 − q1p3 + q2V, (7.11)

and its matter part reads

G(m)1 = 0, (7.12)

G(m)2 =

F (p1)

4q2

[(∂1φ) − π

F (p1)

]2

− F (p1)q3f(φ), (7.13)

G(m)3 = −F (p1)

4q3

[(∂1φ) +

π

F (p1)

]2

+ F (p1)q2f(φ). (7.14)

By means of the Poisson bracket (pi′ = pi(x

′) etc.)

qi, p

′j

= δijδ(x

1 − x1′) (7.15)

the stability of the first class primary constraints (7.5) identifies the Gi of(7.8) as first class secondary constraints. There are no ternary constraints ascan be seen from the Poisson algebra of the Gi

Gi, G

′j

= CijkGkδ(x− x′) , (7.16)

with (Cijk = −Cjik; all non-listed Cijk-components vanish)

C122 = −1,

C133 = 1,

C231 = − ∂V∂p1

+F ′(p1)

(e)F (p1)L(m),

C232 = − ∂V∂p2

,

C233 = − ∂V∂p3

.

(7.17)

In the matterless case and for minimal coupling (F ′ = 0 in (7.17)) the structurefunctions Cijk depend on the momenta only. For the JT model (2.12) withV = Λp1 again the Lie-algebra of SO(1, 2) is reproduced (cf. the observationafter eq. (2.36)). Already without matter the symmetry generated by the Gi =

G(g)i corresponds to a nonlinear (finite W -) algebra A(g). Including also the

(mutually commuting) momenta pi in A(g) = A(g), pi that algebra closesand the Casimir invariant C(g) of the PSM appears as one of the two elementsof the center [192], the second of which can be expressed as ∂1C(g).

It is remarkable that the commutators (7.16) resemble, though, the ones ofan ordinary gauge theory or the Ashtekar approach to gravity [8,9] in the sense

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

that no space-derivatives of the delta functions appear 58 . The usual Hamil-tonian constraints H and the diffeomorphism constraints H1 in an analysis ofthe ADM-type [7] always lead to such derivatives (cf. e.g. [408]). Indeed, Hand H1 can be reproduced by suitable linear combination of the Gi [246,289].

We emphasize that upon quantization we have no ordering problems inthe present formalism 59 . Due to the linear appearance of coordinates in thegeometric parts G

(g)i of the constraints any hermitian version of them is au-

tomatically Weyl ordered 60 . Moreover, the Hamiltonian is hermitian if theconstraints exhibit that property (since the Hamiltonian essentially is just asum over them). This property carries over to Gi since Cijk for minimal cou-pling depend only on the momenta and for non-minimal coupling the onlyaddition consists of the matter Lagrangian. The commutator between struc-ture functions and constraints vanishes: for minimal coupling this is a trivialconsequence of the PSM structure of the Hamiltonian and for non-minimalcoupling the only non-trivial term (present in C231) vanishes as well since itcommutes with G1. Moreover, the commutator of two (Weyl ordered) con-straints again yields the classical expressions (7.17) in Weyl ordered form.Therefore, the Poisson algebra (7.16) can be elevated without problems to acommutator algebra for quantum operators.

As no ternary constraints exist and as all constraints are first class theextended phase space with (anticommuting) ghost fields can be constructedeasily, following the approach of Batalin, Vilkovisky and Fradkin [164,27,163].One first determines the BRST charge Ω which fulfills Ω2 = 1

2Ω,Ω = 0.

Treating qi as canonical variable one obtains two quadruplets of constraint/ca-nonical coordinate/ghost/ghost momentum

(pi, qi, bi, pbi), (Gi,−, ci, pci) , (7.18)

with canonical (graded) brackets

ci, p

ci′

= −δijδ(x1 − x1′) =bi, p

bi

′. (7.19)

In (7.18) no “coordinate” conjugate to the secondary constraints Gi appears,although one could try to construct some quantities which fulfill canonicalPoisson bracket relations with them. These quantities are not needed for theBRST procedure which according to [197] yields

Ω =∫ (

ciGi +1

2cicjCijkp

ck + bipi

)d2x . (7.20)

58 It is also possible to switch to other sets of constraints, e.g. including ∂1C(g) asone of them, thus abelianizing them in the matterless case. This, however, worksonly in a given patch, since the transformation involved breaks down at a horizon(cf. e.g. [197]).59 We are grateful to P. van Nieuwenhuizen for discussions on that point.60 To be more explicit: classical terms of the form pq have a unique hermitian rep-resentation, namely (qp+ pq)/2. This is also their Weyl ordered version.

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

Since the structure functions (7.17) are field dependent it is non-trivial thatthe homological perturbation series stops at rank = 1. In general one wouldexpect the presence of higher order ghost terms (“ghost self interactions”).However, it can be verified easily that Ω as defined in (7.20) is nilpotentby itself. For the matterless case this is a simple consequence of the Poissonstructure: The Jacobi identity (2.36) for the Poisson tensor implies that thehomological perturbation series already stops at the Yang-Mills level [379].It turns out that the inclusion of (dynamical) scalars does not change thisfeature.

The quantity (7.20) generates BRST transformations with anticommut-ing constant parameter δλ by δHc = δλΩ,Hc = 0. It not only leaves thecanonical Hamiltonian density Hc invariant, but also the extended Hamilto-nian density

Hext = Hc + ψ,Ω (7.21)

in which Hc has been supplemented by a (BRST exact) term with the gauge-fixing fermion ψ. In our case also Hc = pci qi,Ω is exact, a well-known featureof reparametrization invariant theories. A useful class of gauge fixing fermionsis given by [204]

ψ = pbiχi (7.22)

where χi are some gauge fixing functions. The class of temporal gauges (3.3)

qi = ai, ai = (0, 1, 0) (7.23)

has turned out to be very convenient for the exact path integration of thegeometric part of the action [292, 204, 281, 285]. It can be incorporated in(7.22) by the choice

χi =1

ǫ(qi − ai) (7.24)

with ǫ being a positive constant. Then (7.21) reduces to

Hext =1

ǫpbibi −

1

ǫ(qi − ai)pi − qiGi − qicjCijkpck + pcibi . (7.25)

It is necessary to perform the limit ǫ→ 0 to impose (7.23) in the path integral.This can be achieved by a redefinition of the canonical momenta

pi → ˆpi = ǫpi, pbi → pbi = ǫpbi , (7.26)

which has unit super-Jacobian in the path integral measure. Taking ǫ→ 0 af-terwards, in terms of the new momenta yields a well-defined extended Hamil-tonian.

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

7.2 Path integral quantization

After that step the path integral in extended phase space becomes

W =∫

(Dqi) (Dpi) (Dqi) (D ˆpi) (Dφ) (Dπ) (Dci) (Dpci) (Dbi) (Dpbi)

× exp[i∫

(L(0) + Jipi + jiqi + σφ) d2x], (7.27)

withL(0) = piqi + ǫ ˆpi ˙qi + πφ+ ǫpbi bi + pci ci −Hext . (7.28)

It turns out to be very useful to introduce sources ji and σ not only for thegeometric variables qi and for the scalar field φ, but also for the momenta pi,denoted by Ji. Integrating out ˆpi and qi yields an effective Lagrangian withqi = ai as required by (7.23). After further trivial integrations with respect tobi and pbi , and finally ci and pci , (7.27) simplifies to

W =∫

(Dqi) (Dpi) (Dφ) (Dπ) detM exp iL(1) (7.29)

withL(1) =

∫d2x (piqi + πφ+ aiGi + Jipi + jiqi + σφ) . (7.30)

In the functional matrix

M =

∂0 −1 0

0 ∂0 0

K p3U(p1) ∂0 + p2U(p1)

(7.31)

the (complicated) contribution K is irrelevant for its determinant:

detM = (det ∂0)2 det (∂0 + p2U(p1)) (7.32)

Indeed, apart from that important contribution to the measure the generatingfunctional of Green functions W with the effective Lagrangian eq. (7.30) isnothing but the “naive” result, obtained by gauge fixing the Hamiltonian Hc.

The present approach to quantization may be questioned because it isnot based directly upon the classical physical theory like the dilaton actionappearing naturally in SRG. However, the classical equivalence argument ofsect. 2.2 in the quantum language simply means to integrate out 61 the torsion-independent part of the spin connection and of the Xa in (2.17). The onlydelicate point is the transformation in the measure of the path integral. Asshown in ref. [281] there exists a gauge (e−0 = 1, e+1 = 1, e+0 = 0) which

61 In the path integral the classical elimination procedure is replaced by first inte-grating the two components of ωµ which appear linearly. The resulting δ-functionsallow the elimination of Xa by means of the relation (2.29).

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

does not produce a Faddeev-Popov-type determinant in the transition to theequivalent dilaton theory (of the form (2.9)). Thus for all physical observables,defined as to be independent of the gauge, that equivalence should also holdat the quantum level.

The (Gaussian) path integral 62 of momenta π in (7.29) is done in thenext step:

W =∫

(Dqi)(Dpi)(Dφ)(det q2)1/2 detM exp iL(2) (7.33)

L(2) =∫d2x

[piqi + q1p2 − q3(V (p1) + U(p1)p2p3)

+ F (p1)((∂1φ)(∂0φ) − q2(∂0φ)2 − q3f(φ)

)+ jiqi + Jipi + σφ

](7.34)

In the only contributing constraint G2, which is now written explicitly, thetotal derivative ∂1p2 has been dropped 63 .

As it stands (7.33) lacks a covariant measure for the final matter integra-tion. If this is not corrected properly, counterterms emerging from a nonco-variant measure may obscure the quantization procedure. The accepted rem-edy [411] (cf. also [172,26]) is to insert the appropriate factor by hand. It arisesfrom the requirement that the inner product

∫d2x

√−gφψ =∫d2x

(φ 4√−g

) (ψ 4√−g

)

be invariant under diffeomorphisms. As a consequence, the Gaussian integralwith the invariant measure yields

∫(Dφ 4

√−g) exp[i∫ √−gφΓφ

]= (det Γ)−1/2 . (7.35)

In the present gauge (3.3) with√−g = (e) = e+1 = q3 this means that

(det q2)1/2 in (7.33) should be replaced by (det q3)

1/2.In the customary approach the next step would be the integral of the

momenta pi. However, in a generic 2D gravity model, including the physicallyrelevant SRG, the p-integrals are not Gaussian. On the other hand, the action(7.34) is linear in the geometric coordinates qi. Even the new determinantin the measure by the identity (u and u are anticommuting scalars and v a

62 An effective action for a general class of gauges where this integration is notpossible has been proposed too [247].63 There exists a shortcut to obtain (7.33) with (7.34) and (7.32) [197]: instead of(7.22) with (7.24) one can use the gauge fixing fermion Ψ = pc2 and straightforwardlyintegrate all ghosts and their momenta, without limiting procedure for a quantityε as in (7.24).

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

commuting one)

(q3)1/2 =

∫(Dv) (Du) (Du) exp i(v2 + uu)q3 (7.36)

may be reexpressed formally as yet another linear contribution (in q3) to theaction. This suggests to perform the qi-integrals first, yielding (functional)δ-functions which, in turn, may be used to get rid of the pi-integrals after-wards 64 .

The vanishing arguments of the three δ-functions for the respective (q1, q2, q3)-integrals yield three differential equations (h = uu+ v2 from (7.36))

∂0p1 − p2 − j1 = 0 , (7.37)

∂0p2 − j2 + F (p1)(∂0φ)2 = 0 , (7.38)

(∂0 + p2U(p1))p3 + V (p1) + F (p1)f(φ) − h− j3 = 0 . (7.39)

The differential operators acting on pi precisely combine to the ones in thedeterminant detM of the measure in (7.29) with (7.32). Therefore, detM willbe cancelled exactly in the subsequent integration of pi. Eqs. (7.37)-(7.39)are the classical (Hamiltonian) differential equations for the momenta (withsources ji). Matter is represented in (7.38), (7.39) by the terms proportionalto Newton’s constant (F ∝ κ).

7.3 Path integral without matter

It is not possible to obtain an exact solution for pi for general matterinteractions. Therefore, the latter can be treated only perturbatively, and inthe first step F → 0 should be considered 65 . Then the solutions of (7.29)-(7.37) can be written as

p1 = B1 = p1 + ∂−10 (p2 + j1) , (7.40)

p2 = B2 = p2 + ∂−10 j2 , (7.41)

p3 = B3 = e−Q[∂−1

0 eQ (j3 − V (p1)) + p3

](7.42)

where ∂0pi = 0 and ∂−10 , ∂−2

0 have to be properly defined one-dimensionalGreen-functions in the genuine realm of quantum theory (see below). In theintegration of (7.39) the differential operator H = (∂0 + p2U(p1)) has beenreexpressed in terms of

Q = ∂−10 (U(B1)B2) (7.43)

64 Historically this exact integrability was realized first for the matterless KV-model[204] where the quadratic dependence on pi allowed the first integration to be the(traditional) Gaussian one. The inverted sequence of integrals has been initiatedin [281].65 For minimally coupled scalars F = const. is independent of pi so that this termcan be taken along one more step. In the end, however, one cannot avoid thatperturbation expansion.

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

as H−1 = e−Q ∂−10 eQ. Proceeding as announced above, after

∫(Dq) (Dp) we

arrive at the exact expression for the generating functional for Green functions

W0(j, J) = exp iL(0)eff , (7.44)

L(0)eff =

∫d2x

[JiBi + L0(j, B)

], (7.45)

where Bi = Bi(j, pi) . Here L(0)eff trivially coincides with the generating func-

tional of connected Green’s function. In (7.45) a new contribution L0(j, B) hasbeen added. It originates from an ambiguity in the first term of the squarebracket. In an expression

∫dx0

∫dy0 Jx0(∂−1

0 )A the symbol ∂−10 means an inte-

gral which when acting upon J contains an undetermined integration constantg(x1). This generates a new term g

∫A. Applying this to J1B1 + J2B2 yields

uninteresting couplings 66 . But from B3 with A = eQ(j3 − V ) together withJ3 an important contribution to the action follows:

L(0) = g eQ (j3 − V ) (7.46)

Indeed that term is the only one to survive in the matterless case at Ji = 0,i.e. for vanishing sources of the momenta. On the other hand, precisely thataction L(0) had been derived in the first exact path integral [204] computedfor the KV-model [250,251]. There Ji ≡ 0 had been taken from the beginning.It also plays a crucial role for the derivation of the solutions for the (classical)e.o.m.-s for the geometric variables which simply follow from the “expectationvalues” in the matterless case

〈qi〉 =1

iW0(0)

δW0

δji

∣∣∣∣∣j=J=0

. (7.47)

These 〈qi〉 indeed coincide with the classical solutions (3.20)-(3.22) when con-stants of integration are adjusted and the gauge (3.3) is assumed. The useful-ness of the sources Ji for the momenta is evident when the Casimir functionC(g) of (3.14) is determined from its expectation value:

⟨C(g)(p)

⟩=

C(g)(

1iδδJ

)W0(J)

W0(0)

∣∣∣∣∣∣J=0

= p3 (7.48)

The last equality in (7.48) follows from introducing the solutions (7.40)-(7.42)

into⟨C(g)(p)

⟩= C(g)(B0

i ), where in B0i = Bi(j = 0) the residual gauge is

fixed so that B02 = p2 = 1, B0

1 = x0. Here, as well as in the solutions qi of(7.47) it is evident that p3 describes the (classical) background.

It should be emphasized that one encounters the unusual situation thatin the matterless case the classical theory is expressed by W0 in a quantum

66 It turns out that the corresponding ambiguous contributions are fixed uniquelyby imposing boundary conditions on the momenta p1 and p2.

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

field theoretical formalism. Therefore, the question arises for the whereaboutsof the e.o.m-s which are the counterparts of (7.37)-(7.39), but which dependon derivatives ∂1pi instead of ∂0pi. These relations disappeared because ofthe gauge fixing, much like the Gauss law disappears in the temporal gaugeA0 = 0 for the U(1) gauge field Aµ. Actually, in the “quantum” formalismof the classical result they reappear as “Ward-identities” by gauge variations(diffeomorphisms, local Lorentz transformations) of W0(j, J). For details refs.[285, 197] can be consulted.

There is one most important lesson to be drawn retrospectively for thematterless case which may have consequences in a more general context thanthe present one of 2D theories of gravity: The exact quantum integral ofthe geometry leading here to the classical theory uses, among others, a pathintegral over all values of q3 in order to arrive at the classical equation for themomenta through the δ-function. However, q3 in the gauge (3.3) is identical tothe determinant (e) =

√−g. Therefore, a summation including negative andvanishing volumes has to be made to arrive at the correct (classical) result.

It is instructive to derive the effective action corresponding to the gener-ating functional (7.45). In terms of the mean fields 〈qi〉, 〈pi〉

〈qi〉 =δL

(0)eff

δji, 〈pi〉 =

δL(0)eff

δJi, (7.49)

the effective action Γ(〈qi〉, 〈pi〉) results from the Legendre transform of L(0)eff ,

Γ(〈qi〉, 〈pi〉) = L(0)eff (j, J) −

∫d2x (ji〈qi〉 + Ji〈pi〉) , (7.50)

where the sources must be expressed through the mean fields. To economizewriting the brackets the notations

〈qi〉 = (ω1, e−1 , e

+1 ) , 〈pi〉 = (X,X+, X−) , (7.51)

imply a simple return to the original geometric variables (cf. (7.2) and (7.3)).A peculiar feature of the first order formalism is that only the last three

of the equations (7.49) are needed:

j1 = ∂0X −X+ , (7.52)

j2 = ∂0X+ , (7.53)

j3 − V (X) = e−Q∂0

(eQX−

)=(∂0 +X+U(X)

)X− . (7.54)

The exact effective action (7.50) for the dilaton gravity models without matterimmediately follows from (7.52)-(7.54):

Γ =∫

Md2x

[ω1X

+ − ω1(∂0X) − e−1 (∂0X+) − e+1 (∂0X

−)

−e+1 (V (X) +X+X−U(X))]± g

∫

∂Mdx1eQX− , (7.55)

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

It has been assumed that the manifold M has the form of a strip M =[r1, r2]×R. The upper (+) sign in front of the surface term 67 corresponds tothe “right” boundary x0 = r2, and the lower (−) sign to x0 = r1.

The volume term in (7.55) is just the classical action in the temporal gauge(cf. first line in (7.34)). Thus all dilaton gravities without matter are locally

quantum trivial [281]. Therefore, all eventual quantum effects are encoded inthe boundary part and are global.

Except in subsection 5.1 and in eq. (7.55) so far complications fromboundary effects were entirely ignored. Their inclusion in the path integralapproach is a highly nontrivial problem and, in general, requires the introduc-tion of non-local operators at the boundary (cf. e.g. [25, 313, 418, 332, 144]).Matterless 2D quantum gravity being a special case of PSM-s, the discussioncan be incorporated into the one of these more general models 68 , if the bulkaction (2.34) is supplemented by

∫

∂M2

f(C)XIAI , (7.56)

where C is the Casimir function (2.45-2.47). A consistent way 69 to implementboundary conditions is fixing f(C) = 0 and δXI = 0 at time-like ∂M2, whichimplies XI |∂M2 = XI(r) with r being the “radius”. In fact, this prescriptionwe had imposed tacitly by dropping all boundary contributions in (7.44) andby choosing fixed functions of x0 (which corresponds to a radial coordinate inour gauge) for the boundary values of pi.

As pointed out in sect. 2.3, gravity theories in D = 2 without matter arespecial examples of PSM-s. Not surprisingly, the exact path integral also isencountered there [220]. Recently also an “almost closed expression” for thepartition function on an arbitrary oriented two-manifold has been presentedas well [221].

It is interesting to compare this result with local perturbative calcula-tions [373, 242, 139, 138]. Results obtained in different gauges must coincideon-shell only. The effective “quantum” actions obtained in these papers indeedvanish on-shell in full agreement with our non-perturbative calculations. Hencethe non-trivial off-shell counterterms appearing in refs. [373,242,139,138] arepure artifacts of the gauges employed. In ref. [66] local quantum triviality ofsome dilaton models has been verified with the conformal field theory tech-

67 The two omitted surface terms (cf. footnote 66) are just∫∂M dx1X+ and∫

∂M dx1X. Since both quantities will be fixed by suitable boundary conditions inthe next section we have already dropped them.68 We are grateful to L. Bergamin and P. van Nieuwenhuizen for discussions on thatmodel.69 We mean consistency as defined in [300]: boundary conditions arise from (1) ex-tremizing the action, (2) invariance of the action and (3) closure of the set of bound-ary conditions under symmetry transformations. In Maxwell theory, e.g., these con-sistency requirements single out electric or magnetic boundary conditions [422].

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

nique. Some authors [87,93] also rely on rather complicated field redefinitionswhich, however, as a rule produce Jacobian factors, making a comparison withthe result [281] very difficult. Finally, ref. [301] should be mentioned whereloop calculations in the presence of the Polyakov term have been performed,although only part of the degrees of freedom was quantized.

7.4 Path integral with matter

7.4.1 General formalism

When the geometry interacts with matter the computation must be re-sumed at the generating functional (7.33). After performing the integrations(Dp)(Dq) as in the matterless case one arrives at

W =∫

(Dφ) exp iL(3), (7.57)

L(3) =∫d2x

[F (B1)(∂0φ)(∂1φ) + JiBi + σφ+ L(ji, Bi)

], (7.58)

L(ji, Bi) = geQ(h+ j3 − V (B1) − F (B1)f(φ)). (7.59)

Here Bi are the solutions for pi from (7.37)-(7.39) and thus are functions ofthe scalar field φ as well. The notation Q and V also indicates the dependenceof these quantities on Bi instead of Bi (cf. (7.43)). Nevertheless, the actionstill is seen to be only linear in h = uu + v2 in L (7.59) and B3 of (7.39).Therefore the identity (7.36) may be used backwards with q3 replaced by

E+1 (J, j, p, f) = J3 e

−Q∂0 eQ ∂−1

0 eQ + g eQ . (7.60)

In the ensuing new version of (7.57)

W =∫

(D φ) (detE+1 )1/2 exp iLeff , (7.61)

Leff =∫d2x

[F (B1)(∂0φ)(∂1φ) + JiBi |h=0 + L |h=0 + σφ

], (7.62)

the measure allows an intuitive interpretation. For physically interesting Greenfunctions with J3 = 0 the penultimate term in (7.60) is nothing but q3 again,however expressed in terms of the sources j and containing the scalar field.Thus this determinant in the measure duely takes into account back reactionsof scalar matter upon the geometry.

This is how far we can get using non-perturbative methods. The finalmatter integration cannot be performed exactly.

7.4.2 Perturbation theory

In the treatment of matter one now may follow the usual steps of per-turbative quantum field theory. In order to avoid cumbersome formulas and,

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

nevertheless, elucidate the basic principles we restrict the discussion to mini-mal coupling (F (p1) = 1) and no local self-interaction (f(φ) = 0).

First the terms quadratic in φ of (7.58) are isolated by expanding Bi andL to this order. Then, they are considered together with the source term σφin a (Gaussian) path integral. The φ-dependence in the measure contributesto higher loop order only. Higher order terms O(φ2n) (n ≥ 2) in (7.62) areinterpreted as vertices and taken outside the integral, with the replacementφ→ 1

iδδσ

. We denote them summarily as Z(φ):

W = exp

[i∫Z

(1

i

δ

δσ

)]W (7.63)

W (j, J, σ) =∫

(Dφ)√

detE+1 exp i

∫[(∂0φ)(∂1φ) −E−

1 (∂0φ)2 + σφ] (7.64)

In the coefficient E−1 the different (nonlocal) contributions from the quadratic

terms in (∂0φ)2 of (7.63) are lumped together. Comparing (7.64) with the pathintegral

W =∫

(Dφ 4√−g) exp i

∫d2x

√−g [1

2gµν(∂µφ)(∂νφ) + σφ] (7.65)

explains our choice of the symbol E−1 , because it is a generalization of the

zweibein component e−1 which for the EF gauge would appear in this place.Here, by construction, E+

1 and E−1 depend on the external sources and not

on the scalar field. A more general form (with F ′(p1) 6= 0) of the “effectivezweibein” will be considered in sect. 8.

A Gaussian integral like (7.65) leads to the inverse square root of a func-tional determinant which in D = 2 may be reexpressed as a Polyakov ac-tion 70 [361] ( = gµν∇µ∂ν)

[ det ]−1/2 = exp iL(Pol) , (7.66)

L(Pol) = − 1

96π

∫∫d2xd2y

√−gRx−1xy Ry . (7.67)

Then the full expression W (7.65), written as (7.66), becomes

W = exp[iL(Pol)

]exp

[− i

2

∫d2x

∫d2y σx∆xyσy

], (7.68)

where the propagator ∆ = ∆(j, J) = [√−g ]−1 by its dependence on j con-

tains the eventual interaction with external zweibeine. For minimally coupled

70 For nonminimal coupling (e.g. F (p1) ∝ p1) its place would be taken by a corre-sponding quantity generalized to depend on the dilaton field p1 = B1 expressed interms of the scalar field and external sources. The effective action proposed in [287]indicates the possible form of such a generalization.

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

scalars (F = 1) it obeys

(∂0∂1 − ∂0E−1 ∂0) ∆xx′ = δ2(x− x′) . (7.69)

The ansatz∆xx′ =

∫

x′′

θxx′′(∂−10 )x′′x′d

2x′′ (7.70)

allows the formal computation of θxx′ as ( P means path ordering)

θxx′ = P−1 ∂−11 P , (7.71)

P (x) = P exp[−∫

xdy1E−

1 (x0, y1) ∂0

]. (7.72)

For the classical expressions in the exact path integral the meaning of ∂−10

as an integration (with undetermined integration constant) is evident. In thequantum case a more careful definition of the Green functions is requiredwhich implies a UV and IR regularization. A suitable definition is (∂0 →∇0 = ∂0 − iµ)

∇−10 = −Θ(y0 − x0) eiµ(x0−y0) . (7.73)

The regularization parameter µ = µ0 − iǫ (µ0 → +0, ǫ → +0) guaranteesproper behavior at x0 − y0 → ±∞. One easily verifies the same propertyin ∇−2

0 xy =∫z ∇−1

0 xz∇−10 zy as well as in higher powers. Only in expressions

involving the classical background like ∇−10 p2 in (7.40) when (7.41) is inserted

this rule must be adapted. With ∇0 p2 = 0 and thus p2 = p2(x1) eiµx

0the

expression ∇−10 p2 diverges. The solution consists in simply going back in these

(classical) terms to the classical interpretation where ∇−10 = ∂−1

0 correspondsto integration.

The formulas for GDTs with nonminimally coupled self interacting scalarscan be derived retaining F 6= 1 and f 6= 0 in (7.38), (7.39). Then the per-turbation expansion in terms of Newton’s constant requires an expansion interms of F already in the solution of these equations. Together with the per-turbation theory outlined in connection with a path integral (7.64) this yieldsrather complicated formulas which, therefore, will not be exhibited here. Onlyin connection with the “virtual BH” of sect. 8 this case will be dealt with.

At the moment there seems to exist only one computation of higher loopeffects for the simpler case of minimally coupled scalars (F = 1) and a cor-responding dilaton theory without kinetic term (U = 0), but the tools areavailable for arbitrary loop calculations. As noted in ref. [284] for that spe-cific class of theories the whole two-loop effect is just a renormalization of thepotential V .

7.4.3 Exact path integral with matter

The JT model (2.12) is an example of a situation where the path integralcan be calculated exactly even in the presence of minimally coupled matterfields [281]. There the integration (Dφ) produces the Polyakov action. Thusthe generating functional for the Green functions reads:

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7 NONPERTURBATIVE PATH INTEGRAL QUANTIZATION

WJT =∫

(Dqi)(Dpi) exp iLJTeff , (7.74)

LJTeff =

∫d2x

[piqi + q1p2 + Λq3p1

]+ L(Pol)(q2, q3) , (7.75)

where source terms j, J, σ and the propagator for the scalar field have beendropped. The crucial observation is that (7.75) is now linear in pi. Therefore,when integrating first over the momenta one obtains three functional deltafunctions which may be used to integrate over qi. Up to this change, the wholeprocedure works as before. For details we refer to ref. [281]. Again, somethinglike local quantum triviality occurs. The action (7.75) already incorporatesall quantum effects, because in this case no higher loop corrections exist, afeature used in refs. [146, 145, 55] to extend the one-loop calculations to allorders of perturbation theory.

The method of exact functional integration described here seems to bea rather general one, although it does not seem possible to formulate generalcriteria of applicability. We just note that in a similar way an exact pathintegral has been calculated in a different context, namely the Bianchi IXreduction of Ashtekar gravity [4]. As a final remark of this section it shouldbe stressed that in gravity theories – in contrast to quantum field theoryin Minkowski space – there is no immediate relation between classical andquantum integrability.

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8 VIRTUAL BLACK HOLE AND S-MATRIX

8 Virtual black hole and S-Matrix

Only the interaction with matter in D = 2 provides continuous physicaldegrees of freedom. Since the asymptotic states depend on the model underconsideration we do not discuss the most general case, but for illustrating themain technical details we focus on an explicit example instead: SRG with anon-minimally coupled massless scalar field. We also select a situation in whichthe existence of an S-matrix in the usual quantum field theoretic sense shouldbe unchallenged, namely gravitational scattering of scalars in asymptoticallyflat space, however, without fixing the background before quantization.

In sect. 7 we have demonstrated that all geometrical degrees of freedomcan be integrated out exactly. This procedure yields effective non-local inter-actions of the remaining fields (scalars in our case). In this section we calculateexplicitly some lowest order effective vertices and ensuing tree level S-matrixelements corresponding to gravitational scattering of s-wave scalars. The re-sults can be interpreted as an exchange of virtual black holes.

The vertices are extracted from the action (7.62) after separation of theinteraction part Z (cf. (7.63)). They appear as complicated nonlocal expres-sions with multiple integrals in x0 from repeated multiplications of the formalobject ∂−1

0 present in Bi, the solutions for pi of eqs. (7.37)-(7.39). There existtwo classes of vertices (we have attached all outer legs in the formulae below),symmetric ones

V (2n)a =

∫dx2

1 . . . dx2nv

(2n)a (x1, . . . , xn) (∂0φ)2x1

. . . (∂0φ)2xn, (8.1)

and non-symmetric ones 71

V(2n)b =

∫dx2

1 . . . dx2nv

(2n)b (x1, . . . , xn) (∂0φ∂1φ)x1

(∂0φ)2x2. . . (∂0φ)2

xn. (8.2)

They have the following properties:• They contain an even number of outer legs. Thus, in addition to a propa-

gator term (cf. e.g. (7.68) for the simpler case of minimally coupled scalars)there are φ4-vertices, φ6-vertices and so on.

• Each pair of outer legs is attached at one point xi to the non-local vertex.• Each outer leg contains one derivative.• Non-locality is inherited from the ∂−1

0 operators in the Bi.• The symmetric vertices originate from the term L in (7.62).• The non-symmetric vertices are produced on the one hand by F (B1)∂0φ∂1φ

in (7.62), on the other hand also L yields such terms. For minimal couplingall non-symmetric vertices vanish.

• The information contained in the tree-graphs is classical. Thus, it must bepossible to extract it by other means. Nevertheless, the path integral seems

71 For the evaluation of the S-matrix one has to permute all external legs and thusleg-exchange symmetry is restored.

109

8 VIRTUAL BLACK HOLE AND S-MATRIX

to be the most adequate language to derive scattering amplitudes 72 .The lowest order tree-graph and the ensuing S-matrix element has been

evaluated in ref. [196] for F (X) = const. . The result was trivial, unless mass-terms f(φ) = m2φ2 had been added (cf. the discussion after eq. (8.23)). There-fore, we will focus in the rest of this section on the (also phenomenologicallymore relevant) case of non-minimal coupling [157].

8.1 Non-minimal coupling, spherically reduced gravity

In principle all effective interactions of the scalars can be extracted byexpanding the non-local action (7.62) in a power series of φ. At each order thenumber of integrations increases, and one has to fix the ranges appropriately.This becomes cumbersome already at the φ4 level. Fortunately, two observa-tions [285] simplify the calculations considerably. First, instead of dealing withcomplicated non-local kernels one may solve corresponding differential equa-tions. All ambiguities are then removed by imposing asymptotic conditions onthe solutions. Second, instead of taking the n-th functional derivative of theaction with respect to bilinear combinations of the scalar, the matter fieldsmay be localized at n different space-time points. This mimics the effect offunctional differentiation.

To be more specific, the symmetric vertex (8.1) may serve as an example

v(2n)a (x1, . . . , xn) ∝

δnLeff

δ((∂0φ(x1))2) . . . δ((∂0φ(xn))2). (8.3)

By its definition, the functional derivative is a response to a small localizedchange of the functional argument ((∂0φ)2 in the present case). Therefore, letus choose a specific matter distribution such that (∂0φ)2 is localized at n− 1points:

(∂0φ)2(x) ∝n∑

k=2

c[k]δ2(x− xk) . (8.4)

Now let X(x1) and E−1 be solutions of the classical field equations in the

presence of localized matter (8.4). In

v(2n)a (x1, . . . , xn) ∝ F (X(x1))E

−1 (x1)[x2, . . . xn]|πc

(8.5)

we have indicated the dependence of X and E1 on the matter distribution.The notation πc after the vertical line means that one has to expand in c[k]

and to select the coefficient in front of the product∏nk=2 c

[k]. The proof of thisstatement with explicit coefficients instead of the proportionality symbol aswell as a corresponding argument for the nonsymmetric vertex (8.2) can befound in refs. [285, 196,157].

72 As in other well-known examples – e.g. the Klein-Nishina formula for relativisticCompton scattering [261] – this formalism seems to be much superior to a classicalcomputation.

110

8 VIRTUAL BLACK HOLE AND S-MATRIX

Qualitatively, the result (8.5) is rather easy to understand. In the classicalaction (∂0φ)2 appears multiplied by F (X)e−1 . Due to local quantum trivialityin the absence of matter it is natural to expect “effective” quantities of thesame nature in the vertices.

The most interesting case is SRG [157], where V = −2, U = −(2p1)−1,

f(φ) = 0, F = p12, ji = Ji = Q = 0. To extract the terms quartic in φ

(n = 2 in (8.1), (8.2)) one has to take a matter distribution localized at onespace-time point (cf. (8.4))

φ0 :=1

2(∂0φ)2 → c0 δ

2(x− y) , (8.6)

φ1 :=1

2(∂0φ)(∂1φ) → c1 δ

2 (x− y) , (8.7)

and to solve the classical e.o.m.-s up to linear order in the constants c0 andc1 which just keep track of the number of sources. The differential equations(7.37)-(7.39), together with classical equations for qi become

∂0p1 = p2,

∂0p2 = p1φ0,

∂0p3 = 2 +p2p3

2p1,

∂0q1 =q3p2p3

2p21

+ φ1 − q2φ0,

∂0q2 = −q1 −q3p3

2p1,

∂0q3 = −q3p2

2p1.

(8.8)

Their solutions, to be substituted back into the action with the inverse re-placement (8.6) and (8.7), are found easily:

p1(x) = x0 − (x0 − y0)c0y0h(x, y), (8.9)

p2(x) = 1 − c0y0h(x, y), (8.10)

q2(x) = 4√p1 +

(8c0y

0√p1 − 2c0y03/2 − c1y

0 + (c1 − 6c0y01/2

)p1

)h(x, y)

(8.11)

q3(x) =1√p1. (8.12)

Here h(x, y) := θ(y0 − x0)δ(x1 − y1) corresponds to one of the possible pre-scriptions introduced in ref. [196] for the boundary values at x0 → ∞. It turnsout that the vertices below are independent of any such choice. The matchingconditions at x0 = y0 follow from continuity properties: p1, q2 and q3 are C0

and ∂0q2(y0 + 0)− ∂0q2(y

0 − 0) = − (c1 − q2(y0)c0) δ(x

1 − y1). The integrationconstant which would produce an asymptotic (i.e. for x0 → ∞) Schwarzschildterm has been fixed to zero. Consistency of integration constants with theset of e.o.m.-s containing ∂1 automatically yields a vanishing Rindler term.Furthermore, it relates the asymptotic Schwarzschild term to the asymptoticvalue for the geometric part of the conserved quantity [285]. Thus, only fourintegration constants can be chosen independently. We fix those in pi and q3.

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8 VIRTUAL BLACK HOLE AND S-MATRIX

Because of our particular choice p3(x0 → ∞) = 0 a BH may appear only at

an intermediate stage (the “virtual black hole”, see below), but should notact asymptotically. Due to the infinite range of gravity this is necessary for aproper S-matrix element if spherical waves are used as asymptotic states forincoming and outgoing scalar particles.

8.2 Effective line element

The arguments of the previous section suggest that matter interacts withsome effective geometry which solves the classical e.o.m.-s in the presence ofexternal sources. Moreover, this geometry can be extracted directly from thevertices. A more formal (but essentially equivalent) way to see this is to cal-culate the vacuum expectation values of q2 and q3 by varying the exact pathintegral (7.61) with respect to j2 and j3 in lowest order of the matter loop ex-pansion and in the presence of external matter field. The method described inthe previous subsection appears to be more straightforward and considerablysimpler.

The matter dependent solutions in the gauge (3.3) with (8.11) and (8.12)define an effective line element

(ds)2 = 2q3dx0(dx1 + q2dx

0)

= 2drdu+K(r, u)(du)2, (8.13)

with the identifications 73

u = 2√

2x1 r =√p1(x0)/2 . (8.14)

In the asymptotic region by our previous residual gauge fixing the Killingnorm K(r, u)|x0>y0 = 1 is constant. The line element (8.13) then appears inoutgoing Sachs-Bondi form. In the VBH region the Killing norm

K(r, u)|x0<y0 =(1 − 2m

r− ar + d

)(1 + O(c0)) , (8.15)

with m = δ(x1 − y1)(c1y0 + 2c0y

03/2)/27/2, a = δ(x1 − y1)(6c0y

01/2 − c1)/23/2

and d = δ(x1 − y1)2c0y0 has two zeros located approximately at r = 2m and

r = 1/a corresponding for positive m and a to a Schwarzschild horizon and aRindler type one.

73 The somewhat unusual role of the coordinates should be noted: x0 is asymptot-ically proportional to r2; thus our Hamiltonian evolves with respect to a “radius”as “time”-parameter. This also implies that e.g. the asymptotic energy density isrelated to the component T11 and not T00 of the energy momentum tensor.

112

8 VIRTUAL BLACK HOLE AND S-MATRIX

8.3 Virtual black hole

The geometric part of the conserved quantity (3.14) in our present nota-tion (3.35) reads

C(g) =p2p3√p1

− 4√p1. (8.16)

As a consequence of the choice of integration constants C(g) vanishes in theasymptotic region x0 > y0. The functions p1 and p3 are continuous, but p2

jumps at x0 = y0. Thus, C(g) is discontinuous. This phenomenon has beencalled “virtual black hole” (VBH) in [196]. It is generic rather than an artifactof our special choice of asymptotic conditions. The reason why we have chosenthis name is simple: The geometric part of the conserved quantity (8.16) isessentially equivalent to the so-called mass aspect function, which is closelyrelated to the BH mass (cf. sect. 5.2). Moreover, inspection of the Killing norm(8.15) reveals that for negligible Rindler acceleration a the Schwarzschild hori-zon corresponds to a BH with precisely that mass. It disappears in the asymp-totic states (by construction), but mediates an interaction between them.

The idea that BH-s must be considered in the S-matrix together withelementary matter fields has been put forward some time ago [402]. The ap-proach [157] reviewed here, for the first time allowed to derive (rather than toconjecture) the appearance of the BH states in the quantum scattering matrixof gravity.

The solutions (8.9) and (8.10) establish

C(g)∣∣∣x0<y0

= 4c0y03/2 ∝ −mV BH . (8.17)

Thus, c1 only enters the Rindler term in the Killing norm, but not the VBHmass (8.17).

i0

i-

i+

ℑ -

ℑ +

y

Fig. 8.1. CP dia-gram of the VBH

The CP diagram corresponding to the line element(8.15) as presented in figure 8.1 needs some explana-tions: first of all, the effective line element is non-localin the sense that it depends not only on one set of co-ordinates (e.g. u, r) but on two (x = (u, r), y = (u0, r0)),where r0 and u0 are related to y0 and y1 like r and u tox0 and x1 in (8.14). As discussed previously, this non-locality was a consequence of integrating out geometrynon-perturbatively. For each choice of y it is possible todraw an ordinary CP-diagram treating u0, r0 as externalparameters. The light-like “cut” in figure 8.1 correspondsto u = u0 and the endpoint labeled by y to the pointx = y. The non-trivial part of our effective geometry isconcentrated on the cut.

We do not want to suggest to take the effective ge-ometry (8.13) at face value – this would be like over-interpreting the role ofvirtual particles in a loop diagram. It is a nonlocal entity and we still have

113

8 VIRTUAL BLACK HOLE AND S-MATRIX

V(4)(x,y)a

x y

∂0 ϕ

q’

∂0 ϕ

q

∂0 ϕ

k’

∂0 ϕ

k

+

V(4)(x,y)b

x y

∂0 ϕ

q’

∂0 ϕ

q

∂1 ϕ

k’

∂0 ϕ

k

Fig. 8.2. Total V (4)-vertex with outer legs

to “sum” (read: integrate) over all possible geometries of this type in orderto obtain the nonlocal vertices and the scattering amplitude. Nonetheless, thesimplicity of this geometry and the fact that all possible configurations aresummed over, are nice qualitative features of this picture.

The localization of “mass” and “Rindler acceleration” on a light-like cut(see fig. 8.1) in (8.13) is not an artifact of an accidental gauge choice, but hasa physical interpretation in terms of the Ricci-scalar [198], the explicit formof which is given by [199]

R(V BH)(u, r; u0, r0) = δ(u− u0)

− δ(r − r0)

(4m0

r2− 4d

r+ 6a0

)

+ Θ(r0 − r)

(6a0

r− 2d

r2

) . (8.18)

As discussed in ref. [199] certain parallels to Hawking’s Euclidean VBHs [214]can be observed, but also essential differences. The main one is our Minkowskisignature which we deem to be a positive feature.

8.4 Non-local φ4 vertices

All integration constants have been fixed by the arguments in the preced-ing paragraphs. The fourth order vertex of quantum field theory is extractedfrom the second line of (7.34) by collecting the terms linear in c0 and c1 re-placing each by φ0 and φ1, respectively. The tree graphs we obtain in that way(cf. fig. 8.2) contain the nonlocal vertices

V (4)a =

∫

x

∫

yφ0(x)φ0(y)

(dq2dc0

p1 + q2dp1

dc0

)∣∣∣∣∣ci=0

=∫

x

∫

yφ0(x)φ0(y)

∣∣∣∣√y0 −

√x0

∣∣∣∣√x0y0

(3x0 + 3y0 + 2

√x0y0

)δ(x1 − y1), (8.19)

and

114

8 VIRTUAL BLACK HOLE AND S-MATRIX

V(4)b = −

∫

x

∫

y

(φ0(y)φ1(x)

dp1

dc0− φ0(x)φ1(y)

dq2dc1

p1

)∣∣∣∣∣ci=0

= −∫

x

∫

yφ0(x)φ1(y)

∣∣∣x0 − y0∣∣∣x0δ(x1 − y1), (8.20)

with∫x :=

∞∫0dx0

∞∫−∞

dx1.

8.5 Scattering amplitude

In terms of the time variable t := r + u the scalar field asymptoticallysatisfies the spherical wave equation. For proper s-waves only the sphericalBessel function

Rk0(r) =sin(kr)

kr(8.21)

survives in the mode decomposition (Dk := 4πk2dk):

φ(r, t) =1

(2π)3/2

∞∫

0

Dk√2kRk0

[a+k e

ikt + a−k e−ikt

]. (8.22)

With a± obeying the commutation relation [a−k , a+k′] = δ(k − k′)/(4πk2), they

will be used to define asymptotic states and to construct the Fock space. Thenormalization factor is chosen such that the Hamiltonian of asymptotic scalarsreads

H(as) =1

2

∞∫

0

Dr[(∂tφ)2 + (∂rφ)2

]=

∞∫

0

Dka+k a

−k k. (8.23)

In ref. [196] we had observed a non-physical feature in the massless casefor (in D = 2) minimally coupled scalars: Either the S-matrix was divergentor – if the VBH was “plugged” by suitable boundary conditions on φ at r = 0– it vanished. This implied an effective decoupling of the plane waves from thegeometry. For massive scalars a finite nonvanishing scattering amplitude hasbeen found.

In the present more physical case of s-waves from D = 4 GR at a firstglance it may seem surprising that the simple additional factor X in front ofthe matter Lagrangian induces fundamental changes in the qualitative behav-ior. In fact, it causes the partial differential equations (8.8) to become coupled,

giving rise to an additional vertex (V(4)b ).

After a long and tedious calculation (for details see refs. [197, 158]) forthe S-matrix element with ingoing modes q, q′ and outgoing ones k, k′,

T (q, q′; k, k′) =1

2

⟨0∣∣∣a−k a

−k′

(V (4)a + V

(4)b

)a+q a

+q′

∣∣∣ 0⟩, (8.24)

having restored 74 the full dependence on the gravitational constant κ =

74 Up to this point the overall factor in (2.2) had been omitted.

115

8 VIRTUAL BLACK HOLE AND S-MATRIX

00.2

0.40.6

0.81

alpha

0

0.2

0.4

0.6

0.8

1

beta

0

100

200sigma

00.2

0.40.6

0.8alpha

Fig. 8.3. Kinematic plot of s-wave cross-section dσ/dα

8πGN , we arrive at

T (q, q′; k, k′) = −iκδ (k + k′ − q − q′)

2(4π)4|kk′qq′|3/2 E3T (8.25)

with the conserved total energy E = q + q′,

T (q, q′; k, k′) :=1

E3

Π ln

Π2

E6+

1

Π

∑

p∈k,k′,q,q′p2 ln

p2

E2

·3kk′qq′ − 1

2

∑

r 6=p

∑

s 6=r,p

(r2s2

), (8.26)

and the momentum transfer function Π = (k+k′)(k−q)(k′−q). The interestingpart of the scattering amplitude is encoded in the scale independent factor T .

With the definitions k = Eα, k′ = E(1 − α), q = Eβ, and q′ = E(1 − β)(α, β ∈ [0, 1], E ∈ R

+) a quantity to be interpreted as a cross-section forspherical waves can be defined [157]:

dσ

dα=

1

4(4π)3

κ2E2|T (α, β)|2(1 − |2β − 1|)(1 − α)(1 − β)αβ

. (8.27)

The kinematic plot fig. 8.3 contains the relevant physical information. The de-pendence of the cross-section on the total incoming energy is trivially given bythe monomial prefactor E2: it vanishes in the IR limit and diverges quadrat-ically in the UV limit. At least the last fact is not surprising, consideringour assumption of energies being small as compared to the Planck energy. Itsimply signals the breakdown of our perturbation theory.

The main results of the detailed discussion [157,199] are:• Poles exist in the case of vanishing momentum transfer (forward scattering).• An ingoing s-wave can decay into three outgoing ones. Although this may

be expected on general grounds, within the present formalism it is possibleto provide explicit results for the decay rate.

116

8 VIRTUAL BLACK HOLE AND S-MATRIX

• Despite the non-locality of the effective theory, the S-matrix is CPT invari-ant at tree level.

• Fig. 8.3 appears to exhibit self-similarity. Indeed, by zooming into the centerof that figure one obtains again an identically-looking plot. However, thisself-similarity is only a leading (and next-to-leading) order effect and breakesdown in the flat regions.

8.6 Implications for the information paradox

Very roughly, the information paradox [18] may be formulated in the fol-lowing way. Imagine a pure quantum state in a non-singular asymptoticallyMinkowski space-time. Let this pure state collapse into a BH which evapo-rates due to the Hawking effect. This effect is only understood for a back-ground which does not change appreciably due to radiation: if it does, it is,nevertheless, assumed that this evaporation proceeds through an (unknown!)final phase so that the BH disappears. The final state of this process willbe Minkowski space filled with thermal radiation which is definitely a mixedquantum state. Therefore, a pure quantum state seems to evolve into a mixedone, contradicting basic laws of quantum mechanics. One reason already hasbeen given why this picture is a rather approximate one. In addition the exactthermal Planck spectrum of the radiation requires infinite time for its for-mation, i.e. radiation can be strictly thermal only if the BH never disappearscompletely. However, the formation of even approximately thermal final statesseems very difficult to master in quantum theory which has prompted severalvery interesting developments in quantum gravity, as e.g. models of stable BHremnants [2] and the S-matrix approach of ref. [394].

The understanding of the evolution of VBHs is crucial for quantum gravity[213]. Their evaporation – or, more exactly, their conversion to mixed stateswould inevitably violate either locality or energy-momentum conservation [20].Since the non-perturbative approach to path integral quantization now alsopredicts VBHs in two dimensions, it is important to understand whether thereis an “information loss” in these models. Of course, there is none at the treelevel discussed above. However, we also have good grounds to believe that thesituation will remain the same in higher loops.

Our first argument is somewhat formal. In the two-dimensional model wewere able to extract the VBH from the degrees of freedom already present inthe theory rather than to be forced to introduce it from the outside. The wholesystem has been quantized in full accord with the general principles of thequantization for systems with constraints. According to general theorems [219]the resulting quantum theory must be unitary, respect causality and energyconservation, and must forbid transitions of pure states to mixed ones, as longas we are able to refer to a Fock space of the asymptotic states.

Our next argument is more physical. BH evaporation is related to thecondition (6.25) which fixes the energy-momentum tensor at the horizon andthus defines the Unruh vacuum state. This condition is clearly not applicable

117

8 VIRTUAL BLACK HOLE AND S-MATRIX

to VBHs. The relevant vacuum state for the scalar field is just the usualMinkowski space vacuum containing no information about VBH states whichmay be formed in quantum scattering. Kruskal coordinates for a VBH cannotbe associated with any real observer. Therefore, the argument that the energy-momentum tensor must be finite at the horizon is not applicable to it. The onlyvacuum state which can be defined by a condition at infinity rather then ona horizon is the Boulware vacuum which does not contain Hawking radiationso that VBHs do not radiate anything to infinity.

It must be admitted that in order to put this argumentation upon a firmbasis one should calculate the next (one-loop) order in the path integral. SinceHawking radiation is a one-loop effect, this order of the perturbation theorywill be actually sufficient.

118

9 CANONICAL QUANTIZATION

9 Canonical quantization

Canonical quantization methods dominated 2D dilaton gravity during itsearly years. They owe their success to the fact that the geometrical sector con-tains no propagating degrees of freedom, and, therefore, the problem reduceseffectively to a quantum mechanical one.

After the extensive discussion of the path integral in the previous sectionwe intend to be brief for this essentially equivalent approach. The prehistoryof canonical quantization of gravity involves the seminal papers of Arnowitt,Deser and Misner [7], Wheeler [435] and DeWitt [121] which led to Misner’s“minisuperspace quantization” program [331], where almost all degrees of free-dom were frozen by symmetry requirements. Kuchar extended these techniquesto “midisuperspace quantization” for the explicit example of cylindrical grav-itational waves [273], i.e. to a system with field degrees of freedom, albeit stillusing symmetry requirements in order to simplify the formalism. This seemsto be the only midisuperspace model that could be treated exactly. The mostnotable example of a non-soluble model is the collapse of spherically symmet-ric matter [36] (cf. [412] for an essential correction to that paper). A canonicaltreatment of a complete Schwarzschild spacetime under somewhat too strongassumptions provided Lund’s proof [308] of the non-existence of an extrinsictime representation for vacuum Schwarzschild BHs [74].

Possibly due to the impact of Hawking’s work on semi-classical radiationof BHs [211] the discussion of genuine quantum gravity effects was postponeduntil the CGHS model [71] rekindled the interest in (exact) quantization ofBH-s [173,276,277,77,174,395,24,175,303,73,75,74,76,326,327,328,415,425].In particular, in ref. [276] an extrinsic time representation for the quantizedSchwarzschild BH allowed to circumvent Lund’s no-go theorem by relaxing itspremises.

As an explicit example for demonstrating the main points we focus onthe CGHS model, the Dirac quantization of which has been studied by Jackiwand collaborators [73, 75, 76, 77, 32], by Mikovic [326, 327, 328], and later alsoby other authors.

Our brief summary follows the work of Kuchar, Romano and Varadara-jan [277]. The starting point is not really the CGHS action (2.8), but its con-formally related one (4.40). In this way, one eliminates the kinetic term of thedilaton field in (2.9) at the cost of a singular conformal transformation (4.39).This action is then cast into canonical form by the standard ADM decomposi-tion. It has to be supplemented by surface terms invoking the requirement offunctional differentiability 75 . However, the boundary action leads to an im-portant caveat: At the left and right infinity corresponding to the asymptoticregions of patch A and patch B in the figures 3.5 and 3.6 arbitrary variations

75 The physically most transparent way to impose it is a careful treatment of asymp-totic conditions on the geometric variables [30] (including lapse and shift; cf. alsosect. 5.1).

119

9 CANONICAL QUANTIZATION

of the lapse are required. Otherwise unwanted “natural” boundary conditionsfor the BH mass emerge which imply vanishing of the BH mass. This problemhas been resolved for the Schwarzschild BH [276], parameterizing the lapsefunction at the boundaries by a proper time function. It turns out that thetotal action

L =∫d2x

(πφ+ πy y + pσσ −NH −N1H1

)+∫dt (τLmL − τRmR) , (9.1)

depends on these two additional parameters τL and τR and on the standardcanonical variables: N is the lapse, N1 the shift,H the Hamiltonian constraint,H1 the momentum constraint, π, φ denotes the matter degree of freedom (thepresence of a single minimally coupled scalar field is assumed), and in thenotation of ref. [277] y, πy, σ, pσ are geometric canonical field variables. In theboundary action the indices L,R refer to “left infinity” and “right infinity”,mL,R are the – conveniently normalized – BH masses. The relative sign betweenthe last two terms originates from the different time orientations one choosesfor the two patches in order to match the behavior of the Killing time T inthe corresponding global diagram.

There exists a canonical transformation mapping the action (9.1) ontoa simpler one (in terms of which the constraints become second order poly-nomials). One has to be particularly careful with the boundary part. In thenew variables the constraints can be solved exactly, because they have thesame form as those of a parametrized massless scalar field propagating on aflat 2D background. The main obstacle in replacing the canonical variables bycorresponding operators is a Schwinger term encountered in the commutatorsof the energy-momentum tensor operators [46]. This anomaly converts theclassical first class constraints into quantum second class ones and thus theimposition of the operator constraints on the states leads to inconsistencies 76 .Kuchar proposed a trick to get rid of that anomaly in the Schrodinger pic-ture [274, 275]: The momentum operators are supplemented by an additionalterm which does not change the canonical commutation relations but whichcancels the anomaly.

To summarize: by performing first a conformal and then a canonical trans-formation the CGHS model was mapped onto a parametrized field theory ona flat background which could be quantized successfully. Clearly this quantumtheory of the parametrized field is a standard unitary quantum field theory,i.e. no information loss is encountered. However, the interesting questions areprecisely those related to the physical spacetime. Thus, one still has to show

76 This is true at least in the Schrodinger picture. In the Heisenberg picture thequantum theory is well-defined and has the same number of degrees of freedom asthe classical one. Indeed, also the Heisenberg e.o.m.-s have the same form as theclassical e.o.m.-s (of course, one has two additional quantum mechanical degrees offreedom from the two parameters in the boundary action, but they are just constantsof motion).

120

9 CANONICAL QUANTIZATION

how to pose such questions in the framework based on the auxiliary flat back-ground. As emphasized by the authors themselves [276, 277] it seems thatdifficult problems reemerge which were avoided so far:• It is not clear how to make sense of the operator version of the physical line

element (ds)2physical = (ds)2

flat exp (−2ρ).• The “correct” operator ordering of the conformal factor is an open question

when ρ is expressed in terms of the auxiliary canonical variables.• The classical dilaton field should remain positive to ensure the correct sig-

nature of the physical metric. In a quantum theory it is highly nontrivial tomaintain this positivity requirement. There have been attempts to clarifythis issue with a 1 + 0 dimensional model [432].

Besides, the presence of an anomaly may add difficulties in implementation ofthe Dirac quantization scheme [76].

If there is no matter field in the model the canonical quantization isespecially simple. The reduced phase space quantization program 77 can becarried through exactly to the very end, i.e. one can solve the constraints andfix the gauge freedom. However, the result is essentially trivial for sphericallyreduced gravity [407, 276, 90, 91], as well as for the other dilaton models [92]:the quantum functional only depends on the ADM mass.

Since the Maxwell field in two dimensions does not add new propagatingdegrees of freedom an extension of the canonical approach to charged BHsmay be done in a rather straightforward manner (cf. [319, 23]).

Other instances where the programme of canonical quantization has beencarried through in essentially quantum mechanical models like the collapseof spherically symmetric (null-)dust are refs. [40, 206, 207, 270, 424, 208]. Anexample for a semiclassical model of BH evolution with time variable is ref.[82].

77 A clear explanation of the reduced phase space quantization can be found in [150].

121

10 CONCLUSIONS AND DISCUSSION

10 Conclusions and discussion

The last decade has seen remarkable progress in the treatment of 2Ddilaton gravity models. Having retraced the main historical lines of this devel-opment in the Introduction we now confront the main results of this field witha list of well-known problems which quantum gravity, in fact, shares with otherquantum theories in which geometry plays a fundamental dynamical role.

First we summarize the main contributions which dilaton gravity has beenable to provide with respect to these questions and which we have describedin some detail in this report.

Dilaton models in D = 2 possess the basic advantage that their geometricpart, in a certain sense, is a “topological” theory, albeit one where the solutionsare not related to a discrete winding number. An important special case is thetheory which arises from spherical reduction of Einstein gravity in D = 4, i.e.also the treatment of Schwarzschild black holes is covered by it. Other relevantmembers are the string-inspired dilaton theory and the Jackiw-Teitelboimmodel.

In the absence of matter the classical solution for all such theories canbe given in closed analytic form, a result which appears more naturally in theEddington-Finkelstein gauge for the metric. In that gauge also a very straight-forward procedure allows the construction of the global solution without thenecessity to introduce explicitly or implicitly global Kruskal-like coordinates.It is a peculiar feature of effective two dimensions of space-time that the ADM-mass, even in the presence of matter interactions, generalizes to an ”absolute”(in space and time!) conserved quantity. Technically many new results arerelated to the complete dynamical equivalence between the standard formula-tion of dilaton theories by an action expressed in terms of metric and dilatonfield on the one hand, and a “first order” (“covariant Hamiltonian”) actionon the other hand. The latter involves auxiliary fields and the geometry isexpressed in Cartan variables (zweibeine and spin connection). This equiva-lent formulation also contains nontrivial torsion and turns out to represent aspecial case of the very general concept of Poisson-Sigma models, a new andrapidly developing field of research with important connections to strings andnon-commutative geometry. Certain generalizations, as e.g. Yang-Mills fieldsor supergravity extensions are covered directly by this formalism.

Strictly speaking, the (semi-classical) treatment of Hawking radiationdoes not represent an application of quantum gravity but it is formulated withrespect to a given classical (Black Hole) background. Nevertheless, in order tojustify other 2D quantum gravity results derived from an effective 2D theory,it should emerge as well from a treatment of the spherically reduced case. Asfar as (in D = 2) minimally coupled scalar fields are concerned all aspectsare well understood. In spherically reduced matter (nonminimal coupling inD = 2) a correct relation between Hawking temperature and Hawking flux hasbeen proposed, however based upon mathematical steps whose justification asyet has not been proved conclusively. For a wide class of twodimensional grav-

122

10 CONCLUSIONS AND DISCUSSION

ity models relations between Hawking temperature and ADM mass can beobtained which differ from the one in Einstein gravity.

The full impact of the advantages from the first order formulation of dila-ton modelsD = 2 is revealed in the path integral quantization of such theories.In the temporal gauge for Cartan variables – corresponding to the Eddington-Finkelstein gauge of the metric – it proved possible to exactly integrate outall geometric degrees of freedom. This intrinsically nonperturbative result isclosely related to the quantum field theoretical “triviality” of generic gravitytheories without matter interactions in D = 2. In this derivation the path in-tegral over all positive and negative “volumes” is an essential ingredient, thusestablishing an important confirmation of the conjecture that this should alsobe the correct procedure in D = 4.

If matter fields are present, still an effective theory is obtained in whichgeometry is treated in a nonperturbative manner. A perturbation expansionin terms of the interactions with matter follows standard quantum field the-oretical methods. It is valid as long as the energies are small as comparedto Planck’s mass. The effective non-local vertices of scalar fields in this for-mulation can be interpreted as the appearance of an intermediate “virtual”Black Hole in certain scattering amplitudes of spherical waves. It should bestressed that this seems to be the first instance where such a virtual BlackHole reflects an intrinsic feature of the theory and is not introduced by anyadditional assumption.

As far as the problem of observables in quantum gravity is concerned, thecomputation of a special (gauge-independent) S-matrix element for sphericallyreduced Einstein gravity seems to be an interesting feature as well. Also someprogress has been reported regarding the final stages of Black Hole evaporationand the intimately connected “quantum information paradoxon”. The veryfact that now a formulation of that system exists in the form of a standardquantum field theory implies that – also at the very end of its existence —a Black Hole does not violate quantum mechanical concepts like unitarity.In that case as in others it has turned out that – at least in D = 2 – theapplication of standard quantum field theoretic techniques can go very far,leading to interesting results without the necessity to infer additional concepts.

This suggests further studies in many directions which have not yet suf-ficiently been covered so far.

At the classical level a more systematic search for 2D gravity theories in-volving matter interactions, but still allowing exact solutions, seems desirable,e.g. for a – perhaps at first only qualitative, but nevertheless exact – descrip-tion of critical behavior in spherical collapse. The same applies for models withadditional abelian or nonabelian gauge fields from which the spherically sym-metric Black Hole with (nonabelian) charges could be studied. Although thegeneral principle to obtain supergravity extensions from 2D dilaton theories isnow available, the new comprehensive approach based upon the Poisson-Sigmastructure of such models has posed many new questions.

Recently a whole new field of scalar-tensor theories in D = 4 (quintes-

123

10 CONCLUSIONS AND DISCUSSION

sence) has been developed. There a dilaton (“Jordan”-) field already appearsin the higher dimension. Certain important aspects of these models shouldbe accessible by 2D-methods when the effective spherically reduced theory isconsidered.

Within the realm of semi-classical problems despite new insight for thetreatment of Hawking radiation starting in the spherically reduced case, stillseveral important questions are open.

Among the possible directions of research in full 2D quantum gravityhigher loop corrections could be investigated. The issue of “quantum” observ-ables is closely related to the treatment of systems with finite boundary andrelated boundary variables.

Possibly also new elements for the long discussion of quantum gravityat the Big Bang (quantum cosmology) could emerge. More immediate conse-quences of the present approach are a generalization of gravitational scatter-ing of scalars described in this report, for scattering off a Black Hole. Anothergeneralization in the quantum case would be the treatment of fermions, eitherdirectly introduced in D = 2 or obtained from D = 4 by reduction. Finally,the virtual black hole phenomenon exists for generic dilaton models. It couldbe interesting to study the S-matrix of gravitational scattering of matter inthe extended context of generalized dilaton theories. The range of technicallyfeasible investigations now certainly has been enlarged substantially.

124

ACKNOWLEDGEMENT

Acknowledgement

We have profited from numerous enlightning discussions with our previ-ous collaborators S. Alexandrov, M. Bordag, M. Ertl, P. Fischer, F. Haider,D. Hofmann, M.O. Katanaev, T. Klosch, S. Lau, H. Liebl, D.J. Schwarz, T.Strobl, G. Tieber, P. Widerin, A. Zelnikov and the members of the Insti-tute for Theoretical Physics at the TU Vienna (especially H. Balasin andL. Bergamin). The exchange of views and e-mails on the quantization partwith P. van Nieuwenhuizen and R. Jackiw is gratefully acknowledged. We alsothank those authors who suggested supplementary references.

One of the authors (W.K.) is especially grateful to R. Jackiw and G. Segrewho in different ways during the 80s kindled his interest in quantum gravity,especially for models in D = 2.

The LATEX-nical support of F. Hochfellner and E. Mossmer has been agreat help for us. Finally, we render special thanks to T. Klosch for letting us“steal” two of his beautiful xfig-pictures.

This work has been supported by project P-14650-TPH of the AustrianScience Foundation (FWF) and by project BO 1112/11-1 of the DeutscheForschungsgemeinschaft (DFG).

Note added after proofs: Due to the broadness of the topics covered ex-plicitly or implicitly in this review the cited literature is necessarily somewhatincomplete. Unfortunately we have omitted some relevant references whichshould have been included: the third part of the series of papers of Klosch andStrobl on classical gravity in 2D [266] which deals with solutions of arbitrarytopology and a series of papers of Chamseddine et al. who was among thepioneers of 2D dilaton gravity, discussing it mostly from a stringy perspec-tive [98, 97, 99, 94, 95, 96]. Additionally, we should mention that quantizationof PSMs (cf. end of sect. 2.3) was done before Hirshfeld and Schwarzweller,e.g. in [376].

Addendum in January 2008: More recent results are contained in anotherreview [448].

125

A SPHERICAL REDUCTION OF THE CURVATURE 2-FORM

A Spherical reduction of the curvature 2-form

In a D-dimensional Pseudo-Riemannian manifold M with Lorentzian sig-nature (+,−,−, ...,−) and spherical symmetry 78 the coordinates describingthe manifold can be separated in a two-dimensional Lorentzian part spanningthe manifold L and a (D − 2)-dimensional Riemannian angular part consti-tuting an SD−2. In adapted coordinates the line element reads

ds2M = gµνdx

µ ⊗ dxν = gαβdxα ⊗ dxβ − Φ2 (xα) gρσdx

ρ ⊗ dxσ, (A.1)

using letters from the beginning of the alphabet (α, β, . . . ; a, b, . . . ) for quan-tities connected with L, letters from the middle of the alphabet (µ, ν, . . . ;m,n, . . . ) for quantities connected with M and letters from the end of thealphabet for quantities connected with SD−2 (ρ, σ, . . . ; r, s, . . . ). Indices willbe lowered and raised with their corresponding metrics.

In the vielbein-formalism 79 ds2L = ηabe

a ⊗ eb, ds2S = δrse

r ⊗ es and com-paring with ds2

M = ηmnem ⊗ en = ds2

L − Φ2ds2S yields

ea = ea,

er = Φer,

ea = ea,

er = Φ−1er.(A.2)

Metricity and torsionlessness for the connection 1-forms on M,L and S leadsto

ωab = ωab, ωrs = ωrs, ωra = (eaΦ) er, ωar = ηabδrs (ebΦ) es, (A.3)

using the relations (A.2).From Cartan’s structure equation (1.25) the curvature 2-form on M fol-

lows:

Rab = Ra

b, (A.4)

Rrs = Rr

s + ηab (eaΦ) (ebΦ) eres, (A.5)

Rar = ηac (ebecΦ) eber + (ebΦ) ωaber, (A.6)

Rra = (ebeaΦ) eber − (ebΦ) ωbae

r, (A.7)

where Rab and Rr

s are the curvature two forms on L and S, respectively.Contracting the vector indices with the form indices and using Rr

s = eresyields the curvature scalar

R = RL − (D − 2) (D − 3)

Φ2[1 + (∇αΦ) (∇αΦ)] − 2

(D − 2

Φ

)(Φ) , (A.8)

78 I.e. the isometry group of the metric has a group isomorphic to SO(D − 1) assubroup with SD−2-spheres as orbits.79 The notation and the meaning of all quantities appearing here is explained insect. 1.2.1.

126

B HEAT KERNEL EXPANSION

where ∇ is the covariant derivative with respect to the metric on L and =

∇α∇α. This – together with√|gM | = Φ(D−2)

√−gL – is the starting point of

spherically reduced gravity formulated by a 2D effective action. Note that thegeneralization to continuous and negative dimensions D is possible in (A.8)which leads to the subclass b = a− 1 of the models of (3.67).

Characteristic classes are independent of the metrical structure since theydepend solely on the topology, but typically they can be expressed as integralsover local quantities using index theorems. As an example we treat Euler andPontryagin class in D = 4. The latter can be expressed as

P4 =1

8π2

∫

MRmnRmn = 0, (A.9)

and it vanishes because RabRab = 0 = RstRst and with (A.6), (A.7) alsoRasRas yields no contribution.

The Euler class

E4 =1

2(4π)2

∫

MRklRmnεklmn (A.10)

is non-trivial in general and can be expressed as a 2D integral over L:

E4 =1

2π

∫

Lεab[Ra

b

(1 + ηcd(ecΦ)(edΦ)

)

+ 2(ηad(ecedΦ)ec + (ecΦ)ωac

) (ηbe(eceeΦ)ec + (ecΦ)ωcb

) ]. (A.11)

B Heat kernel expansion

Some basic properties of the heat kernel expansion are collected herewhich are needed in the main text. More details can be found in the mono-graphs [185, 143,11, 260].

In most of the quantum field theory problems one deals with an operatorof Laplace type. In a suitable basis such an operator can be represented as:

A = −(gµν∇µ∇ν + E) , (B.1)

where ∇µ is a covariant derivative, and E is an endomorphism of a vectorbundle (or, in simpler terms, a matrix valued function). The connection in thecovariant derivative and the matrix E may have gauge and spin indices. Weconsider the oparator A in arbitrary dimension D.

The smeared heat kernel is defined by the equation

K(f, A, t) = Tr(f exp(−tA)) , (B.2)

where f is a function, but more complicated cases with f being a differentialoperator may be considered as well [54]. If the underlying manifold M has no

127

B HEAT KERNEL EXPANSION

boundary, there exists an asymptotic series as t→ +0

K(f, A, t) ≃∞∑

n=0

an(f, A)t−D2

+n , (B.3)

where the coefficients an are locally computable. This means that they can beexpressed as integrals of local polynomials constructed from the Riemanniancurvature, E, gauge field strength, and covariant derivatives. On manifoldswith boundaries half-integer n are also admitted. A very important propertyis that numerical coefficients in front of a monomial depend on the dimen-sion D via an overall factor (4π)−D/2 only [185]. This last statement followsimmediately from writing the general form of such a coefficient on a productmanifold M = M1 ⊗S1 and assuming complete triviality in the S1 direction.

The heat kernel coefficients an are known for n ≤ 5 [414]. We find itinstructive to present here the calculation of a0 and a1 in order to make ourreview self-contained, and to advertise a very powerful method of such calcu-lations. The first step is to write down all possible invariants of an appropriatedimension. The mass dimension of the operator A is given by dimA = +2.Therefore, dim t = −2. The volume element has the dimension −D. All ge-ometric invariants (like e.g. curvature) have positive dimensions. The lowestdimension (−D) involves just the integral of the smearing function over thevolume. This explains why the expansion (B.3) starts with t−D/2. Thus, thefirst two terms in (B.3) must read

a0(f, A) = (4π)−D/2∫

MdDx

√gtr(α0f) , (B.4)

a1(f, A) = (4π)−D/2∫

MdDx

√gtr (f(α1E + α2R)) . (B.5)

where tr denotes the finite-dimensional matrix trace. At this point αi still areunknown constants which will be defined by particular case calculations orthrough functional relations between the heat kernels for different operators.The constant α0 follows from the well-know solution of the heat equation inflat space:

α0 = 1 . (B.6)

Let us consider now how the heat kernel changes under the conformaltransformations of the operator A and by the shift by a function.

d

dǫ

∣∣∣∣∣ǫ=0

an(1, e−2ǫfA) = (D − 2n)an(f, A) . (B.7)

d

dǫ

∣∣∣∣∣ǫ=0

an(1, A− ǫF ) = an−1(F,A) . (B.8)

Here f and F are arbitrary functions. The proof of these two properties ispurely combinatorial. It uses differentiation of an exponential (B.2) and com-

128

B HEAT KERNEL EXPANSION

mutativity under the trace. From the equations (B.6) and (B.8)

α1 = 1 (B.9)

follows. A combination of the two transformations, A(ǫ, δ) = e−2ǫf(A − δF ),allows to prove that for D = 2(n+ 1)

0 =d

dǫ

∣∣∣∣∣ǫ=0

an+1(1, A(ǫ, δ)) ,

0 =d

dδ

∣∣∣∣∣δ=0

d

dǫ

∣∣∣∣∣ǫ=0

an+1(1, A(ǫ, δ)) =d

dǫ

∣∣∣∣∣ǫ=0

d

dδ

∣∣∣∣∣δ=0

an+1(1, A(ǫ, δ))

=d

dǫ

∣∣∣∣∣ǫ=0

an(e−2ǫfF, e−2ǫfA) . (B.10)

The conformal transformations of the individual invariants which may enter(B.10) must be defined. They are perfectly standard in the “geometry” part:

d

dǫ

∣∣∣∣∣ǫ=0

√g = Df

√g ,

d

dǫ

∣∣∣∣∣ǫ=0

R = −2fR− 2(D − 1)∇2f . (B.11)

E is transformed such that the operator A is conformally covariant:

d

dǫ

∣∣∣∣∣ǫ=0

E = −2fE +1

2(D − 2)∇2f . (B.12)

Note, that for the standard conformal (Weyl) transformations the “potential”term E transforms homgeneously, i.e. the second term on r.h.s. of (B.12) isabsent.

Finally, the general expression (B.5) is substituted in the variational equa-tion (B.10) for D = 4. The result

α2 =1

6(B.13)

completes the calculation of a1.Heat kernel methods became standard in quantum field theory after

the famous works by DeWitt [120] 80 where a different calculation schemewas used. The approach we have presented here goes back to the paper byGilkey [184]. This approach appears somewhat simpler, although is less “al-gorithmic” since one has to invent new functional relations appropriate for aparticular problem. The full power of this method has been demostrated on

80 For the first time the heat kernel (proper time) methods were used in quantumtheory by Fock [161].

129

B HEAT KERNEL EXPANSION

manifolds with boundaries [52] (cf. also [419] for minor corrections). With noother method a complicated calculation as the one for a5/2 for mixed boundaryconditions [53] is possible.

The last topic to be addressed is the relation between the heat kernel andthe zeta function of the same operator. It is clear from the definitions (6.18)and (B.2) that

ζ(s|f, A) =1

Γ(s)

∞∫

0

dtts−1K(f, A, t) . (B.14)

This relation can be inverted,

K(f, A, t) =1

2πi

∮dsΓ(s)ζ(s|f, A)t−s , (B.15)

where the integration contour encircles all poles of the integrand. The co-effcient in front of tp in the asymptotic expansion (B.3) corresponds to theresidue of Γ(s)ζ(s|f, A) at the point s = −p. In particular,

aD/2(f, A) = Ress=0(Γ(s)ζ(s|f, A)) = ζ(0|f, A) . (B.16)

For D = 2, A = −∆, E = 0 the equations (B.5), (B.13) and (B.16) providethe relation (6.19) of the main text.

130

REFERENCES

References

[1] L. F. Abbott and S. Deser, “Stability of gravity with a cosmologicalconstant,” Nucl. Phys. B195 (1982) 76.

[2] Y. Aharonov, A. Casher, and S. Nussinov, “The unitarity puzzle and Planckmass stable particles,” Phys. Lett. B191 (1987) 51.

[3] A. Y. Alekseev, P. Schaller, and T. Strobl, “The Topological G/G WZWmodel in the generalized momentum representation,” Phys. Rev. D52 (1995)7146–7160, hep-th/9505012.

[4] S. Alexandrov, I. Grigentch, and D. Vassilevich, “SU(2)-invariant reductionof the 3+1 dimensional Ashtekar’s gravity,” Class. Quant. Grav. 15 (1998)573–580, gr-qc/9705080.

[5] E. Alvarez and J. Conde, “Are the string and Einstein frames equivalent?,”Mod. Phys. Lett. A17 (2002) 413–420, gr-qc/0111031.

[6] G. Amelino-Camelia, D. Bak, and D. Seminara, “Area preserving structureand anomalies in (1+1)- dimensional quantum gravity,” Phys. Lett. B354

(1995) 213–219, hep-th/9505136.

[7] R. Arnowitt, S. Deser, and C. W. Misner in Gravitation: An Introduction toCurrent Research, L. Witten, ed. Wiley, New York, 1962.

[8] A. Ashtekar, “New variables for classical and quantum gravity,” Phys. Rev.Lett. 57 (1986) 2244–2247.

[9] A. Ashtekar, “New Hamiltonian formulation of general relativity,” Phys. Rev.D36 (1987) 1587–1602.

[10] A. Ashtekar and R. Geroch, “Quantum Theory of Gravitation,”Rept.Prog.Phys. 37 (1974) 1211–1256.

[11] I. G. Avramidi, Heat kernel and quantum gravity. Springer, Berlin, 2000.

[12] N. A. Bahcall, J. P. Ostriker, S. Perlmutter, and P. J. Steinhardt, “TheCosmic Triangle: Revealing the State of the Universe,” Science 284 (1999)1481–1488, arXiv:astro-ph/9906463.

[13] R. Balbinot and A. Fabbri, “4D quantum black hole physics from 2Dmodels?,” Phys. Lett. B459 (1999) 112–118, gr-qc/9904034.

[14] R. Balbinot and A. Fabbri, “Hawking radiation by effective two-dimensionaltheories,” Phys. Rev. D59 (1999) 044031, hep-th/9807123.

[15] R. Balbinot and A. Fabbri, “2d black holes and effective actions,”hep-th/0012140.

[16] R. Balbinot, A. Fabbri, V. Frolov, P. Nicolini, P. Sutton, and A. Zelnikov,“Vacuum polarization in the Schwarzschild spacetime and dimensionalreduction,” Phys. Rev. D63 (2001) 084029, hep-th/0012048.

131

REFERENCES

[17] R. Balbinot, A. Fabbri, P. Nicolini, and P. J. Sutton, “Vacuum polarizationin two-dimensional static spacetimes and dimensional reduction,”hep-th/0202036.

[18] T. Banks, “Lectures on black holes and information loss,” Nucl. Phys. Proc.Suppl. 41 (1995) 21–65, hep-th/9412131.

[19] T. Banks and M. O’Loughlin, “Two-dimensional quantum gravity inMinkowski space,” Nucl. Phys. B362 (1991) 649–664.

[20] T. Banks, L. Susskind, and M. E. Peskin, “Difficulties for the evolution ofpure states into mixed states,” Nucl. Phys. B244 (1984) 125.

[21] C. Barbachoux and A. Fabbri, “Semiclassical zero temperature black holes inspherically reduced theories,” hep-th/0201133.

[22] B. M. Barbashov, V. V. Nesterenko, and A. M. Chervyakov, “The solitons insome geometrical field theories,” Theor. Math. Phys. 40 (1979) 572–581.

[23] A. Barvinsky, S. Das, and G. Kunstatter, “Quantum mechanics of chargedblack holes,” Phys. Lett. B517 (2001) 415–420, hep-th/0102061.

[24] A. Barvinsky and G. Kunstatter, “Exact physical black hole states in generic2-D dilaton gravity,” Phys. Lett. B389 (1996) 231–237, hep-th/9606134.

[25] A. O. Barvinsky, “The wave function and the effective action in quantumcosmology: Covariant loop expansion,” Phys. Lett. B195 (1987) 344–348.

[26] F. Bastianelli and O. Corradini, “On mode regularization of theconfiguration space path integral in curved space,” Phys. Rev. D60 (1999)044014, hep-th/9810119.

[27] I. A. Batalin and G. A. Vilkovisky, “Relativistic S matrix of dynamicalsystems with boson and fermion constraints,” Phys. Lett. B69 (1977)309–312.

[28] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer,“Deformation theory and quantization. 1. Deformations of symplecticstructures,” Ann. Phys. 111 (1978) 61.

[29] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer,“Deformation theory and quantization. 2. Physical applications,” Ann. Phys.111 (1978) 111.

[30] R. Beig and N. o Murchadha Ann. Phys. (N.Y.) 174 (1987) 463.

[31] J. Bekenstein, “The quantum mass spectrum of the Kerr black hole,”Lett.Nuovo Cim. 11 (1974) 467.

[32] E. Benedict, R. Jackiw, and H. J. Lee, “Functional Schroedinger and BRSTquantization of (1+1)- dimensional gravity,” Phys. Rev. D54 (1996)6213–6225, hep-th/9607062.

132

REFERENCES

[33] R. Benguria, P. Cordero, and C. Teitelboim, “Aspects of the Hamiltoniandynamics of interacting gravitational gauge and Higgs fields withapplications to spherical symmetry,” Nucl. Phys. B122 (1977) 61.

[34] V. A. Berezin, A. M. Boyarsky, and A. Y. Neronov, “Quantumgeometrodynamics for black holes and wormholes,” Phys. Rev. D57 (1998)1118–1128, gr-qc/9708060.

[35] V. A. Berezin, A. M. Boyarsky, and A. Y. Neronov, “Black hole massspectrum vs spectrum of Hawking radiation,” Phys. Lett. B455 (1999)109–114, gr-qc/9808027.

[36] B. K. Berger, D. M. Chitre, V. E. Moncrief, and Y. Nutku, “Hamiltonianformulation of spherically symmetric gravitational fields,” Phys. Rev. D5

(1972) 2467–2470.

[37] N. Berkovits, S. Gukov, and B. C. Vallilo, “Superstrings in 2D backgroundswith R-R flux and new extremal black holes,” Nucl. Phys. B614 (2001)195–232, hep-th/0107140.

[38] Z. Bern, L. J. Dixon, D. C. Dunbar, M. Perelstein, and J. S. Rozowsky, “Onthe relationship between Yang-Mills theory and gravity and its implicationfor ultraviolet divergences,” Nucl. Phys. B530 (1998) 401–456,hep-th/9802162.

[39] D. Bernard and N. Regnault, “Vertex operator solutions of 2-d dimensionallyreduced gravity,” Commun. Math. Phys. 210 (2000) 177–201,solv-int/9902017.

[40] J. Bicak and K. V. Kuchar, “Null dust in canonical gravity,” Phys. Rev. D56

(1997) 4878–4895, gr-qc/9704053.

[41] A. Bilal and C. G. Callan, “Liouville models of black hole evaporation,”Nucl. Phys. B394 (1993) 73–100, hep-th/9205089.

[42] D. Birmingham, M. Blau, M. Rakowski, and G. Thompson, “Topologicalfield theory,” Phys. Rept. 209 (1991) 129–340.

[43] M. Bojowald, H. A. Kastrup, F. Schramm, and T. Strobl, “Group theoreticalquantization of a phase space S(1) x R+ and the mass spectrum ofSchwarzschild black holes in D space-time dimensions,” Phys. Rev. D62

(2000) 044026, gr-qc/9906105.

[44] S. Bose, L. Parker, and Y. Peleg, “Semiinfinite throat as the end stategeometry of two- dimensional black hole evaporation,” Phys. Rev. D52

(1995) 3512–3517, hep-th/9502098.

[45] D. G. Boulware, “Quantum field theory in Schwarzschild and Rindlerspaces,” Phys. Rev. D11 (1975) 1404.

[46] D. Boulware and S. Deser, “Stress-tensor commutators and Schwingerterms,” J. Math. Phys. 8 (1967) 1468–1477.

133

REFERENCES

[47] R. Bousso, “Quantum global structure of de Sitter space,” Phys. Rev. D60

(1999) 063503, hep-th/9902183.

[48] R. Bousso and S. W. Hawking, “Trace anomaly of dilaton coupled scalars intwo dimensions,” Phys. Rev. D56 (1997) 7788–7791, hep-th/9705236.

[49] R. Bousso and S. W. Hawking, “(Anti-)evaporation of Schwarzschild-deSitter black holes,” Phys. Rev. D57 (1998) 2436–2442, hep-th/9709224.

[50] R. Bousso, “Proliferation of de Sitter space,” Phys. Rev. D58 (1998) 083511,hep-th/9805081.

[51] C. Brans and R. H. Dicke, “Mach’s principle and a relativistic theory ofgravitation,” Phys. Rev. 124 (1961) 925–935.

[52] T. P. Branson and P. B. Gilkey, “The asymptotics of the Laplacian on amanifold with boundary,” Commun. Part. Diff. Eqs. 15 (1990) 245.

[53] T. P. Branson, P. B. Gilkey, K. Kirsten, and D. V. Vassilevich, “Heat kernelasymptotics with mixed boundary conditions,” Nucl. Phys. B563 (1999) 603,hep-th/9906144.

[54] T. P. Branson, P. B. Gilkey, and D. V. Vassilevich, “Vacuum expectationvalue asymptotics for second order differential operators on manifolds withboundary,” J. Math. Phys. 39 (1998) 1040–1049, hep-th/9702178.

[55] M. Brigante, S. Cacciatori, D. Klemm, and D. Zanon, “The asymptoticdynamics of two-dimensional (anti-)de Sitter gravity,” JHEP 03 (2002) 005,hep-th/0202073.

[56] O. Brodbeck and M. Zagermann, “Dimensionally reduced gravity, Hermitiansymmetric spaces and the Ashtekar variables,” Class. Quant. Grav. 17

(2000) 2749–2764, arXiv:gr-qc/9911118.

[57] J. D. Brown, S. R. Lau, and J. W. York, “Action and energy of thegravitational field,” gr-qc/0010024.

[58] J. D. Brown and J. York, James W., “Quasilocal energy and conservedcharges derived from the gravitational action,” Phys. Rev. D47 (1993)1407–1419.

[59] J. Brown, Lower Dimensional Gravity. World Scientific, 1988.

[60] M. Brown and J. Gates, S. James, “Superspace Bianchi identities and thesupercovariant derivative,” Ann. Phys. 122 (1979) 443.

[61] M. Buric and V. Radovanovic, “Quantum corrections for theReissner-Nordstrom black hole,” Class. Quant. Grav. 16 (1999) 3937–3951,gr-qc/9907036.

[62] M. Buric and V. Radovanovic, “ADM mass of the quantum correctedSchwarzschild black hole,” Class. Quant. Grav. 17 (2000) 33–42,gr-qc/9907106.

134

REFERENCES

[63] M. Buric and V. Radovanovic, “Quantum corrections for (anti)-evaporatingblack hole,” Phys. Rev. D63 (2001) 044020, hep-th/0007172.

[64] M. Buric, V. Radovanovic, and A. Mikovic, “One-loop corrections forSchwarzschild black hole via 2D dilaton gravity,” Phys. Rev. D59 (1999)084002, gr-qc/9804083.

[65] M. Buric, M. Dimitrijevic, and V. Radovanovic, “Quantum corrections forBTZ black hole via 2D reduced model,” Phys. Rev. D65 (2002) 064022,hep-th/0108036.

[66] M. Cadoni, P. Carta, and S. Mignemi, “Quantum dilaton gravity as a lineardilaton conformal field theory,” hep-th/0202010.

[67] M. Cadoni and S. Mignemi, “On the conformal equivalence between 2-dblack holes and Rindler space-time,” Phys. Lett. B358 (1995) 217–222,gr-qc/9505032.

[68] M. Cadoni and S. Mignemi, “Cosmology of the Jackiw-Teitelboim model,”gr-qc/0202066.

[69] M. Cadoni, “Trace anomaly and Hawking effect in generic 2D dilaton gravitytheories,” Phys. Rev. D53 (1996) 4413–4420, gr-qc/9510012.

[70] M. Cadoni, “2d extremal black holes as solitons,” Phys. Rev. D58 (1998)104001, hep-th/9803257.

[71] C. G. Callan, Jr., S. B. Giddings, J. A. Harvey, and A. Strominger,“Evanescent black holes,” Phys. Rev. D45 (1992) 1005–1009,hep-th/9111056.

[72] C. G. Callan, Jr., E. J. Martinec, M. J. Perry, and D. Friedan, “Strings inbackground fields,” Nucl. Phys. B262 (1985) 593.

[73] D. Cangemi and R. Jackiw, “Gauge invariant formulations of lineal gravity,”Phys. Rev. Lett. 69 (1992) 233–236, hep-th/9203056.

[74] D. Cangemi and R. Jackiw, “Geometric gravitational forces on particlesmoving in a line,” Phys. Lett. B299 (1993) 24–29, hep-th/9210036.

[75] D. Cangemi and R. Jackiw, “Poincare gauge theory for gravitational forcesin (1+1)- dimensions,” Ann. Phys. 225 (1993) 229–263, hep-th/9302026.

[76] D. Cangemi and R. Jackiw, “Quantal analysis of string inspired lineal gravitywith matter fields,” Phys. Lett. B337 (1994) 271–278, hep-th/9405119.

[77] D. Cangemi, R. Jackiw, and B. Zwiebach, “Physical states in matter coupleddilaton gravity,” Ann. Phys. 245 (1996) 408–444, hep-th/9505161.

[78] S. Capozziello, R. de Ritis, and A. A. Marino, “Some aspects of thecosmological conformal equivalence between *Jordan frame* and *Einsteinframe*,” Class. Quant. Grav. 14 (1997) 3243–3258, gr-qc/9612053.

135

REFERENCES

[79] S. Carlip, “Quantum gravity: A progress report,” Rept. Prog. Phys. 64

(2001) 885, arXiv:gr-qc/0108040.

[80] S. M. Carroll, “Quintessence and the rest of the world,” Phys. Rev. Lett. 81

(1998) 3067–3070, arXiv:astro-ph/9806099.

[81] B. Carter, “Killing Horizons and Orthogonally Transitive Groups inSpace-Time,” J. Math. Phys. 10 (1969) 70.

[82] R. Casadio, “On time evolution of quantum black holes,” Phys. Lett. B511

(2001) 285–290, gr-qc/0102006.

[83] R. Casadio and B. Harms, “Charged dilatonic black holes: String frame vs.Einstein frame,” Mod. Phys. Lett. A14 (1999) 1089–1104, gr-qc/9806032.

[84] S. Cassemiro F. F. and V. O. Rivelles, “Canonical and functionalSchroedinger quantization of two- dimensional dilaton gravity,” Phys. Lett.B452 (1999) 234–239, arXiv:hep-th/9812096.

[85] A. S. Cattaneo and G. Felder, “A path integral approach to the Kontsevichquantization formula,” Commun. Math. Phys. 212 (2000) 591–611,math.qa/9902090.

[86] A. S. Cattaneo and G. Felder, “Poisson sigma models and deformationquantization,” Mod. Phys. Lett. A16 (2001) 179–190, hep-th/0102208.

[87] M. Cavaglia and C. Ungarelli, “Quantum gravity corrections to theSchwarzschild mass,” Phys. Rev. D61 (2000) 064019, hep-th/9912024.

[88] M. Cavaglia, “Geometrodynamical formulation of two-dimensional dilatongravity,” Phys. Rev. D59 (1999) 084011, arXiv:hep-th/9811059.

[89] M. Cavaglia, “A note on Weyl transformations in two-dimensional dilatongravity,” Mod. Phys. Lett. A15 (2000) 2113–2118, hep-th/0011136.

[90] M. Cavaglia, V. de Alfaro, and A. T. Filippov, “Hamiltonian formalism forblack holes and quantization,” Int. J. Mod. Phys. D4 (1995) 661–672,gr-qc/9411070.

[91] M. Cavaglia, V. de Alfaro, and A. T. Filippov, “Quantization of theSchwarzschild Black Hole,” Int. J. Mod. Phys. D5 (1996) 227–250,gr-qc/9508062.

[92] M. Cavaglia, V. de Alfaro, and A. T. Filippov, “Quantization of the stringinspired dilaton gravity and the Birkhoff theorem,” Phys. Lett. B424 (1998)265–270, hep-th/9802158.

[93] M. Cavaglia and A. Fabbri, “Quantum gravitational corrections to black holegeometries,” Phys. Rev. D65 (2002) 044012, hep-th/0108050.

[94] A. H. Chamseddine, “A solution to two-dimensional quantum gravity:Noncritical strings,” Phys. Lett. B256 (1991) 379–386.

136

REFERENCES

[95] A. H. Chamseddine, “Superstrings in arbitrary dimensions,” Phys. Lett.B258 (1991) 97–103.

[96] A. H. Chamseddine, “A study of noncritical strings in arbitrary dimensions,”Nucl. Phys. B368 (1992) 98–120.

[97] A. H. Chamseddine and M. Reuter, “Induced two-dimensional quantumgravity and SL(2,R) Kac-Moody current algebra,” Nucl. Phys. B317 (1989)757–771.

[98] A. H. Chamseddine and D. Wyler, “Gauge theory of topological gravity in(1+1)-dimensions,” Phys. Lett. B228 (1989) 75.

[99] A. H. Chamseddine and D. Wyler, “Topological gravity in(1+1)-dimensions,” Nucl. Phys. B340 (1990) 595–616.

[100] K. C. K. Chan, “Is mass conformally invariant?,” gr-qc/9701029.

[101] K. C. K. Chan, J. D. E. Creighton, and R. B. Mann, “Conserved masses inGHS Einstein and string black holes,” Phys. Rev. D54 (1996) 3892–3899,gr-qc/9604055.

[102] T. Chiba and M. Siino, “Disappearance of black hole criticality insemiclassical general relativity,” Mod. Phys. Lett. A12 (1997) 709–718.

[103] M. W. Choptuik, “Universality and scaling in gravitational collapse of amassless scalar field,” Phys. Rev. Lett. 70 (1993) 9–12.

[104] S. M. Christensen and S. A. Fulling, “Trace anomalies and the Hawkingeffect,” Phys. Rev. D15 (1977) 2088–2104.

[105] T. Christodoulakis, G. A. Diamandis, B. C. Georgalas, and E. C. Vagenas,“Bogoliubov coefficients of 2d charged black holes,” Phys. Lett. B501 (2001)269–276, hep-th/0010097.

[106] T. Christodoulakis, G. A. Diamandis, B. C. Georgalas, and E. C. Vagenas,“Casimir effect in 2d stringy black hole backgrounds,” Phys. Rev. D64

(2001) 124022, hep-th/0107049.

[107] D. Christodoulou, “Global existence of generalized solutions of thespherically symmetric Einstein scalar equations in the large,” Commun.Math. Phys. 106 (1986) 587–621.

[108] D. Christodoulou, “The problem of a selfgravitating scalar field,” Commun.Math. Phys. 105 (1986) 337–361.

[109] D. Christodoulou, “A mathematical theory of gravitational collapse,”Commun. Math. Phys. 109 (1987) 613–647.

[110] D. Christodoulou, “The structure and uniqueness of generalized solutions ofthe spherically symmetric Einstein scalar equations,” Commun. Math. Phys.109 (1987) 591–611.

137

REFERENCES

[111] G. Cognola and S. Zerbini, “On the dimensional reduction procedure,” Nucl.Phys. B602 (2001) 383–398, hep-th/0008061.

[112] J. Cruz, A. Fabbri, and J. Navarro-Salas, “Can conformal transformationschange the fate of 2D black holes?,” Phys. Lett. B449 (1999) 30–38,arXiv:hep-th/9811246.

[113] J. Cruz, A. Fabbri, and J. Navarro-Salas, “On the normalization of Killingvectors and energy conservation in two-dimensional gravity,” Phys. Rev.D60 (1999) 107506, gr-qc/9902084.

[114] J. Cruz, J. M. Izquierdo, D. J. Navarro, and J. Navarro-Salas, “Free fields viacanonical transformations of matter-coupled 2D dilaton gravity models,”Phys. Rev. D58 (1998) 044010, arXiv:hep-th/9704168.

[115] J. Cruz and J. Navarro-Salas, “Solvable models for radiating black holes andarea- preserving diffeomorphisms,” Phys. Lett. B375 (1996) 47–53,hep-th/9512187.

[116] S. P. de Alwis, “Black hole physics from Liouville theory,” Phys. Lett. B300

(1993) 330–335, hep-th/9206020.

[117] S. P. de Alwis and J. Lykken, “2-d gravity and the black hole solution in 2-dcritical string theory,” Phys. Lett. B269 (1991) 264–270.

[118] J. de Boer, F. Harmsze, and T. Tjin, “Nonlinear finite W symmetries andapplications in elementary systems,” Phys. Rept. 272 (1996) 139–214,hep-th/9503161.

[119] S. Deser and D. Seminara, “Counterterms/M-theory corrections to D = 11supergravity,” Phys. Rev. Lett. 82 (1999) 2435–2438, hep-th/9812136.

[120] B. S. DeWitt, Dynamical theory of groups and fields. Gordon and Breach,New York, 1965.

[121] B. S. DeWitt, “Quantum theory of gravity. 1. The canonical theory,” Phys.Rev. 160 (1967) 1113–1148.

[122] E. D’Hoker, D. Freedman, and R. Jackiw, “SO(2,1) invariant quantization ofthe Liouville theory,” Phys. Rev. D28 (1983) 2583.

[123] E. D’Hoker and R. Jackiw, “Liouville field theory,” Phys. Rev. D26 (1982)3517.

[124] E. D’Hoker and R. Jackiw, “Space translation breaking and compactificationin the Liouville theory,” Phys. Rev. Lett. 50 (1983) 1719–1722.

[125] K. Diba and D. A. Lowe, “Near-extremal black hole evaporation inasymptotically flat spacetime,” hep-th/0202005.

[126] R. Dick, “Inequivalence of Jordan and Einstein frame: What is the lowenergy gravity in string theory?,” Gen. Rel. Grav. 30 (1998) 435–444.

138

REFERENCES

[127] R. Dijkgraaf, H. Verlinde, and E. Verlinde, “String propagation in a blackhole geometry,” Nucl. Phys. B371 (1992) 269–314.

[128] P. A. M. Dirac, Lectures on Quantum Mechanics. Belfer Graduate School ofScience, Yeshiva University, New York, 1996.

[129] J. F. Donoghue, “General relativity as an effective field theory: The leadingquantum corrections,” Phys. Rev. D50 (1994) 3874–3888, gr-qc/9405057.

[130] J. S. Dowker, “Conformal anomaly in 2d dilaton-scalar theory,” Class.Quant. Grav. 15 (1998) 1881–1884, hep-th/9802029.

[131] J. S. Dowker and R. Critchley, “Effective Lagrangian and energy momentumtensor in de Sitter space,” Phys. Rev. D13 (1976) 3224.

[132] M. J. Duff, “Twenty years of the Weyl anomaly,” Class. Quant. Grav. 11

(1994) 1387–1404, hep-th/9308075.

[133] M. J. Duff, B. E. W. Nilsson, and C. N. Pope, “Kaluza-Klein supergravity,”Phys. Rept. 130 (1986) 1–142.

[134] A. S. Eddington, “A comparison of Whitehead’s and Einstein’s formulas,”Nature 113 (1924) 192.

[135] A. Einstein, “Die Feldgleichungen der Gravitation,” Sitzungsber. Preuss.Akad. Wiss. Berlin (Math. Phys.) 1915 (1915) 844–847.

[136] A. Einstein, “Zur Allgemeinen Relativitatstheorie,” Sitzungsber. Preuss.Akad. Wiss. Berlin (Math. Phys.) 1915 (1915) 778–786.

[137] S. Elitzur, A. Forge, and E. Rabinovici, “Some global aspects of stringcompactifications,” Nucl. Phys. B359 (1991) 581–610.

[138] E. Elizalde, P. Fosalba-Vela, S. Naftulin, and S. D. Odintsov, “On black holesin the theory of dilatonic gravity coupled to a scalar field,” Phys. Lett. B352

(1995) 235–241, hep-th/9505030.

[139] E. Elizalde, S. Naftulin, and S. D. Odintsov, “The renormalization structureand quantum equivalence of 2- d dilaton gravities,” Int. J. Mod. Phys. A9

(1994) 933–952, hep-th/9304091.

[140] M. Ertl, W. Kummer, and T. Strobl, “General two-dimensional supergravityfrom Poisson superalgebras,” JHEP 01 (2001) 042, arXiv:hep-th/0012219.

[141] M. F. Ertl, M. O. Katanaev, and W. Kummer, “Generalized supergravity intwo dimensions,” Nucl. Phys. B530 (1998) 457–486, hep-th/9710051.

[142] M. Ertl, Supergravity in two spacetime dimensions. PhD thesis, TechnischeUniversitat Wien, 2001. arXiv:hep-th/0102140.

[143] G. Esposito, A. Kamenshchik, and G. Pollifrone, Euclidean quantum gravityon manifolds with boundary. Kluwer, Dordrecht, 1997.

[144] G. Esposito, Quantum gravity in four-dimensions. Nova Science, New York,2001.

139

REFERENCES

[145] A. Fabbri, D. J. Navarro, and J. Navarro-Salas, “Evaporation ofnear-extremal Reissner-Nordstrom black holes,” Phys. Rev. Lett. 85 (2000)2434–2437, hep-th/0004027.

[146] A. Fabbri, D. J. Navarro, and J. Navarro-Salas, “Quantum evolution ofnear-extremal Reissner-Nordstrom black holes,” Nucl. Phys. B595 (2001)381–401, hep-th/0006035.

[147] A. Fabbri and J. Navarro-Salas, “Critical energy flux and mass in solvabletheories of 2d dilaton gravity,” Phys. Rev. D58 (1998) 084011,gr-qc/9805082.

[148] A. Fabbri and J. G. Russo, “Soluble models in 2d dilaton gravity,” Phys.Rev. D53 (1996) 6995–7002, hep-th/9510109.

[149] L. D. Faddeev, “The energy problem in Einstein’s theory of gravitation,”Sov. Phys. Usp. 25 (1982) 130–142.

[150] L. D. Faddeev and R. Jackiw, “Hamiltonian reduction of unconstrained andconstrained systems,” Phys. Rev. Lett. 60 (1988) 1692.

[151] L. D. Faddeev, R. M. Kashaev, and A. Y. Volkov, “Strongly coupledquantum discrete Liouville theory. I: Algebraic approach and duality,”Commun. Math. Phys. 219 (2001) 199–219, hep-th/0006156.

[152] V. Faraoni and E. Gunzig, “Einstein frame or Jordan frame?,” Int. J. Theor.Phys. 38 (1999) 217–225.

[153] M. Fierz, “Uber die physikalische Deutung der erweitertenGravitationstheorie P. Jordans,” Helv. Phys. Acta 29 (1956) 128.

[154] A. T. Filippov, “Exact solutions of 1+1 dimensional dilaton gravity coupledto matter,” Mod. Phys. Lett. A11 (1996) 1691–1704, hep-th/9605008.

[155] A. T. Filippov, “Integrable 1+1 dimensional gravity models,” Int. J. Mod.Phys. A12 (1997) 13–22, gr-qc/9612058.

[156] D. Finkelstein, “Past-Future Asymmetry of the Gravitational Field of aPoint Particle,” Phys.Rev. 110 (1958) 965–967.

[157] P. Fischer, D. Grumiller, W. Kummer, and D. V. Vassilevich, “S-matrix fors-wave gravitational scattering,” Phys. Lett. B521 (2001) 357–363,gr-qc/0105034. Erratum ibid. B532 (2002) 373.

[158] P. Fischer, “Vertices in spherically reduced quantum gravity,” Master’sthesis, Vienna University of Technology, 2001.

[159] I. Z. Fisher, “Scalar mesostatic field with regard for gravitational effects,”Zh. Eksp. Teor. Fiz. 18 (1948) 636–640, gr-qc/9911008.

[160] P. Fiziev and H. Kleinert, “New action principle for classical particletrajectories in spaces with torsion,” Europhys. Lett. 35 (1996) 241–246,hep-th/9503074.

140

REFERENCES

[161] V. A. Fock, “The proper time in classical and quantum mechanics,” Izvestiyaof USSR Academy of Science (Phys.) 4-5 (1937) 551.

[162] P. Forgacs and N. S. Manton, “Space-time symmetries in gauge theories,”Commun. Math. Phys. 72 (1980) 15.

[163] E. S. Fradkin and T. E. Fradkina, “Quantization of relativistic systems withboson and fermion first and second class constraints,” Phys. Lett. B72

(1978) 343.

[164] E. S. Fradkin and G. A. Vilkovisky, “Quantization of relativistic systemswith constraints,” Phys. Lett. B55 (1975) 224.

[165] V. Frolov, D. Fursaev, J. Gegenberg, and G. Kunstatter, “Thermodynamicsand statistical mechanics of induced Liouville gravity,” Phys. Rev. D60

(1999) 024016, hep-th/9901087.

[166] V. Frolov, P. Sutton, and A. Zelnikov, “The dimensional-reduction anomaly,”Phys. Rev. D61 (2000) 024021, hep-th/9909086.

[167] V. Frolov and A. Zelnikov, “Non-minimally coupled massive scalar field in a2D black hole: Exactly solvable model,” Phys. Rev. D63 (2001) 125026,hep-th/0012252.

[168] V. P. Frolov, “Two-dimensional black hole physics,” Phys. Rev. D46 (1992)5383–5394.

[169] V. P. Frolov and D. V. Fursaev, “Thermal fields, entropy, and black holes,”Class. Quant. Grav. 15 (1998) 2041–2074, hep-th/9802010.

[170] V. P. Frolov and I. D. Novikov, Black hole physics: Basic concepts and newdevelopments. Dordrecht, Netherlands: Kluwer Academic, 1998.

[171] V. P. Frolov, W. Israel, and S. N. Solodukhin, “On one-loop quantumcorrections to the thermodynamics of charged black holes,” Phys. Rev. D54

(1996) 2732–2745, hep-th/9602105.

[172] K. Fujikawa, U. Lindstrom, N. K. Nielsen, M. Rocek, and P. vanNieuwenhuizen, “The regularized BRST coordinate invariant measure,”Phys. Rev. D37 (1988) 391.

[173] J. Gegenberg and G. Kunstatter, “Quantum theory of black holes,” Phys.Rev. D47 (1993) 4192–4195, gr-qc/9302006.

[174] J. Gegenberg, G. Kunstatter, and D. Louis-Martinez, “Observables fortwo-dimensional black holes,” Phys. Rev. D51 (1995) 1781–1786,gr-qc/9408015.

[175] J. Gegenberg, G. Kunstatter, and T. Strobl, “Edge states and entropy of 2dblack holes,” Phys. Rev. D55 (1997) 7651–7665, gr-qc/9612033.

[176] L. A. Gergely, “A spherically symmetric closed universe as an example of a2d dilatonic model,” Phys. Rev. D59 (1999) 104014, gr-qc/9902016.

141

REFERENCES

[177] L. A. Gergely, “Spherically symmetric static solution for colliding null dust,”Phys. Rev. D58 (1998) 084030, gr-qc/9809024.

[178] R. Geroch, “A method for generating new solutions of Einstein’s equation.2,” J. Math. Phys. 13 (1972) 394–404.

[179] G. W. Gibbons and S. W. Hawking, “Action integrals and partitionfunctions in quantum gravity,” Phys. Rev. D15 (1977) 2752–2756.

[180] G. W. Gibbons and K. Maeda, “Black holes and membranes in higherdimensional theories with dilaton fields,” Nucl. Phys. B298 (1988) 741.

[181] G. W. Gibbons and M. J. Perry, “Black holes in thermal equilibrium,” Phys.Rev. Lett. 36 (1976) 985.

[182] G. W. Gibbons and M. J. Perry, “The Physics of 2-D stringy space-times,”Int. J. Mod. Phys. D1 (1992) 335–354, hep-th/9204090.

[183] S. B. Giddings, “Quantum mechanics of black holes,” Trieste HEPCosmology (1994) 0530–574, arXiv:hep-th/9412138.

[184] P. B. Gilkey, “The spectral geometry of a Riemannian manifold,” J. Diff.Geom. 10 (1975) 601.

[185] P. B. Gilkey, Invariance theory, the heat equation, and the Atiyah–Singerindex theorem. CRC Press, Boca Raton, 1994.

[186] P. Ginsparg and G. W. Moore, “Lectures on 2-d gravity and 2-d stringtheory,” hep-th/9304011.

[187] D. M. Gitman and I. V. Tyutin, Quantization of fields with constraints.Springer, Berlin, 1990.

[188] M. H. Goroff and A. Sagnotti, “The ultraviolet behavior of Einstein gravity,”Nucl. Phys. B266 (1986) 709.

[189] R. H. Gowdy, “Gravitational waves in closed universes,” Phys. Rev. Lett. 27

(1971) 826–829.

[190] W. Graf, “Geometric dilaton gravity and smooth charged wormholes,” Phys.Rev. D67 (2003) 024002, gr-qc/0209002.

[191] M. B. Green, J. H. Schwarz, and E. Witten, SUPERSTRING THEORY.Cambridge University Press, 1987. Vol. 1: Introduction.

[192] H. Grosse, W. Kummer, P. Presnajder, and D. J. Schwarz, “Novel symmetryof nonEinsteinian gravity in two- dimensions,” J. Math. Phys. 33 (1992)3892–3900, hep-th/9205071.

[193] D. Grumiller, D. Hofmann, and W. Kummer, “2D gravity without testparticles is pointless (Comment on hep-th/0011136),” Mod. Phys. Lett. A16

(2001) 1597–1600, arXiv:gr-qc/0012026.

[194] D. Grumiller, D. Hofmann, and W. Kummer, “Two-dilaton theories in twodimensions,” Annals Phys. 290 (2001) 69–82, arXiv:gr-qc/0005098.

142

REFERENCES

[195] D. Grumiller and W. Kummer, “Absolute conservation law for black holes,”Phys. Rev. D61 (2000) 064006, gr-qc/9902074.

[196] D. Grumiller, W. Kummer, and D. V. Vassilevich, “The virtual black hole in2d quantum gravity,” Nucl. Phys. B580 (2000) 438–456, gr-qc/0001038.

[197] D. Grumiller, Quantum dilaton gravity in two dimensions with matter. PhDthesis, Technische Universitat Wien, 2001. gr-qc/0105078.

[198] D. Grumiller, “The virtual black hole in 2D quantum gravity and itsrelevance for the S-matrix,” Int. J. Mod. Phys. A 17 (2001) 989–992,hep-th/0111138.

[199] D. Grumiller, “Virtual black hole phenomenology from 2d dilaton theories,”Class. Quant. Grav. 19 (2002) 997–1009, gr-qc/0111097.

[200] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlatorsfrom non-critical string theory,” Phys. Lett. B428 (1998) 105–114,hep-th/9802109.

[201] C. Gundlach, “Critical phenomena in gravitational collapse,” Adv. Theor.Math. Phys. 2 (1998) 1–49, arXiv:gr-qc/9712084.

[202] C. Gundlach, “Critical phenomena in gravitational collapse - LivingReviews,” Living Rev. Rel. 2 (1999) 4, arXiv:gr-qc/0001046.

[203] Y. V. Gusev and A. I. Zelnikov, “Two-dimensional effective action for matterfields coupled to the dilaton,” Phys. Rev. D61 (2000) 084010,hep-th/9910198.

[204] F. Haider and W. Kummer, “Quantum functional integration ofnonEinsteinian gravity in d = 2,” Int. J. Mod. Phys. A9 (1994) 207–220.

[205] P. Hajıcek, “Spherically symmetric systems of fields and black holes. 2.Apparent horizon in canonical formalism,” Phys. Rev. D30 (1984) 1178.

[206] P. Hajicek and I. Kouletsis, “Pair of null gravitating shells I. Space ofsolutions and its symmetries,” gr-qc/0112060.

[207] P. Hajicek and I. Kouletsis, “Pair of null gravitating shells II. Canonicaltheory and embedding variables,” gr-qc/0112061.

[208] P. Hajicek, “Quantum theory of gravitational collapse (lecture notes onquantum conchology),” gr-qc/0204049.

[209] K.-J. Hamada, “Quantum theory of dilaton gravity in (1+1)-dimensions,”Phys. Lett. B300 (1993) 322–329, hep-th/9206071.

[210] K.-J. Hamada and A. Tsuchiya, “Quantum gravity and black hole dynamicsin (1+1)- dimensions,” Int. J. Mod. Phys. A8 (1993) 4897–4914,hep-th/9211135.

[211] S. W. Hawking, “Particle creation by black holes,” Commun. Math. Phys. 43

(1975) 199–220.

143

REFERENCES

[212] S. W. Hawking, “Zeta function regularization of path integrals in curvedspace-time,” Commun. Math. Phys. 55 (1977) 133.

[213] S. W. Hawking, “The unpredictability of quantum gravity,” Commun. Math.Phys. 87 (1982) 395.

[214] S. W. Hawking, “Virtual black holes,” Phys. Rev. D53 (1996) 3099–3107,hep-th/9510029.

[215] S. W. Hawking and G. T. Horowitz, “The gravitational Hamiltonian, action,entropy and surface terms,” Class. Quant. Grav. 13 (1996) 1487–1498,gr-qc/9501014.

[216] S. Hawking and G. Ellis, The Large Scale Structure of Space-Time.Cambridge University Press, 1973.

[217] F. W. Hehl, P. Von Der Heyde, G. D. Kerlick, and J. M. Nester, “Generalrelativity with spin and torsion: Foundations and prospects,” Rev. Mod.Phys. 48 (1976) 393–416.

[218] F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Neeman, “Metric affinegauge theory of gravity: Field equations, Noether identities, world spinors,and breaking of dilation invariance,” Phys. Rept. 258 (1995) 1–171,arXiv:gr-qc/9402012.

[219] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems. PrincetonUniversity Press, 41 William Street, Princeton, New Jersey 08540, USA,1992.

[220] A. C. Hirshfeld and T. Schwarzweller, “Path integral quantization of thePoisson-sigma model,” Annalen Phys. 9 (2000) 83–101, hep-th/9910178.

[221] A. C. Hirshfeld and T. Schwarzweller, “The partition function of the linearPoisson-sigma model on arbitrary surfaces,” hep-th/0112086.

[222] D. Hofmann, Quantum Radiation from Black Holes. PhD thesis, TechnischeUniversitat Wien, 2002. gr-qc/0209007.

[223] P. S. Howe, “Super Weyl transformations in two-dimensions,” J. Phys. A12

(1979) 393–402.

[224] P. S. Howe, K. S. Stelle, and P. K. Townsend, “Miraculous ultravioletcancellations in supersymmetry made manifest,” Nucl. Phys. B236 (1984)125.

[225] V. Husain and L. Smolin, “Exactly solvable quantum cosmologies from twoKilling field reductions of general relativity,” Nucl. Phys. B327 (1989) 205.

[226] S. Ichinose, “Weyl anomaly of 2D dilaton-scalar gravity and hermiticity ofsystem operator,” Phys. Rev. D57 (1998) 6224–6229, hep-th/9707025.

[227] S. Ichinose and S. D. Odintsov, “Conformal anomaly in 4d gravity-mattertheories non- minimally coupled with dilaton,” Nucl. Phys. B539 (1999)643–670, hep-th/9802043.

144

REFERENCES

[228] N. Ikeda, “Gauge theory based on nonlinear Lie superalgebras and structureof 2-d dilaton supergravity,” Int. J. Mod. Phys. A9 (1994) 1137–1152.

[229] N. Ikeda, “Two-dimensional gravity and nonlinear gauge theory,” Ann. Phys.235 (1994) 435–464, arXiv:hep-th/9312059.

[230] N. Ikeda and K. I. Izawa, “General form of dilaton gravity and nonlineargauge theory,” Prog. Theor. Phys. 90 (1993) 237–246, hep-th/9304012.

[231] M. Z. Iofa, “Energy in dilaton gravity in canonical approach,”hep-th/9602023.

[232] C. J. Isham, “Conceptual and geometrical problems in quantum gravity,”.Lectures given at 30th Int. Schladming Winter School, Schladming, Austria,Feb 27 - Mar 8, 1991.

[233] N. Ishibashi, M. Li, and A. R. Steif, “Two-dimensional charged black holes instring theory,” Phys. Rev. Lett. 67 (1991) 3336–3338.

[234] W. Israel, “New interpretation of the extended Schwarzschild manifold,”Phys.Rev. 143 (1966) 1016.

[235] W. Israel, “Thermo field dynamics of black holes,” Phys. Lett. A57 (1976)107–110.

[236] V. Iyer and R. M. Wald, “Some properties of Nother charge and a proposalfor dynamical black hole entropy,” Phys. Rev. D50 (1994) 846–864,gr-qc/9403028.

[237] J. M. Izquierdo, “Free differential algebras and generic 2d dilatonic(super)gravities,” Phys. Rev. D59 (1999) 084017, arXiv:hep-th/9807007.

[238] R. Jackiw, “Lower dimensional gravity,” Nucl. Phys. B252 (1985) 343–356.

[239] R. Jackiw, “Another view on massless matter - gravity fields in two-dimensions,” hep-th/9501016.

[240] P. Jordan, Schwerkraft und Weltall : Grundlagen der theoretischenKosmologie. Vieweg, second ed., 1955.

[241] P. Jordan, “The present state of Dirac’s cosmological hypothesis,” Z. Phys.157 (1959) 112–121.

[242] R. Kantowski and C. Marzban, “One loop Vilkovisky-De Witt counter termsfor 2-D gravity plus scalar field theory,” Phys. Rev. D46 (1992) 5449–5457,hep-th/9208015.

[243] D. R. Karakhanian, R. P. Manvelian, and R. L. Mkrtchian, “Area preservingstructure of 2-d gravity,” Phys. Lett. B329 (1994) 185–188, hep-th/9401031.

[244] H. A. Kastrup and T. Thiemann, “Spherically symmetric gravity as acompletely integrable system,” Nucl. Phys. B425 (1994) 665–686,arXiv:gr-qc/9401032.

145

REFERENCES

[245] M. O. Katanaev, “All universal coverings of two-dimensional gravity withtorsion,” J. Math. Phys. 34 (1993) 700–736.

[246] M. O. Katanaev, “Canonical quantization of the string with dynamicalgeometry and anomaly free nontrivial string in two- dimensions,” Nucl. Phys.B416 (1994) 563–605, hep-th/0101168.

[247] M. O. Katanaev, “Effective action for scalar fields in two-dimensionalgravity,” Annals Phys. 296 (2002) 1–50, gr-qc/0101033.

[248] M. O. Katanaev, W. Kummer, and H. Liebl, “Geometric Interpretation andClassification of Global Solutions in Generalized Dilaton Gravity,” Phys.Rev. D53 (1996) 5609–5618, gr-qc/9511009.

[249] M. O. Katanaev, W. Kummer, and H. Liebl, “On the completeness of theblack hole singularity in 2d dilaton theories,” Nucl. Phys. B486 (1997)353–370, gr-qc/9602040.

[250] M. O. Katanaev and I. V. Volovich, “String model with dynamical geometryand torsion,” Phys. Lett. B175 (1986) 413–416.

[251] M. O. Katanaev and I. V. Volovich, “Two-dimensional gravity withdynamical torsion and strings,” Ann. Phys. 197 (1990) 1.

[252] D. I. Kazakov and S. N. Solodukhin, “On Quantum deformation of theSchwarzschild solution,” Nucl. Phys. B429 (1994) 153–176, hep-th/9310150.

[253] V. A. Kazakov and A. A. Tseytlin, “On free energy of 2-d black hole inbosonic string theory,” JHEP 06 (2001) 021, hep-th/0104138.

[254] S. P. Khastgir and A. Kumar, “String effective action and two-dimensionalcharged black hole,” Mod. Phys. Lett. A6 (1991) 3365–3372,hep-th/9109026.

[255] Y. Kiem, C.-Y. Lee, and D. Park, “Exactly solvable charged dilaton gravitytheories in two dimensions,” Class. Quant. Grav. 15 (1998) 2973–2979,arXiv:hep-th/9703044.

[256] S. Kim and H. Lee, “Wormhole as the end state of two-dimensional blackhole evaporation,” Phys. Lett. B458 (1999) 245–251, gr-qc/9907013.

[257] W. T. Kim and J. Lee, “ADM, Bondi mass, and energy conservation intwo-dimensional dilaton gravities,” Int. J. Mod. Phys. A11 (1996) 553–562,hep-th/9502078.

[258] W. T. Kim and J. J. Oh, “Dilaton driven Hawking radiation in AdS(2) blackhole,” Phys. Lett. B461 (1999) 189–195, hep-th/9905007.

[259] W. T. Kim, “AdS(2) and quantum stability in the CGHS model,” Phys. Rev.D60 (1999) 024011, hep-th/9810055.

[260] K. Kirsten, Spectral functions in mathematics and physics. Chapman &Hall/CRC Press, Boca Raton, 2001.

146

REFERENCES

[261] O. Klein and Y. Nishina Z.Physik 52 (1929) 853.

[262] H. Kleinert and A. Pelster, “Autoparallels from a new action principle,”Gen. Rel. Grav. 31 (1999) 1439, gr-qc/9605028.

[263] T. Klosch and T. Strobl, “Classical and quantum gravity in(1+1)-dimensions. Part I: A unifying approach,” Class. Quant. Grav. 13

(1996) 965–984, arXiv:gr-qc/9508020.

[264] T. Klosch and T. Strobl, “Classical and quantum gravity in 1+1 dimensions.Part II: The universal coverings,” Class. Quant. Grav. 13 (1996) 2395–2422,arXiv:gr-qc/9511081.

[265] T. Klosch and T. Strobl, “Explicit global coordinates for Schwarzschild andReissner-Nordstrom solutions,” Class. Quant. Grav. 13 (1996) 1191–1200,arXiv:gr-qc/9507011.

[266] T. Klosch and T. Strobl, “Classical and quantum gravity in 1+1 dimensions.Part III: Solutions of arbitrary topology and kinks in 1+1 gravity,” Class.Quant. Grav. 14 (1997) 1689–1723, hep-th/9607226.

[267] T. Klosch and T. Strobl, “A global view of kinks in 1+1 gravity,” Phys. Rev.D57 (1998) 1034–1044, arXiv:gr-qc/9707053.

[268] D. Korotkin and H. Samtleben, “Canonical quantization of cylindricalgravitational waves with two polarizations,” Phys. Rev. Lett. 80 (1998)14–17, gr-qc/9705013.

[269] D. Korotkin and H. Samtleben, “Yangian symmetry in integrable quantumgravity,” Nucl. Phys. B527 (1998) 657–689, hep-th/9710210.

[270] I. Kouletsis and P. Hajicek, “Pair of null gravitating shells III. Algebra ofDirac’s observables,” gr-qc/0112062.

[271] G. M. Kremer and F. P. Devecchi, “Thermodynamics and kinetic theory ofrelativistic gases in 2-d cosmological models,” Phys. Rev. D65 (2002)083515, gr-qc/0202025.

[272] K. Kuchar, “Canonical methods of quantization,”. In *Oxford 1980,Proceedings, Quantum Gravity 2*, 329-376.

[273] K. Kuchar, “Canonical quantization of cylindrical gravitational waves,”Phys. Rev. D4 (1971) 955–986.

[274] K. Kuchar, “Dirac constraint quantization of a parametrized field theory byanomaly - Free operator representations of space- time diffeomorphisms,”Phys. Rev. D39 (1989) 2263–2280.

[275] K. Kuchar, “Parametrized scalar field on R X S(1): Dynamical pictures,space-time diffeomorphisms, and conformal isometries,” Phys. Rev. D39

(1989) 1579–1593.

[276] K. V. Kuchar, “Geometrodynamics of Schwarzschild black holes,” Phys. Rev.D50 (1994) 3961–3981, arXiv:gr-qc/9403003.

147

REFERENCES

[277] K. V. Kuchar, J. D. Romano, and M. Varadarajan, “Dirac constraintquantization of a dilatonic model of gravitational collapse,” Phys. Rev. D55

(1997) 795–808, gr-qc/9608011.

[278] W. Kummer, “Deformed ISO(2,1) symmetry and non-Einsteinian 2d-gravitywith matter,” in HADRON STRUCTURE ’92, D. Bruncko and J. Urban,eds. September, 1992. Stara Lesna, Czechoslovakia.

[279] W. Kummer, “On the gauge-independence of the S-matrix,” Eur. Phys. J.C21 (2001) 175–179, hep-th/0104123.

[280] W. Kummer and S. R. Lau, “Boundary conditions and quasilocal energy inthe canonical formulation of all 1 + 1 models of gravity,” Annals Phys. 258

(1997) 37–80, arXiv:gr-qc/9612021.

[281] W. Kummer, H. Liebl, and D. V. Vassilevich, “Exact path integralquantization of generic 2-d dilaton gravity,” Nucl. Phys. B493 (1997)491–502, gr-qc/9612012.

[282] W. Kummer, H. Liebl, and D. V. Vassilevich, “Hawking radiation fornon-minimally coupled matter from generalized 2d black hole models,” Mod.Phys. Lett. A12 (1997) 2683–2690, hep-th/9707041.

[283] W. Kummer, H. Liebl, and D. V. Vassilevich, “Comment on: *Trace anomalyof dilaton coupled scalars in two dimensions*,” Phys. Rev. D58 (1998)108501, hep-th/9801122.

[284] W. Kummer, H. Liebl, and D. V. Vassilevich, “Non-perturbative pathintegral of 2d dilaton gravity and two-loop effects from scalar matter,” Nucl.Phys. B513 (1998) 723–734, hep-th/9707115.

[285] W. Kummer, H. Liebl, and D. V. Vassilevich, “Integrating geometry ingeneral 2d dilaton gravity with matter,” Nucl. Phys. B544 (1999) 403–431,hep-th/9809168.

[286] W. Kummer and G. Tieber, “Universal conservation law and modifiedNother symmetry in 2d models of gravity with matter,” Phys. Rev. D59

(1999) 044001, arXiv:hep-th/9807122.

[287] W. Kummer and D. V. Vassilevich, “Effective action and Hawking radiationfor dilaton coupled scalars in two dimensions,” Phys. Rev. D60 (1999)084021, hep-th/9811092.

[288] W. Kummer and D. V. Vassilevich, “Hawking radiation from dilaton gravityin (1+1) dimensions: A pedagogical review,” Annalen Phys. 8 (1999)801–827, gr-qc/9907041.

[289] W. Kummer and P. Widerin, “NonEinsteinian gravity in d=2: Symmetryand current algebra,” Mod. Phys. Lett. A9 (1994) 1407–1414.

[290] W. Kummer and P. Widerin, “Conserved quasilocal quantities and generalcovariant theories in two-dimensions,” Phys. Rev. D52 (1995) 6965–6975,arXiv:gr-qc/9502031.

148

REFERENCES

[291] W. Kummer and D. J. Schwarz, “General analytic solution of R**2 gravitywith dynamical torsion in two-dimensions,” Phys. Rev. D45 (1992)3628–3635.

[292] W. Kummer and D. J. Schwarz, “Renormalization of R**2 gravity withdynamical torsion in d = 2,” Nucl. Phys. B382 (1992) 171–186.

[293] G. Kunstatter, R. Petryk, and S. Shelemy, “Hamiltonian thermodynamics ofblack holes in generic 2-d dilaton gravity,” Phys. Rev. D57 (1998)3537–3547, arXiv:gr-qc/9709043.

[294] H. La, “Area preserving diffeomorphisms and 2-d gravity,” hep-th/9510147.

[295] S. R. Lau, “On the canonical reduction of spherically symmetric gravity,”Class. Quant. Grav. 13 (1996) 1541–1570, arXiv:gr-qc/9508028.

[296] O. Lauscher and M. Reuter, “Is quantum Einstein gravity nonperturbativelyrenormalizable?,” Class. Quant. Grav. 19 (2002) 483–492, hep-th/0110021.

[297] M. M. Leite and V. O. Rivelles, “Topological dilatonic supergravitytheories,” Class. Quant. Grav. 12 (1995) 627–636, hep-th/9410003.

[298] J. P. S. Lemos and P. M. Sa, “The black holes of a general two-dimensionaldilaton gravity theory,” Phys. Rev. D49 (1994) 2897–2908,arXiv:gr-qc/9311008.

[299] H. Liebl, D. V. Vassilevich, and S. Alexandrov, “Hawking radiation andmasses in generalized dilaton theories,” Class. Quant. Grav. 14 (1997)889–904, arXiv:gr-qc/9605044.

[300] U. Lindstrom, M. Rocek, and P. van Nieuwenhuizen, “Open String BoundaryConditions and Bulk Tachyons.” in preparation.

[301] F. C. Lombardo and F. D. Mazzitelli, “Influence functional in twodimensional dilaton gravity,” Phys. Rev. D58 (1998) 024009,gr-qc/9712091.

[302] F. C. Lombardo, F. D. Mazzitelli, and J. G. Russo, “Energy-momentumtensor for scalar fields coupled to the dilaton in two dimensions,” Phys. Rev.D59 (1999) 064007, gr-qc/9808048.

[303] D. Louis-Martinez, J. Gegenberg, and G. Kunstatter, “Exact Diracquantization of all 2-d dilaton gravity theories,” Phys. Lett. B321 (1994)193–198, gr-qc/9309018.

[304] D. Louis-Martinez and G. Kunstatter, “On Birckhoff’s theorem in 2-ddilaton gravity,” Phys. Rev. D49 (1994) 5227–5230.

[305] D. Louis-Martinez and G. Kunstatter, “Two-dimensional dilaton gravitycoupled to an Abelian gauged field,” Phys. Rev. D52 (1995) 3494–3505,gr-qc/9503016.

149

REFERENCES

[306] D. Louis-Martinez, “Dirac quantization of two-dimensional dilaton gravityminimally coupled to N massless scalar fields,” Phys. Rev. D55 (1997)7982–7984, hep-th/9611031.

[307] J. Louko and B. F. Whiting, “Hamiltonian thermodynamics of theSchwarzschild black hole,” Phys. Rev. D51 (1995) 5583–5599,gr-qc/9411017.

[308] F. Lund Phys.Rev. D8 (1973) 3247.

[309] J. Maldacena, “The large N limit of superconformal field theories andsupergravity,” Adv. Theor. Math. Phys. 2 (1998) 231–252, hep-th/9711200.

[310] G. Mandal, A. M. Sengupta, and S. R. Wadia, “Classical solutions oftwo-dimensional string theory,” Mod. Phys. Lett. A6 (1991) 1685–1692.

[311] R. B. Mann, “Conservation laws and 2-d black holes in dilaton gravity,”Phys. Rev. D47 (1993) 4438–4442, hep-th/9206044.

[312] R. B. Mann, A. Shiekh, and L. Tarasov, “Classical and quantum propertiesof two-dimensional black holes,” Nucl. Phys. B341 (1990) 134–154.

[313] V. N. Marachevsky and D. Vassilevich, “Diffeomorphism invariant eigenvalueproblem for metric perturbations in a bounded region,” Class. Quant. Grav.13 (1996) 645–652, gr-qc/9509051.

[314] R. L. Marsa and M. W. Choptuik, “Black hole scalar field interactions inspherical symmetry,” Phys. Rev. D54 (1996) 4929–4943,arXiv:gr-qc/9607034.

[315] E. J. Martinec, “Superspace geometry of fermionic strings,” Phys. Rev. D28

(1983) 2604.

[316] M. D. McGuigan, C. R. Nappi, and S. A. Yost, “Charged black holes intwo-dimensional string theory,” Nucl. Phys. B375 (1992) 421–452,hep-th/9111038.

[317] A. J. M. Medved, “Near-extremal spherically symmetric black holes inarbitrary-dimensional spacetimes,” hep-th/0112056.

[318] A. J. M. Medved, “Reissner-Nordstrom near extremality from a Jackiw-Teitelboim perspective,” hep-th/0111091.

[319] A. J. M. Medved and G. Kunstatter, “Hamiltonian thermodynamics ofcharged black holes,” Phys. Rev. D59 (1999) 104005, hep-th/9811052.

[320] A. J. M. Medved and G. Kunstatter, “Quantum corrections to thethermodynamics of charged 2-d black holes,” Phys. Rev. D60 (1999) 104029,hep-th/9904070.

[321] A. J. M. Medved and G. Kunstatter, “One-loop corrected thermodynamics ofthe extremal and non- extremal spinning BTZ black hole,” Phys. Rev. D63

(2001) 104005, hep-th/0009050.

150

REFERENCES

[322] E. W. Mielke, F. Gronwald, Y. N. Obukhov, R. Tresguerres, and F. W. Hehl,“Towards complete integrability of two-dimensional Poincare gauge gravity,”Phys. Rev. D48 (1993) 3648–3662, hep-th/9304043.

[323] S. Mignemi, “Black hole solutions in generalized two-dimensional dilationgravity theories,” Phys. Rev. D50 (1994) 4733–4736.

[324] S. Mignemi and D. L. Wiltshire, “Spherically symmetric solutions indimensionally reduced space-times,” Class. Quant. Grav. 6 (1989) 987.

[325] A. Mikovic and V. Radovanovic, “One-loop effective action for sphericalscalar field collapse,” Class. Quant. Grav. 15 (1998) 827–838,hep-th/9706066.

[326] A. Mikovic, “Exactly solvable models of 2-d dilaton quantum gravity,” Phys.Lett. B291 (1992) 19–25, hep-th/9207006.

[327] A. Mikovic, “Two-dimensional dilaton gravity in a unitary gauge,” Phys.Lett. B304 (1993) 70–76, hep-th/9211082.

[328] A. Mikovic, “Hawking radiation and back reaction in a unitary theory of 2-Dquantum gravity,” Phys. Lett. B355 (1995) 85–91, hep-th/9407104.

[329] A. Mikovic, “General solution for self-gravitating spherical null dust,” Phys.Rev. D56 (1997) 6067–6070, gr-qc/9705030.

[330] A. Mikovic and V. Radovanovic, “Loop corrections in the spectrum of 2dHawking radiation,” Class. Quant. Grav. 14 (1997) 2647–2661,gr-qc/9703035.

[331] C. Misner in Magic without Magic: John Archibald Wheeler, A Collection ofEssays in Honor of his 60th Birthday, J. Klauder, ed. Freeman, SanFrancisco, 1972.

[332] I. G. Moss and P. J. Silva, “BRST invariant boundary conditions for gaugetheories,” Phys. Rev. D55 (1997) 1072–1078, gr-qc/9610023.

[333] V. Mukhanov, A. Wipf, and A. Zelnikov, “On 4-D Hawking radiation fromeffective action,” Phys. Lett. B332 (1994) 283–291, hep-th/9403018.

[334] M. Nakahara, Geometry, Topology and Physics. IOP Publishing, Bristol,1990.

[335] C. R. Nappi and A. Pasquinucci, “Thermodynamics of two-dimensional blackholes,” Mod. Phys. Lett. A7 (1992) 3337–3346, gr-qc/9208002.

[336] D. J. Navarro and J. Navarro-Salas, “Baecklund transformations in 2ddilaton gravity,” Mod. Phys. Lett. A13 (1998) 2049–2056, hep-th/9807003.

[337] M. Navarro, “Generalized symmetries and invariant matter couplings intwo-dimensional dilaton gravity,” Phys. Rev. D56 (1997) 2384–2389,arXiv:gr-qc/9702040.

151

REFERENCES

[338] H. Nicolai, D. Korotkin, and H. Samtleben, “Integrable classical andquantum gravity,” hep-th/9612065.

[339] H. Nicolai and H. Samtleben, “Integrability and canonical structure of d = 2,N = 16 supergravity,” Nucl. Phys. B533 (1998) 210–242, hep-th/9804152.

[340] A. Nijenhuis, “Jacobi-type identities for bilinear differential concomitants ofcertain tensor fields,” Proc. Kon. Ned. Akad. Wet. Amsterdam A58 (1955)390.

[341] S. Nojiri and I. Oda, “Charged dilatonic black hole and Hawking radiation intwo- dimensions,” Phys. Lett. B294 (1992) 317–324, hep-th/9206087.

[342] S. Nojiri and S. D. Odintsov, “Trace anomaly and non-local effective actionfor 2d conformally invariant scalar interacting with dilaton,” Mod. Phys.Lett. A12 (1997) 2083–2088, hep-th/9706009.

[343] S. Nojiri and S. D. Odintsov, “Trace anomaly induced effective action for 2Dand 4D dilaton coupled scalars,” Phys. Rev. D57 (1998) 2363–2371,hep-th/9706143.

[344] S. Nojiri and S. D. Odintsov, “Quantum (in)stability of 2d charged dilatonblack holes due to back-reaction of dilaton coupled scalars,” Phys. Rev. D59

(1999) 044003, hep-th/9806055.

[345] S. Nojiri and S. D. Odintsov, “Quantum dilatonic gravity in d = 2, 4 and 5dimensions,” Int. J. Mod. Phys. A16 (2001) 1015–1108, hep-th/0009202.

[346] G. Nordstrom, “On the energy of the gravitation field in Einstein’s theory,”Proc. Kon. Ned. Akad. Wet. 20 (1916) 1238.

[347] Y. V. Novozhilov and D. V. Vassilevich, “Induced quantum conformalgravity,” Phys. Lett. B220 (1989) 36.

[348] Y. N. Obukhov and F. W. Hehl, “Black holes in two dimensions,”hep-th/9807101.

[349] S. D. Odintsov and I. L. Shapiro, “One loop renormalization oftwo-dimensional induced quantum gravity,” Phys. Lett. B263 (1991)183–189.

[350] S. D. Odintsov and I. L. Shapiro, “Perturbative finiteness of 2-d inducedquantum gravity,” Mod. Phys. Lett. A7 (1992) 437–440.

[351] A. Ori, “Evaporation of a two-dimensional charged black hole,” Phys. Rev.D63 (2001) 104016, gr-qc/0102067.

[352] A. Palatini Rend. Circ. Mat. Palermo 43 (1919) 203.

[353] D. Park and Y. Kiem, “General Static Solutions of 2-dimensionalEinstein-Dilaton- Maxwell-Scalar Theories,” Phys. Rev. D53 (1996)5513–5520, hep-th/9601166.

152

REFERENCES

[354] Y.-C. Park and A. Strominger, “Supersymmetry and positive energy inclassical and quantum two-dimensional dilaton gravity,” Phys. Rev. D47

(1993) 1569–1575, arXiv:hep-th/9210017.

[355] A. W. Peet, “The Bekenstein formula and string theory (N-brane theory),”Class. Quant. Grav. 15 (1998) 3291–3338, hep-th/9712253.

[356] H. Pelzer and T. Strobl, “Generalized 2d dilaton gravity with matter fields,”Class. Quant. Grav. 15 (1998) 3803–3825, arXiv:gr-qc/9805059.

[357] R. Penrose, “Gravitational collapse: The role of general relativity,” Riv.Nuovo Cim. 1 (1969) 252–276.

[358] R. Penrose and W. Rindler, Spinors and Space-Time I. CambridgeUniversity Press, 1984.

[359] Supernova Cosmology Project Collaboration, S. Perlmutter et al.,“Measurements of Omega and Lambda from 42 High-Redshift Supernovae,”Astrophys. J. 517 (1999) 565–586, arXiv:astro-ph/9812133.

[360] J. Polchinski, String theory. Cambridge University Press, 1998. Vol. 1: AnIntroduction to the Bosonic String.

[361] A. M. Polyakov, “Quantum geometry of bosonic strings,” Phys. Lett. B103

(1981) 207–210.

[362] A. M. Polyakov, “Quantum gravity in two-dimensions,” Mod. Phys. Lett. A2

(1987) 893.

[363] B. Ratra and P. J. E. Peebles, “Cosmological consequences of a rollinghomogeneous scalar field,” Phys. Rev. D37 (1988) 3406.

[364] H. Reissner, “Uber die Eigengravitation des elektrischen Feldes nach derEinsteinschen Theorie,” Ann. Phys. 50 (1916) 106.

[365] A. G. Riess, “The Case for an Accelerating Universe from Supernovae,” ThePublications of the Astronomical Society of the Pacific 112 (Oct., 2000)1284–1299.

[366] Supernova Search Team Collaboration, A. G. Riess et al., “ObservationalEvidence from Supernovae for an Accelerating Universe and a CosmologicalConstant,” Astron. J. 116 (1998) 1009–1038, astro-ph/9805201.

[367] A. G. Riess et al., “The Farthest Known Supernova: Support for anAccelerating Universe and a Glimpse of the Epoch of Deceleration,”Astrophys. J. 560 (2001) 49–71, astro-ph/0104455.

[368] V. O. Rivelles, “Topological two-dimensional dilaton supergravity,” Phys.Lett. B321 (1994) 189–192, hep-th/9301029.

[369] M. D. Roberts, “Scalar field counterexamples to the cosmic censorshiphypothesis,” Gen. Rel. Grav. 21 (1989) 907–939.

153

REFERENCES

[370] M. Rocek, P. van Nieuwenhuizen, and S. C. Zhang, “Superspace pathintegral measure of the n=1 spinning string,” Ann. Phys. 172 (1986) 348.

[371] C. Rovelli, “Loop quantum gravity,” Living Rev. Rel. 1 (1998) 1,gr-qc/9710008.

[372] J. G. Russo, L. Susskind, and L. Thorlacius, “The Endpoint of Hawkingradiation,” Phys. Rev. D46 (1992) 3444–3449, hep-th/9206070.

[373] J. G. Russo and A. A. Tseytlin, “Scalar tensor quantum gravity intwo-dimensions,” Nucl. Phys. B382 (1992) 259–275, arXiv:hep-th/9201021.

[374] J. G. Russo, L. Susskind, and L. Thorlacius, “Black hole evaporation in 1 + 1dimensions,” Phys. Lett. B292 (1992) 13–18, hep-th/9201074.

[375] I. Sachs and S. N. Solodukhin, “Horizon holography,” Phys. Rev. D64 (2001)124023, hep-th/0107173.

[376] P. Schaller and T. Strobl, “Quantization of field theories generalizing gravityYang- Mills systems on the cylinder,” in Integrable models and strings,pp. 98–122. 1994. gr-qc/9406027. Helsinki 1993.

[377] P. Schaller and T. Strobl, “Canonical quantization of nonEinsteinian gravityand the problem of time,” Class. Quant. Grav. 11 (1994) 331–346,arXiv:hep-th/9211054.

[378] P. Schaller and T. Strobl, “Poisson sigma models: A generalization of 2-dgravity Yang- Mills systems,” in Finite dimensional integrable systems,pp. 181–190. 1994. hep-th/9411163. Dubna.

[379] P. Schaller and T. Strobl, “Poisson structure induced (topological) fieldtheories,” Mod. Phys. Lett. A9 (1994) 3129–3136, hep-th/9405110.

[380] H.-J. Schmidt, “Scale invariant gravity in two-dimensions,” J. Math. Phys.32 (1991) 1562–1566.

[381] H.-J. Schmidt, “The classical solutions of two-dimensional gravity,” Gen.Rel. Grav. 31 (1999) 1187–1210, gr-qc/9905051.

[382] V. Schomerus, “D-branes and deformation quantization,” JHEP 06 (1999)030, hep-th/9903205.

[383] J. Schouten, “On the differential operators of first order in tensor calculus,”Convegno di Geom. Diff. 1 (1954) 7.

[384] A. S. Schwarz, “The partition function of degenerate quadratic functionaland Ray-Singer invariants,” Lett. Math. Phys. 2 (1978) 247–252.

[385] A. S. Schwarz, “The partition function of a degenerate functional,”Commun. Math. Phys. 67 (1979) 1–16.

[386] K. Schwarzschild, “On the gravitational field of a mass point according toEinstein’s theory,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. )1916 (1916) 189–196, arXiv:physics/9905030.

154

REFERENCES

[387] N. Seiberg and E. Witten, “String theory and noncommutative geometry,”JHEP 09 (1999) 032, hep-th/9908142.

[388] A. E. Sikkema and R. B. Mann, “Gravitation and cosmology intwo-dimensions,” Class. Quant. Grav. 8 (1991) 219–236.

[389] S. N. Solodukhin, “Black hole solution in 2-D gravity with torsion,” JETPLett. 57 (1993) 329–334.

[390] S. N. Solodukhin, “Topological 2-D Riemann-Cartan-Weyl gravity,” Class.Quant. Grav. 10 (1993) 1011–1021.

[391] S. N. Solodukhin, “Two-dimensional black hole with torsion,” Phys. Lett.B319 (1993) 87–95, hep-th/9302040.

[392] S. Solodukhin, “On exact integrability of 2-D Poincare gravity,” Mod. Phys.Lett. A9 (1994) 2817–2824, hep-th/9404034.

[393] S. Solodukhin, “Exact solution of 2-D Poincare gravity coupled to fermionmatter,” Phys. Rev. D51 (1995) 603–608, hep-th/9404045.

[394] C. R. Stephens, G. ’t Hooft, and B. F. Whiting, “Black hole evaporationwithout information loss,” Class. Quant. Grav. 11 (1994) 621–648,gr-qc/9310006.

[395] T. Strobl, “Dirac quantization of gravity Yang-Mills systems in (1+1)-dimensions,” Phys. Rev. D50 (1994) 7346–7350, hep-th/9403121.

[396] T. Strobl, Poisson structure induced field theories and models of 1+1dimensional gravity. PhD thesis, Technische Universitat Wien, 1994.hep-th/0011248.

[397] T. Strobl, “Gravity in two spacetime dimensions,” hep-th/0011240.Habilitation thesis.

[398] T. Strobl, “Target-superspace in 2d dilatonic supergravity,” Phys. Lett.B460 (1999) 87–93, arXiv:hep-th/9906230.

[399] A. Strominger, “Les Houches lectures on black holes,”arXiv:hep-th/9501071. Talk given at NATO Advanced Study Institute.

[400] L. Susskind, “The World as a hologram,” J. Math. Phys. 36 (1995)6377–6396, hep-th/9409089.

[401] P. Sutton, “The dimensional-reduction anomaly in spherically symmetricspacetimes,” Phys. Rev. D62 (2000) 044033, hep-th/0003290.

[402] G. ’t Hooft, “The scattering matrix approach for the quantum black hole: Anoverview,” Int. J. Mod. Phys. A11 (1996) 4623–4688, gr-qc/9607022.

[403] G. ’t Hooft, “Dimensional reduction in quantum gravity,” in Salamfestschrift.World Scientific, 1993. gr-qc/9310026.

[404] G. ’t Hooft and M. J. G. Veltman, “One loop divergencies in the theory ofgravitation,” Annales Poincare Phys. Theor. A20 (1974) 69–94.

155

REFERENCES

[405] C. Teitelboim, “Gravitation and Hamiltonian structure in two space-timedimensions,” Phys. Lett. B126 (1983) 41.

[406] J. Teschner, “Liouville theory revisited,” Class. Quant. Grav. 18 (2001)R153–R222, hep-th/0104158.

[407] T. Thiemann and H. A. Kastrup, “Canonical quantization of sphericallysymmetric gravity in Ashtekar’s selfdual representation,” Nucl. Phys. B399

(1993) 211–258, arXiv:gr-qc/9310012.

[408] T. Thiemann, “Introduction to modern canonical quantum generalrelativity,” gr-qc/0110034.

[409] P. Thomi, B. Isaak, and P. Hajıcek, “Spherically symmetric systems of fieldsand black holes. 1. Definition and properties of apparent horizon,” Phys.Rev. D30 (1984) 1168.

[410] G. Tieber, Gravitationsmodelle mit Materie in zwei Dimensionen und ihreSymmetrien. PhD thesis, Technische Universitat Wien, 1997. in german.

[411] D. J. Toms, “The functional measure for quantum field theory in curvedspace-time,” Phys. Rev. D35 (1987) 3796.

[412] W. G. Unruh, “Notes on black hole evaporation,” Phys. Rev. D14 (1976)870.

[413] E. C. Vagenas, “Are extremal 2-d black holes really frozen?,” Phys. Lett.B503 (2001) 399–403, hep-th/0012134.

[414] A. E. M. van de Ven, “Index-free heat kernel coefficients,” Class. Quant.Grav. 15 (1998) 2311, hep-th/9708152.

[415] M. Varadarajan, “Classical and quantum geometrodynamics of 2-d vacuumdilatonic black holes,” Phys. Rev. D52 (1995) 7080–7088, gr-qc/9508039.

[416] M. Varadarajan, “Quantum gravity effects in the CGHS model of collapse toa black hole,” Phys. Rev. D57 (1998) 3463–3473, gr-qc/9801058.

[417] G. G. Varzugin, “The inverse scattering method for cylindrical gravitationalwaves,” gr-qc/0001024.

[418] D. V. Vassilevich, “On gauge invariant boundary conditions for 2-d gravitywith dynamical torsion,” Mod. Phys. Lett. A10 (1995) 2239–2244,hep-th/9504011.

[419] D. V. Vassilevich, “Vector fields on a disk with mixed boundary conditions,”J. Math. Phys. 36 (1995) 3174–3182, gr-qc/9404052.

[420] D. V. Vassilevich, “Spectral branes,” JHEP 03 (2001) 023, hep-th/0102091.

[421] D. V. Vassilevich and A. Zelnikov, “Discrete symmetries of functionaldeterminants,” Nucl. Phys. B594 (2001) 501–517, hep-th/0009084.

156

REFERENCES

[422] D. V. Vassilevich, “QED on curved background and on manifolds withboundaries: Unitarity versus covariance,” Phys. Rev. D52 (1995) 999–1010,gr-qc/9411036.

[423] C. Vaz and L. Witten, “Quantum naked singularities in 2d dilaton gravity,”Nucl. Phys. B487 (1997) 409–441, hep-th/9604064.

[424] C. Vaz, L. Witten, and T. P. Singh, “Toward a quantization of null dustcollapse,” gr-qc/0112024.

[425] S. N. Vergeles, “Canonical quantization of two-dimensional gravity,” Sov.Phys. JETP Lett. 117 (2000) 3, gr-qc/0102001.

[426] H. Verlinde, “Black holes and strings in two dimensions,” in Trieste SpringSchool on Strings and Quantum Gravity, pp. 178–207. April, 1991.

[427] M. S. Volkov and D. V. Gal’tsov, “Gravitating non-Abelian solitons andblack holes with Yang-Mills fields,” Phys. Rept. 319 (1999) 1–83,hep-th/9810070.

[428] R. M. Wald, General Relativity. The University of Chicago Press, 1984.

[429] R. M. Wald, “Black hole entropy in Nother charge,” Phys. Rev. D48 (1993)3427–3431, gr-qc/9307038.

[430] M. Walker, “Block diagrams and the extension of timelike two-surfaces,” J.Math. Phys. 11 (1970) 2280.

[431] L.-M. Wang and P. J. Steinhardt, “Cluster abundance constraints onquintessence models,” Astrophys. J. 508 (1998) 483–490,arXiv:astro-ph/9804015.

[432] G. Watson and J. R. Klauder, “Generalized affine coherent states: A naturalframework for quantization of metric-like variables,” J. Math. Phys. 41

(2000) 8072–8082, quant-ph/0001026.

[433] C. Wetterich, “Cosmology and the fate of dilatation symmetry,” Nucl. Phys.B302 (1988) 668.

[434] J. Wheeler, “Geometrodynamics and the issue of the final state,” inRelativity, Groups and Topology, C. DeWitt and B. DeWitt, eds., p. 316.Gordon and Breach, 1964.

[435] J. Wheeler in Batelle Recontres: 1967 Lectures in Mathematics and Physics,C. DeWitt and J. Wheeler, eds. Benjamin, 1968.

[436] C. M. Will, “The confrontation between general relativity and experiment,”Living Rev. Rel. 4 (2001) 4, gr-qc/0103036.

[437] A. W. Wipf, “Tunnel determinants,” Nucl. Phys. B269 (1986) 24.

[438] E. Witten, “Some exact multipseudoparticle solutions of classical Yang- Millstheory,” Phys. Rev. Lett. 38 (1977) 121.

157

REFERENCES

[439] E. Witten, “Nonabelian bosonization in two dimensions,” Commun. Math.Phys. 92 (1984) 455–472.

[440] E. Witten, “Topological quantum field theory,” Commun. Math. Phys. 117

(1988) 353.

[441] E. Witten, “Topological sigma models,” Commun. Math. Phys. 118 (1988)411.

[442] E. Witten, “Quantum field theory and the Jones polynomial,” Commun.Math. Phys. 121 (1989) 351.

[443] E. Witten, “On string theory and black holes,” Phys. Rev. D44 (1991)314–324.

[444] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys.2 (1998) 253–291, hep-th/9802150.

[445] O. B. Zaslavsky, “Exactly solvable models of two-dimensional dilaton gravityand quantum eternal black holes,” Phys. Rev. D59 (1999) 084013,hep-th/9804089.

[446] O. B. Zaslavsky, “Semi-infinite throats at finite temperature and staticsolutions in exactly solvable models of 2d dilaton gravity,” Phys. Lett. B459

(1999) 105–111, hep-th/9904184.

[447] I. Zlatev, L.-M. Wang, and P. J. Steinhardt, “Quintessence, cosmiccoincidence, and the cosmological constant,” Phys. Rev. Lett. 82 (1999)896–899, arXiv:astro-ph/9807002.

[448] D. Grumiller and R. Meyer, “Ramifications of lineland,” Turk. J. Phys. 30

(2006) 349–378, hep-th/0604049.

158

TUW–03–18

The Classical Solutions of the DimensionallyReduced Gravitational Chern-Simons Theory

D. Grumiller∗, W. Kummer†

∗ † Institut fur Theoretische Physik, Technische Universitat Wien

Wiedner Hauptstr. 8–10, A-1040 Wien, Austria

Abstract

The Kaluza-Klein reduction of the 3d gravitational Chern-Simons termto a 2d theory is equivalent to a Poisson-sigma model with fourdimensionaltarget space and degenerate Poisson tensor of rank 2. Thus two constantsof motion (Casimir functions) exist, namely charge and energy. The ap-plication of well-known methods developed in the framework of first ordergravity allows to construct all classical solutions straightforwardly and todiscuss their global structure. For a certain fine tuning of the values ofthe constants of motion the solutions of hep-th/0305117 are reproduced.Possible generalizations are pointed out.

∗e-mail: grumil@hep.itp.tuwien.ac.at†e-mail: wkummer@tph.tuwien.ac.at

1 INTRODUCTION

1 Introduction

As shown by the authors of ref. [1]1 the gravitational Chern-Simons term can bereduced by a Kaluza-Klein like ansatz (3.28), decomposing the 3d metric into a2d metric gµν , a U(1) gauge field A = Aµdxµ and a scalar φ. Invoking conformalinvariance, φ has been set to 1. The resulting 2d action (cf. eq. (3.35))2

L [gµν , Aµ] =1

8π2

∫

M2

d2x√−g

(

FR − F 3)

, (1)

depends on the 2d curvature scalar R and on the abelian dual field strengthF = −2 ∗dA. It thus represents a 2d field theory of gravity interacting with thegauge field 1-form A.

Classical solutions have been constructed locally in ref. [1], labelled by aconstant of motion c whereby another constant of motion has been fixed to acertain value. As far as curvature is concerned this discussion has been exhaus-tive; however, as will be shown in this work, isocurvature solutions exists witha different number (and different types) of Killing horizons. The main purposeof this note is to elevate the discussion to a global level, i.e. to construct allpossible Carter-Penrose (CP) diagrams. A condensed version of the results isplotted in fig. 2.

An action like (1) is equivalent to a first order gravity action which, in turn,is a special case of a Poisson-sigma model (PSM) [3]

L =1

4π2

∫

M2

[Xa(D ∧ e)a + Xd ∧ ω + Y d ∧ A + ǫV(X, Y )] , (2)

with target space coordinates Y, X, X+, X−, gauge field 1-forms A, ω, e−, e+

and

V(X, Y ) =1

2

(

XY − X3)

. (3)

The notation of ref. [2] is used.3 In addition to the Cartan variables an abeliangauge field 1-form A is present and a new target space coordinate Y which actsas Lagrange multiplier for gauge curvature. Theories of that type are known fora long time [4]. Actually the transition from (2) to (1) is very easy. Variationof Y in (2) yields X = −2 ∗ dA = F where F is precisely the dual field strengthin (1). Because this equation is linear in X , the re-insertion of X into thevariational principle is permitted. A similar argument allows the replacement

1If not stated otherwise all cross-references of the type (x.y) refer to formulas in that paper.2Signs have been adjusted in order to agree with the notation of ref. [2]; to compare with

ref. [1] the relations R = −r and F = f are helpful.3ea is the zweibein one-form, ǫ = e+ ∧ e− is the volume two-form. The one-form ω

represents the spin-connection ωab = εa

bω with the totally antisymmetric Levi-Civita symbolεab (ε01 = +1). With the flat metric ηab in light-cone coordinates (η+− = 1 = η−+, η++ =0 = η−−) the first (“torsion”) term of (2) is given by Xa(D ∧ e)a = ηabX

b(D ∧ e)a =X+(d − ω) ∧ e− + X−(d + ω) ∧ e+. Signs and factors of the Hodge-∗ operation are definedby ∗ǫ = 1. The target space coordinates X, Xa can be interpreted as Lagrange multipliers forgeometric curvature and torsion, respectively.

160

2 ALL CLASSICAL SOLUTIONS

of the the spin connection by its dependent form (cf. (9) below) because thefirst term of (2) requires vanishing torsion. The second term with the dependentspin connection in R = 2 ∗ dω immediately leads to the first term of (1). In(2) the corresponding Poisson tensor has rank 2, apart from one point in targetspace, namely Xa = X = Y = 0. Therefore, the number of Casimir functionsis two (in physical terms they correspond to the conserved total charge c andenergy C(g)). A reformulation (2) is advantageous because powerful tools existto deal with PSMs at the classical and quantum level [3, 5]. Further details onfirst order gravity and a more comprehensive list of references can be found inref. [2].

With eq. (2) as the starting point all classical solutions can be determinedwith ease. The solution discussed in ref. [1] is found to be a special case wherethe Casimir functions c and C(g) are related in a special manner. In general eachsolution is labelled by the constant values of c and C(g) and is valid in a certainpatch of coordinates. From those patches all global solutions can be found thestructure of which is summarized in fig. 2. Finally, possible generalizations arepointed out.

2 All classical solutions

The equations of motion (EOM) for the action (2) read:

dY = 0 , (4)

dX + X−e+ − X+e− = 0 , (5)

(d ± ω)X± ± 1

2(X3 − XY )e± = 0 , (6)

dA + ǫ1

2X = 0 , (7)

dω − ǫ1

2(3X2 − Y ) = 0 , (8)

(d ± ω)e± = 0 . (9)

The action (2) is mapped into −L by the Z2 transformation X → −X , X± →−X±, A → −A.4 An important distinguishing feature as compared to dilatongravity coupled to an abelian gauge field is the term XY present in (3) becauseit is linear in Y . By contrast a typical abelian gauge theory with F 2 in thesecond order form would require a term proportional to Y 2 in (3), as can bechecked easily.

The integration of (4) immediately yields the first Casimir function, Y = c =const. which may be interpreted as “charge”. The second, geometric one (cf. e.g.(3.23) of ref. [2]), the “energy”, is obtained by multiplying eqs. (6) respectively

4In the second order approach the same discrete symmetry has been observed in ref. [1](cf. the comment below (4.49)).

161

2 ALL CLASSICAL SOLUTIONS

by X−, X+, adding them and inserting (5):

C(g) = X+X− − 1

8X4 +

Y

4X2 . (10)

Eq. (7) implies X = −2 ∗ dA, thus the dual field strength F is determined bythe “dilaton” field X . The last equation (9) entails the condition of vanishingtorsion and can be used to solve for the spin-connection ω = ηabe

a ∗ d ∧ eb.

2.1 Constant dilaton vacua

For X+ = 0 = X− eq. (5) implies X = const. From (6) it can be deducedimmediately that only three solutions are possible: a Z2 symmetric one (X = 0)and two non-symmetric ones (X = ±√

c, c > 0). The solutions for the curvaturescalar R = −c resp. R = 2c from (8) indicate (A)dS space (cf. (4.50) and (4.51)).The corresponding line element can be presented as5

(ds)2 = 2dudx +

(

R

2x2 + Ax + B

)

(du)2 , (11)

with some integration constants A, B which have to be fixed appropriately. Theyare neither defined by the first Casimir c (which enters R) nor by the secondone C(g). The latter vanishes for the symmetric solution and becomes equal toC(g) = c2/8 for the non-symmetric ones. The global structure is the same as theone of the Jackiw-Teitelboim (JT) model [6], namely (A)dS space.

2.2 Generic solutions

All other classical solutions can be constructed in the usual manner [4, 7]. Ina patch where X+ 6= 0 one obtains6 the line element in Eddington-Finkelstein(EF) gauge

(ds)2 = 2dudX + K(X ; C(g), c)(du)2 , K(X ; C(g), c) = 2C(g) − c

2X2 +

1

4X4 .

(12)Evidently there is always a Killing vector7 kα∂α = ∂/∂u with associated Killingnorm gαβkαkβ = K(X ; C(g), c). The curvature scalar becomes

R = d2K/dX2 = −c + 3X2 . (13)

5In fact such solutions exist if X+ = 0 = X− in generic 2d gravity theories (2) whena more general potential V(X+X−, X, Y ) permits one or more solutions to the algebraicequation V(0, X, c) = 0. There are as many distinct vacua as there are solutions to thatequation. Curvature is given by R = −2∂V/∂X. Even if V depends on X+X−, ω remainsthe Levi-Civita connection.

6If X+ = 0 and X− 6= 0 then the same procedure can be applied with + ↔ −. If bothX+ = 0 = X− in an open region we have the constant dilaton vacuum discussed above.

7This is a general feature of 2D first order gravity actions (2), albeit it is not a feature ofgeneric 2D gravity. This property was also noted in appendix A of ref. [1].

162

3 GLOBAL PROPERTIES

Obviously, solutions with constant curvature are only possible for the constantdilaton vacuum. With the coordinate redefinition (cf. eq. (4.52))

X =:√

c tanh y , y :=

(√c

2z

)

, (14)

curvature transforms to

R = −c + 3c tanh2 y = 2c− 3c

cosh2 y. (15)

This is consistent with (4.53). With the Ansatz du = αdt+β(z)dz and (14) theline element (12) can be brought into diagonal form:

(ds)2 =1

cosh4 y(1 + δ) (dt)2 − (dz)2

1 + δ, δ :=

(

8C(g)/c2 − 1)

cosh4 y . (16)

In the special case c2 = 8C(g) eq. (16) coincides with eq. (4.54).Whenever a diagonal gauge of this type is chosen for a geometry exhibiting

Killing horizons coordinate singularities appear. As a consequence the line el-ement (16) acquires a coordinate singularity at X = ±√

c. Therefore, the lineelement in EF gauge (12) is a more suitable starting point for a discussion ofthe global structure because it allows for an extension across Killing horizons.

3 Global properties

Applying well-known methods [8, 9] the first step of a global discussion is toconstruct the building blocks of the CP diagrams. The second step is to findtheir consistent geodesic extensions. In a third step solutions of more compli-cated topology can be arranged [10]. Finally, one can try to identify patches ina nontrivial way in order to obtain kink solutions [11].

3.1 Building blocks

The basic patches are represented by CP diagrams derived from the metric inEF form (12), together with their mirror images (the flip corresponds essentiallyto a change from ingoing to outgoing EF gauge or vice versa). They determinethe set of building blocks from which the global CP diagram is found in a nextstep by geodesic extension.

The Killing norm K in (12) has the form of a Higgs potential. Its four zerosare given by

X1,2,3,4h = ±

√

c ±√

c2 − 8C(g) . (17)

Only for real zeros a Killing horizon emerges. There are several possibilitiesregarding the number and type of Killing horizons. For positive c any numberfrom 0 to 4 is possible, for negative or vanishing c just 0, 1 or 2 horizons canarise. In all CP diagrams bold lines correspond to the curvature singularities

163

3 GLOBAL PROPERTIES

encountered at X → ±∞. Dashed lines are Killing horizons (multiply dashedlines are extremal ones). The lines of constant X are depicted as ordinarylines. The triangular shape of the outermost patches is a consequence of theasymptotic behavior (X → ±∞) of the Killing norm. The singularities arenull complete (because X diverges) but incomplete with respect to non-nullgeodesics, because the “proper time” (cf. eq. (3.50) of ref. [2]; A = const.)

τ =

∫ X

dX ′/√

|A − K(X ′)| = const. −O(

1

X

)

, (18)

does not diverge at the boundary. This somewhat counter intuitive feature hasbeen witnessed already for the dilaton black hole [12]. Regarding this propertythe singularities differ essentially from the ones in the JT model which arecomplete with respect to all geodesics.

“Time” and “space” in conformal coordinates should be plotted in the ver-tical resp. horizontal direction. Therefore, all diagrams below except B0 shouldbe considered rotated clockwise by 45o.

No horizons If K has no zeros no Killing horizons arise. This happens forpositive c provided that 8C(g) > c2 and for c ≤ 0 if C(g) > 0. Modulo complete-ness properties this diagram is equivalent to the one of the JT model when nohorizons are present (cf. e.g. fig. 9 in ref. [9]).

B0:

One extremal horizon This scenario can only happen for c ≤ 0 (if theinequality is saturated the zero in the Killing norm is of fourth order, otherwisejust second order). Additionally, C(g) must vanish. The horizon is located atX = 0.

B1a: B1b:

Two horizons For negative C(g) and arbitrary c two horizons arise at X =

±√

c +√

c2 − 8C(g). Modulo completeness properties this diagram is equivalentto the one of the JT model when two horizons are present.

B2a:

Two extremal horizons This special case appears for c > 0 and c2 = 8C(g).The square patch in the middle corresponds to the nontrivial solution discussedin ref. [1].

164

3 GLOBAL PROPERTIES

B2b:

Two horizons and one extremal horizon If c > 0 and C(g) = 0 an extremalhorizon at X = 0 is present. The two non-extremal ones are located at X =±√

2c. This building block will generate non-smooth CP diagrams due to theappearance of extremal and non-extremal horizons (cf. fig. 3 of ref. [9] and thediscussion on that page).

B3:

Four horizons For c > 0 and c2 > 8C(g) > 0 four horizons are present givenby (17).

B4:

3.2 Maximal extensions

The boundaries of each building block are either geodesically complete (infiniteaffine parameter with respect to all geodesics) or incomplete otherwise. Looselyspeaking, when in the latter case a curvature singularity is encountered no con-tinuation is possible. For an incomplete boundary without such an obstructionappropriate gluing of patches provides a geodesic extension. Identifying overlap-ping squares and triangles of each type of building block in this manner the fullCP diagram is constructed. Generically basic patches with 3 or more horizonsproduce 2d webs rather than onedimensional ribbons as global CP diagrams.Here, as a nontrivial consequence of the triangular shape at both ends of thebuilding blocks, with the diagonal oriented in the same direction, the allowedtopologies drastically simplify to a ribbon-like structure.8

B0 already coincides with its maximal extension. The one of B4 is depictedin fig. 1. All other global diagrams with a smaller number of horizons can beobtained from this one by contracting appropriate patches and by adding dashedlines if extremal horizons are present. For instance, the one horizon cases B1a

and B1b can be obtained by eliminating all square patches and adding eitherone or three dashed lines.

There are up to three types of vertex points in these diagrams: verticesbetween the singularities along the border, vertices where lines X = const.

8Such a structure is rather typical for theories with charge and mass. The most prominentexample is the Reissner-Nordstrom black hole.

165

3 GLOBAL PROPERTIES

Figure 1: Maximally extended CP diagram for the four horizon scenario

from 4 adjacent patches meet (“sources” or “sinks” for Killing fields) and ver-tices which are similar to the bifurcation 2-sphere of Schwarzschild spacetime.Their (in)completeness properties follow from (18) for A = 0 (so-called “special

geodesics”): τ =∫ X

dX ′|K(X ′)|−1/2. Thus, the vertices at the boundary areincomplete. All other vertices are incomplete if no extremal horizon is present,because (18) remains finite for A = 0 only at nondegenerate horizons.

Of course, as in the Reissner-Nordstrom case, one can identify periodically(e.g. by gluing together the left hand side with the right hand side in fig. 1).Mobius-strip like identifications are possible as well.

3.3 Kinks

From a global point of view the “kink” solution discussed in ref. [1] consists ofthe two symmetry breaking constant dilaton vacuum solutions in the regions|X | >

√c and the square patch of B2b inbetween.

Such a patching in general induces a matter shock wave at the connectingboundary. For C1 solutions no patching of that kind is possible in the frameworkof PSMs [13] simply because either X+ or X− becomes discontinuous (in oneregion it is non-vanishing, in the others it is identical to zero).

It is illustrative to discuss in more detail what happens if one joins (11) to(12). By adjusting A and B the Killing norm can be made C2. Hence curvaturebecomes continuous. Nevertheless, the discontinuity of X+ in eq. (6) impliesthe existence of matter at the horizon (the version of (6) with matter is givenby eq. (3.8) of ref. [2]) with a localized energy-momentum 1-form

T + :=δL(m)

δe−=

(

δ(x −√

c) − δ(x +√

c))

dx , T− :=δL(m)

δe+= 0 , (19)

where L(m) is the induced matter action. The coordinate x is the same as in(11). It coincides with X for X2 ≤ c.

This problem is not evident if the coordinate system (16) is used because thematter sources are pushed to z = ±∞. But patching at a coordinate singularitylike the one at these points is difficult to interpret. It is therefore not quite clear

166

4 OUTLOOK

in what sense the solution presented in ref. [1] can be considered as kink froma global point of view.

Actually general methods exist which allow the construction of kink solutionstaking the global diagrams as a starting point [10, 11]. As noted above theribbon-like CP diagrams related to B0-B4 allow for periodic identifications. Ifthey are performed in a nontrivial manner as in fig. 9 of ref. [11] this providesone way kink solutions may appear. It could be rewarding to study them at thelevel of 2D dilaton gravity in the first order formulation in order to learn moreabout non-trivial sectors of Chern-Simons theory in 3D.

4 Outlook

The solution (4.52)-(4.54) of ref. [1] has been reproduced in the framework ofthe first order approach to 2d gravity with the following generalizations: It isembedded into a larger patch of the geometry because the coordinate X in (12)is not bounded by

√c as opposed to (4.52). Moreover, a second Casimir function

is present and only for a special tuning between both Casimirs, c2 = 8C(g), thesolution (4.54) is reproduced; otherwise, more general solutions emerge with upto 4 Killing horizons. Their global properties have been discussed. A summaryof these results is contained in the “phase-space” plot fig. 2.

A straightforward generalization of the formulation (2) would be the con-sideration of arbitrary V(X+X−, X, Y ) instead of the special case (3). For allthese models one Casimir function (corresponding to the total charge) becomesY = c, while the other one is in general more complicated and related to thetotal energy. Possible applications of such models are twofold: if Y appears atleast quadratically in V it can be eliminated from the EOM obtained by varyingwith respect to Y (not necessarily uniquely); in this case it represents the dualfield strength (possibly with some coupling to the dilaton X). Such a situationis encountered e.g. for potentials of the type V = V(X+X−, X) + F (X)Y 2 in-cluding the spherically reduced Reissner-Nordstrom black hole. If, however, Yappears only linearly in the form V = V(X+X−, X)+ F (X)Y as in the presentcase then the “dilaton” X (or a function thereof) determines the dual fieldstrength. This implies an interesting “gauge curvature to geometric curvature”coupling in the action which is explicit in the second order formulation (1).

Further generalizations are conceivable, e.g. the coupling to matter fieldsthus making the theory nontopological. In that case the virtual black holephenomenon should be present [14] and interesting results can be derived withinthe path integral formalism [15].

Indeed, powerful methods to study these models classically, semi-classicallyand at the quantum level already do exist [2].

167

4 OUTLOOK

c2 =8C (g)

K(X)

X

K(X)

X

X

K(X)

K(X)

X

X

K(X)

C(g)

c

Figure 2: The phase space of building blocks for general CP diagrams. Thewhite, dark gray and light gray region contains all CP diagrams with four, twoand zero non-extremal Killing horizons, respectively. Bold lines in the phase di-agram correspond to CP diagrams containing one or two extremal horizons (andpossibly additional non-extremal ones). The point at the center corresponds tothe special case c = 0 = C(g) with an extremely extremal horizon (with fourthorder zero in the Killing norm). The solution found in ref. [1] corresponds tothe curved bold line separating the white from the light gray region. In theCP diagrams bold, dashed and ordinary lines correspond to curvature singular-ities, non-extremal Killing horizons and X = const. lines, respectively (only thenon-extremal cases are depicted). The Killing norm as a function of X also hasbeen plotted in the five non-extremal regions (in the extremal limit zeros canbe located at some of the extrema).

168

REFERENCES

Acknowledgement

This work has been supported by project P-14650-TPH of the Austrian ScienceFoundation (FWF). We are grateful to R. Jackiw, T. Strobl and D. Vassilevichfor helpful correspondence. DG renders special thanks to C. Bohmer for supportwith xfig.

References

[1] G. Guralnik, A. Iorio, R. Jackiw, and S. Y. Pi, “Dimensionally reducedgravitational Chern-Simons term and its kink,” hep-th/0305117.

[2] D. Grumiller, W. Kummer, and D. V. Vassilevich, “Dilaton gravity in twodimensions,” Phys. Rept. 369 (2002) 327–429, hep-th/0204253.

[3] N. Ikeda and K. I. Izawa, “General form of dilaton gravity and nonlineargauge theory,” Prog. Theor. Phys. 90 (1993) 237–246, hep-th/9304012;N. Ikeda, “Two-dimensional gravity and nonlinear gauge theory,” Ann.

Phys. 235 (1994) 435–464, arXiv:hep-th/9312059; P. Schaller andT. Strobl, “Poisson structure induced (topological) field theories,” Mod.

Phys. Lett. A9 (1994) 3129–3136, hep-th/9405110; “Poisson sigmamodels: A generalization of 2-d gravity Yang- Mills systems,” in Finite

dimensional integrable systems, pp. 181–190. 1994. hep-th/9411163.Dubna.

[4] W. Kummer and P. Widerin, “Conserved quasilocal quantities andgeneral covariant theories in two-dimensions,” Phys. Rev. D52 (1995)6965–6975, arXiv:gr-qc/9502031.

[5] A. S. Cattaneo and G. Felder, “A path integral approach to theKontsevich quantization formula,” Commun. Math. Phys. 212 (2000)591–611, math.qa/9902090.

[6] B. M. Barbashov, V. V. Nesterenko, and A. M. Chervyakov, “The solitonsin some geometrical field theories,” Theor. Math. Phys. 40 (1979) 15–27and 572–581; J. Phys. A13 (1979) 301–312; E. D’Hoker and R. Jackiw,“Liouville field theory,” Phys. Rev. D26 (1982) 3517; C. Teitelboim,“Gravitation and Hamiltonian structure in two space-time dimensions,”Phys. Lett. B126 (1983) 41; E. D’Hoker, D. Freedman, and R. Jackiw,“SO(2,1) invariant quantization of the Liouville theory,” Phys. Rev. D28

(1983) 2583; E. D’Hoker and R. Jackiw, “Space translation breaking andcompactification in the Liouville theory,” Phys. Rev. Lett. 50 (1983)1719–1722; R. Jackiw, “Another view on massless matter - gravity fieldsin two- dimensions,” hep-th/9501016.

[7] T. Klosch and T. Strobl, “Classical and quantum gravity in(1+1)-dimensions. Part I: A unifying approach,” Class. Quant. Grav. 13

(1996) 965–984, arXiv:gr-qc/9508020.

169

REFERENCES

[8] M. Walker, “Block diagrams and the extension of timelike two-surfaces,”J. Math. Phys. 11 (1970) 2280.

[9] T. Klosch and T. Strobl, “Classical and quantum gravity in 1+1dimensions. Part II: The universal coverings,” Class. Quant. Grav. 13

(1996) 2395–2422, arXiv:gr-qc/9511081.

[10] T. Klosch and T. Strobl, “Classical and quantum gravity in 1+1dimensions. Part III: Solutions of arbitrary topology and kinks in 1+1gravity,” Class. Quant. Grav. 14 (1997) 1689–1723, hep-th/9607226.

[11] T. Klosch and T. Strobl, “A global view of kinks in 1+1 gravity,” Phys.

Rev. D57 (1998) 1034–1044, arXiv:gr-qc/9707053.

[12] M. O. Katanaev, W. Kummer, and H. Liebl, “On the completeness of theblack hole singularity in 2d dilaton theories,” Nucl. Phys. B486 (1997)353–370, gr-qc/9602040.

[13] M. Bojowald and T. Strobl, “Classical solutions for Poisson sigma modelson a Riemann surface,” hep-th/0304252.

[14] D. Grumiller, W. Kummer, and D. V. Vassilevich, “The virtual black holein 2d quantum gravity,” Nucl. Phys. B580 (2000) 438–456,gr-qc/0001038; P. Fischer, D. Grumiller, W. Kummer, and D. V.Vassilevich, “S-matrix for s-wave gravitational scattering,” Phys. Lett.

B521 (2001) 357–363, gr-qc/0105034; Erratum ibid. B532 (2002) 373;D. Grumiller, “Virtual black hole phenomenology from 2d dilatontheories,” Class. Quant. Grav. 19 (2002) 997–1009, gr-qc/0111097;D. Grumiller, W. Kummer, and D. V. Vassilevich, “Virtual black holes ingeneralized dilaton theories and their special role in string gravity,”hep-th/0208052, to be published in EPJC.

[15] W. Kummer, H. Liebl, and D. V. Vassilevich, “Exact path integralquantization of generic 2-d dilaton gravity,” Nucl. Phys. B493 (1997)491–502, gr-qc/9612012; “Integrating geometry in general 2d dilatongravity with matter,” Nucl. Phys. B544 (1999) 403–431,hep-th/9809168; D. Grumiller, Quantum dilaton gravity in two

dimensions with matter. PhD thesis, Technische Universitat Wien, 2001.gr-qc/0105078; D. Grumiller, W. Kummer, and D. V. Vassilevich,“Positive specific heat of the quantum corrected dilaton black hole,”hep-th/0305036; D. Grumiller, “Three functions in dilaton gravity: Thegood, the bad and the muggy,” hep-th/0305073.

170

TUW–03–24ESI 1381

Supersymmetric black holes in 2D dilatonsupergravity: baldness and extremality

L. Bergamin∗, D. Grumiller† and W. Kummer‡

Institut fur Theoretische PhysikTechnische Universitat Wien

Wiedner Hauptstr. 8–10, A-1040 WienAustria

Abstract

We present a systematic discussion of supersymmetric solutions of2D dilaton supergravity. In particular those solutions which retain atleast half of the supersymmetries are ground states with respect to thebosonic Casimir function (essentially the ADM mass). Nevertheless,by tuning the prepotential appropriately, black hole solutions mayemerge with an arbitrary number of Killing horizons. The absence ofdilatino and gravitino hair is proven. Moreover, the impossibility ofsupersymmetric dS ground states and of nonextremal black holes isconfirmed, even in the presence of a dilaton. In these derivations theknowledge of the general analytic solution of 2D dilaton supergravityplays an important role. The latter result is addressed in the moregeneral context of gPSMs which have no supergravity interpretation.

Finally it is demonstrated that the inclusion of non-minimally cou-pled matter, a step which is already nontrivial by itself, does notchange these features in an essential way.

∗e-mail: bergamin@tph.tuwien.ac.at†e-mail: grumil@hep.itp.tuwien.ac.at‡e-mail: wkummer@tph.tuwien.ac.at

1 Introduction

In the mid 1990s, during and after the “second string revolution”, BPS(Bogomolnyi-Prasad-Sommerfield [1]) black holes (BHs) [2,3] have attractedmuch interest because in particular they allow to derive the BH entropyby counting D-brane microstates exploiting string dualities (for reviews cf.e.g. [4]). We define a BPS BH as a supergravity (SUGRA) solution respect-ing half of the supersymmetries and exhibiting at least one Killing horizonin the bosonic line element.

A key reference for the properties of supersymmetric solutions in 2D dila-ton SUGRA is the work of Park and Strominger [5]. As shown by theseauthors for the SUGRA version of the CGHS model [6], as well as for aSUGRA extended generalized 2D dilaton theory, a certain vacuum solutioncan be defined with vanishing fermions. A specific solution, which still re-tained one supersymmetry, was constructed for the CGHS-related model. Inthe generic case the existence of such a solution was proven, but it was notconstructed.

Until quite recently a systematic study of all supersymmetric solutionsin 2D dilaton SUGRA theories has not been possible. Recently two of thepresent authors have shown [7, 8] that the superfield formulation of [5] canbe identified with the one of a certain subclass of graded Poisson-Sigmamodels (gPSMs). General gPSMs are fermionic extensions of bosonic PSMswhich, in the present case, are taken to be 2D dilaton theories of gravity [9].This subclass of gPSMs has been dubbed minimal field SUGRA (MFS) inref. [8]. The important consequence of this equivalence is that the knownanalytic solution in the MFS formulation [8] represents the full solution fordilaton SUGRA [5], including all solutions with nonvanishing fermionic fieldcomponents.

This permits us to attack in a systematic manner the problem of 2DSUGRA solutions which retain at least one supersymmetry, the main goalof our present work. It turns out that, although some information can beobtained from the symmetry relations, the knowledge of the full solution isa necessary input. This is of particular importance regarding the eventualexistence of fermionic hair for BHs1.

In doing this we are also able to incorporate SUGRA invariant matter,where a special previous model [11] is extended to generalized dilaton theoriesand also transcribed into the more convenient MFS formulation. Althoughno general analytic solution is possible when matter is included, we show

1An early attempt to prove the validity of this no-hair conjecture for the CGHS modelcan be found in [10].

172

that there are no nonextremal BH solutions of 2D SUGRA respecting halfof the supersymmetries. This result is not unexpected: “Pure” SUGRA—i.e. not deformed by a (super-)dilaton field—does not permit nonextremalsupersymmetric BH solutions according to a simple but elegant argument2

due to Gibbons [2]. Acutally, in 2D dilaton SUGRA a more direct proofis possible, which avoids the continuation to Euclidian signature: First, weshow that the body of the Casimir function (a quantity in gPSM theoriesrelated to the energy of the system) has to vanish and then we prove thatthese ground state solutions cannot provide simple zeros of the Killing norm.En passant all solutions respecting at least half of the supersymmetries areclassified including the ones of ref. [5].

The paper is organized as follows: After a short review of MFS mod-els (sect. 2) in sect. 3 all ground states with unbroken supersymmetry arediscussed, which turn out to be constant dilaton vacua. Such vacua ap-pear nontrivially in some, but not all dilaton theories. Sect. 4 is devotedto a classification of solutions respecting half of the supersymmetries. Allof them imply a vanishing body for the bosonic Casimir function. In sect.5 it is demonstrated that for such ground states the Killing norm has tobe positively semi-definite and thus only extremal horizons may exist. Thisstatement even can be extended to general gPSM theories. Sect. 6 discussesthe coupling of MFS to matter degrees of freedom and sect. 7 generalizes theresult of sect. 5 to the one including matter.

The notation is explained in app. A. Equations of motion and someaspects of the exact solution are contained in app. B. The subject of app.C are some key formulas [8] relating MFS to the superfield formulation of2D dilaton SUGRA [5], which are needed for the construction of the mattercouplings.

2 Minimal field SUGRA (MFS)

General 2D dilaton SUGRA can be formulated in terms of a gPSM [9, 13].Its action with target space variables XI , gauge fields AI and Poisson tensorP IJ

SgPSM =

∫

M

dXI ∧AI +1

2P IJAJ ∧ AI (2.1)

2A crucial ingredient is the continuation to the Euclidian domain and the observationthat the absence of conical singularities enforces boundary conditions of thermal quantumfield theory, which are not compatible with supersymmetry [12] implying T = 0 and henceextremality.

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is invariant under the symmetry transformations

δXI = P IJεJ , δAI = −dεI −(

∂IPJK

)

εK AJ (2.2)

as a consequence of the graded non-linear Jacobi identity

P IL∂LPJK + g-perm (IJK) = 0 . (2.3)

Not every gPSM may be used to describe 2D SUGRA. A subclass of ap-propriate models (called minimal field SUGRA, MFS) has been identifiedin [7, 8] for N = (1, 1) SUGRA. It contains a dilatino χα and a gravitinoψα, both of which are Majorana spinors (XI = (φ,Xa, χα), AI = (ω, ea, ψα),Y = X++X−−):

P aφ = Xbǫba P αφ= −1

2χβγ∗β

α (2.4)

P ab =

(

V + Y Z − 1

2χ2

(V Z + V ′

2u+

2V 2

u3

)

)

ǫab (2.5)

P αb =Z

4Xa(χγaγ

bγ∗)α

+iV

u(χγb)α (2.6)

P αβ = −2iXcγαβc +

(

u+Z

8χ2

)

γ∗αβ (2.7)

V , Z and the prepotential u are functions of the dilaton field φ and obey therelation

V = −1

8

[

(u2)′ + u2Z]

. (2.8)

This Poisson tensor leads to the action

SMFS =

∫

M

(

φdω+XaDea+χαDψα+ǫ

(

V +Y Z− 1

2χ2

(V Z + V ′

2u+

2V 2

u3

)

)

+Z

4Xa(χγaγ

bebγ∗ψ) +iV

u(χγaeaψ)

+ iXa(ψγaψ) − 1

2

(

u+Z

8χ2

)

(ψγ∗ψ))

. (2.9)

An important class of simplified models is described by the special choiceZ = 0. Following the nomenclature of [8] it is called MFS0 and barredvariables are used. As can be verified easily the action (2.9) and SMFS0 arerelated by a conformal transformation of the fields (Q′(φ) = Z(φ))

φ = φ , Xa = e−12Q(φ)Xa , χα = e−

14Q(φ)χα , (2.10)

ω = ω +Z

2

(

Xbeb +1

2χβψβ

)

, ea = e12Q(φ)ea , ψα = e

14Q(φ)ψα . (2.11)

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The symmetry transformations (2.2) for fermionic ǫ with (2.4)-(2.7) become:

δφ =1

2(χγ∗ε) (2.12)

δXa = −Z4Xb(χγbγ

aγ∗ε) −iV

u(χγaε) (2.13)

δχα = 2iXc(εγc)α −

(

u+Z

8χ2

)

(εγ∗)α (2.14)

δω =Z ′

4Xb(χγbγ

aγ∗ε)ea + i(V

u

)′(χγaε)ea +

(

u′ +Z ′

8χ2

)

(εγ∗ψ) (2.15)

δea =Z

4(χγaγ

bγ∗ε)eb − 2i(εγaψ) (2.16)

δψα = −(Dε)α +Z

4Xa(γaγ

bγ∗ε)αeb +iV

u(γbε)αeb +

Z

4χα(εγ∗ψ) (2.17)

By eliminating Xa and the torsion dependent part of the spin connection anew action (MFDS) in terms of dilaton, dilatino, zweibein and gravitino isobtained3

SMFDS =

∫

d2x e

(

1

2Rφ+ (χσ) + V − 1

4uχ2

(

V Z + V ′ + 4V 2

u2

)

− 1

2Z

(

∂mφ∂mφ+1

2(χγ∗ψ

m)∂mφ+1

2ǫmn∂nφ(χψm)

)

− iV

uǫmn(χγnψm) +

u

2ǫmn(ψnγ∗ψm)

)

. (2.18)

Its supersymmetry transformations read

δφ =1

2(χγ∗ε) , (2.19)

δχα = −2iǫmn(

∂nφ+1

2(χγ∗ψn)

)

(εγm)α −(

u+Z

8χ2

)

(εγ∗)α , (2.20)

δema =

Z

4(χγaγbγ∗ε)emb − 2i(εγaψm) , (2.21)

δψmα = −(Dε)α +iV

u(γmε)α +

Z

4

(

∂nφ(γmγnε)α +1

2(ψmγ

nχ)(γnγ∗ε)α

)

.

(2.22)

The action (2.18) has been shown to be equivalent [8] to the general dilatonsuperfield SUGRA of Park and Strominger [5] (cf. app. C).

3Quantities with a tilde refer to the dependent spin connection ωa = ǫmn∂nema −iǫmn(ψnγaψm), i.e. σα = ∗(Dψ)α and R = 2 ∗ dω. The quantity σα is the fermionicpartner of the curvature scalar R.

175

3 Both supersymmetries unbroken

From the MFS supersymmetry transformations in sect. 2 one can read offdifferent conditions for solutions respecting full supersymmetry. From (2.12)and (2.16) follows4 χ+ = χ− = ψ+ = ψ− = 0. The two terms in (2.14) arelinearly independent and thus X++ = X−− = u = 0 as well. In addition(2.17) implies that the transformation parameters must be covariantly con-stant: (Dε)α = 0. For a solution where both supersymmetries are unbrokenthe Poisson tensor (2.4)-(2.7) vanishes identically.

The equations of motion imply that the dilaton φ has to be a constant.It is restricted to a solution of the equation u = 0. Such constant dilatonvacua (CDV) are, for instance, encountered [14] in the “kink” solution of thedimensionally reduced gravitational Chern-Simons term [15].

We recall in app. B that a key ingredient of the solution is the con-served Casimir function. At this point its additive ambiguity can be fixed:supersymmetry covariance requires that solutions respecting both supersym-metries have vanishing Casimir function. This means that eventual additiveconstants in (B.10),(B.11) are absent.

Positivity of energy would imply u2 ≥ 8Y because the Casimir functionCB is related to the negative ADM mass (cf. section 5 of ref. [16]). If thisinequality is saturated the ground state is obtained. Eq. (B.10) in particularimplies that all CDV solutions with u = 0 have vanishing body of the Casimirfunction.5

As an illustration we consider a two parameter family of models (theso-called “ab-family”) encompassing most of the relevant ones [17]. Amongother solutions, BHs immersed in Minkowski, Rindler or (A)dS space can bedescribed. This family is defined by (2.9) with

Z(φ) = −aφ, u(φ) = cφα , α, a, c ∈ R . (3.1)

Supersymmetry restricts the constant B = c2(b+ 1)/4 in the potential (2.8)

V (φ) = −B2φa+b , α =

a+ b+ 1

2(3.2)

to B > 0 if b > −1 and to B < 0 if b < −1. The curvature scalar of theground state geometry is given by

R =bc2

2(α− a)X2(α−1) . (3.3)

4Conventions and light cone coordinates are summarized in app. A.5CDV solutions with u 6= 0 must obey u′/u = − 1

2Z, which leads to CS = 0 while

CB 6= 0. Clearly, they cannot respect both supersymmetries.

176

The Minkowski Ground State (MGS) condition6 reads α = a, Rindler spacefollows for b = 0 and (A)dS means α = 1. In the latter case supersymmetryrestricts the curvature scalar R = c2(1 − a)2/2 to positive values, which inour notation implies AdS.

For fully supersymmetric solutions of the ab-family the only possible val-ues for the dilaton are φ = 0 or |φ| = ∞ (depending on the value of α),unless c = 0, in which case the prepotential u vanishes identically.

4 One supersymmetry unbroken

4.1 Casimir function CB = 0

The symmetry transformation δφ = 0 of the dilaton from (2.12) implies

χ+ε+ = χ−ε− . (4.1)

The vanishing of (2.14) and (2.16) leads to

uε− = −2√

2X++ε+ , uε+ = −2√

2X−−ε− , (4.2)

εγaψ = −iZ8ǫa

beb(χε) . (4.3)

In (4.2) terms proportional to χ2ε have to vanish as a consequence of (4.1).Eqs. (4.2) require

Y =1

8u2 , (4.4)

which in turn implies that the body of the Casimir function (B.10) vanishes.In this sense, BPS like states are always ground states. Notice that eq. (4.4)remains valid in the case u = 0, implying that at least one component of Xa

vanishes as well and vice versa.

It is worthwhile emphasizing that (4.4) corresponds to a vanishing deter-minant

∆ = det

(

−2√

2X++ −u−u −2

√2X−−

)

(4.5)

of the bosonic part of (2.7) (cf. [9]). ∆ = 0 must hold for any solution thatrespects at least one supersymmetry.

6It means simply that for vanishing bosonic Casimir function the bosonic line elementis diffeomorphic to the one of Minkowski space.

177

4.2 Classification of solutions

Vanishing fermions

We first consider the case of vanishing dilatino χα and gravitino ψα. Thenall three quantities u,Xa have to be nonvanishing or otherwise the solutionwith full supersymmetry of sect. 3 is recovered. With the conditions (4.2)and hence also (4.4) the variations (2.12)-(2.16) vanish, while (2.17) equalzero represents two differential equations for ε+, resp. ε−. By inserting theexplicit solution (sect. 6 of ref. [8]) it can be checked that both are identical.A straightforward calculation, without using any further restrictions on thedifferent variables involved, yields

dε+ +

(

dX++

2X++− (

u′

u+

1

2Z) dφ

)

ε+ = 0 , (4.6)

possessing the general solution (Q is defined in (B.12))

ε+ = e12Q u√

X++ε . (4.7)

Here ε is a spinorial integration constant. ε− is obtained via (4.2). Thus,all solutions without fermion fields, exhibiting one supersymmetry, leave alinear combination of the two supersymmetries unbroken, in agreement withthe discussion in ref. [5].

There exists one special case of (4.6) where an even simpler solution exists:If Z = 0 (MFS0 in the parlance of ref. [8]) and if in addition u is a constant,the differential equation (4.6) simply says that the symmetry parameter iscovariantly constant. This model is generalized teleparallel dilaton gravity.As u is simply a cosmological constant, it drops out in the constraint algebra,and therefore this case has been referred to as rigid supersymmetry in ref. [9].

Nonvanishing fermions and no-hair theorem

Further supersymmetric solutions are possible for nonvanishing fermions, asituation not considered in ref. [5], but relevant for the question of fermionichair7.

Assuming that at least one component of χα is different from zero, e.g.χ+ 6= 0, eq. (4.1) can have the chiral solution ε+ = χ− = 0, while ε− 6= 0provides the remaining supersymmetry, or it can relate ε+ to ε− by meansof χ+ 6= 0 and χ− 6= 0.

7No-hair theorems are a recurrent theme in BH physics (ref. [18] and refs. therein).

178

Solutions of the first type are almost trivial, because they require u =X−− = 0. Since the dilaton must not be constant (or else the CDV casewould be recovered) this implies that u must vanish identically, and not justat a certain value of φ. All fermions must be of one chirality, in particularε+ = χ− = ψ− = 0. The quantities X++, e++, ω, χ

+ must be covariantlyconstant and may contain soul contributions. The dilaton has to fulfill thelinear dilaton vacuum condition dφ = const. Only e−− and ε− are slightlynontrivial and can be deduced from

(De)−− = Zµe−− , (Dε)− =Z

2µε− , µ = (X++e++ +

1

2χ+ψ+) . (4.8)

The bosonic line element is flat and obviously the Casimir function is iden-tically zero.

For the remaining class of solutions with both components of χ and of εdifferent from zero the body of the Casimir function still vanishes, but thesoul can be non-vanishing as from (2.13) and (B.11):

Z = −u′

uCS =

1

32eQu′χ2 (4.9)

This is equivalent to the MGS condition mentioned in section 3 below (3.3).Thus, only MGS models are allowed and since the solution has to be theground state, the bosonic part of the geometry is trivially Minkowski space.

Therefore, a solution with nonvanishing fermions must have a trivial

bosonic background (this feature is true also for the first type). Consequentlythere exist no BPS BHs with fermionic hair.

There remains a technical subtlety about the nonchiral solutions. As bothcomponents of ε are non-zero, (4.2) implies that all interesting cases have u,X++ and X−− different from zero. Actually, the solutions presented in [8] donot cover this case, because the one for C 6= 0 ((6.9)-(6.13) of [8]), dependingon C−1, cannot be used in the present case, as the inverse of a pure soulis ill-defined. The solution for C = 0 ((6.17)-(6.21) of [8]) depends on anarbitrary fermionic gauge potential A = − df , which is the gauge potentialassociated with the additional fermionic Casimir function (eq. (6.15) of [8])appearing in that case. Analyzing the e.o.m.-s (app. B and cf. also [9]) forCB = 0 but CS 6= 0, this fermionic Casimir function does no longer appear,but the solution remains valid if the gauge potential A = − df obeys theconstraint χ2A = 0.

Finally (2.17) again leads to the differential equation

Dε+ −(

Z

16

u2

X++e−− − Z

2X++e++

)

ε+ = 0 . (4.10)

179

Inserting the solution described above all trilinear and higher spinorial termsare found to vanish. Thus, the solution of (4.10) reduces to (4.7).

5 No nonextremal BPS black holes

5.1 MFS models

It has been shown in section 4 that the body of the bosonic Casimir functionhas to vanish for all solutions respecting at least half of the supersymmetries.We will focus first on BPS solutions where either X++ 6= 0 or X−− 6= 0 (orboth).

In the bosonic case ground state solutions C = 0 imply for the line element(cf. e.g. eq. (3.26), (3.27) of [16], ⊗ denotes the symmetrized tensor product)

(ds)2 = 2 dF ⊗ (dr − eQ(φ)w(φ)dF ) , dr = eQ(φ) dφ . (5.1)

Obviously, there exists always a Killing vector ξµ∂µ = ∂F the norm of whichis given by K = −2eQ(φ)w(φ). By choosing the function w(φ) appropriately,nonextremal Killing horizons are possible, e.g. for Q = 0,−2w = 1 − 2m/φa Schwarzschild-like BH emerges.

In SUGRA, however, (B.12) implies a negative (semi-)definite w andhence the Killing norm K can only have zeros of even degree:

K(φ) = −2eQ(φ)w(φ) =

(

1

2u(φ)eQ(φ)

)2

≥ 0 (5.2)

Thus, if a Killing horizon exists it has to be an extremal one, which confirmsthe general proof using thermal field theory arguments [2]. For instance,with u = 2(

√φ−M), Z(φ) = −1/(2φ) the line element reads (ds)2 = 2 dF ⊗

(dr + (1 − M/r)2 dF ), which is the two-dimensional part of an extremalReissner-Nordstrom BH8.

One can trivially generalize this result to all ground state solutions whichare not CDV, i.e. non-supersymmetric solutions of 2D dilaton SUGRA with

8In dilaton (super)gravity the number and types of horizons can be adjusted by se-lecting a certain behavior of the functions w and (to a lesser extent) Q, which enterthe Killing norm (5.2). In many cases of physical relevance extremality is induced bytuning of certain charges/constants of motion, but we emphasize that the explicit pres-ence of additional (gauge) fields by no means is necessary for extremality. For instance,the Reissner-Nordstrom BH can be constructed either from spherically reduced Einstein-Maxwell theory by tuning the two Casimir functions (mass and charge) accordingly, but itis also possible to provide an effective description where one of these constants, the charge,enters as a parameter of the action rather than a constant of motion.

180

vanishing body of the Casimir function and non-constant dilaton, becausethe key inequality (5.2) still holds.

Finally, the simpler CDV case X++ = 0 = X−− shall be addressed. Asshown in the previous two sections CDV implies that both supersymmetriesare either broken or unbroken. The equations of motion imply a vanishingbody of torsion and a constant body of curvature. Thus – as in the bosoniccase (cf. sect. 2.1 in [14]) – only (A)dS, Rindler9 or Minkowski spaces are pos-sible. However, supersymmetry provides again an obstruction: curvature isproportional to (u′)2 and thus again the dS case, together with the possibilityof nonextremal Killing horizons, is ruled out by supersymmetry.10

5.2 Generic gPSM gravity

The question of nonextremal BPS BHs may be addressed in a more generalcontext, namely for gPSM gravity that does not belong to the MFS class.Here we consider generic Poisson tensors with local Lorentz invariance imple-mented as in eq. (2.4). In addition the fermionic extension P αβ must have fullrank almost everywhere in the space of solutions with the notable exceptionof those which still obey eq. (4.1) and where consequently the determinant∆ (analogous to (4.5)) vanishes. These solutions will be called BPS becausethey still respect half of the fermionic symmetries.

In a generic gPSM (2.7) is replaced by P αβ = vαβ + χ−χ+vαβ2 with

vαβ =

(√2X++(u− u) −u

−u√

2X−−(u+ u)

)

, (5.3)

where u, u and u are functions of φ and Y [9]. vαβ2 is determined by the

Jacobi identity (2.3). Also in the bosonic potential

P ab = ǫab(

v(φ, Y ) + χ−χ+v2(φ, Y ))

(5.4)

the body is an independent function, while v2 again follows from the Jacobiidentity. Vanishing determinant of (5.3) implies

Y =u2

2(u2 − u2). (5.5)

9If curvature is non-vanishing the Rindler term can always be absorbed by a linearredefinition of the coordinates r → r′ = αr + β, F → F ′ = F/α, α 6= 0. For vanishingcurvature the Minkowski term can always be absorbed by a similar redefinition.

10In fact, there is one very trivial possibility that remains for a CDV solution withvanishing curvature which allows for the existence of exactly one nonextremal Killinghorizon: the line element (ds)2 = 2 dF ⊗ (dr + br dF ), b 6= 0 contains a nonextremal(Rindler) horizon at r = 0. However, this is neither a BPS state nor a true BH solution.

181

On the other hand, the Killing norm is proportional to Y (cf. eq. (36) in [19])and thus nonextremal horizons are possible if u and u are both non-zero andfield-dependent11 (e.g. u = a + bφ, u + u = a + bφ, u − u = c; a, b, c ∈ R).However, following the arguments of ref. [2] all BPS BHs should be extremalin generalized gPSM gravity theories as well.12

This apparent contradiction is resolved by investigating singularity ob-structions on the Poisson tensor (cf. sect. 3 in ref. [9] and ref. [7]). Solvingthe Jacobi identity (2.3) with u, u, u and v as a given input, all remainingfunctions P aβ, v2 in (5.4) and vαβ

2 are proportional to ∆−1. Only for veryspecial relations among the four free functions the inverse powers of ∆ canbe removed. It turns out that these relations imply extremality of eventualhorizons appearing in BPS solutions. Consequently, “BPS states” of general-ized gPSM gravity theories with nonextremal horizons are singular solutionsof the equations of motion13.

6 Extension with conformal matter

We will prove in the next section that the conclusions of the sect. 5.1 do notchange when conformal matter is coupled to the dilaton SUGRA system. Asthese conclusions rely on the details of the symmetry transformations andthe conserved quantities, in a first step the extension to MFS with matterfields is introduced in this section. To this end the close relation betweenMFS and the models obtained from superspace [8] is used. In superspacenon-minimally coupled conformal matter is described by the Lagrangian

S(m) =1

4

∫

d2xd2θ EP (Φ)DαMDαM . (6.1)

Here P (Φ) is a function of the dilaton superfield Φ (cf. (C.5)) and for theθ-expansion of the matter multiplet M we write14

M = f − iθλ +1

2θ2H . (6.2)

11These states in general are not ground-states in the sense of sect. 3.12The constraints from gPSM symmetries are first class and free of ordering prob-

lems [7, 20, 21]. Therefore, on the constraint surface the unbroken fermionic symmetrystill commutes with the Hamiltonian, which is the central ingredient in the argument byGibbons.

13Similar states with singularities in the gravitino sector at the horizon had been foundin 4D supergravity as well, cf. the discussion in ref. [2].

14Whenever the distinction between superfield components and MFS fields is important,underlined symbols are used for the former ones. However, for simplicity this is omittedin most formulas of this section, as the matter action is invariant under the redefinition(C.18), while all other identifications are trivial.

182

Integrating out superspace one arrives at (cf. app. C)

S(m) =

∫

d2x e

[

P (φ)(1

2(∂mf∂mf + iλγm∂mλ+H2)

+ i(ψnγmγnλ)∂mf +

1

4(ψnγmγnψ

m)λ2)

+1

4P ′(φ)

(

i(λγ∗χ)H − (χγ∗γmλ)∂mf − Fλ2

)

− 1

32P ′′(φ)χ2λ2

]

.

(6.3)

The action (6.3) depends on the auxiliary fields H from the matter multipletand F from the dilaton multiplet. H can be integrated out without detailedknowledge of the geometric part of the action. To integrate out F , however,u(Φ) and Z(Φ) in (C.1) must be specified15.

6.1 Matter extension at Z = 0

A particulary simple situation is realized by choosing Z = 0 (following thenotation of [8], barred variables are used for this special case throughout).Then the action (C.1) is bilinear in A and F and the elimination condition(C.12) is modified according to A = −u′/2 + P ′λ2/4, F = −u/2. The partof the action independent of the matter field retains the form (C.13) withZ = 0, while (6.3) after elimination of all auxiliary fields becomes:

S(m) =

∫

d2x e

[

P(1

2(∂mf∂mf + iλγm∂mλ) + i(ψnγ

mγnλ)∂mf

+1

4(ψnγmγnψ

m)λ2)

+u

8P ′λ2 − 1

4P ′(χγ∗γ

mλ)∂mf

− 1

32

(

P ′′ − 1

2

[

P ′]2

P

)

χ2λ2

]

(6.4)

The symmetry transformations of the matter fields f and λα after eliminationof H read (it should be noted [8] that the symmetry parameters ε and ε aredifferent in general)

δf = i(ελ) , δλα =(

∂mf + i(ψmλ))

(γmε)α − 1

4

P ′

P(λγ∗χ)εα , (6.5)

15As F appears in a term ∝ P ′ the restriction to Z = 0 in the following is not necessarywhen considering minimally coupled matter.

183

while the zweibein, the dilaton and the dilatino still transform according to(C.14), (C.16) and (C.17) resp. The transformation rule for the gravitinochanges, as it depends on the auxiliary field A:

δψm

α= −( ˜Dε)α +

i

4

(

u′ − 1

2P ′λ2

)

(εγm)α (6.6)

When working with the MFS formulation of the geometric part, it isadvantageous to formulate the matter action (6.4) in terms of differentialforms as well:

S(m) =

∫

M

[

P(1

2df ∧ ∗ df +

i

2λγae

a ∧ ∗ dλ+ i ∗ (ea ∧ ∗ df)eb ∧ ∗ψγaγbλ

+1

4∗ (eb ∧ ∗ψ)γaγbea ∧ ∗ψλ2

)

+u

8P ′λ2ǫ

− 1

4P ′(χγ∗γ

aλ)ea ∧ ∗ df − 1

32

(

P ′′ − 1

2

[

P ′]2

P

)

χ2λ2ǫ

]

(6.7)

For the special case Z = 0 discussed so far, the identification (C.18)between the variables of MFS and of the superfield formulation (C.1) in[5] becomes trivial. Thus, replacing ε in (6.5) and (6.6) by ε, the action(2.18) with Z = 0 together with (6.4) is invariant under (2.19)-(2.21), (6.5)and (6.6). The special case of minimal coupling (P (φ) = 1) of this matterextension of a gPSM based dilaton SUGRA model has already been obtainedin ref. [11] using Noether techniques. But the above derivation using theequivalence of this theory to a superspace formulation has definite advantagesas it straightforwardly generalizes to more complicated matter actions.

As a first step we should derive from the result obtained so far the matterextension of MFS0 (MFS0 indicates MFS for Z = 0). This step is necessary asonly the first order formalism in terms of a gPSM allows the straightforwardtreatment of the model at the classical as well as at the quantum level. Alsoin the present context the MFS formulation is much superior. The MFSaction is different from (2.18), which was obtained after elimination of Xa

and (the torsion dependent part of) ω. Now these auxiliary variables mustbe re-introduced together with the matter coupling. Second, we would liketo extend the matter coupling to the most general MFS (2.9). So far, wearrived at a matter extension for the special case Z = 0 only. The generalmatter coupling (Z 6= 0) will be obtained by the use of a certain dilatondependent conformal transformation (cf. [8, 9]). It is argued in the end thatthe same result could also have been derived in a different way.

184

The discussion of a consistent matter extension of MFS0 considerablysimplifies by observing that the action (6.4) or (6.7) does not change whenthe independent variables Xa and ω are re-introduced. This is trivial for ω, as(6.4) does not contain the dependent spin connection ˜ω. The independenceof Xa is obvious as well [8, 9]: The elimination condition of Xa dependson derivatives acting onto the dilaton field and the matter action does notcontain such terms. Thus the matter extension of MFS0 must be of theform16

Stot(X, A, f , λ) = SMFS0(X, A) + S(m)(ea, ψα, φ, f , λ) . (6.8)

Not completely trivial is the derivation of the correct supersymmetry trans-formations. However, it is important to realize that (6.8) already is invariantup to equations of motion17 of Xa and ω: As these e.o.m.-s are linear in Xa

and ω, the elimination of these fields “commutes” with the symmetry trans-formation. Therefore there exists a simple and systematic way to modify theMFS0 symmetry laws (2.12)-(2.17) such that (6.8) together with (6.5) is againinvariant. In an abstract notation the behavior of (6.8) under (2.12)-(2.17)and (6.5) may be written as

δStot(X, A, f , λ) = NaX(X, A, f , λ; ε) · (Xa-e.o.m.)

+ (ω-e.o.m.) ∧Nω(X, A, f , λ; ε) .(6.9)

Of course, the two field-dependent quantities NaX and Nω multiplying the

e.o.m.-s must vanish in the absence of matter fields:

NaX(X, A, f = 0, λ = 0; ε) = 0 , Nω(X, A, f = 0, λ = 0; ε) = 0 . (6.10)

They are used to modify the symmetry transformations of Xa and ω by

δXa = δMFS0Xa −Na

X , δω = δMFS0ω −Nω , (6.11)

where the transformations (2.12)-(2.17) with Z = 0 have been renamed δMFS0 .The explicit calculation of Na

X and Nω is straightforward. As S(m) de-pends on the MFS0 fields φ, ea and ψα only, the variations (2.12), (2.16) and(2.17) within (6.4) lead to potential non-invariance. But (2.12) and (2.16)are equivalent to the supersymmetry transformations of these fields withinthe superfield formulation (cf. (C.6), (C.9)). There remains the supersym-metry transformation of the gravitino. Again, most terms are equivalent tothe superspace formulation (remember that Z = 0!), except for the covariant

16XI = (φ,Xa, χα), AI = (ω, ea, ψα)17We denote the equations of motion according to the field which has been varied. Thus

the Xa-e.o.m. refers to (B.7), while the ω-e.o.m. to (B.3).

185

derivative (Dε)α. Here the dependent spin connection ω has been replacedby the independent one. As the independent part of the spin connection iseliminated by means of the Xa-e.o.m. (B.7) this leads to contributions toNa

X . A further source of non-invariance is the modification of the gravitinotransformation in (6.6). This yields another contribution to Na

X from thecovariant derivative acting on ψ, but also to Nω from the term ∝ Xaψγaψ.The latter contributions are proportional to P ′(φ) and vanish in the case ofminimal coupling. Putting all terms together one finds

δXa = δMFS0Xa +

i

2P (εγmγaγ∗λ)∂mf +

1

4P (εγmγaγ∗ψm)λ2

− i

16P ′(χγaǫ)λ2 ,

(6.12)

δω = δMFS0ω − 1

4P ′(ǫγ∗ψ − ǫα ∗ ψα)λ2 . (6.13)

With these new transformation laws the action (6.8) is finally fully invariantunder supersymmetry, while local Lorentz invariance and diffeomorphisminvariance are manifest.

6.2 Matter extension at Z 6= 0

To extend the matter couplings to the general MFS (Z 6= 0 in (2.9)) we usethe conformal transformation (2.10) and (2.11) of sect. 2. The matter actionis invariant under those transformations of the fields, when the new matterfields are defined as

f = f , λ = e−14Q(φ)λ . (6.14)

After the combined transformations (2.10), (2.11) and (6.14) an action withgeneral MFS as geometrical part coupled to conformal matter is obtained.S(m) in (6.4) by construction is invariant under the conformal transformationand therefore that equation, after dropping all bars, is the correct matterextension of MFS. Of course, the new action

Stot(X,A, f, λ) = SMFS(X,A) + S(m)(ea, ψα, φ, f, λ) (6.15)

is invariant under the old ε-transformations that act on the barred variables

δStot

(

X(X), A(A, X), f(f), λ(λ, X))

= 0 , (6.16)

but we have to be careful with the new transformations δ (depending on ε), asthe transformation parameters themselves change under a conformal trans-formation as well. The importance of this behavior for the understanding

186

of gPSM based SUGRA has been pointed out in [8]. Conformal transforma-tions represent a special case of target space diffeomorphisms in the (g)PSMformulation (cf. sect. 4.1 of [8]). Under such transformations the variablesand symmetry parameters change as

δXI = δXI(X) , (6.17)

δAI = δAI(A,X) + e.o.m.-s , (6.18)

εI =∂XJ

∂XIεJ , (6.19)

where the indices are the ones used in the gPSM formulation (2.1). Eq. (6.19)together with (2.10) and (2.11) for a pure supersymmetry transformationyield (cf. sect. 5.2 in ref. [8], esp. eq. (5.8)):

ε = (εφ, εa, εα) = (0, 0, εα)conformal transformation−−−−−−−−−−−−→ ε = (

Z

4(χε), 0, εα) (6.20)

Thus for the general MFS the symmetries (6.5) are modified by a localLorentz transformation with field-dependent parameter εφ = (1/4)Zχε. Thesymmetry law of f remains unchanged under both, the conformal transfor-mation (6.14) and the additional local Lorentz transformation (6.20), as thisfield is invariant under these symmetries. However, for λ the local Lorentztransformation and the supersymmetry transformation of the conformal fac-tor in (6.14) add up to the new contributions displayed in eq. (6.27) below.

Still the action (6.15) is not invariant under (6.26), (6.27) and (2.12)-(2.17): First, the modified laws of ψ (6.6), Xa (6.12) and ω (6.13) shouldbe rewritten in terms of the MFS variables. But as none of these extensionsgenerates derivatives onto the conformal factors, this boils down to rewritethese transformation rules in terms of variables without bars. Second, thee.o.m.-s appearing on the r.h.s. of (6.18) may necessitate further modifica-tions of the MFS symmetries. As the conformal transformations (2.10) and(2.11) depend on the dilaton field only, under supersymmetry transforma-tions discussed so far the action (6.15) is invariant up to e.o.m.-s of ω. Thesenew non-invariant terms originate from the variation of the gravitino (cf. eq.(4.8) of ref. [8] and comments below this equation). Indeed a straightforwardcalculation shows that

δψα = − dǫα +1

4Z dφ ǫα + . . . , δψα = − dǫα + . . . , (6.21)

where the dots indicate terms which do not contain derivatives. The equationof motion of ω involved here drops out in the geometric part, but obviously

187

not in the matter extension. However, from the above equation together withthe formulation of S(m) in (6.7), supersymmetry can be restored analogouslyto the arguments leading to (6.12). It can be read off from (6.21) that thenew (matter-field dependent) piece to the transformation of ω is obtained byreplacing ψα in (6.7) by 1/4Zǫα:

δω = (6.13)− 1

4ZP

(

i(εγaγbλ)ema ∂mf(∗eb)+

1

2(εγaγbψm)em

a (∗eb)λ2)

. (6.22)

It is useful to summarize what we have obtained in sect. 6: The gPSMbased MFS models of eq. (2.9) can be extended by the coupling of matterfields. The complete action (6.15) is given by the sum of (2.9) and (6.4). Thesupersymmetry transformations (2.12), (2.14), (2.16) for φ, χα and ea are notchanged by the matter coupling. Eq. (2.17) for the gravitino receives newcontributions from the elimination of the auxiliary fields in superspace, whileδXa and δω are changed by re-introducing the auxiliary fields of the gPSMformulation. Thus, the complete list of supersymmetry transformations isgiven by (2.12)-(2.17) plus new contributions from the matter fields,

δ(m)Xa =

i

2P (εγmγaγ∗λ)∂mf

+1

4P (εγmγaγ∗ψm)λ2 − i

16P ′(χγaǫ)λ2 ,

(6.23)

δ(m)ω = −1

4P ′(ǫγ∗ψ − ǫα ∗ ψα)λ2

− 1

4ZP

(

i(εγaγbλ)ema ∂mf(∗eb) +

1

2(εγaγbψm)em

a (∗eb)λ2)

,

(6.24)

δ(m)ψα = iP ′

8λ2(γbε)αeb , (6.25)

together with the transformations of the matter fields

δf = i(ελ) , (6.26)

δλα =(

∂mf + i(ψmλ))

(γmε)α

− 1

8Z

(

(χγ∗ε)λα + (χε)(γ∗λ)α

)

− P ′

4P(λγ∗χ)εα .

(6.27)

Of course, we could have used the relation to the general Park-Stromingermodel SFDS of eq. (C.13) to the MFS (cf. (C.18) and also ref. [8]) insteadof the conformal transformation of MFS0 to derive the matter coupling at

188

Z 6= 0. Thus we may eliminate Xa and ω in (6.15) and by this procedurearrive at the matter action (6.4) coupled to MFDS. Using the techniquesdeveloped in [8] this equivalence follows almost trivially. Considering thesymmetry transformations we note that we find again (cf. (C.19))

∆λα = −1

4Z(χε)(γ∗λ)α (6.28)

in agreement with the result derived in [8].One might wonder whether, on a different route, it is possible to derive

a different matter extension of gPSM based SUGRA, where modifications ofthe transformation laws of XI and AI do not occur. The answer is negative,as long as this extension shall preserve both, local Lorentz invariance andsupersymmetry. Indeed, the commutator of two local supersymmetry trans-formations is a local Lorentz transformation δφ plus a “local translation” δa.Invariance under strict gPSM symmetry transformations would imply thatthe matter action is invariant under δa, which, except for rigid supersymme-try, cannot be fulfilled.

7 Supersymmetric ground states with matter

The matter extension of MFS derived in the previous section allows thediscussion of supersymmetric ground states including matter fields. Thefully supersymmetric states are trivial: The geometric variables obey thesame constraints as derived already in section 3, the matter fields must obeyf = const. and λ = 0.

More involved are the states with one supersymmetry: Eq. (6.26) leadsto

λ+ε+ = −λ−ε− (7.1)

which, in analogy to (4.1), implies λ2ε ≡ 0. Furthermore, as (2.12), (2.14)and (2.16) did not receive new matter-field dependent contributions, therelations (4.1), (4.2) and (4.3) still hold. Of course, this still implies (4.4),but the geometric part of the Casimir function is no longer conserved (seediscussion below).

As a consequence of (7.1) and (4.1)-(4.3) the vanishing of δλα in (6.27)reduces to

δλα = (γmε)α∂mf = 0 . (7.2)

It is straightforward to check that all matter-field dependent modificationsin (6.23)-(6.25) vanish due to (7.1) and (7.2). Thus the matter couplings donot change the classification of the solutions as given in section 4 as well asthe results of section 5.

189

In order to understand the condition (7.2) on the matter field configura-tions it is advantagous to reformulate it as

f++ε+ = 0 , f−−ε− = 0 , (7.3)

where f±± = ∗(e±±∧df) are (anti-)selfdual field configurations of the scalarfield. Thus the chiral solution of (4.1) and (7.1) with χ− = λ− = ε+ = 0admits selfdual scalar fields while the anti-chiral one allows anti-selfdual f .The third solution with ε+ 6= 0 and ε− 6= 0 is compatible with static f , only.

Even in the presence of matter a conserved quantity can be constructed.Its physical relevance is displayed in the close relationship to energy defini-tions well-known from General Relativity, such as ADM-, Bondi- and quasi-local mass (for details we refer to sect. 5 of [16] and references therein).

The conservation law dC = 0 in the presence of matter fields is modifiedby analogy to the pure bosonic case (cf. [22]) according to

e−Q dC +X−−W++ +X++W−− +1

8(u′ +

1

2uZ)(χ−W− − χ+W+) = 0

(7.4)

W±± =δ

δe∓∓S(m) W± =δ

δψ∓S(m) (7.5)

In the presence of matter (−W±±) appear on the r.h.s. of the e.o.m-s (B.4),(−W±) on the r.h.s. of (B.5). Eq. (7.4) results from a suitable linear com-bination of the e.o.m-s (B.3)-(B.5), C now is only part of a total conservedquantity, which also contains a matter contribution e−Q dC(m) from the Wterms.

A straightforward calculation from eq. (6.4) yields

W±± = ±e±±[

P(

f++f−− + i ∗ (e++ψλ)f−− + i ∗ (e−−ψλ)f+++

1

2∗ (ψe−−)α ∗ (ψe++)α(λλ)

)

+u

8P ′λ2 +

1

32

(

P ′′ − 1

2

P ′2

P

)

χ2λ2

]

+ df[

P(

f±± + i ∗ (e±±ψλ))

− i

2√

2P ′χ±λ±

]

+ P[

i(ψλ)f±± − 1

2ψ ∗ (ψe±±)(λλ) ∓ 1√

2λ± dλ±

]

,

(7.6)

190

W∓ = P

[

−iλ±(e++f−− + e−−f++ ± df) − 1

2ψ∓(λλ)

± 1

2

(

e++ ∗ (ψ∓e−−) + e−− ∗ (ψ∓e

++))

(λλ)

]

.

(7.7)

Now also the question may be posed about the meaning of the restriction(4.4) within that generalized conservation law. The body of dC in (7.4)vanishes trivially due to that equation. The restriction to (anti-)selfdual orstatic f as derived from (7.3) ensures that the body of (7.4) vanishes withoutimposing further constraints on the fields. Thus the result of sect. 5 for thematterless case continues to hold if non-minimally coupled conformal matteris included.

8 Conclusions

In our present work we present the complete classification of all BPS BHsin 2D dilaton SUGRA coupled to conformal matter. The use of a first orderformulation as suggested from the graded Poisson-Sigma model approach forthe geometric part of the action plays a crucial role in the calculations. As nomatter extension thereof had been considered in the literature, its derivationis an important result on its own. For future reference we compile the MFSaction non-minimally coupled to conformal matter (with coupling functionP (φ)) at this place:

S =

∫

M

[

φdω+XaDea + χαDψα + ǫ

(

V + Y Z − 1

2χ2

(V Z + V ′

2u+

2V 2

u3

)

)

+Z

4Xa(χγaγ

bebγ∗ψ) +iV

u(χγaeaψ) + iXa(ψγaψ) − 1

2

(

u+Z

8χ2

)

(ψγ∗ψ)

+ P(1

2df ∧ ∗ df +

i

2λγae

a ∧ ∗ dλ+ i ∗ (ea ∧ ∗ df)eb ∧ ∗ψγaγbλ

+1

4∗ (eb ∧ ∗ψ)γaγbea ∧ ∗ψλ2

)

+u

8P ′λ2ǫ

− 1

4P ′(χγ∗γ

aλ)ea ∧ ∗ df − 1

32

(

P ′′ − 1

2

[

P ′]2

P

)

χ2λ2ǫ

]

(8.1)

This action is invariant under the supersymmetry transformations (2.12)-(2.17) supplemented by (6.23)-(6.27) and (6.26),(6.27).

Starting from this action it has been shown that all BPS like states havevanishing body of the Casimir function and thus are ground states. Solutions

191

with vanishing fermions allow a non-trivial bosonic geometry, but all Killinghorizons were found to be extremal. On the other hand, the geometry ofsolutions with non-vanishing fermions must be Minkowski space and conse-quently there exist no supersymmetric BHs with dilatino or gravitino hair.The impossibility of supersymmetric dS ground states has been reproducedfor our class of models and the absolute conservation law—the modificationof the Casimir function in presence of matter fields—has been calculatedexplicitely.

Note added in proofs: While proof reading an e-print appeared [25]which allows a nice application of some of the current paper’s methods. Thatstudy is based upon 2D type 0A string theory and among other issues anupper bound on the number q ≤ 16πe < 12 of electric and magnetic D0branes is derived (eq. (4.7) of [25]). The same bound immediately followsfrom reality of the prepotential

u(φ) ∝√

1 − (q2/(16π))(lnφ/φ)

or, equivalently, from semi-negativity of w(φ) in (B.12). Note that our dilatonφ is related to the dilaton Φ in [25] by φ = exp (−2Φ). In addition, as a simpleconsequence of the conservation of the Casimir function (B.9) we agree onthe result for the ADM mass (eq. (3.9) of [25]).

Acknowledgement

This work has been supported by projects P-14650-TPH and P-16030-N08of the Austrian Science Foundation (FWF). We thank P. van Nieuwenhuizenand Th. Mohaupt for helpful discussions on dS vacua and BPS like blackholes, resp. We are grateful to D. Vassilevich and V. Frolov for asking veryrelevant questions. This work has been completed in the hospitable atmo-sphere of the International Erwin Schrodinger Institute. We would like tothank one of the referees for suggesting important improvements.

A Notations and conventions

These conventions are identical to [9, 23], where additional explanations canbe found.

Indices chosen from the Latin alphabet are commuting (lower case) orgeneric (upper case), Greek indices are anti-commuting. Holonomic coor-dinates are labeled by M , N , O etc., anholonomic ones by A, B, C etc.,

192

whereas I, J , K etc. are general indices of the gPSM. The index φ is used toindicate the dilaton component of the gPSM fields:

Xφ = φ Aφ = ω (A.1)

The summation convention is always NW → SE, e.g. for a fermion χ:χ2 = χαχα. Our conventions are arranged in such a way that almost everybosonic expression is transformed trivially to the graded case when usingthis summation convention and replacing commuting indices by general ones.This is possible together with exterior derivatives acting from the right, only.Thus the graded Leibniz rule is given by

d (AB) = AdB + (−1)B (dA)B . (A.2)

In terms of anholonomic indices the metric and the symplectic 2×2 tensorare defined as

ηab =

(

1 00 −1

)

, ǫab = −ǫab =

(

0 1−1 0

)

, ǫαβ = ǫαβ =

(

0 1−1 0

)

.

(A.3)

The metric in terms of holonomic indices is obtained by gmn = ebne

amηab and

for the determinant the standard expression e = det eam =

√− det gmn is

used. The volume form reads ǫ = 12ǫabeb ∧ ea; by definition ∗ǫ = 1.

The γ-matrices are used in a chiral representation:

γ0α

β=

(

0 11 0

)

γ1α

β=

(

0 1−1 0

)

γ∗αβ = (γ1γ0)α

β=

(

1 00 −1

)

(A.4)

Covariant derivatives of anholonomic indices with respect to the geometricvariables ea = dxmeam and ψα = dxmψαm include the two-dimensional spin-connection one form ωab = ωǫab. When acting on lower indices the explicitexpressions read (1

2γ∗ is the generator of Lorentz transformations in spinor

space):

(De)a = dea + ωǫabeb (Dψ)α = dψα − 1

2ωγ∗α

βψβ (A.5)

Light-cone components are very convenient. As we work with spinors ina chiral representation we can use

χα = (χ+, χ−) , χα =

(

χ+

χ−

)

. (A.6)

193

For Majorana spinors upper and lower chiral components are related by χ+ =χ−, χ− = −χ+, χ2 = χαχα = 2χ−χ+. Vectors in light-cone coordinates aregiven by

v++ =i√2(v0 + v1) , v−− =

−i√2(v0 − v1) . (A.7)

The additional factor i in (A.7) permits a direct identification of the light-cone components with the components of the spin-tensor vαβ = i√

2vcγαβ

c .This implies that η++|−− = 1 and ǫ−−|++ = −ǫ++|−− = 1.

B E.o.m.-s and conserved quantity

The equations of motion for a generic gPSM are

dXI + P IJAJ = 0 , (B.1)

dAI +1

2(∂IP

JK)AKAJ = 0 . (B.2)

Consequently, the ones for the MFS action (2.9) become:

dφ−Xbǫbaea +

1

2(χγ∗ψ) = 0 (B.3)

DXa + ǫabeb

(

V + Y Z − 1

2χ2

(V Z + V ′

2u+

2V 2

u3

)

)

−Z4Xb(χγbγ

aγ∗ψ) − iV

u(χγaψ) = 0 (B.4)

Dχα +Z

4Xa(χγaγ

bγ∗)αeb +

iV

u(χγa)αea

+2iXa(ψγa)α −

(

u+Z

8χ2

)

(ψγ∗)α = 0 (B.5)

dω + ǫ

(

V ′ + Y Z ′ − 1

2χ2

(

(V Z + V ′

2u

)′+

(2V 2

u3

)′)

)

+Z ′

4Xb(χγbγ

aeaγ∗ψ) + i(V

u

)′(χγaeaψ) − 1

2

(

u′ +Z ′

8χ2

)

(ψγ∗ψ) = 0 (B.6)

Dea + ηabǫXbZ +

Z

4(χγaγ

bebγ∗ψ) + i(ψγaψ) = 0 (B.7)

Dψα − ǫχα

(V Z + V ′

2u+

2V 2

u3

)

+Z

4Xa(γaγ

bebγ∗ψ)α+

iV

u(γaeaψ)α − Z

8χα(ψγ∗ψ) = 0 (B.8)

194

We re-emphasize that V, Z and the prepotential u are related by (2.8).The full analytic solution of MFS has been given in sect. 6 of [8]. Each

solution is characterized by a certain value of the Casimir function, a quantityconserved in space and time. It consists of a bosonic part (body) and afermionic one (soul):

C = CB + CS (B.9)

CB = eQ(φ)Y + w(φ) = eQ(φ)

(

Y − 1

8u2(φ)

)

(B.10)

CS =1

16eQχ2(u′ +

1

2uZ) (B.11)

In this equation the (logarithm of the) integrating factor and the conformallyinvariant combination of the bosonic potentials

Q(φ) :=

∫ φ

Z(φ′)dφ′ , w(φ) :=

∫ φ

eQ(φ′)V (φ′)dφ′ = −1

8eQ(φ)u2(φ) ≤ 0

(B.12)

have been introduced. In [8] the solutions for C 6= 0 (eqs. (6.9)-(6.13)) andC = 0 (eqs. (6.17)-(6.21)) have been given which are not reproduced here.

C Dilaton SUGRA in superspace

The action for a general dilaton SUGRA in superspace [5] may be writtenas18

SSFDS =

∫

d2xd2θ E(

ΦS − 1

4Z(Φ)DαΦDαΦ +

1

2u(Φ)

)

, (C.1)

with19 [24]

E = e(

1 + iθγaψa+

1

2θ2(A+ ǫabψ

bγ∗ψa

) , (C.2)

S = A + 2θγ∗σ − iAθγaψa

+1

2θ2

(

ǫmn∂nωm − A(A+ ǫabψbγ∗ψa

) − 2iψaγaγ∗σ)

,(C.3)

Dα = ∂α + i(γaθ)α∂a . (C.4)

18In ref. [5] the first term in the brackets was chosen as EJ(Φ)S. If a global fieldredefinition J(Φ) → Φ is not possible, these models are not equivalent globally to MFS [8].

19Except for the zweibein, components of superfields are denoted by underlined variablesto distinguish them from the fields in the gPSM approach.

195

Quantities with a tilde are defined in analogy to footnote 3. Φ is the dilatonsuperfield with component expansion

Φ = φ+1

2θγ∗χ+

1

2θ2F . (C.5)

The supersymmetry transformations of the component fields of the superde-terminant are given by

δema = −2i(εγaψ

m) , δem

a = 2i(εγmψa) , (C.6)

δψm

α = −(

(Dε)α +i

2A(εγm)α

)

, (C.7)

δA = −2(

(εγ∗σ) − i

2Aem

a(εγaψ

m))

, (C.8)

while the ones of the dilaton superfield read:

δφ = −1

2εγ∗χ (C.9)

δχα

= −2(γ∗ε)αF + i(γ∗γbε)α(ψ

bγ∗χ) − 2i(γ∗γ

mε)α∂mφ (C.10)

δF = i(εγaψa)F − i

2

(

εγmγ∗(Dmχ))

+ (ελm)(

(ψmγ∗χ) − 2∂mφ

)

(C.11)

Integrating out superspace and eliminating the auxiliary fields A and F usingtheir equations of motion

F = −u2, A = −1

2(u′ + uZ) +

1

8Z ′χ2 . (C.12)

one arrives at the action

SSFDS =

∫

d2x e

(

1

2Rφ+ (χσ) − 1

2Z

(

∂mφ∂mφ− i

4χγm∂mχ

− (ψnγmγnγ∗χ)∂mφ

)

− 1

8

(

(u2)′ + u2Z)

+u

2ǫmn(ψ

nγ∗ψm

)

+i

2u′(ζγ∗χ) +

1

8

(

u′′ +1

4uZ ′ +

1

2Z(ψ

nγmγnψ

m))

(χχ)

)

,

(C.13)

196

while the symmetry transformations of the remaining fields take the form

δema = −2i(εγaψ

m) , δem

a = 2i(εγmψa) , (C.14)

δψm

α = −(Dε)α +i

4

(

u′ + uZ − 1

8Z ′(χχ)

)

(εγm)α , (C.15)

δφ = −1

2εγ∗χ , (C.16)

δχα

= u(γ∗ε)α + i(γ∗γbε)α(ψ

bγ∗χ) − 2i(γ∗γ

mε)α∂mφ . (C.17)

In ref. [8] it has been shown that this action is equivalent to the action (2.18)of MFDS if the identifications

ψα

m= ψα

m +i

8Z(φ)ea

mǫab(χγb)α , φ = φ , χ = χ , (C.18)

are made. The supersymmetry transformations are equivalent up to a localLorentz transformation with field dependent parameter:

ε = ε ∆ = δMFDS − δSFDS =Z

2χε δφ (C.19)

References

[1] E. B. Bogomolny, “Stability of classical solutions,” Sov. J. Nucl. Phys.

24 (1976) 449;

M. K. Prasad and C. M. Sommerfield, “An exact classical solution forthe ’tHooft monopole and the Julia-Zee dyon,” Phys. Rev. Lett. 35(1975) 760–762.

[2] G. W. Gibbons, “Supersymmetric soliton states in extendedsupergravity theories.,” in Unified theories of elementary particles,P. Breitenlohner and H. Durr, eds., vol. 160 of Lecture Notes in

Physics, pp. 145–151. Springer, Berlin, 1982.

[3] G. W. Gibbons and C. M. Hull, “A Bogomolny bound for generalrelativity and solitons in N=2 supergravity,” Phys. Lett. B109 (1982)190;

K. P. Tod, “All metrics admitting supercovariantly constant spinors,”Phys. Lett. B121 (1983) 241–244.

[4] D. Youm, “Black holes and solitons in string theory,” Phys. Rept. 316(1999) 1–232, hep-th/9710046;

197

R. D’Auria and P. Fre, “BPS black holes in supergravity: Dualitygroups, p-branes, central charges and the entropy,” hep-th/9812160;

T. Mohaupt, “Black hole entropy, special geometry and strings,”Fortsch. Phys. 49 (2001) 3–161, hep-th/0007195.

[5] Y.-C. Park and A. Strominger, “Supersymmetry and positive energy inclassical and quantum two-dimensional dilaton gravity,” Phys. Rev.

D47 (1993) 1569–1575, arXiv:hep-th/9210017.

[6] C. G. Callan, Jr., S. B. Giddings, J. A. Harvey, and A. Strominger,“Evanescent black holes,” Phys. Rev. D45 (1992) 1005–1009,hep-th/9111056.

[7] L. Bergamin and W. Kummer, “Graded poisson sigma models anddilaton-deformed 2d supergravity algebra,” JHEP 05 (2003) 074,hep-th/0209209.

[8] L. Bergamin and W. Kummer, “The complete solution of 2D superfieldsupergravity from graded Poisson-Sigma models and the superpointparticle,” Phys. Rev. D68 (2003) 104005, hep-th/0306217.

[9] M. Ertl, W. Kummer, and T. Strobl, “General two-dimensionalsupergravity from Poisson superalgebras,” JHEP 01 (2001) 042,arXiv:hep-th/0012219.

[10] J. Gamboa and Y. Georgelin, “The no hair conjecture in 2-d dilatonsupergravity,” Phys. Rev. D48 (1993) 4713–4719, gr-qc/9306022.

[11] J. M. Izquierdo, “Free differential algebras and generic 2d dilatonic(super)gravities,” Phys. Rev. D59 (1999) 084017,arXiv:hep-th/9807007.

[12] L. Girardello, M. T. Grisaru, and P. Salomonson, “Temperature andsupersymmetry,” Nucl. Phys. B178 (1981) 331;

H. Matsumoto, M. Nakahara, Y. Nakano, and H. Umezawa,“Supersymmetry at finite temperature,” Phys. Rev. D29 (1984) 2838;

D. Buchholz and I. Ojima, “Spontaneous collapse of supersymmetry,”Nucl. Phys. B498 (1997) 228–242, hep-th/9701005.

[13] N. Ikeda and K. I. Izawa, “General form of dilaton gravity andnonlinear gauge theory,” Prog. Theor. Phys. 90 (1993) 237–246,hep-th/9304012;

198

N. Ikeda, “Two-dimensional gravity and nonlinear gauge theory,” Ann.

Phys. 235 (1994) 435–464, arXiv:hep-th/9312059; “Gauge theorybased on nonlinear Lie superalgebras and structure of 2-d dilatonsupergravity,” Int. J. Mod. Phys. A9 (1994) 1137–1152;

P. Schaller and T. Strobl, “Poisson structure induced (topological) fieldtheories,” Mod. Phys. Lett. A9 (1994) 3129–3136, hep-th/9405110;“Poisson sigma models: A generalization of 2-d gravity Yang- Millssystems,” in Finite dimensional integrable systems, Eds. A.N. Sissakianand G.S. Pogosyan, pp. 181–190. 1995. hep-th/9411163. Dubna;

T. Strobl, “Target-superspace in 2d dilatonic supergravity,” Phys. Lett.

B460 (1999) 87–93, arXiv:hep-th/9906230.

[14] D. Grumiller and W. Kummer, “The classical solutions of thedimensionally reduced gravitational Chern-Simons theory,” Annals

Phys. 308 (2003) 211–221, hep-th/0306036.

[15] G. Guralnik, A. Iorio, R. Jackiw, and S. Y. Pi, “Dimensionally reducedgravitational Chern-Simons term and its kink,” Annals Phys. 308(2003) 222–236, hep-th/0305117.

[16] D. Grumiller, W. Kummer, and D. V. Vassilevich, “Dilaton gravity intwo dimensions,” Phys. Rept. 369 (2002) 327–429, hep-th/0204253.

[17] M. O. Katanaev, W. Kummer, and H. Liebl, “On the completeness ofthe black hole singularity in 2d dilaton theories,” Nucl. Phys. B486(1997) 353–370, gr-qc/9602040.

[18] V. Frolov and I. Novikov, Black Hole Physics. Kluwer AcademicPublishers, 1998.

[19] T. Klosch and T. Strobl, “Classical and quantum gravity in(1+1)-dimensions. Part I: A unifying approach,” Class. Quant. Grav.

13 (1996) 965–984, arXiv:gr-qc/9508020.

[20] H. Grosse, W. Kummer, P. Presnajder, and D. J. Schwarz, “Novelsymmetry of nonEinsteinian gravity in two- dimensions,” J. Math.

Phys. 33 (1992) 3892–3900, hep-th/9205071.

[21] L. Bergamin, D. Grumiller, and W. Kummer, “Quantization of 2Ddilaton supergravity with matter”, preprint TUW-04-04, LU-ITP04-004.

199

[22] D. Grumiller and W. Kummer, “Absolute conservation law for blackholes,” Phys. Rev. D61 (2000) 064006, gr-qc/9902074.

[23] M. Ertl, Supergravity in two spacetime dimensions. PhD thesis,Technische Universitat Wien, 2001. arXiv:hep-th/0102140.

[24] P. S. Howe, “Super Weyl transformations in two-dimensions,” J. Phys.

A12 (1979) 393–402.

[25] J. Davis, L. A. P. Zayas and D. Vaman, “On Black HoleThermodynamics of 2-D Type 0A,” hep-th/0402152.

200

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VIRTUAL BLACK HOLES AND THE S-MATRIX

D. GRUMILLER∗

Institute for Theoretical Physics

Leipzig University

Augustusplatz 10-11, D-04103 Leipzig, Germany

A brief review on virtual black holes is presented, with special emphasis on phe-nomenologically relevant issues like their influence on scattering or on the specific heatof (real) black holes. Regarding theoretical topics results important for (avoidance of)information loss are summarized.

After recalling Hawking’s Euklidean notion of virtual black holes and a Minkowskiannotion which emerged in studies of 2D models, the importance of virtual black holes forscattering experiments is addressed. Among the key features is that virtual black holestend to regularize divergences of quantum field theory and that a unitary S-matrixmay be constructed. Also the thermodynamical behavior of real evaporating black holesmay be ameliorated by interactions with virtual black holes. Open experimental andtheoretical challenges are mentioned briefly.

1. Introduction

The definition of “Virtual Black Holes” needs two ingredients, namely “Virtual”

and “Black Holes”.

One of the basic lessons of Quantum Field Theory (QFT) is the prediction of

virtual particles. “Virtual” means, roughly speaking, that the particle is sufficiently

off the mass shell. Probably the most spectacular macroscopic physical consequence

of virtual particles is the Casimir effect [1]: in the simplest setup with two infinitely

large parallel conducting planes the latter induce boundary conditions upon the

quantum fields which change the spectrum of virtual particles. Consequently, the

vacuum energy in the configuration with the plates is smaller than in the configu-

ration without, and thus an attractive force between the plates is generated. It can

be measured with great accuracy and experiments coincide well with theoretical

predictions (for a review cf. [2]). Also in scattering experiments virtual particles

can mediate measurable interactions between real ones, although the former do not

enter the asymptotic states by definition in contrast to the latter. The classic ex-

ample is the Lamb shift [3]. Moreover, every unstable particle is “slightly virtual”.

It is not very clear where to draw the line between various degrees of virtuality

(for instance, stable: no decay; metastable: decay width Γ very small, sharp Breit-

Wigner resonance; unstable: large decay width but still small as compared to mass

m; almost virtual: decay width comparable to mass; virtual: far off the mass shell),

∗E-mail: grumiller@itp.uni-leipzig.de

201

February 8, 2010 9:8 WSPC/Guidelines main

202 D. Grumiller

but the rough definition above will be sufficient for the present work. A nice ex-

ample displaying various degrees of virtuality is provided by toponium [4] (particle

properties are taken from [5]). Toponium is built from a top and an anti-top. Mass

and decay width of the top quark are mt ≈ 178.0 ± 4.3 GeV and Γt ≈ 1.5 GeV ,

respectively; thus, mt ≫ Γt and the top quark, if it existed freely, would be consid-

ered as unstable particle (but probably not as metastable one because mt ≫ Γt is

not valid). For toponium bound states the relevant energy scale is the Bohr energy

EB = mtα2s (natural units c = ~ = 1 are used in this work), where αs ≈ 0.11. But

since EB ≈ Γ toponium bound states are “almost virtual”. The decay products are

unstable by themselves (in the preferred channel W bosons and b quarks are pro-

duced together with their anti-particles) and eventually decay into metastable and

stable particles. Radiative corrections to all these processes are governed by vir-

tual particles. While it may be a somewhat semantic issue whether virtual particles

should be considered as “real”, they are definitely “real” as far as their relevance

to Nature is concerned.1

Regarding the second ingredient, there is no reasonable doubt that Black Holes

(BHs) are real objects in Nature (see [7]; for a review on BH binaries cf. e.g. [8]). As

QFT tells us that for each real object a corresponding virtual one should exist there

is no question as to the existence of Virtual Black Holes (VBHs). So in principle

VBHs are of interest for physics. However, it is less obvious that they are of practical

relevance to experiments, especially to those accessible in the near future. After all,

macroscopic BHs are so massive that “virtuality” becomes as irrelevant as it does for

stars like our sun – the impact of virtual stars on physical experiments is negligible.

While for physics of real BHs the most relevant objects are macroscopic ones, for

the physics of VBHs the microscopic ones dominate.

Besides purely experimental issues there is considerable theoretical interest con-

cerning VBHs. After all, “virtual” implies, at least to some extent, the application

of QFT-like methods, while “BH” implies that the theory to be quantized should

be General Relativity or one of its generalizations. It is well known that quantiza-

tion of gravity is a difficult task (for recent reviews cf. e.g. [9]). Thus, conclusions

drawn from the study of VBHs may lead to valuable insight into quantum gravity.

In particular, there is the famous information paradox (for reviews cf. e.g. [10,11]).2

So a natural question to ask is whether VBHs lead to information loss, and if they

do, what are the consequences e.g. for scattering of ordinary particles.

This paper is organized as follows: in Section 2 two notions of VBHs are recalled,

starting with Hawking’s Euklidean version [13] and ending with our Minkowskian

one [14]. For technical reasons the Minkowskian definition is restricted to the context

1In fact, there seems to be so much “reality” involved that sometimes even issues like the partondistribution of virtual photons are discussed [6].2The information paradox recently attracted some attention beyond the physics community dueto the “betrayal” of one of the most prominent and persistent members of the “Information lossparty”, S. Hawking, who appears to have joined the “Unitarity party” [12].

February 8, 2010 9:8 WSPC/Guidelines main

Virtual black holes and the S-matrix 203

of 2D dilaton gravity, which contains (among many other models) the Schwarzschild

BH. It is fair to say that this review naturally is biased towards the second definition.

Section 3 is devoted to VBHs in scattering experiments and implications for the

problem of information loss. Quantum corrections to thermodynamical observables

are the topic of Section 4. After a brief review of thermodynamics in 2D dilaton

gravity the quantum corrected specific heat of the Witten BH/CGHS model is

presented. Some open points are addressed in the concluding Section 5, which also

mentions the possibility of VBHs in Loop Quantum Gravity and String Theory.

Although most of this work has the character of a review there are a few new

comments and results in Sections 3 and 4. In particular, the formula for the specific

heat (31) to the best of my knowledge is new, as well as the related brief discussion

of Hawking-Page like transitions in generic 2D dilaton gravity. Finally, it should be

mentioned that in order to be able to appreciate some of the technical points in

Sections 2.2, 3.2 and 4 a review on 2D dilaton gravity [15] may be recommended.

2. Definitions of VBHs

2.1. Hawking’s Euklidean version

Ever since John Wheeler’s proposal of “space-time foam” [16] physicists have toyed

with the idea of quantum induced topology fluctuations. This has culminated not

only in spin-foam models, which are considered as a serious candidate for quantum

gravity (cf. e.g. [17]), but also in Hawking’s bubble approach of VBHs [13].

In that paper Hawking finally abandoned the wormhole picture of spacetime

foam [18] and went back to an earlier idea, referred to as “quantum bubble pic-

ture” [19]. So instead of regarding the first homology group as relevant, as for

topologies that are multiply connected by wormholes, the second homology group

was considered. The corresponding Betti number just counts the number of 2-

spheres that may not be shrunk to zero (cf. e.g. [20]). Hawking then argues that for

simply connected 4-manifolds for mathematical and physical reasons only S2 × S2

has to be taken into account.3 The question then arises, what has S2 × S2 to do

with a VBH? It is answered by analogy to electrodynamics: in an external electric

field pair creation may occur and one way to describe this process is by gluing in

a sufficiently smooth way the Minkowskian solution of an electron and a positron

accelerating away from each other to the Euklidean solution describing a virtual

electron-positron pair (see Fig. 1).

3Actually, the argument is not very explicit in the original work, but it appears that Hawkinginvokes Wall’s theorem [21] which states that after taking the connected sum with sufficientlymany copies of S2 ×S2 any two simply connected 4-manifolds with isomorphic intersection formsbecome diffeomorphic to each other. Consequently, simply connected 4-manifolds may be builtby gluing (copies of) three basic units which Hawking calls “bubbles”: projective planes (CP 2),Kummer-Kahler-Kodaira surfaces (K3) and VBHs (S2 × S2). Hawking dismisses CP 2 because itdoes not allow spin structure and K3 because it contributes to anomalies and helicity changingprocesses.

February 8, 2010 9:8 WSPC/Guidelines main

204 D. Grumiller

ee+

.Euk

Min.

−

Fig. 1. e+e− pair creation by tunneling through Euklidean space

The final ingredient to Hawking’s construction is the Ernst solution [22] describ-

ing the pair creation of charged BHs in an external electric or magnetic field. By

analytic continuation to the Euklidean domain one finds the topology of S2×S2 mi-

nus a point. Analogy to electromagnetism suggests correspondence to a BH loop in

a spacetime asymptotic to R4. Because S2×S2 minus a point is the topological sum

of S2×S2 and R4 Hawking concludes that the S2×S2 bubbles found by topological

considerations summarized above may be interpreted as VBH loops. Elaborations

on Euklidean VBHs include the study of loss of quantum coherence [23], of higher

spin fields in VBH backgrounds [24], of quantum evolution in spacetime foam [25], of

effects relevant to neutrino-oscillations [26] and of non-standard Kaon-dynamics [27]

(the latter being inspired also by earlier work [28]).

Physical consequences The almost purely topological considerations above led

Hawking to present several surprising physical implications [13]: he argued in favor

of loss of quantum coherence4 (see also [29]) and derived several consequences from

it: the fact that the θ angle of QCD is zero, non-existence of fundamental scalar fields

(and thus the prediction that the Higgs particle does not exist unless it is composed,

for instance, of fermions) and the suggestion that at end of BH evaporation the

Planck size remnant eventually disappears into the sea of VBHs. The last point is

less relevant experimentally. But theoretically it appears to imply that 2D models

of BH evaporation cannot describe the disappearance of BHs in a way that is non-

singular. Another crucial remark is that in Hawking’s picture VBHs may only be

created in pairs. While a similar property holds for e+e− – for a good reason,

namely charge conservation, which applies even to virtual particles – it is slightly

difficult to comprehend why VBHs may not be produced in singles – after all, there

does not seem to be any violation of global charges when a single BH is produced,

be it real or virtual. Because it is an interesting task by itself, VBHs in Minkowski

space will be considered next, where it is found that no such restriction arises.

4Because of this, together with Hawking’s U-turn regarding information loss [12] VBHs may soonshare the fate of wormholes to be abandoned by one of their creators.

February 8, 2010 9:8 WSPC/Guidelines main

Virtual black holes and the S-matrix 205

2.2. VBHs in Minkowski space

Although clearly VBHs by the very definition of “virtual” do not satisfy classical

field Eqs., Hawking’s Euklidean definition of VBHs reminds of instantons (which

are non-singular solutions of the classical field Eqs. with finite Euklidean action,

cf. e.g. [30]), because in both cases topology of an Euklidean configuration plays a

crucial role. It is well-known that instantons may also be described in Minkowski

space (although it is not necessarily a convenient way to describe them [30]), so

naturally the question emerges whether VBHs have a Minkowskian counterpart.

As shown below this turns out to be the case. For simplicity, the discussion will be

restricted to 2D dilaton gravity which contains, among other models, spherically

reduced gravity, i.e., the phenomenologically relevant Schwarzschild BH. Another

reason to restrict to 2D is that, as Hawking and Ross state in ref. [23], “one can

neither calculate the scattering in a general metric, nor integrate over all metrics”

(in D=4). Fortunately, in D=2 one can, as shown in the extensive work starting

from the two basic papers of Kummer and Schwarz [31].

2.2.1. A brief review of classical 2D dilaton gravity

The purpose of this brief collection of well-known results is merely to fix the no-

tation.5 For background information and refs. the extensive review [15] may be

consulted (for earlier reviews cf. refs. [32]). Thus, without further ado definitions

will be listed: ea = eaµdxµ is the dyad 1-form dual to Ea – i.e. ea(Eb) = δa

b . Latin

indices refer to an anholonomic frame, Greek indices to a holonomic one. The 1-

form ω represents the spin-connection ωab = εa

bω with the totally antisymmetric

Levi-Civita symbol εab (ε01 = +1). With the flat metric ηab in light-cone coordi-

nates (η+− = 1 = η−+, η++ = 0 = η−−) it reads ε±± = ±1. The torsion 2-form

is given by T± = (d ± ω) ∧ e±. The curvature 2-form Rab can be represented by

the 2-form R defined by Rab = εa

bR, R = d ∧ ω. The volume 2-form is denoted by

ǫ = e+ ∧ e−. Signs and factors of the Hodge-∗ operation are defined by ∗ǫ = 1. The

quantities ω, ea are called “Cartan variables”. Since the Einstein-Hilbert action∫

M2

R ∝ (1− γ) yields just the Euler number for a surface with genus γ one has to

generalize it appropriately. The simplest idea is to introduce a Lagrange multiplier

for curvature, X , also known as “dilaton field”, and an arbitrary potential thereof,

V (X), in the action∫

M2

(XR + ǫV (X)). Having introduced curvature it is natural

to consider torsion as well. By analogy the first order gravity action [33]

L(1) =

∫

M2

(XaT a + XR + ǫV(XaXa, X)) (1)

can be motivated where Xa are the Lagrange multipliers for torsion. It encompasses

essentially all known dilaton theories in 2D. Actually, for most practical purposes

5Signs of mass M and curvature scalar r have been fixed conveniently such that M > 0 for positivemass configurations and r > 0 for dS. This is the only difference to the notations used in ref. [15].

February 8, 2010 9:8 WSPC/Guidelines main

206 D. Grumiller

the potential takes the form

V(XaXa, X) = V (X) + X+X−U(X) . (2)

The action (1) is equivalent to the frequently used second order action [34, 40]

L(2) =

∫

M2

d2x√−g

[

X−r

2− U(X)

2(∇X)2 + V (X)

]

, (3)

with the same functions U, V as in (2). The curvature scalar r and covariant deriva-

tive ∇ are associated with the Levi-Civita connection related to the metric gµν , the

determinant of which is denoted by g. If ω is torsion-free r ∝ ∗R.

It is useful to introduce the following combinations of U, V :

I(X) := exp

∫ X

U(y) dy , w(X) :=

∫ X

I(y)V (y) dy (4)

The integration constants may be absorbed, respectively, by rescalings and shifts

of the mass M . Under dilaton dependent conformal transformations Xa → Xa/Ω,

ea → eaΩ, ω → ω + Xaea d ln Ω/ dX Eq. (1) is mapped to a new action of the

same type with transformed potentials U , V . Thus, it is not invariant. It turns out

that only the combination w(X) as defined in (4) remains invariant, so conformally

invariant quantities may depend on w only. Note that I is positive apart from

eventual boundaries (typically, I may vanish in the asymptotic region and/or at

singularities). It can be shown that there is always a conserved quantity (dM = 0),

M = −X+X−I(X) − w(X) . (5)

The classical solutions are labelled by this constant of motion. In the absence of

matter there are no propagating physical degrees of freedom.

The line element in Eddington-Finkelstein gauge reads

ds2 = 2 du dX − 2I(X)(M + w(X)) du2 , (6)

with dX := I(X) dX . Evidently there is always a Killing vector k · ∂ = ∂/∂u with

associated Killing norm k · k = −2I(M + w). Since I 6= 0 Killing horizons are

encountered at X = Xh where Xh is a solution of

w(Xh) + M = 0 . (7)

In the simple conformal frame I = 1 the curvature scalar may be expressed as

r = 2w′′ . (8)

Note that the independence of curvature from the mass M and from I(X) is an

artifact of the conformal frame chosen.

For sake of completeness it should be mentioned that in addition to the 1-

parameter family of solutions, labelled by M , isolated solutions may exist, so-called

constant dilaton vacua, which have to obey X = XCDV = const. with w′(XCDV ) =

0. The corresponding geometry has constant curvature, i.e., only Minkowski, Rindler

or (A)dS are possible spacetimes for constant dilaton vacua. Incidentally, for the

generic case (6) the value of the dilaton on an extremal Killing horizon is also

subject to these two constraints.

February 8, 2010 9:8 WSPC/Guidelines main

Virtual black holes and the S-matrix 207

2.2.2. A brief review of quantum dilaton gravity with matter

Adding a matter action for a scalar field φ, coupled to the dilaton via F (X),

L(m) =1

2

∫

M2

F (X)dφ ∧ ∗dφ =

∫

M2

d2x√−gL(m) , (9)

to (1) makes the theory non-topological. Hamiltonian analysis yields primary and

secondary first class constraints. The latter (Gi) form a non-trivial algebra with

respect to the Poisson-bracket [35, 36]:6

G1(x1), G2(x

1′)

= −G2 δ(x1 − x1′) , (10)

G1(x1), G3(x

1′)

= G3 δ(x1 − x1′) , (11)

G2(x1), G3(x

1′)

= − dVdX i

Gi δ(x1 − x1′) +d lnF

dXL(m)G1 δ(x1 − x1′) . (12)

Here, X i denotes (X, X+, X−) and x1 is one of the world-sheet coordinates (the one

that has not been used as “time”). The simpler case F = const. (minimal coupling)

had already been studied before in ref. [38]. A BRST analysis reveals that the BRST

charge is nilpotent at “Yang-Mills level”, i.e., without higher order ghost terms;

thus, retrospectively, one may use instead the simpler Faddeev-Popov prescription.

Another crucial observation is that the geometric part of the constraints is linear in

the Cartan variables. Together with a gauge-fixing fermion that implies Eddington-

Finkelstein gauge

ω0 = 0 , e−0 = 1 , e+0 = 0 , (13)

these features allow an exact path integral quantization, i.e., schematically7

W (sources) =

∫

Deaµ Dωµ DX i D(ghosts)Dφ

× exp

[

i

∫

d2x(

Leff(eaµ, ωµ, X i, ghosts, φ) + sources

)

]

, (14)

of all fields but matter without introducing a fixed background geometry. Thus, the

quantization procedure is non-perturbative and background independent. However,

there are ambiguities coming from integration constants the fixing of which selects

a certain asymptotics of spacetime; two of them are trivial while the third one

6It should be observed that the algebra closes on δ, rather than δ′; nevertheless, by combiningthe constraints linearly in a certain way one may obtain an algebra closing with δ′, namely theVirasoro algebra (times an abelian one corresponding to Lorentz transformations, cf. e.g. [37]).7The term “ghosts” denotes the whole ghost and gauge-fixing sector. It should be noted thatthe path integral (14) involves positive and negative values of the dilaton and both orientationssign e+

1 = ±. Further details on the quantization procedure may be found in appendix E of [36]and in Section 7 of [15].

February 8, 2010 9:8 WSPC/Guidelines main

208 D. Grumiller

essentially determines the ADM mass (whenever this notion makes sense).8 Thus,

background independence holds only in the bulk but fails to hold in the asymptotic

region; we regard this actually as an advantage for describing scattering processes

because there is no “background independent asymptotic observer”.

What one ends up with is a generating functional for Green functions depending

solely on the matter field φ, the corresponding source σ and on the integration

constants mentioned in the previous paragraph [43]:9

W (σ) =

∫

(Dφ) exp (iLeff) , (15)

Leff =

∫

d2x[

F (X)∂0φ∂1φ − gw′(X) + σφ]

. (16)

The constant g is an effective coupling which turns out to be inessential and may be

absorbed by a redefinition of the unit of length. For minimal coupling (F = const.)

w′ is the only source for matter vertices. It is a non-polynomial function, in general.

Moreover, the quantity X, which is the quantum version of the dilaton X , depends

not only on integration constants but also non-locally on matter; to be more precise,

it depends non-locally on (∂0φ)2. Thus, in general the effective action (16) is non-

local and non-polynomial in the matter field.

2.2.3. Emergence of VBHs

A consequence of the quantization procedure sketched above is the possibility to

reconstruct geometry from matter. That is, if one had an exact solution to the

effective Eqs. of motion following from (16), one obtained not only the behavior

of the matter field but simultaneously the geometry on which it propagates by

solving relatively simple constraints. In general (15), (16) cannot be treated exactly

but only perturbatively. Despite of the perturbative treatment no a priori split

of geometry into background and fluctuations is invoked. Rather, to each order in

perturbation theory geometry may be reconstructed self-consistently up to the same

order, including back reactions. In the following it will be outlined briefly how to

obtain contributions from the lowest non-trivial order in matter without going into

technical details [14], i.e., how to obtain the non-local 4-point vertices and the

corresponding VBH geometries.10

8The issue of mass is slightly delicate in gravity. For a clarifying discussion in 2D see [39]. One ofthe key ingredients is the existence of the conserved quantity M in (5) [40] which has a deeperexplanation in the context of first order gravity [41] and PSMs [33]. A recent mass definitionextending the range of applicability of [39] may be found in appendix A of [42].9(Dφ) denotes path integration with proper measure. In the context of VBHs questions regard-ing the measure and source terms for geometry are mostly irrelevant. Therefore, the generatingfunctional for Green functions simplifies considerably as compared to the exact case [15, 38].10 In a perturbative treatment of (15), (16) vertices with an arbitrary number of external ∂0φlegs are created; in addition, there may be a single ∂1φ leg provided F (X) 6= const. Note that thetotal number of external legs always is even. Thus, to lowest non-trivial order in a perturbative

February 8, 2010 9:8 WSPC/Guidelines main

Virtual black holes and the S-matrix 209

Of course, one can apply the straightforward but somewhat tedious standard

methods to derive their Feynman rules [38]. Fortunately, there is an equivalent,

albeit much easier, way to derive the Feynman rules, namely by considering matter

localized in the following way

(∂0φ)(∂1φ) = c1δ(2)(x − y) , (∂0φ)2 = c0δ

(2)(x − y) , (17)

and by solving the classical Eqs. of motion up to lowest order in ci. It can be

shown [15,38] that this mimics the effects of functional differentiation. So in short,

instead of taking the nth functional derivative of the generating functional (15)

with respect to bilinear combinations of the scalar field – the brute force method to

obtain the Feynman rules for the vertices – one may localize matter on n points. For

lowest order one point is sufficient. Consequently, it turns out that the conserved

quantity (5), which in the absence of matter determines the BH mass, no longer is

constant. In fact, it is not even local due to interactions with matter but rather a

function depending on two points xi, yi on the world-sheet [43]:

M → Meff(x, y) = M − 2c0F (y0)(M + w(y0))θ(y0 − x0)δ(y1 − x1) . (18)

Thus, even if M = 0 there may be intermediate states with Meff 6= 0. These

intermediate states have been called VBHs in ref. [14].

Why is it justified to infer the production of VBHs from (18)? First of all,

Meff is an off-shell quantity because clearly the field configuration in Eq. (17) does

not satisfy the Eqs. of motion. Moreover, this quantity does not influence directly

the asymptotics x0 → ∞ because of the θ-function. So the attribute “virtual”

is adequate. In addition, the classical interpretation of M is as mass of a BH,

so M plays a role similar to the charge in electrodynamics (there, when off-shell

particles with a certain charge appear they are referred to as virtual ones). To settle

this issue convincingly one has to reconstruct geometry as outlined above and to

check whether or not it corresponds to something resembling a BH. Using a simple

coordinate transformation dr ∝ I(x0) dx0, du ∝ dx1 the general result is

ds2VBH = 2 dr du + K(r, u; r0, u0) du2 , (19)

with some complicated expression for K(r, u; r0, u0) that may be found explicitly

in ref. [43]. For spherically reduced gravity (19) simplifies to [44]

ds2VBH = 2 dr du +

(

1 − 2M(r, u; r0, u0)

r− a(r, u; r0, u0)r + d(r, u; r0, u0)

)

du2 .

(20)

Remarkably, this looks like the Schwarzschild metric with a Rindler term. Therefore

also the notion of “black hole” is justified. The quantities M (essentially given by

expansion in powers of φ there are two 4-point vertices, one with four ∂0φ legs and one with three∂0φ legs and a ∂1φ leg. Non-locality implies their dependence on two sets of coordinates, x, y.

February 8, 2010 9:8 WSPC/Guidelines main

210 D. Grumiller

(18)), a and d are localized11 on the cut u = u0 with compact support r < r0.

The non-local vertices consist of integrals over both sets of coordinates with an

integrand containing both pairs of external matter legs at different points x and

y, and a non-local kernel producing the VBHs (cf. Fig. 4 below). For instance, the

integrated vertex with no ∂1φ leg reads

V (4)sym =

∫

x

∫

y

(∂0φ(x))2 V (4)a (x, y) (∂0φ(y))2 . (21)

A similar expression holds for the integrated vertex with a ∂1φ leg, with the kernel

denoted by V(4)b . The explicit form of the kernels V

(4)a , V

(4)b for spherically reduced

gravity may be found in ref. [44], while the general case is derived in [43].

i0

i-

i+

ℑ -

ℑ +

y

Fig. 2. CP diagram of a single VBH; the point y corresponds to u = u0, r = r0 in (20)

A Carter-Penrose (CP) diagram corresponding to the coherent sum of all VBHs

can be constructed as follows (cf. figs. 2-3; all symbols have their standard meaning,

i.e., i0, i± and I ± are spatial, time-like and light-like infinity, respectively) [45,46]:

• Take Minkowski spacetime (or whatever corresponds to the geometry im-

plied by the boundary conditions imposed on the auxiliary fields X i) and

draw N different points in its CP diagram; see left diagram of Fig. 3.

• Draw N copies of this CP diagram and add one light like cut to each

(always ending at a different point); remove the other N − 1 points; the

line element is given by (20); see middle diagram of Fig. 3. Note: each of

these CP diagrams is equivalent to the one depicted in Fig. 2 with varying

endpoint y, which is nothing but the CP diagram associated with a single

VBH with line element (20).

11The localization of “mass” and “Rindler acceleration” on a light-like cut is not an artifact ofan accidental gauge choice, but has a physical interpretation in terms of the Ricci-scalar [45].Incidentally, the Ricci-scalar is such that the Einstein-Hilbert action in D=4 vanishes for all VBHconfigurations.

February 8, 2010 9:8 WSPC/Guidelines main

Virtual black holes and the S-matrix 211

• Glue together all CP diagrams at I ± and i0 (which is a common boundary

to all these diagrams); see right diagram of Fig. 3.

• Take the limit N → ∞. Thus, the full CP diagram consists of infinitely

many layers, each of which resembling Fig. 2, the only difference being the

end point y. Asymptotically all layers coincide. This is a pictorial realization

of Everett’s “Many world interpretation”.

... ...

Fig. 3. Constructing the CP diagram of all VBHs

One should not take the effective geometry at face value – this would be like over-

interpreting the role of virtual particles. Nonetheless, the simplicity of this geometry

and the fact that all possible configurations are summed over are nice qualitative

features of this picture. Because all VBH geometries coincide asymptotically the

boundaries of the diagram, i0, i± and I ±, behave in a classical way12, thus enabling

one to construct an ordinary Fock space like in fixed background QFT. Heuristically,

the more one zooms into geometry the less classical it becomes.

The situation is complementary to Kuchar’s proposal of geometrodynamics13 of

BHs: while we have fixed boundary conditions for X i (and hence a fixed ADM mass)

but a “smeared geometry” (in the sense that a continuous spectrum of asymptoti-

cally equivalent VBHs contributes to the S-matrix), Kuchar encountered a “smeared

mass” (obeying a Schrodinger Eq.) but an otherwise fixed geometry [47].

Below it will be shown how the VBHs described above enter the S-matrix to-

gether with some consequences which are observable, at least in principle.

3. VBHs in scattering experiments

3.1. Low scale quantum gravity

In the past years the possibility of BH production at future colliders [48], like LHC

[49], and in cosmic rays has been studied in great detail (for reviews cf. e.g. [50]).

A necessary ingredient to experimental verification is the assumption of low-scale

12Clearly the boundary conditions imposed play a crucial role in this context. They produceeffectively a fixed background, but only at the boundary.13This approach considers only the matterless case and thus a full comparison to our results isnot possible.

February 8, 2010 9:8 WSPC/Guidelines main

212 D. Grumiller

quantum gravity, where “low” refers to about 1 TeV [51]. If one considers this

scenario seriously one should also contemplate the possibility of VBH production.

In fact, even if it turned out that the scale explored at LHC is slightly below the

quantum gravity scale, and thus real BHs may not be produced with a rate sufficient

for detection, in principle effects from VBH production could still be accessible

experimentally. As compared to the excitement caused by real BH production the

number of studies devoted to VBH production is small. Apart from considerations

regarding proton decay14 the only work I am aware of is an unpublished e-print

[55].15 Thus, regarding VBH production at future colliders or in cosmic rays it

seems that there are many issues of potential phenomenological interest awaiting

to be discovered.

3.2. S-matrix for s-wave gravitational scattering

The idea that BHs must be considered in the S-matrix together with elementary

matter fields has been put forward some time ago [56]. The approach [14, 36, 38,

43–46, 57–59] reviewed here, for the first time allowed to derive (rather than to

conjecture) the appearance of VBH states in the quantum scattering matrix of

gravity and to predict consequences for certain physical observables.

Qualitatively it is clear what has to be done in order to obtain the S-matrix:

Take all possible VBHs of Fig. 3 and sum them coherently with proper weight

factors and suitably attached external legs of scalar fields. To be more precise,

one has to take the vertex (21), calculate the kernel V(4)a and perform a mode

decomposition of the scalar field (which is well-defined because the asymptotic

region allows the construction of a standard Fock space), thus introducing creation

and annihilation operators a±k obeying [a−

k , a+k′ ] ∝ δ(k − k′). Then do the same for

the non-symmetric vertex and calculate the amplitude for scattering of two ingoing

modes with momenta q, q′ into two outgoing ones with momenta k, k′:

T (q, q′; k, k′) ∝ 〈0|a−k a−

k′

(

V (4)sym + V

(4)non−sym

)

a+q a+

q′ |0〉 (22)

This had been done quantitatively [44] in a straightforward but rather lengthy

calculation [36,57]. The physical model behind these detailed calculations is Einstein

gravity in D=4, minimally coupled to a massless Klein-Gordon field, truncated to

the s-wave sector. Therefore, spherical reduction may be applied and a 2D model of

the type discussed above emerges. Thus, one is able to study gravitational scattering

of matter s-waves in the framework reviewed above. For that model to lowest non-

14Since BHs may be responsible for the violation of global quantities such as baryon or leptonnumber [52] there was some concern that VBHs might rule out the possibility of TeV rangequantum gravity due to proton decay [53] which was refuted in [54].15In ref. [55] a Hawking temperature is assigned to VBHs and effects from Hawking radiation arecalculated. This is hard to justify for genuine VBHs but might apply to “nearly virtual” BHs.

February 8, 2010 9:8 WSPC/Guidelines main

Virtual black holes and the S-matrix 213

trivial order the tree-graph S-matrix (22) is given by

T (q, q′; k, k′) = − iκδ (k + k′ − q − q′)

2(4π)4|kk′qq′|3/2E3T , (23)

with the total energy E = q + q′, κ = 8πGN ,

T (q, q′; k, k′) :=1

E3

[

Π lnΠ2

E6

+1

Π

∑

p∈k,k′,q,q′

p2 lnp2

E2

(

3kk′qq′ − 1

2

∑

r 6=p

∑

s6=r,p

(

r2s2)

)]

, (24)

and the momentum transfer function

Π(k, q, k′) = (k + k′)(k − q)(k′ − q) . (25)

Here are some remarks regarding the result (23)-(25):

Scale independence The interesting part of the scattering amplitude is encoded

in the scale independent (!) factor T in (24). This issue will be addressed in more

detail below. Note that scale invariance does not apply to the full amplitude (23).

Forward scattering The forward scattering poles occurring for Π = 0 should

be noted. Their appearance may have been anticipated on general grounds from

classical scattering theory.

Simplicity As a brief glance at the details shows [36, 57] there are actually two

contribution to the amplitude which have to be added (cf. footnote 10 and see

Fig. 4). Each of them is not only vastly more complicated than (24) but also diver-

gent. These somewhat miraculous cancellations urgently ask for some explanation.

The one we have found to be convincing is gauge-independence of the S-matrix.

Thus, the complicated expressions for single Feynman-diagrams are an artifact of

our gauge choice (6) which has been a prerequisite for the exact path integration

over geometric degrees of freedom, auxiliary fields and ghosts. It remains a challenge

to find a simpler derivation of (24).

V(4)a (x,y)

x y

∂0 φ

q’

∂0 φ

q

∂0 φ

k’

∂0 φ

k

+

V(4)b (x,y)

x y

∂0 φ

q’

∂0 φ

q

∂1 φ

k’

∂0 φ

k

Fig. 4. The total V (4)-vertex (with outer legs) contains a symmetric contribution V(4)a and

(for non-minimal coupling) a non-symmetric one V(4)b . The shaded blobs depict the intermediate

interactions with VBHs.

February 8, 2010 9:8 WSPC/Guidelines main

214 D. Grumiller

Scattering on self-energy and decay of s-waves Physically the s-waves of the

massless Klein-Gordon field are scattered on their own gravitational self-energy. By

rearrangement of the outer legs also a decay of an ingoing s-wave into three outgoing

ones is possible and the corresponding decay rate may be calculated [36, 44, 57].

CPT invariance By switching from outgoing to ingoing Eddington-Finkelstein

gauge it has been argued in ref. [45] that the amplitude is CPT-invariant. This is a

non-trivial feature because one might expect CPT violation from interactions with

VBHs on general grounds and because the effective action (16) is non-local.

Cross section With the definitions

k = Eα , k′ = E(1 − α) q = Eβ , q′ = E(1 − β) , (26)

where α, β ∈ [0, 1] and E ∈ R+, a quantity to be interpreted as a cross-section for

spherical waves can be defined [44]:

dσ

dα=

1

4(4π)3κ2E2|T (α, β)|2

(1 − |2β − 1|)(1 − α)(1 − β)αβ. (27)

The forward scattering poles are clearly visible in Fig. 5.

00.2

0.40.6

0.81

alpha

0

0.2

0.4

0.6

0.8

1

beta

0

100

200sigma

00.2

0.40.6

0.8alpha

Fig. 5. Kinematic plot of s-wave cross-section dσ/dα for constant E

Pseudo self-similarity Another property discovered and discussed in ref. [45] is

the apparent equivalence of completely different kinematical sectors of the scat-

tering amplitude, i.e., if one zooms into the central region of Fig. 5 one obtains

another plot which is (almost) identical to that figure. Self-similarity is broken only

at next-to-next-to-leading order in an expansion around a reference point close to

a forward scattering pole.

February 8, 2010 9:8 WSPC/Guidelines main

Virtual black holes and the S-matrix 215

VBHs regularize QFT Often it is claimed that gravity may regularize (some

of) the divergences of QFT (for the classical example invoking the self-energy of

a charged point particle cf. e.g. the first chapter in ref. [60]). It is thus of inter-

est that VBHs confirm this hope. To this end consider a generic amplitude in

a scattering process in QFT. Typically, in D dimensions one would expect from

energy-momentum conservation the appearance of two δ-functions in the ampli-

tude, δ(E−E′)δ(p−p′), where p is a D-1 vector, the sum over all ingoing momenta;

similarly, p′ is the sum over all outgoing momenta and E (E′) is the sum over all

ingoing (outgoing) energies. For massless particles in 2D the singular expression

δ(E−E′)δ(0) emerges.16 However, it turns out that for scattering based upon fully

quantized gravity (15) one of the δ-functions is absent, essentially because of the

non-local nature of the interactions mediated by VBHs. Thus, the amplitude (23)

contains only one δ-function and consequently it is finite.

Unitarity There is a somewhat formal argument why any S-matrix derived from

(15) is unitary (cf. also Section 8.6 of [15]): as the quantization procedure has

been in full accord with the general principles of quantization for systems with

constraints, according to general theorems [62] the resulting quantum theory must

be unitary, respect causality and energy conservation, and must forbid transitions

of pure states into mixed ones, as long as one is able to refer to a Fock space of the

asymptotic states. In the example presented above this was indeed the case. This

formal argument is backed up by physical considerations: the evaporation of real

BHs may be considered as a consequence of a boundary condition imposed on the

energy-momentum tensor at the horizon, thus defining the (Unruh) vacuum state

(cf. e.g. [11]). Such a condition clearly is not applicable to VBHs – global (Kruskal)

coordinates for a VBH cannot be associated with any real observer. Therefore, the

relevant vacuum state is just the usual Minkowski space vacuum which contains no

information about BHs, neither real nor virtual ones. Incidentally, the only vacuum

state which may be defined by a condition at infinity rather than on the horizon is

the Boulware vacuum which does not contain Hawking radiation, so that VBHs do

not radiate anything to infinity.

Symmetry properties of the amplitude It has been noted already in [36, 44]

that, somewhat surprisingly, the interesting part of the amplitude, T , is scale inde-

pendent, i.e., T (q, q′; k, k′) = T (λq, λq′; λk, λk′) for any non-vanishing λ ∈ R. Here I

would like to elaborate on that observation and to discuss also other symmetry prop-

erties of the amplitude. To this end it is helpful to introduce the dimensionless quan-

tities α, β together with the scale E defined in (26). In terms of these, together with

16Similar problems arise for the mass zero propagator in 2D [61]. Note that the intermediatedivergencies mentioned in the paragraph “Simplicity” above resemble this type of divergency asthey occur for any values of ingoing (outgoing) momenta, but they are “milder” than δ(0), namelyln 0 [36,57]. So already for individual Feynman diagrams gravity attenuates the divergences, whilefor the gauge independent amplitude it eliminates them completely.

February 8, 2010 9:8 WSPC/Guidelines main

216 D. Grumiller

the abbreviations A = α(1 − α), B = β(1 − β), one gets Π(α, β, E) = E3 (A − B).

Plugging this into (24) it is simple to show its independence from the scale E by

collecting all terms containing E: 3 lnE2[A − B + (2AB(1 − A − B) − 12 (2A(1 −

2B) + 2B(1 − 2A)))/(A − B)] = 0. Thus, it is established that the interesting part

of the scattering amplitude, T as defined in (24), does not depend on E at all but

only on the kinematic factors α and β. The discussion in ref. [63] suggests to take

this scale independence seriously and to look for further symmetries.17 Boosts of all

modes E → γE, p → γp (where p = k, q, k′, q′,) just amount to an energy rescaling

because particles are massless so the Lorentz angle does not change; therefore, the

analysis above applies and boosts are a symmetry of the amplitude. There seem to

be no further continuous symmetries. However, there are discrete symmetries: one

may permute within the set of ingoing and/or outgoing particles. Consistently, (23)

is invariant under α ↔ (1 − α) and β ↔ (1 − β). This residual invariance may be

fixed trivially by restricting α ∈ [0, 1/2], β ∈ [0, 1/2]. Exchanging in- with outgoing,

i.e., α ↔ β swaps the sign of T and therefore the full amplitude (23) gets complex

conjugated, as expected for a theory exhibiting CPT invariance.

4. On the specific heat of BHs

From a thermodynamical point of view the specific heat of the father of all BHs,

the Schwarzschild BH, is quite remarkable, namely negative. This leads one to ask

the question how quantum corrections due to VBHs influence the specific heat. To

this end it is recalled briefly how to obtain the specific heat from entropy for generic

dilaton gravity (for a brief review on the latter see above). The final Subsection is

devoted to VBH corrections of the (inverse) specific heat for the CGHS model.

4.1. BH thermodynamics in 2D dilaton gravity

Before being able to appreciate the relevance of quantum corrections to the specific

heat it is worthwhile to collect a few classical results first.

Hawking temperature There are many ways to calculate the Hawking temper-

ature, some of them involving the coupling to matter fields, some of them being

purely geometrical. Because of its simplicity we will restrict ourselves to a cal-

culation of the geometric Hawking temperature as derived from surface gravity

(cf. e.g. [64]).18 The latter can be calculated by taking the normal derivative d/ dX

17I am grateful to N. Pinamonti and D. Vassilevich for discussions on this subject after a talk ofthe former at the MPI Leipzig. It should be pointed out that the SL(2, R) symmetry regardingthe energy spectrum found in ref. [63] relies on scale reparameterization of the total scatteringamplitude, which does not apply to (23).18If defined in this way Hawking temperature turns out to be independent of the conformal frame.Although for the main application below only asymptotically flat spacetimes are encountered itshould be noted that identifying Hawking temperature with surface gravity is somewhat naive forspacetimes which are not asymptotically flat. But the difference is just a redshift factor and for

February 8, 2010 9:8 WSPC/Guidelines main

Virtual black holes and the S-matrix 217

of the Killing norm K(X ; M) evaluated on one of the Killing horizons X = Xh,

where Xh is a solution of K(Xh; M) = 0 = (M + w(Xh)), thus yielding

TH =1

2π

∣

∣

∣w′(X)

∣

∣

∣

X=Xh

. (28)

The numerical prefactor in (28) can be changed e.g. by a redefinition of the Boltz-

mann constant. It has been chosen in accordance with refs. [15, 66].

Entropy In 2D dilaton gravity there are various ways to calculate the Beken-

stein-Hawking entropy [67]. Using two different methods Gegenberg, Kunstatter

and Louis-Martinez were able to calculate the entropy for rather generic 2D dila-

ton gravity [68]. Later, Cadoni and Mignemi confirmed their result for a particular

model using the Cardy formula and counting microstates19 [74]. Finally, Carlip red-

erived the general result by applying CFT methods [75] and later also by virtue of

the Cardy formula [76]. These considerations are based upon earlier observations

regarding near horizon conformal symmetry for the Schwarzschild BH [77]. Surpris-

ingly enough, the naive derivation which employs only the thermodynamic relation

dS = dM/T yields the correct result: Entropy equals the dilaton field evaluated at

the Killing horizon,20

S = 2πXh . (29)

Specific heat By virtue of Cs = T dS/ dT the inverse specific heat reads

C−1s =

1

2π

d

dXlnw′(X)

∣

∣

∣

∣

X=Xh

. (30)

It is also independent of the conformal frame, but in the simple frame I = const.

an intriguing reformulation exists:

Cs =8π2s

−rhTH , (31)

quantities like entropy or specific heat actually (28) is the relevant quantity as it coincides withthe period of Euklidean time (cf. e.g. [65]).19As opposed to String Theory, where the microstates are D-branes (for reviews cf. [69]), or toLoop Quantum Gravity, where the microstates are quanta of area (for a review cf. e.g. [70]), it isfair to say that in the context of 2D dilaton gravity it is not quite clear what these microstatesactually are – cf. e.g. the recent discussion in ref. [71]. If one employs Hod’s conjecture [72] one is led

to consider quasi-normal modes in the limit of high damping. This has been performed recently byKettner, Kunstatter and Medved [73] (cf. Eq. (36) in that ref.). Their result is remarkable insofaras it is rather insensitive to geometry and depends solely on the scale set by surface gravity andon the way matter is coupled to the dilaton field. Thus, taking Hod’s conjecture seriously, itappears that one cannot avoid to conclude that the microstates in 2D dilaton gravity are builtfrom matter degrees of freedom. This is in accordance with the theory being topological in theabsence of matter, but it does not explain why the derivations of entropy which do not employmatter at all work so well. So alternatively, one might conclude that Hod’s conjecture is notapplicable (to 2D dilaton gravity). I thank G. Kunstatter for correspondence on [73].20Up to a multiplicative constant which may be absorbed by a redefinition of Newton’s constant.

February 8, 2010 9:8 WSPC/Guidelines main

218 D. Grumiller

where rh is given by (8) evaluated on the horizon. The sign s = ±1 is positive if w′

is negative on the horizon, otherwise it is positive.21 In all examples below s = +1.

Thus, the sign of rh defines whether the specific heat is positive (e.g. for AdS) or

negative (e.g. for dS). On a curious sidenote it is mentioned that (31) behaves like

an electron gas at low temperature with Sommerfeld constant γ = 8π2s/(−rh).

However, this analogy does not go too far because the (Planck version of the) third

law of thermodynamics is not fulfilled necessarily, i.e., entropy need not vanish as

TH → 0 – for instance, the only model of the whole ab-family in Fig. 6 which obeys

the third law, and consequently S = Cs, is the Jackiw-Teitelboim model.

Model w(X) Xh = S/(2π) TH Cs

Schwarzschild BH [78] −λ√

X M2/λ2 λ2/(4πM) −λ2/(4πT 2H )

Witten BH/CGHS [79] −λX M/λ λ/(2π) ∞Jackiw-Teitelboim [80] −λX2

p

M/λ√

λM/π 2π2TH/λ

ab-family [81] −λXb+1 (M/λ)1/(b+1) α(M/λ)b/(b+1) 2πb−1(TH/α)1/b

Schwarzschild-AdS [82] −λ√

X(1 + X/ℓ2) soluble alg. λ4π

p

1/Xh(1 + 3Xh/ℓ2) −4πXh1+3Xh/ℓ2

1−3Xh/ℓ2

Reduced CS [83] −λ(X2 − c)2q

c +p

M/λ 2λπ

p

M/λq

c +p

M/λ π2

λTH/(3X2

h − c)

Fig. 6. Table of examples. Note the (irrelevant) scale factor λ > 0 and the abbreviation α =λ(b+1)/(2π). For simplicity X is assumed to be positive. In the penultimate example Xh is givenuniquely by the real root of a cubic Eq. In the last example all expressions refer to the outermosthorizon. In the first five examples horizons exist iff M > 0, in the last one iff M ≥ 0.

Free energy et al. Once entropy S is known as a function of temperature T and

energy M due to the absence of pressure it is straightforward to calculate other

thermodynamical quantities of interest. For instance, the free energy is given by

F = M − TS; the Euklidean action follows from I = F/T ; the partition function

is given by Z = e−I . If more than one horizon is present one can assign an entropy

to each of them, but of course the thermodynamical discussion of the whole system

becomes more complicated.

Hawking-Page like phase transition In their by now classic paper on ther-

modynamics of BHs in AdS, Hawking and Page found a critical temperature sig-

nalling a phase transition between a BH phase and a pure AdS phase [82]. This

has engendered much further research, mostly in the framework of the AdS/CFT

correspondence (for a review cf. [84]). This transition is displayed most clearly by

a change of the specific heat from positive to negative sign: for Schwarzschild-AdS

according to Fig. 6 the critical value of Xh is given by Xch = ℓ2/3. For Xh > Xc

h the

21If w′ vanishes on the horizon then TH = 0 and s is irrelevant, unless simultaneously w′′ = 0.In that special case – which arises if and only if the Killing norm has at least a triple zero – it isbetter to use (30).

February 8, 2010 9:8 WSPC/Guidelines main

Virtual black holes and the S-matrix 219

specific heat is positive, for Xh < Xch it is negative.22 By analogy, a similar phase

transition may be expected for other models with corresponding behavior of Cs.

For instance, the last example in Fig. 6 exhibits also a critical value of Xh, namely

Xch = ±

√

c/3. Notably, this value can never be reached for outer horizons, but may

be reached for inner ones.

By virtue of the reformulation (31) a candidate for a Hawking-Page like tran-

sition arises at a certain critical value of M ∈ (0,∞) such that rh = 0 at the

(non-extremal) Killing horizon.

Summary The function w(x) + M encodes all thermodynamical properties dis-

cussed above: its zeros yield the value of the dilaton at horizons and thus entropy, its

first derivative is proportional to the Hawking temperature, and its second deriva-

tive (together with the first) determines the specific heat. If w′′ vanishes on the

horizon for a finite value of M a Hawking-Page like phase transition may occur.

4.2. Quantum corrections to the specific heat

In general, quantum corrections are, well, corrections to some classical result in the

sense that the dominant contribution is classical. However, there exist instances

where quantum corrections become pivotal and compete with (or even beat the)

classical contribution.

The table in Fig. 6 reveals that for the CGHS model the inverse specific heat

vanishes classically. Thus, any non-vanishing contribution to C−1s must be purely of

quantum origin. Therefore, it is of some interest to study these corrections in more

detail. This has been undertaken23 in ref. [87] by coupling geometry to a single mass-

less scalar field and peforming path integration over geometry non-perturbatively

as outlined in Section 2.2. It has been found that the interaction with matter in-

duced VBHs24 effectively amounts to a shift of the Killing norm from its classical

22Actually, in the original work [82] Hawking and Page did not invoke the specific heat directly.The consideration of the specific heat as an indicator for a phase transition is in accordance withthe discussion in ref. [85].23Several years before our derivation Zaslavskii has performed a comparable calculation [86].Although the specific heat is not calculated explicitly in that work it is a trivial excercise toextract it from the quantum corrected expressions for mass and temperature. Comparing his

result to ours agreement is found up to an overall sign. Extensive discussions and cross-checkingsof pesky signs have not revealed any obvious sign error in either of the publications. It should bementioned, however, that there is actually a difference between our calculation and Zaslavskii’sconcerning the boundary conditions imposed: while he used Hartle-Hawking boundary conditionswe have employed Unruh boundary conditions. It is not clear whether this difference is responsiblefor the relative sign. In any case, the important conclusion remains unchallenged by such details:interactions with VBHs produce crucial corrections to the inverse specific heat of the CGHS BH.24Classically, for the CGHS model VBHs have no observable effect [43]. Note that the VBHinterpretation need not be adopted – indeed, neither [86] nor [87] mention this notion explicitly –but it is in the spirit of the present work.

February 8, 2010 9:8 WSPC/Guidelines main

220 D. Grumiller

value Kc = 1 − (M/λ)e−2λr (with X = exp (2λr)) to

Kq = 1 − M

λe−2λr +

M

48πλe−4λr . (32)

This implies an effective shift of w from its classical value wc = −λX to

w(X) = wc

(

1 +M

48πλX2

)

, (33)

leaving I = Ic = 1/(2λX) uncorrected. The results above are valid provided M ≫λ. Here are some consequences of (and remarks to) the result (32):

Positive specific heat The most dramatic implication of quantum corrections

arises for the inverse specific heat: while it vanishes according to the standard

analysis (see Fig. 6), it turns out to be positive when interactions with VBHs are

taken into account,

Cs =96π2

λ2M2 . (34)

This implies that quantum effects tend to stabilize the system (see, however, foot-

note 23).

Violation of area law The behavior of the Killing norm (32) implies that the

horizon is shifted to slightly smaller values of r due to quantum corrections from

VBHs. Therefore, Hawking’s area theorem [88] is violated.

Corrections to radiation loss Applying the 2D Stefan-Boltzmann law yields to

leading order a decrease of the BH mass linear in “time”, proportional to T 2H . This

is modified according to

M(t) ≈ M0 −π

6(T 0

H)2(t − t0) +λ

24πln

M(t)

M0, (35)

where t > t0, M0 = M(0) and T 0H = λ/(2π) (in accordance with Fig. 6). Terms of

higher order in λ/M have been neglected. The last term in (35) is the one which is

due to corrections from VBHs. Note that M(t) may be expressed in terms of the

Lambert W-function [89].

Conformal non-invariance The quantum corrections crucially depend on the

conformal frame – for instance, if one uses the simple conformal frame I = const.

then no quantum corrections to the Killing norm arise. The feature of classical con-

formal invariance of certain quantities but quantum non-invariance is in accordance

with the general discussion in [90]. Fortunately, for the CGHS model there exists

a preferred way to choose a conformal frame, a so-called Minkowski ground state

February 8, 2010 9:8 WSPC/Guidelines main

Virtual black holes and the S-matrix 221

frame;25 it is this frame that has been used in the derivation of (32).26

Logarithmic entropy corrections By a simple thermodynamical calculation

based upon (32) corrections to the entropy have been calculated in ref. [59]:

S = S0 −1

24lnS0 + O(1) , (36)

where S0 = 2πM/λ (cf. Fig. 6). The logarithmic behavior is in qualitative agreement

with the one found in the literature by various methods [91] (cf. [92] for a brief and

recent summary); the factor 1/24 is in accordance with [93].

5. Outlook

Several interesting physical consequences from interactions with VBHs can be de-

duced, both in the Euklidean and in the Minkowskian approach, as reviewed in

Sections 2-4. I conclude briefly with a couple of open issues.

Experimental challenges Having established that real BHs are part of Nature

it seems natural to consider experiments sensitive to virtual ones. Although the

hope has been expressed in Section 3.1 that such experiments may be feasible if

the quantum gravity scale is at low energies, a thorough study of VBHs in that

framework still is lacking. By the same token that real BHs may lead to observable

effects at near future colliders or in high energy cosmic rays one could argue that

VBHs will imply observable consequences. This might play a pivotal role if the

quantum gravity scale turns out to be low, but not low enough to yield convincing

evidence for real BHs.

Regarding a verification of the scattering amplitude (23) prospects do not look

too promising: one would need a system where all forces but gravity can be ne-

glected, which is spherically symmetric and which consists solely of massless scalar

particles. Only the last condition may be dropped with ease in the framework pre-

sented here, while the first two are crucial ingredients. The only system which comes

to my mind exhibiting similar features consists of rapidly expanding or collapsing

spherical shells (not necessarily thin ones), like the s-wave part of a supernova.

So in conclusion, it seems difficult to invent an experimental setup which manages

to unravel the interesting kinematical features hidden in (24), besides the forward

scattering poles, but undoubtedly it is a very interesting challenge.

25The definition is as follows: for vanishing value of the mass M geometry must be Minkowskispace. This leads to I ∝ X−a with a = b + 1 for the ab-family in Fig. 6. Hence, for the CGHSmodel a = 1.26Note that (32) would allow to consider the following correction as natural I(X) = Ic(X)(1 −1/(48πX)). However, this “quantum correction of the conformal frame” is problematic because itinduces a singularity in the conformal factor at 48πX = 1.

February 8, 2010 9:8 WSPC/Guidelines main

222 D. Grumiller

Theoretical challenges One of the reasons why VBHs are so interesting from a

theoretical point of view is the puzzle of information loss which arises for real BHs.

While Hawking argued some time ago in favor of VBH-induced information loss,

studies in 2D revealed that no such information loss occurs. Of course, this does

not solve the information paradox for macroscopic BHs; but it shows that micro-

scopic (virtual) BHs enter the S-matrix just like any other particle, a conjecture

put forward some time ago by ’tHooft [56]. Nevertheless, there is a link between

these microscopic studies and macroscopic considerations like in refs. [94]: in both

cases non-locality plays a crucial role. While for (microscopic) VBHs non-locality

led to a finite result for the S-matrix (23), for macroscopic evaporating BHs the

violation of the “locality bound” [94] contradicts the assumption of independent

Hilbert spaces for the interior and the exterior of a BH and thus information need

not be lost.

Several generalizations of the results are obvious: for instance, it would be in-

teresting to study quantum corrections to the specific heat generically, in partic-

ular for the Schwarzschild BH. Extensions to SUGRA exhibit also the VBH phe-

nomenon [95], but so far no amplitudes or corrections to specific heat have been

calculated. Certainly it would be gratifying to extend the 2D study of Minkowskian

VBHs and their effects on the S-matrix to more complicated systems in D=4, i.e.,

to drop the assumption of spherical symmetry.

Finally, it is fair to ask whether VBHs exist beyond the Euklidean and the

Minkowskian path integral approach addressed in this work. In the context of Loop

Quantum Gravity (LQG) there has been recent progress in describing quantum

horizons for spherically symmetric configurations [96]. Also, it has been found in

that ref. that a binary degree of freedom exists, essentially the orientation of the

spherically symmetric isolated horizon. This is a promising step towards VBHs

as described in Section 2 because also in the path integral (14) we have summed

over positive orientations (e+1 > 0) and negative ones (e+

1 < 0). However, the

boundary conditions imposed in the asymptotic region uniquely select one of these

orientations for the effective line element (20). Thus, although in intermediate states

both orientations are possible, for the fiducial observer in the asymptotic region one

orientation is selected. It would be interesting to observe similar features in LQG.

Thus, as a next step one could consider an isolated quantum horizon together with

an asymptotic region which should be chosen to be essentially the same for all

spin network configurations, e.g. a flat one or dS space. Then, one would have an

asymptotic region behaving almost classically, while the bulk part of geometry, in

particular the horizon, still behaved in a quantum way. In this manner, if it turned

out to be possible to relax the condition on spherical symmetry, even graviton-

graviton scattering with intermediate VBHs might be described by LQG.

For String Theory the answer is affirmative: as the Witten BH/CGHS model

follows from strings in D=2 and VBHs arise for that model they may be expected

to be a generic feature of String Theory. A concrete realization of VBHs in a more

general framework of String Theory, i.e., not restricted to the Witten BH, could be

February 8, 2010 9:8 WSPC/Guidelines main

Virtual black holes and the S-matrix 223

an interesting subject for the future.

Acknowledgements

This work has been supported by an Erwin-Schrodinger fellowship granted by the

Austrian Science Foundation (FWF), project J-2330-N08. I am grateful to my long-

time collaborators on 2D gravity, W. Kummer and D. Vassilevich, for numerous

stimulating discussions and to P. Fischer for joining our efforts and for providing

valuable input. I thank S. Giddings, G. Landsberg and D.J. Schwarz for helpful cor-

respondence and M. Bojowald for sending me a draft of ref. [96] prior to submission

to the arXiv. Last but not least I render special thanks to D. Ahluwalia-Khalilova

for the kind invitation to compile this review.

References

1. H. B. G. Casimir, “On the attraction between two perfectly conducting plates,”Kon. Ned. Akad. Wetensch. Proc. 51 (1948) 793–795.

2. M. Bordag, U. Mohideen, and V. M. Mostepanenko, “New developments in theCasimir effect,” Phys. Rept. 353 (2001) 1–205, quant-ph/0106045.

3. J. Schwinger (Ed.), “Quantum electrodynamics”, Dover Publications, 1958.4. V. S. Fadin and V. A. Khoze, “Production of a pair of heavy quarks in e+ e-

annihilation in the threshold region,” Sov. J. Nucl. Phys. 48 (1988) 309–313; M. J.Strassler and M. E. Peskin, “The Heavy top quark threshold: QCD and the Higgs,”Phys. Rev. D43 (1991) 1500–1514; W. Kummer and W. Modritsch, “Relativisticbound state equation for unstable fermions and the toponium width,” Phys. Lett.B349 (1995) 525–532, hep-ph/9501406.

5. Particle Data Group Collaboration, K. Hagiwara et al., “Review of particlephysics,” Phys. Rev. D66 (2002) 010001; D0 Collaboration, V. M. Abazov et al.,“A precision measurement of the mass of the top quark,” Nature 429 (2004)638–642, hep-ex/0406031.

6. G. A. Schuler and T. Sjostrand, “Parton distributions of the virtual photon,” Phys.Lett. B376 (1996) 193–200, hep-ph/9601282.

7. R. Schodel et al., “A Star in a 15.2 year orbit around the supermassive black holeat the center of the Milky Way,” Nature 419 (2002) 694–696.

8. J. E. McClintock and R. A. Remillard, “Black hole binaries,” astro-ph/0306213.9. S. Carlip, “Quantum gravity: A progress report,” Rept. Prog. Phys. 64 (2001) 885,

arXiv:gr-qc/0108040; L. Smolin, “How far are we from the quantum theory ofgravity?,” hep-th/0303185; E. Alvarez, “Quantum gravity,” gr-qc/0405107.

10. J. Preskill, “Do black holes destroy information?,” hep-th/9209058; D. N. Page,“Black hole information,” hep-th/9305040; T. Banks, “Lectures on black holes andinformation loss,” Nucl. Phys. Proc. Suppl. 41 (1995) 21–65, hep-th/9412131;G. ’t Hooft, “Black holes, Hawking radiation, and the information paradox,” Nucl.Phys. Proc. Suppl. 43 (1995) 1–11.

11. V. Frolov and I. Novikov, Black Hole Physics. Kluwer Academic Publishers, 1998.12. As there is no e-print at the time of writing this article cf. the summary on

J. Baez’ webpage http://math.ucr.edu/home/baez/week207.html.13. S. W. Hawking, “Virtual black holes,” Phys. Rev. D53 (1996) 3099–3107,

hep-th/9510029.

February 8, 2010 9:8 WSPC/Guidelines main

224 D. Grumiller

14. D. Grumiller, W. Kummer, and D. V. Vassilevich, “The virtual black hole in 2dquantum gravity,” Nucl. Phys. B580 (2000) 438–456, gr-qc/0001038.

15. D. Grumiller, W. Kummer, and D. V. Vassilevich, “Dilaton gravity in twodimensions,” Phys. Rept. 369 (2002) 327–429, hep-th/0204253.

16. J. Wheeler, “Geometrodynamics and the issue of the final state,” in Relativity,Groups and Topology, C. DeWitt and B. DeWitt, eds., p. 316. Gordon and Breach,1964.

17. J. C. Baez, “An introduction to spin foam models of BF theory and quantumgravity,” Lect. Notes Phys. 543 (2000) 25–94, gr-qc/9905087; C. Rovelli, “Loopquantum gravity,” Living Rev. Rel. 1 (1998) 1, gr-qc/9710008.

18. S. W. Hawking, “Wormholes in space-time,” Phys. Rev. D37 (1988) 904–910; S. R.Coleman, “Why there is nothing rather than something: A theory of thecosmological constant,” Nucl. Phys. B310 (1988) 643.

19. S. W. Hawking, “Space-time foam,” Nucl. Phys. B144 (1978) 349–362.20. M. Nakahara, Geometry, Topology and Physics. IOP Publishing, Bristol, 1990.21. C. T. C. Wall, “On simply-connected 4-manifolds,” J. London Math. Soc. 39

(1964) 141–149.22. F. J. Ernst, “Removal of the nodal singularity of the C-metric,” J. Math. Phys. 17

(1976) 515.23. S. W. Hawking and S. F. Ross, “Loss of quantum coherence through scattering off

virtual black holes,” Phys. Rev. D56 (1997) 6403–6415, hep-th/9705147.24. T. Prestidge, “Higher spin field equations in a virtual black hole metric,” Phys.

Rev. D58 (1998) 124022, hep-th/9802028.25. L. J. Garay, “Quantum evolution in spacetime foam,” Int. J. Mod. Phys. A14

(1999) 4079–4120, gr-qc/9911002.26. F. Benatti and R. Floreanini, “Massless neutrino oscillations,” Phys. Rev. D64

(2001) 085015, hep-ph/0105303.27. F. Benatti and R. Floreanini, “Non-standard neutral kaons dynamics from infinite

statistics,” Annals Phys. 273 (1999) 58–71, hep-th/9811196.28. J. R. Ellis, J. S. Hagelin, D. V. Nanopoulos, and M. Srednicki, “Search for

violations of quantum mechanics,” Nucl. Phys. B241 (1984) 381; P. Huet andM. E. Peskin, “Violation of CPT and quantum mechanics in the K0 - anti-K0system,” Nucl. Phys. B434 (1995) 3–38, hep-ph/9403257; J. R. Ellis, J. L. Lopez,N. E. Mavromatos, and D. V. Nanopoulos, “Precision tests of CPT symmetry andquantum mechanics in the neutral kaon system,” Phys. Rev. D53 (1996)3846–3870, hep-ph/9505340.

29. T. Banks, L. Susskind, and M. E. Peskin, “Difficulties for the evolution of purestates into mixed states,” Nucl. Phys. B244 (1984) 125; W. G. Unruh and R. M.Wald, “On evolution laws taking pure states to mixed states in quantum fieldtheory,” Phys. Rev. D52 (1995) 2176–2182, hep-th/9503024.

30. R. Rajaraman, SOLITONS AND INSTANTONS. AN INTRODUCTION TOSOLITONS AND INSTANTONS IN QUANTUM FIELD THEORY.North-Holland, Amsterdam, 1982.

31. W. Kummer and D. J. Schwarz, “General analytic solution of R**2 gravity withdynamical torsion in two-dimensions,” Phys. Rev. D45 (1992) 3628–3635;“Renormalization of R**2 gravity with dynamical torsion in d = 2,” Nucl. Phys.B382 (1992) 171–186.

32. J. Brown, Lower Dimensional Gravity. World Scientific, 1988; T. Strobl, “Gravityin two spacetime dimensions,” hep-th/0011240. Habilitation thesis; S. Nojiri andS. D. Odintsov, “Quantum dilatonic gravity in d = 2, 4 and 5 dimensions,” Int. J.

February 8, 2010 9:8 WSPC/Guidelines main

Virtual black holes and the S-matrix 225

Mod. Phys. A16 (2001) 1015–1108, hep-th/0009202.33. P. Schaller and T. Strobl, “Poisson structure induced (topological) field theories,”

Mod. Phys. Lett. A9 (1994) 3129–3136, hep-th/9405110.34. J. G. Russo and A. A. Tseytlin, “Scalar tensor quantum gravity in

two-dimensions,” Nucl. Phys. B382 (1992) 259–275, arXiv:hep-th/9201021; S. D.Odintsov and I. L. Shapiro, “One loop renormalization of two-dimensional inducedquantum gravity,” Phys. Lett. B263 (1991) 183–189.

35. M. O. Katanaev, “Effective action for scalar fields in two-dimensional gravity,”Annals Phys. 296 (2002) 1–50, gr-qc/0101033.

36. D. Grumiller, Quantum dilaton gravity in two dimensions with matter. PhD thesis,Technische Universitat Wien, 2001. gr-qc/0105078.

37. M. O. Katanaev, “All universal coverings of two-dimensional gravity with torsion,”J. Math. Phys. 34 (1993) 700–736.

38. W. Kummer, H. Liebl, and D. V. Vassilevich, “Integrating geometry in general 2ddilaton gravity with matter,” Nucl. Phys. B544 (1999) 403–431, hep-th/9809168.

39. H. Liebl, D. V. Vassilevich, and S. Alexandrov, “Hawking radiation and masses ingeneralized dilaton theories,” Class. Quant. Grav. 14 (1997) 889–904,arXiv:gr-qc/9605044.

40. T. Banks and M. O’Loughlin, “Two-dimensional quantum gravity in Minkowskispace,” Nucl. Phys. B362 (1991) 649–664; V. P. Frolov, “Two-dimensional blackhole physics,” Phys. Rev. D46 (1992) 5383–5394; R. B. Mann, “Conservation lawsand 2-d black holes in dilaton gravity,” Phys. Rev. D47 (1993) 4438–4442,hep-th/9206044.

41. H. Grosse, W. Kummer, P. Presnajder, and D. J. Schwarz, “Novel symmetry ofnonEinsteinian gravity in two- dimensions,” J. Math. Phys. 33 (1992) 3892–3900,hep-th/9205071.

42. D. Grumiller and D. Mayerhofer, “On static solutions in 2d dilaton gravity withscalar matter,” gr-qc/0404013.

43. D. Grumiller, W. Kummer, and D. V. Vassilevich, “Virtual black holes ingeneralized dilaton theories (and their special role in string gravity),” EuropeanPhys. J. C30 (2003) 135–143, hep-th/0208052.

44. P. Fischer, D. Grumiller, W. Kummer, and D. V. Vassilevich, “S-matrix for s-wavegravitational scattering,” Phys. Lett. B521 (2001) 357–363, gr-qc/0105034.Erratum ibid. B532 (2002) 373.

45. D. Grumiller, “Virtual black hole phenomenology from 2d dilaton theories,” Class.Quant. Grav. 19 (2002) 997–1009, gr-qc/0111097.

46. D. Grumiller, “Deformations of the Schwarzschild black hole,” gr-qc/0311011.Invited talk at MG X.

47. K. V. Kuchar, “Geometrodynamics of Schwarzschild black holes,” Phys. Rev. D50

(1994) 3961–3981, arXiv:gr-qc/9403003.48. P. C. Argyres, S. Dimopoulos, and J. March-Russell, “Black holes and

sub-millimeter dimensions,” Phys. Lett. B441 (1998) 96–104, hep-th/9808138;T. Banks, M. Dine, and A. E. Nelson, “Constraints on theories with large extradimensions,” JHEP 06 (1999) 014, hep-th/9903019; T. Banks and W. Fischler, “Amodel for high energy scattering in quantum gravity,” hep-th/9906038;R. Emparan, G. T. Horowitz, and R. C. Myers, “Black holes radiate mainly on thebrane,” Phys. Rev. Lett. 85 (2000) 499–502, hep-th/0003118.

49. S. B. Giddings and S. Thomas, “High energy colliders as black hole factories: Theend of short distance physics,” Phys. Rev. D65 (2002) 056010 hep-ph/0106219;S. Dimopoulos and G. Landsberg, “Black holes at the LHC,” Phys. Rev. Lett. 87

February 8, 2010 9:8 WSPC/Guidelines main

226 D. Grumiller

(2001) 161602, hep-ph/0106295.50. M. Cavaglia, “Black hole and brane production in TeV gravity: A review,” Int. J.

Mod. Phys. A18 (2003) 1843–1882, hep-ph/0210296; G. Landsberg, “Black holesat future colliders and beyond: A review,” hep-ph/0211043; “Black holes at futurecolliders and in cosmic rays,” hep-ex/0310034.

51. N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, “The hierarchy problem andnew dimensions at a millimeter,” Phys. Lett. B429 (1998) 263–272,hep-ph/9803315; I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali,“New dimensions at a millimeter to a Fermi and superstrings at a TeV,” Phys.Lett. B436 (1998) 257–263, hep-ph/9804398; N. Arkani-Hamed, S. Dimopoulos,and G. R. Dvali, “Phenomenology, astrophysics and cosmology of theories withsub-millimeter dimensions and TeV scale quantum gravity,” Phys. Rev. D59

(1999) 086004, hep-ph/9807344.52. Y. B. Zeldovich, “A New Type Of Radioactive Decay: Gravitational Annihilation

Of Baryons,” Phys. Lett. A59, 254 (1976); “A Novel Type Of Radioactive Decay:Gravitational Baryon Annihilation,” (In Russian), Zh. Eksp. Teor. Fiz. 72, 18(1977); S. W. Hawking, D. N. Page and C. N. Pope, “The Propagation Of ParticlesIn Space-Time Foam,” Phys. Lett. B86, 175 (1979); D. N. Page, “ParticleTransmutations In Quantum Gravity,” Phys. Lett. B95, 244 (1980).

53. F. C. Adams, G. L. Kane, M. Mbonye and M. J. Perry, “Proton decay, black holes,and large extra dimensions,” Int. J. Mod. Phys. A16 (2001) 2399, hep-ph/0009154.

54. A. B. Kobakhidze, “Proton stability in TeV-scale GUTs,” Phys. Lett. B514 (2001)131, hep-ph/0102323.

55. Y. Uehara, “Virtual black holes at linear colliders,” hep-ph/0205068.56. G. ’t Hooft, “The scattering matrix approach for the quantum black hole: An

overview,” Int. J. Mod. Phys. A11 (1996) 4623–4688, gr-qc/9607022.57. P. Fischer, “Vertices in spherically reduced quantum gravity,” Master’s thesis,

Vienna University of Technology, 2001; D. Grumiller, 2001. DocumentedMathematica notepad available fromhttp://www.teilchen.at/grumiller/projects/myself/thesis/s4.nb.

58. D. Grumiller, “Three functions in dilaton gravity: The good, the bad and themuggy,” in Proceedings of International Workshop on Mathematical Theories andtheir Applications, S. Moskaliuk, ed., pp. 56–96, TIMPANI. Cernivtsi, Ukraine,2004. hep-th/0305073.

59. D. Grumiller and W. Kummer, “How to approach quantum gravity: Backgroundindependence in 1+1 dimensions,” in Symmetries beyond the standard model,N. Mankoc Borstnik, H. B. Nielsen, C. D. Froggatt and D. Lukman, eds.,pp. 184–196 Portoroz, Slovenia, 2003. gr-qc/0310068.

60. A. Ashtekar, “Lectures on nonperturbative canonical gravity,”. Singapore,Singapore: World Scientific (1991) 334 p. (Advanced series in astrophysics andcosmology, 6).

61. H. Balasin, W. Kummer, O. Piguet, and M. Schweda, “On the regularization of themass zero 2-D propagator,” Phys. Lett. B287 (1992) 138–144.

62. M. Henneaux and C. Teitelboim, “Quantization of gauge systems,” PrincetonUniversity Press, 1992.

63. V. Moretti and N. Pinamonti, “Aspects of hidden and manifest SL(2,R) symmetryin 2D near- horizon black-hole backgrounds,” Nucl. Phys. B647 (2002) 131–152,gr-qc/0207072.

64. R. M. Wald, “The thermodynamics of black holes,” Living Rev. Rel. 4 (2001) 6,gr-qc/9912119.

February 8, 2010 9:8 WSPC/Guidelines main

Virtual black holes and the S-matrix 227

65. G. W. Gibbons and S. W. Hawking, eds., Euclidean quantum gravity. Singapore:World Scientific, 1993.

66. W. Kummer and D. V. Vassilevich, “Hawking radiation from dilaton gravity in(1+1) dimensions: A pedagogical review,” Annalen Phys. 8 (1999) 801–827,gr-qc/9907041.

67. J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D7 (1973) 2333–2346;S. W. Hawking, “Particle creation by black holes,” Commun. Math. Phys. 43

(1975) 199–220.68. J. Gegenberg, G. Kunstatter, and D. Louis-Martinez, “Observables for

two-dimensional black holes,” Phys. Rev. D51 (1995) 1781–1786, gr-qc/9408015.69. D. Youm, “Black holes and solitons in string theory,” Phys. Rept. 316 (1999)

1–232, hep-th/9710046; R. D’Auria and P. Fre, “BPS black holes in supergravity:Duality groups, p-branes, central charges and the entropy,” hep-th/9812160;T. Mohaupt, “Black hole entropy, special geometry and strings,” Fortsch. Phys. 49

(2001) 3–161, hep-th/0007195.70. A. Ashtekar, J. C. Baez, and K. Krasnov, “Quantum geometry of isolated horizons

and black hole entropy,” Adv. Theor. Math. Phys. 4 (2000) 1–94, gr-qc/0005126.71. S. Carlip, “Horizon constraints and black hole entropy,” hep-th/0408123.72. S. Hod, “Bohr’s correspondence principle and the area spectrum of quantum black

holes,” Phys. Rev. Lett. 81 (1998) 4293, gr-qc/9812002.73. J. Kettner, G. Kunstatter, and A. J. M. Medved, “Quasinormal modes for single

horizon black holes in generic 2-d dilaton gravity,” gr-qc/0408042.74. M. Cadoni and S. Mignemi, “Entropy of 2d black holes from counting microstates,”

Phys. Rev. D59 (1999) 081501, hep-th/9810251.75. S. Carlip, “Entropy from conformal field theory at killing horizons,” Class. Quant.

Grav. 16 (1999) 3327–3348, gr-qc/9906126.76. S. Carlip, “Near-horizon conformal symmetry and black hole entropy,” Phys. Rev.

Lett. 88 (2002) 241301, gr-qc/0203001.77. S. Carlip, “Black hole entropy from conformal field theory in any dimension,” Phys.

Rev. Lett. 82 (1999) 2828–2831, hep-th/9812013; S. N. Solodukhin, “Conformaldescription of horizon’s states,” Phys. Lett. B454 (1999) 213–222, hep-th/9812056.

78. K. Schwarzschild, “On the gravitational field of a mass point according toEinstein’s theory,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1916

(1916) 189–196, arXiv:physics/9905030.79. G. Mandal, A. M. Sengupta, and S. R. Wadia, “Classical solutions of

two-dimensional string theory,” Mod. Phys. Lett. A6 (1991) 1685–1692; S. Elitzur,A. Forge, and E. Rabinovici, “Some global aspects of string compactifications,”Nucl. Phys. B359 (1991) 581–610; E. Witten, “On string theory and black holes,”Phys. Rev. D44 (1991) 314–324; R. Dijkgraaf, H. Verlinde, and E. Verlinde,“String propagation in a black hole geometry,” Nucl. Phys. B371 (1992) 269–314;C. G. Callan, Jr., S. B. Giddings, J. A. Harvey, and A. Strominger, “Evanescentblack holes,” Phys. Rev. D45 (1992) 1005–1009, hep-th/9111056.

80. C. Teitelboim, “Gravitation and Hamiltonian structure in two space-timedimensions,” Phys. Lett. B126 (1983) 41; R. Jackiw, “Lower dimensional gravity,”Nucl. Phys. B252 (1985) 343–356.

81. M. O. Katanaev, W. Kummer, and H. Liebl, “On the completeness of the blackhole singularity in 2d dilaton theories,” Nucl. Phys. B486 (1997) 353–370,gr-qc/9602040.

82. S. W. Hawking and D. N. Page, “Thermodynamics of black holes in anti-de Sitterspace,” Commun. Math. Phys. 87 (1983) 577.

February 8, 2010 9:8 WSPC/Guidelines main

228 D. Grumiller

83. G. Guralnik, A. Iorio, R. Jackiw, and S. Y. Pi, “Dimensionally reducedgravitational Chern-Simons term and its kink,” Ann. Phys. 308 (2003) 222–236,hep-th/0305117; D. Grumiller and W. Kummer, “The classical solutions of thedimensionally reduced gravitational Chern-Simons theory,” Ann. Phys. 308 (2003)211–221, hep-th/0306036; L. Bergamin, D. Grumiller, A. Iorio, and C. Nunez,“Chemistry of Chern-Simons Supergravity: reduction to a BPS kink, oxidation toM-theory and thermodynamical aspects,” hep-th/0409273.

84. O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, “Large N fieldtheories, string theory and gravity,” Phys. Rept. 323 (2000) 183–386,hep-th/9905111.

85. J. D. Brown, J. Creighton, and R. B. Mann, “Temperature, energy and heatcapacity of asymptotically anti-de Sitter black holes,” Phys. Rev. D50 (1994)6394–6403, gr-qc/9405007.

86. O. B. Zaslavsky, “Quantum corrections to temperature and mass of 1+1 dilatonicblack holes and the trace anomaly,” Phys. Lett. B375 (1996) 43–46.

87. D. Grumiller, W. Kummer, and D. V. Vassilevich, “Positive specific heat of thequantum corrected dilaton black hole,” JHEP 07 (2003) 009, hep-th/0305036.

88. S. W. Hawking, “Gravitational radiation from colliding black holes,” Phys. Rev.Lett. 26 (1971) 1344–1346; S. W. Hawking, “Black holes in general relativity,”Commun. Math. Phys. 25 (1972) 152–166.

89. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “Onthe Lambert W Function,” Adv. Comp. Math. 5 (1996) 329–359. More informationon the Lambert W function is available at the webpagehttp://kong.apmaths.uwo.ca/∼rcorless/frames/PAPERS/LambertW/.

90. E. E. Flanagan, “The conformal frame freedom in theories of gravitation,” Class.Quant. Grav. 21 (2004) 3817, gr-qc/0403063.

91. R. B. Mann and S. N. Solodukhin, “Universality of quantum entropy for extremeblack holes,” Nucl. Phys. B523 (1998) 293–307, hep-th/9709064; R. K. Kaul andP. Majumdar, “Logarithmic correction to the Bekenstein-Hawking entropy,” Phys.Rev. Lett. 84 (2000) 5255–5257, gr-qc/0002040; S. Carlip, “Logarithmiccorrections to black hole entropy from the Cardy formula,” Class. Quant. Grav. 17

(2000) 4175–4186, gr-qc/0005017.92. A. Ghosh and P. Mitra, “A bound on the log correction to the black hole area

law,” gr-qc/0401070.93. T. M. Fiola, J. Preskill, A. Strominger, and S. P. Trivedi, “Black hole

thermodynamics and information loss in two- dimensions,” Phys. Rev. D50 (1994)3987–4014, hep-th/9403137; R. C. Myers, “Black hole entropy in two-dimensions,”Phys. Rev. D50 (1994) 6412–6421, hep-th/9405162; J. D. Hayward, “Entropy inthe RST model,” Phys. Rev. D52 (1995) 2239–2244, gr-qc/9412065.

94. S. B. Giddings and M. Lippert, “Precursors, black holes, and a locality bound,”Phys. Rev. D65 (2002) 024006 hep-th/0103231; “The information paradox andthe locality bound,” Phys. Rev. D69 (2004) 124019 hep-th/0402073.

95. L. Bergamin, D. Grumiller, and W. Kummer, “Quantization of 2d dilatonsupergravity with matter,” JHEP 05 (2004) 060, hep-th/0404004.

96. M. Bojowald and R. Swiderski, “Spherically symmetric quantum horizons,”gr-qc/0410147.

Preprint typeset in JHEP style - HYPER VERSION LU-ITP-2005/001

An action for the exact string black hole

D. Grumiller

Institute for Theoretical Physics, University of Leipzig

Augustusplatz 10-11, D-04109 Leipzig, Germany

E-mail: grumiller@itp.uni-leipzig.de

Abstract: A local action is constructed describing the exact string black hole dis-

covered by Dijkgraaf, Verlinde and Verlinde in 1992. It turns out to be a special 2D

Maxwell-dilaton gravity theory, linear in curvature and field strength. Two constants

of motion exist: mass M ≥ 1, determined by the level k, and U(1)-charge Q ≥ 0, de-

termined by the value of the dilaton at the origin. ADM mass, Hawking temperature

TH ∝√

1 − 1/M and Bekenstein–Hawking entropy are derived and studied in detail.

Winding/momentum mode duality implies the existence of a similar action, arising

from a branch ambiguity, which describes the exact string naked singularity. In the

strong coupling limit the solution dual to AdS2 is found to be the 5D Schwarzschild

black hole. Some applications to black hole thermodynamics and 2D string theory

are discussed and generalizations – supersymmetric extension, coupling to matter

and critical collapse, quantization – are pointed out.

Keywords: Black Holes in String Theory, 2D Gravity, Sigma Models.

Dedicated to Wolfgang Kummer on occasion of his Emeritierung

Contents

1. Introduction 230

2. Recapitulation of 2D dilaton gravity 233

2.1 Geometry and actions in 2D 233

2.2 Coupling to an abelian gauge field 235

2.3 All classical solutions 236

3. The action 238

3.1 Statement of the main result 238

3.2 Proof of equivalence to DVV 240

3.2.1 All classical solutions derived from the action 241

3.2.2 Coordinate transformations to the ESBH 242

3.2.3 The dilaton field 243

3.3 Discussion and reformulations of the action 244

4. Thermodynamical properties 247

4.1 Mass 247

4.2 Temperature 249

4.3 Entropy 251

5. Conclusions and generalizations 252

5.1 Supplementary thermodynamical considerations 252

5.2 Applications in 2D string theory 255

5.3 Supersymmetrization, critical collapse and quantization 258

A. No-go recap 261

B. The art of gauging constants 262

1. Introduction

The exact string black hole (ESBH) was discovered by Dijkgraaf, Verlinde and Ver-

linde (DVV) more than a decade ago [1]. The construction of an action for it which

does not display non-localities or higher order derivatives is a challenging open prob-

lem in the context of 2D string theory. The purpose of this paper is to solve it.

– 230 –

There are several advantages of having such an action available: the main point

of the ESBH is its non-perturbative aspect, i.e., it is believed to be valid to all

orders in the string-coupling α′. Thus, a corresponding action will capture non-

perturbative features of string theory and will allow, among other things, for the first

time a thorough discussion of ADM mass, Hawking temperature and Bekenstein–

Hawking entropy of the ESBH which otherwise requires some ad-hoc assumption.

Moreover, once an action is at our disposal an exact path integral quantization may

be performed. A more detailed exposition of these issues and other applications will

be postponed until the conclusions.

At the perturbative level actions approximating the ESBH are known: to lowest

order in α′ an action emerges the classical solutions of which describe the Witten

BH [2–4], which in turn inspired the CGHS model [5], a 2D dilaton gravity model

with scalar matter that has been used as a toy model for BH evaporation. Pushing

perturbative considerations further Tseytlin was able to show that up to 3 loops

the ESBH is consistent with sigma model conformal invariance [6]; in the super-

symmetric case this holds even at 4 loops [7]. In the strong coupling regime the

ESBH asymptotes to the Jackiw-Teitelboim (JT) model [8]. The exact conformal

field theory (CFT) methods used in [1], based upon the SL(2, R)/U(1) gauged Wess-

Zumino-Witten model, imply the dependence of the ESBH solutions on the level k.

A different (somewhat more direct) derivation leading to the same results for dilaton

and metric was presented in [9] (see also [10]). For a comprehensive history and more

references ref. [11] may be consulted.

In the notation of [12] for Euclidean signature the line element of the ESBH

discovered by DVV is given by

ds2 = f 2(x) dτ 2 + dx2 , (1.1)

with

f(x) =tanh (bx)

√

1 − p tanh2 (bx). (1.2)

Physical scales are adjusted by the parameter b ∈ R+ which has dimension of inverse

length. The corresponding expression for the dilaton,

φ = φ0 − ln cosh (bx) − 1

4ln (1 − p tanh2 (bx)) , (1.3)

contains an integration constant φ0. Additionally, there are the following relations

between constants, string-coupling α′, level k and dimension D of string target space:

α′b2 =1

k − 2, p :=

2

k=

2α′b2

1 + 2α′b2, D − 26 + 6α′b2 = 0 . (1.4)

For D = 2 one obtains p = 8/9, but like in the original work [1] we will treat general

values of p ∈ (0; 1) and consider the limits p → 0 and p → 1 separately: for p = 0

– 231 –

one recovers the Witten BH; for p = 1 the JT model is obtained. Both limits exhibit

singular features: for all p ∈ (0; 1) the solution is regular globally, asymptotically flat

and exactly one Killing horizon exists. However, for p = 0 a curvature singularity

(screened by a horizon) appears and for p = 1 space-time fails to be asymptotically

flat. In the present work exclusively the Minkowskian version of (1.1)

ds2 = f 2(x) dτ 2 − dx2 , (1.5)

will be needed. The maximally extended space-time of this geometry has been stud-

ied by Perry and Teo [13] and by Yi [14].

Winding/momentum mode duality implies the existence of a dual solution, the

Exact String Naked Singularity (ESNS), which can be acquired most easily by re-

placing bx → bx + iπ/2, entailing in all formulas above the substitutions

sinh → i cosh , cosh → i sinh . (1.6)

This concludes the brief recollection of the main properties of the ESBH/ESNS

relevant for the present work. The task is now clear, if slightly ambitious: we are

seeking an action the classical solutions of which yield (1.2)-(1.5). Some prejudices

concerning the action may be helpful: It has to be a 2D action. It may depend on

the scale parameter b, but not on the constant φ0 which should emerge as a constant

of motion. It should functionally depend on the metric, the dilaton and eventual

auxiliary fields. It should be diffeomorphism invariant and local Lorentz invariant.

The absence of non-localities and non-polynomial derivative interactions is crucial.

It would be splendid if no propagating physical degrees of freedom (PPDOF) were

present and marvellous if the action described not only the ESBH but also, by some

“simple” duality transformation, the ESNS. Last but not least one has to circumvent

the no-go result of ref. [15] by relaxing at least one of its premises.

This paper is organized as follows: for sake of self-containment section 2 recalls

some of the main results of Maxwell-dilaton gravity in 2D in the first order formalism:

various formulations of the action (subsection 2.1), coupling to abelian gauge fields

(subsection 2.2), and how to obtain all classical solutions (subsection 2.3). This

section may be skipped by readers familiar with that formalism. Section 3 contains

the main part of the paper: the presentation of the action (subsection 3.1), the

proof of equivalence of line element and dilaton to the ESBH (subsection 3.2), and a

discussion of the action as well as different representations thereof (subsection 3.3).

Equipped with such an action thermodynamical properties may be discussed with

ease (section 4): (ADM) mass (subsection 4.1), (Hawking) temperature (subsection

4.2), and (Bekenstein–Hawking) entropy (4.3). The extensive conclusions in section

5 reveal physical features, applications and generalizations, and compare with the

literature. The appendices are devoted to historical remarks. They are recommended

to readers interested in a bottom-up construction of the action who may wish to

– 232 –

consult them before reading the main statement in section 3: the no-go result is

recapitulated in appendix A and the crucial idea of introducing an abelian Maxwell

field is put into historical context in appendix B.

2. Recapitulation of 2D dilaton gravity

The purpose of this brief summary of well-known results is to provide a self-contained

introduction to dilaton gravity in the first order formalism and to fix the notation.1

For background information and additional references the extensive review [16] may

be consulted. Supplementary material providing relations to non-linear algebras may

be found in appendix B.

2.1 Geometry and actions in 2D

For various reasons, some of which will become apparent while obtaining all classical

solutions, it is very convenient to employ the first order formalism: ea = eaµdxµ is

the dyad 1-form dual to Ea – i.e. ea(Eb) = δab . Latin indices refer to an anholonomic

frame, Greek indices to a holonomic one. The 1-form ω represents the spin-connection

ωab = εa

bω with the totally antisymmetric Levi-Civita symbol εab (ε01 = +1). With

the flat metric ηab in light-cone coordinates (η+− = 1 = η−+, η++ = 0 = η−−) it

reads ε±± = ±1. The torsion 2-form is given by T± = (d±ω) ∧ e±. The curvature

2-form Rab can be represented by the 2-form R defined by Ra

b = εabR, R = dω. The

volume 2-form is denoted by ǫ = e+∧e−. Signs and factors of the Hodge-∗ operation

are defined by ∗ǫ = 1. Since the Einstein-Hilbert action∫

M2R ∝ (1 − γ) yields just

the Euler number for a surface with genus γ one has to generalize it appropriately to

generate equations of motion (EOM). The simplest idea is to introduce a Lagrange

multiplier for curvature, X, also known as “dilaton field”, and an arbitrary potential

thereof, V (X), in the action∫

M2(XR + ǫV (X)). Having introduced curvature it is

natural to consider torsion as well. By analogy the first order gravity action [17]

S(1) =

∫

M2

[XaTa + XR + ǫV(XaXa, X)] (2.1)

can be motivated where Xa are the Lagrange multipliers for torsion. It encompasses

essentially all known dilaton theories in 2D. Actually, for most practical purposes

the potential takes the simpler form

V(XaXa, X) = X+X−U(X) + V (X) . (2.2)

The action (2.1) is equivalent to the frequently used second order action [18–20]

S(2) =

∫

M2

d2x√−g

[

X−r

2− U(X)

2(∇X)2 + V (X)

]

, (2.3)

1The sign of the curvature scalar r has been fixed conveniently such that r > 0 for dS2. This is

the only difference to the notations used in ref. [16].

– 233 –

with the same functions U, V as in (2.2). The curvature scalar r and covariant

derivative ∇ are associated with the Levi-Civita connection related to the metric

gµν , the determinant of which is denoted by g. If ω is torsion-free r ∝ ∗R. In the

absence of matter there are no PPDOF.

There is another intriguing re-interpretation of (2.1): defining XI = (X, X+, X−)

and AI = (ω, e−, e+) the Poisson-sigma model (PSM) action emerges [17]

S(PSM) =

∫

M2

[

dXI ∧ AI +1

2P IJAJ ∧ AI

]

, (2.4)

provided the Poisson tensor is chosen as

P IJ =

0 X+ −X−

−X+ 0 VX− −V 0

. (2.5)

Being a Poisson tensor it is not only anti-symmetric but it also fulfills the Jacobi

identity

P IL∂LP JK + perm (IJK) = 0 . (2.6)

Such a reformulation is advantageous because e.g. the existence of a Casimir function

may be deduced immediately from (2.5). It turns out that this Casimir function is

related to a conserved quantity, “the mass”, which has been found in previous second

order studies of dilaton gravity [19,21,22] as well as in the first order formulation [23].

The PSM perspective on 2D dilaton gravity is summarized in [24].

Finally, it should be mentioned that in the second order formalism often the

dilaton field φ, with

X = e−2φ , (2.7)

is employed. This brings (2.3) into the well-known form

S(2′) = −1

2

∫

M2

d2x√−g e−2φ

[

r + U(φ) (∇φ)2 + V (φ)]

, (2.8)

where the new potentials U , V are related to the old ones via

U = 4e−2φU(e−2φ) , V = −2e2φV (e−2φ) . (2.9)

Two prominent examples are the Witten BH with

U(X) = − 1

X, V (X) = −2b2X , → U(φ) = −4 , V = +4b2 , (2.10)

and the JT model2 with

U(X) = 0 , V (X) = −b2X , → U(φ) = 0 , V = +2b2 . (2.11)

The scale parameter b ∈ R+ defines the physical units and is essentially irrelevant.

2Here it is presented only for negative cosmological constant, i.e., AdS2. The dS2 case may be

obtained by changing the sign in the definition of V in (2.11).

– 234 –

2.2 Coupling to an abelian gauge field

It is straightforward to generalize (2.1) to a Maxwell-dilaton first order action,

S(MD1) =

∫

M2

[XaTa + XR + BF + ǫV(XaXa, X, B)] , (2.12)

where B is an additional scalar field and F = dA is the field strength 2-form, being

the differential of the gauge field 1-form A. Variation with respect to A immediately

establishes a constant of motion,3

dB = 0 , → B = Q , (2.13)

where Q is some real constant, the U(1) charge. Variation with respect to B may

produce a relation that allows to express B as a function of the dilaton and the dual

field strength ∗F . This need not be the case, however.

The result (2.13) implies that the solution of the remaining EOM reduces to the

case without Maxwell field. One just has to replace B by its on-shell value Q in the

potential V. Before discussing how to solve these remaining equations in the next

subsection, two examples will be provided.

Example 1: Standard Maxwell-dilaton models Suppose that V = X+X−U(X)+

V (X) + 12f(X)B2. Then, variation with respect to B gives

B = − ∗Ff(X)

. (2.14)

Inserting this back into the action yields the standard geometric part plus the fol-

lowing term

S(add) = −1

2

∫

M2

∗FF

f(X), (2.15)

which is an ordinary Maxwell term with nonminimal coupling to the dilaton via f(X).

Alternatively, one can use the on-shell condition B = Q, thus obtaining a purely

geometric action with an effective potential V = X+X−U(X) + V (X) + Q2

2f(X).

A typical example is spherically reduced gravity, i.e., the Reissner-Nordstrom BH,

where f(X) ∝ 1/X.

Example 2: Specific non-standard models For V = V (X) + f(X)B variation

with respect to B yields

f(X) = − ∗ F . (2.16)

Provided f(X) is invertible the dilaton X may be expressed on-shell as a function

of the dual field strength. For the Kaluza-Klein reduced gravitational Chern-Simons

theory [25] this observation turned out to be pivotal for a successful application of

first order gravity methods [26] and supersymmetrization [27].3In the PSM language adding an abelian gauge field means adding another row and column of

zeros to the Poisson tensor (2.5). Thus, its rank is unchanged and the dimension of its kernel is

increased by 1. Therefore, an additional Casimir function exists: the constant of motion in (2.13).

– 235 –

2.3 All classical solutions

It is useful to introduce the following combinations of the potentials U and V :

I(X) := exp

∫ X

U(y) dy , w(X) :=

∫ X

I(y)V (y) dy (2.17)

The integration constants may be absorbed, respectively, by rescalings and shifts of

the “mass”, see equation (2.23) below. Under dilaton dependent conformal transfor-

mations Xa → Xa/Ω, ea → eaΩ, ω → ω + Xaea d lnΩ/ dX equation (2.1) is mapped

to a new action of the same type with transformed potentials U , V . Hence, it is not

invariant. It turns out that only the combination w(X) as defined in (2.17) remains

invariant, so conformally invariant quantities may depend on w only. Note that I is

positive apart from eventual boundaries (typically, I may vanish in the asymptotic

region and/or at singularities). One may transform to a conformal frame with I = 1,

solve all EOM and then perform the inverse transformation. Thus, it is sufficient to

solve the classical EOM for U = 0,

dX + X−e+ − X+e− = 0 , (2.18)

(d±ω)X± ∓ e±V (X, B) = 0 , (2.19)

(d±ω)e± = 0 , (2.20)

which is what we are going to do now. Note that the equation containing dω is

redundant, while the equations from the Maxwell sector may be treated as in section

2.2; therefore, they have not been displayed.

Let us start with an assumption: X+ 6= 0 for a given patch.4 If it vanishes

a (Killing) horizon is encountered and one can repeat the calculation below with

indices + and − swapped everywhere. If both vanish in an open region by virtue of

(2.18) a constant dilaton vacuum emerges, which will be addressed separately below.

If both vanish on isolated points the Killing horizon bifurcates there and a more

elaborate discussion is needed [28]. The patch implied by this assumption is a “basic

Eddington-Finkelstein patch”, i.e., a patch with a conformal diagram which, roughly

speaking, extends over half of the bifurcate Killing horizon and exhibits a coordinate

singularity on the other half. In such a patch one may redefine e+ = X+Z with

a new 1-form Z. Then (2.18) implies e− = dX/X+ + X−Z and the volume form

reads ǫ = e+ ∧ e− = Z ∧ dX. The + component of (2.19) yields for the connection

ω = − dX+/X+ + ZV (X, B). One of the torsion conditions (2.20) then leads to

dZ = 0, i.e., Z is closed. Locally, it is also exact: Z = du. It is emphasized that,

4To get some physical intuition as to what this condition could mean: the quantities Xa, which

are the Lagrange multipliers for torsion, can be expressed as directional derivatives of the dilaton

field by virtue of (2.18) (e.g. in the second order formulation a term of the form XaXa corresponds

to (∇X)2). For those who are familiar with the Newman-Penrose formalism: for spherically reduced

gravity the quantities Xa correspond to the expansion spin coefficients ρ and ρ′ (both are real).

– 236 –

besides the two Casimir functions, this is the only integration needed! After these

elementary steps one obtains already the conformally transformed line element in

Eddington-Finkelstein (EF) gauge

ds2 = 2e+e− = 2 du dX + 2X+X− du2 , (2.21)

which nicely demonstrates the power of the first order formalism. In the final step

the combination X+X− has to be expressed as a function of X. This is possible by

noting that the linear combination X+×[(2.19) with − index] + X−×[(2.19) with +

index] together with (2.18) establishes a conservation equation,

d(X+X−) + V (X, B) dX = d(X+X− + w(X, B)) = 0 . (2.22)

Thus, there is always a conserved quantity (dC(g) = 0), which in the original confor-

mal frame reads

C(g) = X+X−I(X) + w(X, B) , (2.23)

where the definitions (2.17) have been inserted. It should be noted that the two

free integration constants inherent to the definitions (2.17) may be absorbed by

rescalings and shifts of C(g), respectively. Therefore, any mass definition based upon

the conserved quantity C(g) is incomplete without fixing the scale and the ground

state geometry.5 The classical solutions are labelled by this mass. Finally, one has to

transform back to the original conformal frame (the relevant conformal factor reads

Ω = I(X)). The line element (2.21) by virtue of (2.23) may be written as

ds2 = 2I(X) du dX − 2I(X)(w(X, B) − C(g)) du2 . (2.24)

Evidently there is always a Killing vector K ·∂ = ∂/∂u with associated Killing norm

K · K = −2I(w − C(g)). Since I 6= 0 Killing horizons are encountered at X = Xh

where Xh is a solution of

w(Xh, B) − C(g) = 0 . (2.25)

It is recalled that (2.24) is valid in a basic EF patch, e.g., an outgoing one. By redoing

the derivation above, but starting from the assumption X− 6= 0 one may obtain an

ingoing EF patch. Global issues will be addressed specifically for the ESBH and the

ESNS in section 3.2.1.

For sake of completeness it should be mentioned that in addition to the family

of solutions, labelled by M and Q, isolated solutions may exist, so-called constant

dilaton vacua, which have to obey X = XCDV = const. with V (XCDV , B) = 0.

The rank of the Poisson tensor (2.5) vanishes on these solutions. The corresponding

geometry has constant curvature, i.e., only Minkowski, Rindler or (A)dS2 are possible

spacetimes for constant dilaton vacua.

5This has been clarified for a large class of dilaton gravity models in ref. [29]. Appendix A of

ref. [30] provides a generalization to arbitrary dilaton gravity models.

– 237 –

3. The action

3.1 Statement of the main result

In this paper it will be proven that the first order Maxwell-dilaton gravity action6

SESBH =

∫

M2

[

XaTa + ΦR + BF + ǫ

(

X+X−U(Φ) + V (Φ))]

, (3.1)

with potentials U, V to be defined below, describes the ESBH as well as the ESNS,

i.e., on-shell the metric gµν = ηab eaµ eb

ν and the dilaton X = exp (−2φ) are given by

(1.2)-(1.6). Regarding the latter, the relation

(Φ − γ)2 = arcsinh 2γ (3.2)

in conjunction with the definition

γ :=X

B(3.3)

may be used to express the auxiliary dilaton field Φ in terms of the “true” dilaton

field X and the auxiliary field B. The two branches of the square root function

correspond to the ESBH (main branch) and the ESNS (second branch), respectively.

Henceforth the notation

Φ± = γ ± arcsinh γ (3.4)

will be employed, where + refers to the ESBH and − to the ESNS. This applies to

all expressions encountered below. If a quantity appears without such a lower index

it is the same for ESBH and ESNS.7 The potentials read

V = −2b2γ , U± = − 1

γN±(γ), (3.5)

with an irrelevant scale parameter b ∈ R+ and

N±(γ) = 1 +2

γ

(

1

γ±√

1 +1

γ2

)

. (3.6)

Note that N+N− = 1. This completes the definition of all terms appearing in the

action (3.1), so what remains to be discussed, apart from the actual proof of equiv-

alence, are the constants of motion.

6The notation is explained in section 2. Here is a brief summary: the 2-forms T± = (d±ω)∧ e±,

R = dω and F = dA are torsion, curvature, and abelian field strength, respectively and depend on

the gauge field 1-forms e± (“Zweibein”), ω (“spin connection”) and A (“Maxwell field”). The scalar

fields X±, Φ and B are Lagrange multipliers for these 2-forms and appear also in the potential, the

last term in (3.1), which is multiplied by the volume 2-form ǫ = e+ ∧ e−.7In order to avoid confusion with light cone indices, which are also denoted by ±, from now on

light cone indices will always appear as upper ones unless stated otherwise, but from the context

the meaning should be clear anyhow.

– 238 –

On general grounds it is known that two constants of motion exist.8 One of

them is just the field B as can be seen easily from varying (3.1) with respect to the

gauge field A, while the other one is the quantity defined in (2.23). They may be

interpreted, respectively, as U(1) charge (cf. (2.13)),

B =1

√

k(k − 2)e−2φ0 =: Q , (3.7)

and mass (cf. (2.23)),

C(g) = −bk =: −2bM . (3.8)

Note that Q ≥ 0 and M ∈ [1,∞). The restriction on Q to positive values is necessary

in order to ensure positivity of Φ for positive X. The restriction on M is not inherent

to the model9 but a consequence of CFT: k may not be smaller than 2 if the string-

coupling α′ is positive (cf. (1.4)). Scaling and shift ambiguity that exist for any

dilaton gravity model are fixed in the next subsection. At the moment k and φ0 are

just some convenient integration constants, but the nomenclature is not accidental:

k will turn out to be the level and φ0 the value of the dilaton at the origin.

As heuristic support for the claim that (3.1) is indeed correct one may consider

the (singular) limits γ → 0 (JT limit) and γ → ∞ (Witten BH limit). In the first

case for the ESBH branch the effective dilaton Φ = 2γ + O(γ3) immediately reveals

the proper potential V (Φ) = −b2Φ + . . . (cf. (2.11)). The ESNS branch becomes

singular, U− ∝ 1/γ3, and is discarded for the time being, while the ESBH branch

yields U+ ∝ γ. However, since this potential is multiplied by X+X− which is also

small in that limit,10 it may be dropped to leading order and the result is

X+X−U∣

∣

γ≪1= O(γ3) , V |γ≪1 = −b2Φ + O(γ3) . (3.9)

In the second case the limit reads Φ± = γ(1 ±O(ln γ/γ)) and the quantities N± =

1±O(1/γ) approach unity. Thus, the ESBH and the ESNS branch coincide, i.e., the

Witten BH becomes a self-dual model, and the potentials read

X+X−U∣

∣

γ≫1≈ −X+X−

Φ, V |γ≫1 ≈ −2b2Φ . (3.10)

These limits may be compared with (2.11) and (2.10), respectively. However, because

both limits are singular this it merely a weak consistency check.

To prove that (3.1) is indeed the correct action one has to study its classical

solutions. It will be shown below that all of them globally coincide with the ones of

8In the PSM language this statement is particularly easy to derive: the number of constants of

motion is given by the dimension of the kernel of the underlying Poisson tensor. For dilaton gravity

this amounts to 1, for dilaton gravity coupled to an abelian gauge field it amounts to 2.9See, however, section 4 below: Reality of the Hawking temperature (4.6) also implies k ≥ 2

without appealing to CFT.10This follows immediately from the EOM (2.18): if dX = O(ǫ) and e± = O(1) then X± = O(ǫ).

– 239 –

DVV for any values of k, φ0 (and b) and for both branches ±. On the other hand, for

non-negative Q and M ≥ 1, (3.1) does not contain any solution in its spectrum that

is not in the DVV family,11 so it is, up to equivalence transformations, “the action

for the exact string BH”.

3.2 Proof of equivalence to DVV

For practically all applications (see sections 4 and 5 for details) the most interesting

part of the construction of the action (3.1) is the result itself. That is the rea-

son why the presentation I have chosen turns history around and starts with the

result in the previous subsection and proceeds to prove its validity in the current

one. The drawback of this is that the reader gets little insight into the actual con-

struction of the action which has been achieved in a bottom-up manner rather than

the top-down way presented here. Therefore, it is recalled at this point that the

reader interested in a bottom-up construction is invited to consult the appendices.

In particular, appendix A recalls the no-go result, including the pivotal feature of

dilaton-shift invariance, and provides some first hints how to circumvent it; appendix

B demonstrates the crucial idea of “integrating in” an abelian BF term in order to

promote what otherwise would be a parameter of the action to a constant of motion,

namely the constant φ0. This is a vital requirement because otherwise there would

be a one-parameter family of actions, labelled by φ0, and dilaton-shift invariance

would not map a solution to another one, but move within that parameter family.

Equipped with the considerations of the appendices one may now reverse-engineer

the action from the knowledge of all classical solutions, see section 2.3 for details. To

follow this reverse-engineering one may essentially read this subsection backwards.

Regardless of how the (rather irrelevant) scaling- and shift-ambiguity contained in

the functions I and w (cf. (2.17)) is fixed one obtains uniquely12 the potentials U

and V as presented in (3.5).

11There are some isolated solutions, so-called “constant dilaton vacua”, X = X+ = X− = 0,

B = Q which yield AdS2 with curvature scalar r = −4b2 dγ/ dΦ = −2b2 for the ESBH branch and

a singular result for the ESNS branch, equivalent to the JT limit. This can be understood easily

because V = 0, the necessary condition for a constant dilaton vacuum, implies γ = 0, the limit

invoked to obtain (3.9). Therefore, these isolated solutions do not reveal new geometries, but they

do yield a new solution for the dilaton, namely X = 0.12If one weakens the requirements and one considers not just the ESBH/ESNS but a conformal

equivalence class (such that all members of this class are related by a conformal transformation with

a conformal factor which is regular globally but which may be singular in the asymptotic region

Φ → ∞) then only a certain combination of the two potentials remains unique, namely the function

w as defined in (2.17), calculated below in (3.13). For certain applications the consideration of such

equivalence classes may be sufficient. The most convenient representant of this class typically is the

one where the transformed potential U vanishes. The corresponding potential V is written below

in (3.30).

– 240 –

3.2.1 All classical solutions derived from the action

We now proceed to obtain globally all classical solutions derived from the action

(3.1) and to demonstrate that they coincide with the ones obtained by DVV.

As a first step one can eliminate the pair A, B and replace the latter by its on-

shell value (3.7) in all other EOM. As the potential in (3.1) does not depend on B

explicitly the abelian field strength vanishes on-shell, F = 0. Thus, not only the

role of B but also the one of A is solely of auxiliary nature. The remaining EOM

obtained by variation with respect to Φ, X±, ω and e± may be solved like in section

2.3. To this end one has to determine first the functions I±, w± as defined in (2.17).

By plugging U± into the left definition (2.17) and exploiting the relation

dΦ±

dγ= 1 ± 1

√

γ2 + 1(3.11)

one obtains13

I± =1

2b

1

1 ±√

γ2 + 1, (3.12)

where the multiplicative ambiguity from the integration constant inherent to the

left definition (2.17) has been fixed by introducing the scale factor 1/(2b). It is

emphasized that it is completely irrelevant how to fix this factor; the choice in (3.12)

has been made for later convenience.

Now this result and the other potential V may be plugged into the right definition

(2.17). This implies

w± = ∓b√

γ2 + 1 + w0 (3.13)

It turns out to be helpful to fix the integration constant w0 in (3.13) such that

w± = − 1

2I±. (3.14)

This can be achieved with

w0 = −b . (3.15)

Again the choice for (3.15) is merely dictated by convenience. Plugging (3.12) and

(3.14) into the general solution (2.24) promotes it to

ds2± =

2 du dΦ±

2b(

1 ±√

γ2 + 1) +

1 +C(g)

b(

1 ±√

γ2 + 1)

du2 . (3.16)

This is precisely the line element of the ESBH (+) and the ESNS (−), as will be

made explicit in section 3.2.2.

13For I− the multiplicative factor has been chosen to be negative. Consequently, I− is manifestly

negative. The positive side effect of this choice is that one may still consider the change between

ESBH and ESNS as a switch of branches of the square root function.

– 241 –

X=0

i

I

I_

+

0

Bi

I

I_

+

0

X=

0Figure 1: CP diagrams of EF patches for the ESBH (left) and the ESNS (right).

The line element (3.16) covers a basic EF patch (see fig. 1; the points i0, I+, I−

and B denote spatial infinity, future light-like infinity, past light-like infinity and the

bifurcation point, respectively; the Killing horizon is denoted by the dashed line, the

boundary X = 0, which is regular for the ESBH and singular for the ESNS, by a bold

line; curved lines are X = const. hypersurfaces). The global Carter-Penrose (CP)

diagram may be obtained by well-known methods [28, 31]: the basic idea is that by

simple mirror flips (e.g. by assuming that X− 6= 0 instead of assuming X+ 6= 0 in a

given patch) one achieves almost a universal covering. The only exceptional points14

are those where X+ = 0 = X−. Around each of these points one may use a Kruskal-

like gauge to get an open region containing them. For the ESNS no such complication

arises as there is no horizon, while for the ESBH exactly one such point exists for

each solution. Consequently, the CP diagram of the ESBH is very similar to the one

of the 2D part of the Schwarzschild BH, except that there is no singularity inside

the BH region for the ESBH. Although straightforward, rather than performing the

construction of the maximally extended space-time as outlined in this paragraph one

may consult refs. [13, 14].

3.2.2 Coordinate transformations to the ESBH

Some simple transformations are invoked in order to bring (3.16) to a more familiar

form. With the Casimir C(g) as parametrized in (3.8), the definition

w± := ±w := ±√

γ2 + 1 , (3.17)

and the relationsdΦ±

dw±

=

√

w± + 1

w± − 1, (3.18)

the line element (3.16) may be written as

ds2± =

2 du dw

2b√

w2 − 1+

(

1 − k

1 ± w

)

du2 . (3.19)

14In the language of general relativity these points are known as bifurcation points B, e.g. the

bifurcation 2-sphere in the CP diagram of the Schwarzschild BH.

– 242 –

An alternative, even simpler, representation of the line element is provided by the

transformation w = cosh (2bR):

ds2± = 2 du dR + ξ±(R) du2 (3.20)

with the Killing norm

ξ±(R) = 1 − 2M

1 ± cosh (2bR). (3.21)

This is equivalent to the one by DVV, where + refers to the ESBH and − to its

dual solution, the ESNS; this can be seen easily for the ESBH: by virtue of the

coordinate transformation15 dτ =√

1 − 2/k (du+dR/ξ+(R)) and cosh (2bR) = ((k−2) cosh2 (bx) + 1) one obtains with

dR =√

1 − 2/k f(x) dx (3.22)

the line element

ds2+ = f 2(x) dτ 2 − dx2 , (3.23)

where

f(x) =tanh (bx)

√

1 − p tanh2 (bx). (3.24)

This is identical to (1.5) with (1.2).

3.2.3 The dilaton field

At this point it is recalled that it is crucial for the ESBH action to produce both,

the dilaton field (1.3) and the line element (1.1), (1.2). It is possible, for instance,

to construct actions which yield the correct dilaton field but only approximate the

appropriate line element (for a concrete example cf. appendix B.1 in [15]).

Having established that (3.1) produces the correct line element, the dilaton field

remains to be discussed. The “true” dilaton field X may be obtained from (3.3)

together with (3.7) in terms of γ. For the ESBH one may now relate γ with the

coordinate x by virtue of the transformations (3.17)-(3.22):

√

X2/B2 + 1 =√

γ2 + 1 = w = cosh (2bR) = (k − 2) cosh2 (bx) + 1 (3.25)

On-shell this leads to

X = e−2φ0 cosh2 (bx)

√

(1 − 2/k) + 2/k cosh−2 (bx) . (3.26)

15Note that this last coordinate transformation exhibits the usual coordinate singularity on a

Killing horizon ξ+(R) = 0 that arises always in the transition from EF gauge to diagonal gauge.

Besides this expected singularity there are no further ones for k > 2. Note that all previous

coordinate redefinitions were well-defined globally. An analogous transformation can be applied for

the ESNS.

– 243 –

Undoing the exponential representation implied by (2.7) and inserting p = 2/k

amounts to

φ = φ0 − ln cosh (bx) − 1

4ln (1 − p tanh2 (bx)) , (3.27)

which is equivalent to (1.3).

3.3 Discussion and reformulations of the action

The main result (3.1) displays several unexpected features: while in retrospect it

may seem obvious that an abelian BF term is capable to circumvent the no-go

result of [15], it is slightly surprising that neither U nor V depend explicitly on B

if expressed in terms of Φ. This is in contrast to both examples in section 2.2 and

also in contrast to the Cangemi-Jackiw formulation of Verlinde’s first order version

of the conformally transformed Witten BH, which motivated the introduction of

an abelian BF term for the ESBH in the first place (cf. appendix B). It suggests

strongly that the auxiliary dilaton field Φ, rather than the “true” dilaton field X,

should be taken as primary degree of freedom, in which case the abelian BF term

decouples completely and may be integrated out without leaving a trace in the action.

Therefore, the constant φ0 – or, equivalently, the U(1) charge B = Q – must not play

any physical role in the absence of matter. Indeed, as will be shown in section 4,

neither mass, nor temperature, nor entropy depend on it; the same holds for all other

quantities of physical interest discussed below, like Killing-norm, curvature, specific

heat, evaporation time or free energy.16 Additionally, the strictly monotonous but

non-algebraic relation (3.3)-(3.4) between the dilaton fields Φ and X is not something

that could have been anticipated a priori. The explicit form of V and U is far less

surprising, but the relation between ESBH and ESNS via N+N− = 1 is interesting

and maybe deserves a deeper explanation.

A parenthetical remark concerns the use of the first order formulation to derive

the action (3.1): as shown in section 2.3 and as witnessed by several precedents (for

a recent example compare e.g. [25] with [26]) the first order formulation seems to

be the most adequate language to describe 2D dilaton gravity. The current paper

may also be considered as a demonstration of this assertion. But of course, as both

versions of the theory are physically equivalent, somebody might have performed

analogous steps to arrive directly at a second order action. The next paragraphs are

devoted to the second order formalism.

16It is somewhat ironic that the crucial step to circumvent the no-go result of [15] has been the

introduction of an abelian BF term which in retrospect, upon proper identification of what should

be regarded as primary degree of freedom in the dilaton sector, decouples completely from the

theory and may be integrated out again. However, as soon as matter is coupled to the system the

value of φ0 (and hence the field B) may be of physical significance – for instance, φ0 may play the

role of a relative coupling constant between geometric and matter action; this issue is addressed at

the end of section 5.3.

– 244 –

There are three second order formulations of the action (3.1)-(3.6) which may be

useful in various contexts, albeit the first one, (3.28) below, appears to be superior

to the other two due to its simplicity and because it invokes directly the dilaton field

Φ rather than X. We will focus exclusively on the ESBH, but again the results for

the ESNS follow straightforwardly from switching to the second branch of the square

root function. Eliminating the auxiliary fields Xa and the spin connection ω yields

S(2.1)ESBH = −1

2

∫

M2

d2x√−g

[

Φ r − 2B f + U(Φ)(∇Φ)2 − 2V (Φ)]

, (3.28)

where g is the determinant of the metric gµν with respect to which the covariant

derivative ∇ is torsion free and metric compatible. The curvature scalar r is multi-

plied by the auxiliary dilaton Φ which obeys the relations (3.2)-(3.4). The auxiliary

field B on-shell is constant, cf. (3.7), while the 2-form field strength is (Hodge) dual

to f , ∗F = f . The potentials U, V in (3.28) are equivalent to the ones in the first

order formulation, (3.5) with (3.6). Inserting explicitly the dilaton X and using the

exponential representation X = exp (−2φ) leads to the action

S(2.2)ESBH = −1

2

∫

M2

d2x√−g

e−2φ

B

[

(

1 + Be2φarcsinhe−2φ

B

)

r − 2B2e2φ f

−(

1 +1

√

e−4φ/B2 + 1

)2B2e4φ

N+(e−2φ/B)

(

∇e−2φ

B

)2

+ 4b2

]

, (3.29)

Note that variation with respect to the Maxwell-field still results in B = const.

on-shell.

It may also be of use to transform (3.1) to a conformal frame where U = 0 be-

cause some of the physical observables may depend only on the linear combination

w in (2.17) which is conformally invariant. It is emphasized that two conformally

related theories are inequivalent, in general, especially because the conformal factor

necessarily becomes singular for Φ → ∞. Fixing the multiplicative constant inher-

ent in I such that I = 1/(2b), the property IV = IV = dw/ dΦ implies for the

transformed potentials

V = − 2b2γ

1 +√

γ2 + 1, U = 0 , (3.30)

to be inserted into (3.1) or into (3.28) instead of V and U , respectively. Also the

transformed version of (3.29) simplifies considerably:

S(2.3)ESBH = −1

2

∫

M2

d2x√−ge−2φ

[

(

1

B+ e2φarcsinh

e−2φ

B

)

r − 2Be2φ f

+4b2

B +√

e−4φ + B2

]

, (3.31)

– 245 –

This may be a convenient starting point for the construction of a Born-Infeld like

action: variation with respect to B establishes a non-differential equation in B in

terms of φ, r and f . Plugging the solution (which need not be unique) back into

(3.31) then produces an action which is highly non-linear in curvature r and field

strength f . However, such non-linearities are undesirable because thermodynamical

discussion (which involves the evaluation of boundary terms), supersymmetrization

and quantization are impaired, if not rendered impossible. A comparable action of

this type, even without Maxwell field, may be constructed as follows:17 in section

2 of ref. [32] it was shown that the Witten BH may be transformed into the ESBH

with some non-linear field redefinitions of metric and dilaton, containing curvature

and derivative terms of the dilaton field non-polynomially. Applying the same field

redefinitions to the leading order action (which describes the Witten BH) yields

an action for the ESBH which inherits these non-polynomialities. By contrast, the

actions (3.1), (3.28), (3.29) and (3.31) are all linear in curvature and contain no higher

derivatives than second ones. This difference is crucial for most of the subsequent

applications. In fact, one may consider linearity in curvature as a conditio sine qua

non for a profitable non-perturbative action.

It has been argued in section 3.1 that the weak and strong coupling limits cor-

rectly produce the Witten BH and JT model, respectively. As a simple cross-check

it is now studied to what extent this holds at the level of solutions, i.e., whether

or not curvature as derived e.g. from (3.20) approaches the appropriate limits. The

curvature scalar r is given by (minus) the second derivative of the Killing norm

(3.21):

r± = −d2ξ±dR2

= 8b2Mcosh2 (2bR) − 2 ∓ cosh (2bR)

(1 ± cosh (2bR))3 (3.32)

Nota bene: r+ remains bounded for all R ∈ (−∞,∞) and r− is singular at R = 0.

If evaluated at the Killing horizon r+ reduces to

r+|R=Rhorizon= 4b2

(

1 − 3

2M

)

. (3.33)

In the JT limit R becomes very small, 2bR = ε ≪ 1, and mass goes to M → 1.

Curvature then simplifies considerably:

r+ = −2b2(1 + O(ε)) , r− ∝ ε−4(1 + O(ε)), . (3.34)

As expected the ESBH tends to AdS2 while the ESNS becomes singular. In the

Witten BH limit R and M become large, 2bR ≫ 1, M ≫ 1, and curvature reduces

to

r± = ±16b2Me−2bR(1 + O(e−2bR)) , (3.35)

17I am grateful to Arkady Tseytlin for providing this argument.

– 246 –

which correctly describes the Witten BH. Self-duality is apparent in (3.35). Regard-

ing the global structure it is recalled that for the ESBH the line X = 0 is regular

while for the ESNS it is singular (cf. fig. 1). However, in the limiting cases these

properties change: the Witten BH has the same CP diagram as the ESBH, except

that the line X = 0 is singular; on the other hand, the JT model implies a CP

diagram similar to the one of the ESNS, except that the line X = 0 is regular and

space-time is not asymptotically flat; instead it is AdS2, so globally the CP diagram

has the form of a vertical strip rather than triangular shape. These discontinuous

changes of the causal structure in the limiting cases k = 2 and k = ∞ (or, from (1.4),

α′ = ∞ and α′ = 0, respectively) concur with prior observations on the singularity

of these limits.

Let us now consider the dual to the JT model in detail. For small γ one obtains

N−|γ≪1 ≈γ2

4, Φ−|γ≪1 ≈

γ3

6, (3.36)

and consequently the potentials read

U−|γ≪1 ≈ − 2

3Φ−

, V |γ≪1 ≈ −2b2(6Φ−)1/3 . (3.37)

This is a special case of the so-called ab-family [33] with a = 2/3 and b = −1/3.

Because of a = b+1 it is also a Minkowskian ground state model. Somewhat surpris-

ingly, spherical reduction of the 5D Einstein-Hilbert action produces the same model.

Thus, winding/momentum mode duality connects AdS2 with the 5D Schwarzschild

BH.

There are two more points worthwhile to address: It is an unexpected result

that mass as given by (3.8) is determined by the level k rather than by the value

of the dilaton at the origin, φ0 – on the issue of mass cf. sections 4.2 and 5.2. The

introduction of an abelian gauge field has been a crucial input to circumvent the

no-go result of ref. [15]. For details on this idea the appendices A and B may be

consulted.

4. Thermodynamical properties

Now that a suitable action for the ESBH exists one may employ it to calculate ther-

modynamical quantities of interest: ADM mass, Hawking temperature and Beken-

stein–Hawking entropy. In each case first a simple and then a more elaborate

derivation will be provided or at least sketched. A comparison with previous re-

sults [2, 12, 13, 29, 34, 35] will be postponed until section 5.

4.1 Mass

We follow the prescription of the appendix of [30] to obtain the ADM mass which

slightly generalizes the procedure presented in [29]. The “zeroth step” is evident

– 247 –

from (3.12)-(3.13) and just amounts to the definition of the functions I and w, where

already a convenient scale factor has been included by hand in the definition of I. The

“first step”, i.e., the identification of the ground state geometry, has been performed

in (3.15), implying a Minkowskian ground state. Such an identification is not just

convenient, but necessary if by “mass” we mean “ADM mass”. The asymptotic

region to be obtained in the “second step” is located at X → ∞ (or alternatively

γ → ∞ or Φ → ∞); the scaling ambiguity already has been fixed, and (3.14) implies

that mass is measured in units of the asymptotic Killing time. The ADM mass is

given by

MADM = −C(g) = bk =2b

p=

1 + 2α′b2

α′b= 2bM . (4.1)

In this manner the mass defined in (3.8) is already the ADM mass up to the scale

factor introduced by hand in (3.12).

One may derive (4.1) also by considerations a la Gibbons-Hawking from a bound-

ary term [36]. In the context of 2D dilaton gravity this procedure is described in

detail e.g. in section 5 of [16]. Following it for the ESBH the result is (cf. (5.10)

in [16])

MADM = limR→∞

√

ξ(R)(

1 −√

ξ(R))

∂RΦ , (4.2)

where ξ(R) = ξ+(R) may be read off from (3.21). Actually, the least trivial term in

(4.2) is the last one: the auxiliary dilaton field Φ enters here because it is the one

which multiplies the Ricci scalar in the action – and for the calculation of the ADM

mass from a boundary term only the contribution ΦR to the action is of importance

(to prevent notationally induced hazard: R refers here to the curvature 2-form, while

in the rest of this paragraph it denotes the coordinate introduced in (3.20)). For large

values of R the relations

∂RΦ ≈ be2bR ,√

ξ(R) ≈ 1 − 2Me−2bR (4.3)

imply (4.1) when plugged into (4.2). If the action (3.1) is multiplied by some overall

constant κ then also the boundary term and hence the right hand side in (4.2) are

multiplied by κ, so MADM → κMADM. One has to bear in mind this simple fact when

applying different conventions for the action.

It should be emphasized that the ADM mass depends only on the parameter b

and the string-coupling α′. In particular, it does not depend on φ0. For the Witten

BH mass is usually identified with some function of φ0 – but the Witten BH is an

exceptional point at the edge of the ESBH family. It is easy to comprehend where

this apparent discrepancy comes from: the action (3.1) is dilaton shift invariant

according to (B.10), while the geometric part of the CGHS action (which describes

the Witten BH) is not dilaton shift invariant; rather, only the EOM are. The same

considerations apply to the corresponding boundary terms and thus in the former

case the mass becomes independent from φ0 while in the latter it does not. For sake

– 248 –

of clarity it pays to plug the limit (3.10) together with the on-shell value of B, (3.7),

into the general action (3.1):

SWBH =1

Q

∫

M2

[

XaTa + XR − ǫ

(

X+X−

X+ 2b2X

)]

, (4.4)

where X± = QX±. Apart from an overall factor of 1/Q this is the geometric part of

the CGHS model describing the Witten BH. As we have seen above an overall factor

just rescales the ADM mass, which is the reason for the aforementioned absence of

φ0 in (4.1). It is stressed that there does not seem to be a “bottom-up” way to fix

the overall factor (which may neither depend on k nor on φ0) in front of (3.1), but as

its only classical effect is the rescaling of physical units this is of limited relevance.

Because it is nice to work with a dimensionless mass in most of the subsequent

considerations the quantity M ∈ [1,∞) will be used, so one has to bear in mind the

scale factor of 2b if one would like to express these results in terms of the ADM mass.

It is recalled that the lower boundary value, M = 1, corresponds to the JT limit,

while the upper one, M → ∞, corresponds to the Witten BH limit.

4.2 Temperature

Hawking temperature as derived naively from surface gravity

TH =1

4π

∣

∣

∣

∣

dξ

dR

∣

∣

∣

∣

R=Rhorizon

, (4.5)

with (3.21) and (4.1) yields the mass-to-temperature law

TH =b

2π

√

1 − 1

M. (4.6)

Thus, for the Witten BH, M → ∞, one gets TH → b/(2π), equivalent to the standard

behavior found in the literature. For low masses, M → 1, the temperature TH

vanishes, being consistent with the fact that JT does not exhibit a horizon.

To put the result (4.6) on firmer grounds an alternative derivation [37] is pre-

sented which involves a minimally coupled massless scalar field T as “carrier” of the

Hawking quanta, the action of which, viz.

SKG =1

2

∫

M2

d2x√−ggµν (∂µT ) (∂νT ) , (4.7)

has to be added to the geometric part (3.1). Vacuum polarization effects determine

the semi-classical energy momentum tensor

Tµν =2√−g

δW

δgµν, (4.8)

– 249 –

where W is the one-loop effective action for the scalar field on the classical back-

ground manifold with metric gµν given by (3.20), (3.21). The trace anomaly18

T µµ = − 1

24πr (4.9)

together with the conservation equation

∇µTµν = 0 , (4.10)

then implies a non-vanishing flux component of the energy-momentum tensor. For a

review on this method in the context of 2D dilaton gravity cf. ref. [38]. In conformal

gauge (all ± indices in this subsection refer to the light cone)

ds2 = 2e2ρ dx+ dx− , ρ =1

2ln ξ =

1

2ln

(

1 − 2M

1 + cosh (2bR)

)

, (4.11)

the conservation equation (4.10) yields

∂+T−− + ∂−T+− − 2(∂−ρ)T+− = 0 . (4.12)

Taking into account staticity, the expression for the curvature scalar, r = −ξ′′ (prime

denotes differentiation with respect to R), as well as T µµ = 2e−2ρT+−, establishes for

the flux component

T−− =1

96π

[

2ξξ′′ − (ξ′)2]

+ t0 , (4.13)

where t0 is an integration constant. It is fixed by the (Unruh) requirement19

T−−|R=Rhorizon= 0 . (4.14)

Consequently, the asymptotic flux is given by

T asymptotic−− = t0 =

1

96π(ξ′)2

∣

∣

∣

∣

R=Rhorizon

. (4.15)

By virtue of the 2D Stefan-Boltzmann law,

T asymptotic−− =

π

6T 2

H , (4.16)

one then derives (4.5), so this method gives the same result for the Hawking tem-

perature as the purely geometric one above.

18The notation of section 6 of ref. [16] is used, with the important exception that the Ricci scalar

there, R, corresponds to −r in the present work.19This condition ensures sufficient regularity of the energy-momentum tensor at the Killing hori-

zon in global (Kruskal) coordinates. Other choices select different vacua, e.g. t0 = 0 produces

Boulware.

– 250 –

4.3 Entropy

Simple thermodynamic considerations establish that entropy S is proportional to the

dilaton field evaluated on the Killing horizon [39], where “the dilaton field” again

refers to the factor multiplying the Ricci scalar in the action. Thus, for the ESBH

one has to evaluate Φ at the Killing horizon. With the same convention for the

Boltzmann constant as in [39] the result is

S = 2πΦ|2M=1+

√γ2+1

= 4π

(

√

M(M − 1) +1

2arcsinh (2

√

M(M − 1))

)

. (4.17)

This may be understood most easily from dS = dMADM/TH = 4π dM/√

1 − 1/M ,

which upon integration yields

S = S0 + 4π(

√

M(M − 1) + arctanh√

1 − 1/M)

. (4.18)

Setting S0 = 0 and using simple trigonometric identities for hyperbolic functions,

(4.18) is seen to be equivalent to (4.17). In the Witten BH limit it simplifies to

S|M≫1 = 4πM (1 + O(ln (M)/M)) ≈ SLO = 4πM , (4.19)

while for the JT limit one gets

S|M=1+ε = 8π√

ε (1 + O(ε)) . (4.20)

Therefore, as might have been anticipated, for k → 2 entropy vanishes. For k → ∞it is worthwhile to display also the next to leading order term:

S|M≫1 = SLO + 2π ln SLO + O(1) (4.21)

Hence, only in the weak coupling regime (α′ ≪ 1) the expected [40, 41] qualitative

behavior of entropy (4.21) is recovered, while in the strong coupling regime (α′ ≫ 1)

no logarithmic corrections to entropy do arise.

A different derivation of entropy is provided by counting of microstates with CFT

methods using the Cardy formula for the asymptotic density of states (cf. e.g. [42]).

We will sketch here a more recent study [43] tailor made for 2D dilaton gravity.

The main feature of this approach is the imposition of a “stretched horizon”, i.e., a

surface which is “almost null”, the “almost” being parametrized by a small parameter

ǫ. Translated to our notation this implies X+X− = O(ǫ) ≪ 1. A Hamiltonian

analysis is then performed with the boundary condition that a (stretched) horizon

must exist. The constraints generate a Virasoro algebra with central charge c = O(ǫ)

– 251 –

which vanishes in the limit ǫ → 0. Thus, at first glance the Cardy formula20 [44]

S = π

√

c

6∆ , (4.22)

where ∆ is the Eigenvalue of the Virasoro operator L0 for which entropy S is being

calculated, appears to produce a vanishing entropy. However, ∆ turns out to be

proportional to 1/ǫ. To be more concrete, the results of [43] are21

c = 24πǫ Φh , ∆ =Φh

πǫ(1 + O(ǫ)) , (4.23)

where Φh is the dilaton Φ restricted to the Killing horizon. Inserting (4.23) into

(4.22) confirms (4.17). It is emphasized that in all approaches the specific form of U

and V is essentially irrelevant; only the ΦR term in the action matters.

5. Conclusions and generalizations

The main result of this work is the action for the ESBH/ESNS, (3.1)-(3.6). It may be

considered as a non-perturbative generalization of the geometric part of the CGHS

model [5] valid for all values of the string-coupling α′, while retaining the pivotal

property of linearity in curvature. It is worthwhile mentioning that the potentials

U, V in (3.5) are unique. The form of (3.1) suggests to take the auxiliary dilaton Φ

more seriously and to treat it as a “genuine” dilaton field. In that case the BF term

decouples and may be eliminated. Such an action (or, alternatively, its second order

cousin (3.28)) may be a useful starting point for adding matter degrees of freedom,

supersymmetrization, quantization, etc. Subsequently a few possible applications

are pointed out and some speculations are presented, although the list by no means

is meant to be exhaustive.

5.1 Supplementary thermodynamical considerations

Comparing the results of section 4 with available ones in the literature [2, 12, 13, 29,

34, 35] almost exclusively disagreement is found. This should not come as a shock,

because the derivation of the ADM mass and of the entropy crucially depends on

the knowledge of the correct ΦR term in the action, hitherto unavailable. In most

20In [43] it has been assumed that the lowest Eigenvalue of L0, denoted by ∆g, vanishes. One

can weaken this requirement and assume that it scales with O(ǫ2) and thus is small as compared

to c which scales with O(ǫ). Therefore, instead of c/6 − 4∆g to leading order only c/6 is present

in (4.22). Similarly, the term ∆ − c/24 to leading order is just given by ∆. Note that there is a

relative factor of 1/2 as compared to [43] because the action (3.28) contains a relative factor of 1/2

as compared to (4) in [43] for 16πG = 1.21Various approaches yield different results for c and ∆. However, in their product these ambi-

guities always seem to disappear.

– 252 –

previous derivations it was assumed explicitly or implicitly that the corresponding

term in the action reads XR, with X as given by (3.3) off-shell and by (3.26) on-

shell. Only regarding Hawking temperature there is some agreement:22 the result

(4.6) coincides with equation (3.12) in ref. [12] and with equation (2.10) in ref. [14].

This is to be expected because for the derivation of Hawking temperature via surface

gravity only knowledge about the metric is needed, but not about the action. So

the first obvious application of the action (3.1) was a reliable derivation of various

thermodynamical quantities, as performed in section 4, which finally clarified what

is the ADM mass (4.1), mass-to-temperature law (4.6) and entropy (4.17) of the

ESBH.

It is worthwhile mentioning that the ADM mass need not constitute the most

natural mass definition. It is beneficial if the ground state geometry is Minkowski

space. But if, say, a BPS solution exists it is often more convenient to shift the mass

such that M(BPS) = 0 and masses of non-BPS solutions are measured with respect

to it.23 Also the ESBH has a ground state geometry which is not Minkowski space,24

namely AdS2 for M = 1. The mass definition

MAdS := MADM − 2b = 2b(M − 1) (5.1)

yields MAdS = 0 for the ground state geometry. In the Witten BH limit the difference

between MAdS and MADM is negligible. For certain applications the “AdS-mass” (5.1)

may be more appropriate than the ADM mass (4.1). From (4.2) it may be derived

easily that for the ESNS the ADM mass is negative because the second term in the

right equation of (4.3) changes sign. This implies a mass gap of 4b between the

ESBH and the ESNS spectra.

The naively defined specific heat of BHs sometimes exhibits remarkable behavior

– for instance, for the Schwarzschild BH it is negative. Therefore, it is worthwhile to

consider the specific heat for the ESBH,

Cs :=dMADM

dTH=

16π2

bM2 TH . (5.2)

So the ESBH behaves like an electron gas at low temperature (TH → 0, M → 1)

with Sommerfeld constant γ = 16π2/b. Of particular relevance is the large mass

22Although there is no universal agreement among the previous literature on that issue. For

instance, [1] gets (translated to our notation) TH ∝√

M − 1, while [13] obtains TH ∝ 1/√

M .23E.g. for the Reissner Nordstrom BH the ADM mass of the extremal solution is given by

MADM = |Q|, so shifting to M := MADM − |Q| yields M = 0 for extremal solutions. For M ≫ Q

the difference between the two definitions is negligible.24Only analytic continuation to k < 2 may lead to Minkowski space, which emerges from the limit

k → 0. Pushing this further to negative k yields C(g) < 0. In that case there are Killing horizons for

the “ESNS”, but not for the “ESBH”. The range k ∈ (2,∞) corresponds to D ∈ (−∞, 26), while

k ∈ [0, 2) implies D ∈ [29,∞) and k ∈ (−∞, 0] leads to D ∈ (26, 29]. Clearly, k = 2 (or D = 26) is

an exceptional case. Amusingly, the Minkowski space solution formally requires D = 29.

– 253 –

limit, because the uncorrected Witten BH has a vanishing inverse specific heat and

thus a finite result is a non-trivial effect:

Cs|M≫1 =2π

b2M2

ADM (1 + O(1/MADM)) (5.3)

It is positive and proportional to the mass squared. Strikingly, up to a numerical

factor of 24π this is precisely the result of [45] obtained by completely different

methods which apply to the quantum corrected Witten BH only. In this fashion

leading order quantum corrections of the field theoretic approach in [45] qualitatively

reveal the same behavior as the stringy corrections implicit in the ESBH.

The Hawking evaporation according to (4.6) implies a loss of mass and thus the

ESBH evolves to another ESBH with lower value of k; once k = 2 is reached Hawking

radiation stops, suggesting AdS2 as endpoint of Hawking evaporation. From a string

theoretical point of view it appears to be difficult to interpret what an evaporation

to a lower level k means – whether this is a defect of the description of the ESBH via

the action (3.1) or a new feature predicted by it.25 But as in the case of the CGHS

model [5] one may “forget” about the origin of the action and treat it as a model

on its own. Then, there is absolutely no interpretational problem with evaporation

to lower k. The time interval ∆t to evaporate from an initial mass M to some final

mass M0 according todM

dt= −π

6T 2

H (5.4)

upon integration yields

∆t =24π

b2

(

M − M0 + lnM − 1

M0 − 1

)

. (5.5)

Therefore, evaporation to AdS2 takes infinitely long, concurrent with its role as

ground state geometry. For large masses the result (5.5) essentially coincides with

the one derived in [45]. On a more speculative side note, it may be rewarding to

consider the exceptional solutions mentioned in footnote 11, viz., the constant dilaton

vacua, as possible end points of the evaporation process. Their geometry is equivalent

to the one of the ground state, AdS2, but the dilaton X vanishes identically. It has

been argued in a different context that at the end point of Hawking evaporation

a phase transition to a constant dilaton vacuum may occur [46] and it would be

excellent to see this falsified or confirmed in a string theoretical derivation.

It could be of interest to determine other quantities, like the free energy F . One

can derive from it e.g. the Euclidean action I = F/TH and the partition function

25A possibility to avoid that temperature (and hence the level k) is a fixed quantity rather than a

free parameter (as required for evaporation) has been addressed below equation (3.12) in [12]: one

may assume that temperature is changed by varying the number of extra massless “matter” fields

that can be added to the system, thereby changing the effective central charge. See also section

5.2.

– 254 –

Z = exp (−I). Taking into account that AdS2 is the ground state geometry the

definition

F := MAdS − THS = MADM − 2b − THS (5.6)

seems to be preferred. Plugging in the results for mass (4.1), temperature (4.6) and

entropy (4.17) yields

F = −b

√

1 − 1

Marcsinh

(

2√

M(M − 1))

. (5.7)

Therefore, F is negative apart from the limiting case M = 1 where F is zero. There

is no extremum of F in the range M ∈ [1,∞). It might be worthwhile to perform a

more elaborate analysis of free energy, e.g. in analogy to [12]. Finally, it could be an

interesting exercise to put the ESBH into a cavity of finite size in order to achieve

an equilibrium and to carry out a thermodynamical study of the combined system

ESBH plus radiation. For recent reviews on BH thermodynamics cf. e.g. [47] and

refs. therein.

5.2 Applications in 2D string theory

It is natural to inquire about implications for 2D string theory. First, the role of k as

a constant of motion is assessed critically, next the absence of higher derivative terms

in the action is highlighted and contrasted with perturbative results, and finally a

list of miscellaneous remarks and speculations is presented.

In string theory the level k typically is a fixed quantity, while in the theory con-

structed in the present work it emerges as a constant of motion, essentially the ADM

mass. This apparent discrepancy urgently asks for some explanation. Therefore, let

us first try to reinterpret k as a parameter of the action: reverting the arguments in

appendix B in general it is possible to integrate out certain fields replacing them by

their on-shell values. For instance, in the paragraph containing (B.3)-(B.7) a con-

stant of motion stemming from an abelian BF -term is converted into a parameter of

the action (and vice versa). However, this particular manoeuvre is impossible for the

constant of motion corresponding to the mass. So we have to live with the fact that

k is not fixed in the action (3.1) or one of its reformulations. I will now try to argue

that one should not only accept this conclusion but embrace it. There is actually a

physical reason why k defines the mass: in the presence of matter the conservation

equation dC(g) = 0 acquires a matter contribution,

dC(g) + W (m) = 0 , (5.8)

where W (m) = dC(m) is an exact 1-form defined by the energy-momentum tensor

(cf. section 5 of [16] or ref. [48]). In a nutshell, the addition of matter deforms

the total mass which now consists of a geometric and a matter part, C(g) and C(m),

respectively. Coming back to the ESBH, the interpretation of k as mass according

– 255 –

to the preceding discussion implies that the addition of matter should “deform” k.

But this is precisely what happens: adding matter will in general change the central

charge and hence the level k. Thus, from an intrinsically 2D dilaton gravity point

of view the interpretation of k as mass is not only possible but favoured (see also

footnote 25).

The absence of higher order derivatives in (3.1) has been a crucial ingredient

for its construction and especially for thermodynamical applications, but it may be

difficult to comprehend from a perturbative point of view; after all, an expansion

in powers of α′ yields arbitrary powers in curvature in the β functions and con-

sequently also in corresponding effective actions [49]. So how is it possible that

non-perturbatively curvature appears only linearly? To get some insight, consider a

simple model (2.3) with U = 0 and V ∝ Xn with n 6= 0, 1. On-shell, the equation

r ∝ Xn−1 modulo branch ambiguities allows to eliminate the dilaton field thus ob-

taining an action non-linear in curvature, with a Lagrangean proportional to rn/(n−1).

If V is not a monomial but a more complicated function similar considerations apply

and one obtains an action which may contain an arbitrary Laurent series in r. Know-

ing just (some terms of) the Laurent series, it may be difficult to induce the simpler

non-perturbative expression linear in r – but if the latter is available one may deduce

the perturbative results, although it will be a somewhat tedious task. By analogy,26

it may be difficult to induce from 3- or 4-loop results the non-perturbative action

(3.1), but it should be doable to derive these perturbative results from (3.1). Be-

cause it has been shown in the present work that all classical solutions of the action

(3.1)-(3.6) coincide globally (!) with the ESBH/ESNS by DVV, it is as reliable as

the ESBH itself. So if there were reasons to doubt the validity of the latter of course

also the former is obsolete; on the other hand, if one trusts the ESBH – the working

hypothesis of the current work – one may equally trust (3.1).

Here are some additional loose ends:

• It would be gratifying to get a better understanding of the (winding/momentum

mode) duality between AdS2 (cf. (2.11)) and Schwarzschild in 5D (cf. (3.37)).

To this end one may take advantage of the strong coupling limit α′ → ∞.

• From (3.33) it is evident that k = 2M = 3 is special insofar as curvature

vanishes at the horizon; for k > 3 (k < 3) it is positive (negative). Incidentally,

26It is really just an analogy but its message hopefully is transparent: in 2D dilaton gravity

various formulations of the same model may exist and equivalence between two seemingly different

models with different field content may not be easy to spot just by looking at the actions. The

safest check is a global comparison of all solutions (a local comparison is inadequate since all 2D

geometries are locally conformally flat). If they coincide the models are classically equivalent. The

procedure in the text above eliminates the dilaton field which is not desirable for comparison with

results from string theory, but if one follows it from (3.4) and (3.5) it is clear that arbitrary powers

in r will arise.

– 256 –

in the SL(2)/U(1) CFT the same value of k separates two regions: the CFT

exhibits a normalizable zero mode which for k ≤ 3 becomes non-normalizable

(cf. section 7.3 in ref. [50]). This may be either coincidence or of importance

for recent discussions on (non)existence of BHs in 2D string theory [50–52].

• The construction of a Born-Infeld like action in terms of dilaton, metric and

Maxwell field as outlined below (3.31) and a comparison with the non-linear

action only in terms of dilaton and metric described there may be of interest;

however, for such an action the caveats mentioned in the same paragraph apply

and limit its pertinence.

• A connection with the 3D charged black strings of Horne and Horowitz [53] has

been spelled out in [14]: dimensional reduction leads to a 2D model exhibiting

one PPDOF, which does not describe solely the ESBH. But its static solutions

coincide with the ESBH. It could be useful to check whether the action (3.1)

may arise from a different kind of reduction of 3D strings.

• Some applications require spacetime to be asymptotically AdS, so one may

study a conformally transformed version of the ESBH which for large Φ ap-

proaches AdS2, the ground state solution according to previous discussion. If

the asymptotic value for curvature,

rasy = 2I−1 d

dΦ

(

I−1 d

dΦ(Iw)

)

, (5.9)

is required to be constant and the conformally invariant function w in (2.17)

behaves asymptotically like w = −b Φβ with β 6= 1 then the non-invariant

function must behave asymptotically as I ∝ Φβ−2, which may be achieved by

an appropriate Φ-dependent conformal factor that is regular globally except

for the asymptotic region Φ → ∞. More concretely, any two metrics gµν and

gµν of the form (2.24) which differ only by the non-invariant function, I and

I, respectively, are connected by a conformal transformation gµν = Ω2gµν with

Ω2 = I/I. If the same procedure is applied for β = 1 then (5.9) not only leads

to constant but to vanishing curvature. Unfortunately this applies to the ESBH

as seen from (3.13) which for large Φ yields w+ = −b Φ + . . . . Nevertheless,

with the asymptotic choice

Ω|Φ→∞∝ 1

ln Φ(5.10)

for the conformal factor the ESBH may be transformed from the asymptotically

flat frame (3.20), (3.21) to an asymptotically AdS2 frame. The transformed

version of (3.12) for large Φ behaves as I ∝ 1/(Φ ln2 Φ) + . . . . The proportion-

ality constant in (5.10) determines the asymptotic value of rasy.

– 257 –

5.3 Supersymmetrization, critical collapse and quantization

Supersymmetrization [54–56] is possible if, and only if, the potential V in (2.2) may

be expressed in terms of a pre-potential u such that

V (Φ) = −1

8

(

u2(Φ)U(Φ) +d

dΦu2(Φ)

)

. (5.11)

Once the pre-potential is defined one may apply a standard machinery [56, 57] to

obtain the supersymmetry transformations and all classical solutions including the

BPS states. Because it is not the purpose of the present work to review these

techniques the focus is solely on the pre-potential. Provided the relation (3.14) holds

the pre-potential always exists.27 For the ESBH (+)/ESNS (−) it reads

u±(Φ±) = −4w±(Φ±) = 4b(

1 ±√

γ2 + 1)

. (5.12)

It may be checked that the relations du2/ dΦ = 32b2γ and u2U = −16b2γ are valid

and thus the correct potential V = −2b2γ is recovered from (5.11). For sake of

consistency the limiting cases may be studied at the level of the pre-potential. The

weak coupling limit

u±|γ≫1 ≈ ±4b γ ≈ ±4b Φ (5.13)

implies the correct pre-potential of the Witten BH, as expected. The strong coupling

limit yields

u+|γ≪1 ≈ 8b + 2b γ2 ≈ 8b + b Φ2+/2 , u−|γ≪1 ≈ −2b γ2 ≈ −2b (6Φ−)2/3 . (5.14)

In u+ the next-to-leading order term has to be considered because otherwise a wrong

result for V is obtained since the leading order term is constant and therefore does

not contribute to du2+/ dΦ+. Although the pre-potential u+ is not the one of the

JT model (which is linear in the dilaton), nevertheless to leading order V as derived

from u+ displays the correct (linear) behavior. It is reassuring that indeed u− is the

pre-potential for the spherically reduced 5D Schwarzschild BH.28

Coupling to matter degrees of freedom makes the theory non-topological in gen-

eral. Integrability is lost apart from certain special cases (like the Witten BH or

27This statement probably is not completely obvious although its derivation is simple: Insert-

ing into the definitions (2.17), (5.11) and taking the positive root yields u(Φ) =√

−8w(Φ)/I(Φ)

(cf. e.g. (B.12) in ref. [57]). For generic w and I a real pre-potential need not exist. However,

if (3.14) is true then the argument of the square root is non-negative. Note that under the shift

u2(Φ) → u2(Φ)+β/I(Φ) for arbitrary β ∈ R the potential V is invariant. Thus, for a given bosonic

model the pre-potential is not unique. This ambiguity corresponds to the freedom to choose w0 in

(3.13). It has been fixed conveniently in (5.12). I am grateful to L. Bergamin for correspondence

on this subject.28For spherically reduced gravity from D dimensions (D > 3) the pre-potential reads u(Φ) ∝

Φ(D−3)/(D−2). Inserting D = 5 essentially yields u− in (5.14).

– 258 –

JT). Thus, dynamics is richer but also more complicated to describe. Already the

addition to (3.1) of the action for a single massless scalar field T ,

SgenKG =

∫

M2

d2x√−g F (Φ) gµν (∂µT ) (∂νT ) , (5.15)

which generalizes (4.7) slightly by allowing for nonminimal coupling to the dilaton

via F (Φ), is capable to introduce a new physical phenomenon: critical collapse. For

instance, the spherically reduced Einstein-massless-Klein-Gordon model (V = const.,

U = −1/(2Φ), F ∝ Φ) leads to the famous Choptuik-scaling [58]

MBH ∝ (p − p∗)γ , (5.16)

where p ∈ [0, 1] is a free parameter characterizing a one-parameter family of initial

data with the property that for p < p∗ a BH never forms while for p > p∗ a BH

always forms with mass MBH determined by (5.16) for p sufficiently close to p∗.

The critical parameter p∗ ∈ (0, 1) may be found by elaborate numerical analysis

and depends on the specific family under consideration; but the critical exponent

γ ≈ 0.37 is universal, albeit model dependent. Other systems may display a different

critical behavior, cf. the review ref. [59]. The critical solution p = p∗ in general

exhibits remarkable features, e.g. discrete or continuous self-similarity and a naked

singularity. It may be interesting to perform similar numerical studies for the ESBH

coupled e.g. to a scalar field (4.7), (5.20) or (5.15), to obtain critical exponents and

to study their dependence on the level k.

A particular example of coupling to matter, namely to the tachyon, is now ad-

dressed with possible implications for 2D type 0A/0B string theory. For the matrix

model description of 2D type 0A/0B string theory cf. [60,61] (for an extensive review

on Liouville theory and its relation to matrix models and strings in 2D cf. [62]; some

earlier reviews are refs. [63]; the matrix model for the 2D Euclidean string BH has

been constructed in [64]). Although some of the subsequent considerations may have

implications for matrix models their thorough discussion will not be attempted in

the present work. Recently [61,65] the low energy effective action for 2D type 0A/0B

string theory in the presence of RR fluxes has been studied from various aspects. For

sake of definiteness henceforth focus will be on 2D type 0A with an equal number q

of electric and magnetic D0 branes. The corresponding action in the second order

formulation reads (remember (2.7))

S0A =

∫

M2

d2x√−g

[

Xr

2− (∇X)2

2X+ 2b2X − b2q2

8π

]

+ ST , (5.17)

where T denotes the tachyon. The translation into the first order form is straight-

forward [30],

I(X) =1

X, w(X) = −2b2X +

b2q2

8πln X , (5.18)

– 259 –

because of

U(X) = − 1

X, V (X) = −2b2X +

b2q2

8π. (5.19)

The action defining the tachyon sector up to second order in T is given by

ST =1

2

∫

M2

d2x√−g [F (X)gµν(∂µT )(∂νT ) + f(X, T )] , (5.20)

with

F (X) = X , f(T , X) = b2T 2

(

X − q2

2π

)

. (5.21)

The geometric part in (5.17) generalizes the Witten BH due to the inclusion of RR

fluxes (compare (5.19) with (2.10)). The tachyon action (5.20) introduces a PPDOF,

the tachyon field. This suggests a twofold generalization of (3.1): one may simply

add to V in (3.5) a term corresponding to RR fluxes – it might be just a constant

proportional to q2 as in the Witten BH limit (5.19) or a more complicated function

depending on Φ which only for k → ∞ approaches this constant. In principle the

term could depend on B as well, thus breaking dilaton-shift invariance discussed in

(B.10); in that case it is possible to circumvent the introduction of q in the action by

adjusting φ0 in (3.7) appropriately.29 The second possibility involves no guess work

and consists of the addition of the tachyon action (5.20) to (3.1) with X replaced

by γB, where γ can be expressed as a function of Φ by inverting (3.4). This may

be an interesting model on its own.30 If B is replaced by its on-shell value (3.7) it

is evident that the constant φ0 now plays the role of a relative coupling constant

between geometry and the tachyon sector. Therefore, as soon as the tachyon enters

the game the abelian BF term in (3.1) ceases to be of purely auxiliar nature and

acquires a physical status.

Finally, if one treats it as a model from scratch one may wish to quantize the

ESBH action (3.1) and/or its supersymmetrized version with pre-potential (5.12),

possibly supplemented by matter degrees of freedom. This was performed for generic

supersymmetric models31 in ref. [68], so in a sense the ESBH had been quantized

before its action was constructed.

29As demonstrated in appendix B the parameter q (or bq) may be eliminated from the action

by “integrating in” a Maxwell field. Rather than introducing a new one the existing one may be

exploited, e.g. by adding to V a term quadratic in B.30A remark is in order: according to the discussion in section 3 of [32] the one-loop form of

the tachyon β-function is an exact result, provided metric and dilaton of the ESBH are the exact

solutions derived from equating the corresponding β-functions to zero. Because the solutions of

(3.1) reproduce the ESBH (without involving non-linear or non-local field redefinitions), equation

(5.21) may be considered as an exact result for the effective Tachyon action, at least for q = 0.31Some “bosonic references” on path integral quantization of generic dilaton gravity with matter

are [16, 66]. In the absence of matter the theory is locally quantum trivial [67].

– 260 –

Acknowledgments

I would like to thank Dima Vassilevich for collaboration on the no-go result regarding

an action for the ESBH, as well as for helpful discussion. I am grateful to Sergei

Alexandrov for useful correspondence and for drawing Dima’s and my attention to

the ESBH 3 years ago. I thank Arkady Tseytlin for pointing out and discussing [32].

I am deeply indebted to Wolfgang Kummer for introducing me to the subject of 2D

dilaton gravity 7 years ago. This work has been supported by an Erwin-Schrodinger

fellowship, project J-2330-N08 of the Austrian Science Foundation (FWF). Some

ideas presented in this work emerged during a visit at MIT in December 2004, and I

am grateful to the CTP group for support and for an inspiring atmosphere, in partic-

ular to Mauro Brigante, Daniel Freedman, Alfredo Iorio, Roman Jackiw and Carlos

Nunez. The final preparations of this paper have been performed just before the

“Wolfgangfest” in January 2005 at the Vienna University of Technology, supported

by project P-16030-N08 of the FWF.

A. No-go recap

The no-go result obtained in [15] relies upon the following assumption: the starting

point has been an action (2.1) with some generic function V, i.e., not restricted to

the simpler form (2.2). Then it has been noted that the integration constant φ0

enters only the dilaton field (1.3), but not the metric (1.5). Therefore, a symmetry

property exists which proved very important: a constant shift of the dilaton φ maps

a solution to another one of the same model. It has been shown next that this allows

only two classes of V:

V(1) = V (X+X−) , V(2) = XU

(

X+X−

X2

)

(A.1)

For V(1) dilaton shift invariance acted additively, i.e., X ∝ φ, while for V(2) it acted

multiplicatively, i.e., X ∝ exp (−2αφ) with some non-vanishing α ∈ R. The first

possibility has been excluded immediately,32 while the second one required further

investigation. Actually, prospects for V(2) did not seem bad at first glance, because

both the Witten BH and the JT model are of that form (with U(Z) = −Z −2b2 and

U(Z) = −b2, respectively). Nevertheless, by constructing the classical solutions and

comparing with the ESBH it could be shown that for no value of α the whole family

of ESBH solutions may be produced. An approximative model found in this way was

interesting on its own and mimicked several important properties of the ESBH, but

it was not “the real stuff”. Still, several of the technical details spelled out in [15]

32While the conclusion is correct that no potential of type V(1) reproduces the ESBH, the argu-

ment in [15] is a bit too simple. But, along the lines of the no-go result for V(2) one can provide a

more elaborated argument that leads to the same conclusions for V(1).

– 261 –

turned out to be very profitable for the construction of (3.1), in particular the ones

contained in section 4.1 and appendix B of that work. Finally, it has been suggested

in the outlook that the consideration of either non-localities or matter degrees of

freedom could help to circumvent the no-go result. The introduction of the latter in

general would imply additional PPDOF and thus a qualitative change as compared

to the “pure” dilaton gravity case, where no PPDOF are present.

There is an important exception to the “rule” that adding matter implies adding

PPDOF: in two dimensions gauge fields do not carry PPDOF, which weakly suggests

to add some gauge field(s) to (2.1) in order to circumvent the no-go result without

destroying the topological nature of the theory. However, there are infinitely many

possibilities of adding gauge fields, so a supplementary selection criterion is needed.

This is provided by the considerations below.

B. The art of gauging constants

Suppose that V in (2.2) depends on a real parameter λ, V = V (X, λ). It is possible

to eliminate it by “integrating in” an abelian BF term such that on-shell B = λ

and the potential reads V = V (X, B). If one encounters more parameters one can

introduce a different abelian BF term for each of them. This implies an additional

abelian gauge symmetry for each parameter eliminated in this way. As a simple

example the spherically reduced Schwarzschild BH may be considered. Its potentials

read

U(X)(SSBH) = − 1

2X, V (X)(SSBH) = −λ2 . (B.1)

The scale parameter λ is not very relevant because it just defines the physical units

of the surface area. Still, one my opt to eliminate it from the action, which is possible

by integrating in a BF term. Thus, the spherically reduced Schwarzschild BH may

be derived from a parameter free action of type (2.12) with33

V(SSBH)(X+X−, X, B) = −X+X−

2X− B2 , (B.2)

where on-shell B = λ.

This technique to the best of my knowledge was first introduced by Cangemi

and Jackiw [69] while formulating the conformally transformed string inspired CGHS

model as a gauge theory based upon the centrally extended Poincare algebra,

[Pa, Pb] = ǫabλI , [Pa, J ] = ǫabPb , [I, Pa] = [I, J ] = 0 , (B.3)

where Pa are generators of translation, J generates boosts and λI := Z is the central

extension. Introducing the connection A = ωJ +eaPa+AZ and Lagrange multipliers

33A brief digression: It seems that this additional U(1) gauge symmetry arising in the

Schwarzschild BH has not been addressed in detail before in the literature. While it is a some-

what trivial feature it may still be of use.

– 262 –

XA = (X, Xa, B) transforming under the coadjoint representation, the non-abelian

BF theory

S(CG) =

∫

XAFA , (B.4)

with FA being the components of the curvature 2-form F = dA+[A,A]/2 = dω J +

(dea + εabω ∧ eb)Pa + (dA − ǫ)Z, yields a particular Maxwell-dilaton gravity model

(2.12) where34

V(CG) = −B , (B.5)

a pre-cursor of which had been proposed earlier by Verlinde [71], based upon the

non-extended version of (B.3),

[Pa, Pb] = 0 , [Pa, J ] = ǫabPb , (B.6)

leading to an action which in opposition to (B.4) explicitly depends on the parameter

λ,

S(V ) =

∫

[

XAFA + 2λǫ]

, (B.7)

where XA and AA are the same as before except for the now absent Maxwell-

component, B and A, respectively.

One can convince oneself immediately that integrating out A in (B.4) yields

dB = 0, and setting B = −2λ produces (B.7). Thus, turning the argument around,

one may “integrate in” a pair A, B in order to eliminate a parameter from the

potential V. This generalizes readily to arbitrary theories of 2D dilaton gravity.

Therefore, the appearance of a parameter in the action, like φ0 from (1.3), need not

present a problem because with the trick above it can be converted into a constant

of motion at the cost of introducing an abelian BF term.

With this lesson in mind we reconsider the no-go result (see appendix A) and

the crucial ingredient to it, dilaton shift invariance (implemented multiplicatively).

For the ESBH there are three parameters: k, φ0 and b. One (combination) of them

must emerge as “mass”, while the other ones may be either parameters of the action

or, by integrating in BF terms, constants of motion. We choose to keep b as a

parameter of the action and thus have to consider a single BF term only. Starting

with a Lagrangian of the form (2.12) one may require global invariance of the classical

EOM under

X → λX , Xa → λXa , B → λB , (B.8)

34Compare e.g. with (36) of ref. [70], noting that χA there corresponds to XA here. The con-

struction of the sentence around this footnote – a trifle too long for the hasty reader as it starts

before (B.4) and ends after (B.7) – is inspired by the vigor of Wolfgang’s grammatical skills and a

tribute to the good old days when sentences were allowed to frolic for a while before converging to

a full stop.

– 263 –

while the gauge fields ω, ea, A do not transform. This restricts to potentials of the

type

V(3) = XU

(

XaXa

B2,

X

B

)

, (B.9)

with some arbitrary two argument function U , as opposed to the more constrained

form of V(2) in (A.1). But this means actually that V(3) may be considered as

arbitrary function of XaXa and X as long as appropriate factors of B are attached.

Thus, one of the restrictions that has been pivotal to the no-go result no longer

is present. As it turned out, the lack of this restriction already was sufficient to

construct the action (3.1). However, dilaton shift invariance is implemented in a

slightly different way there:

X → λX , Xa → Xa , B → λB , A → λ−1A (B.10)

Therefore, the XaTa term is invariant (rather than being multiplied with λ as implied

by (B.8)) and hence for consistency all other terms in the action should be invariant,

too. This is possible if X appears only in the combination X/B, which is indeed the

case. Hence, dilaton shift invariance does not only leave the EOM invariant, but also

the action (3.1) and is therefore a global symmetry.

A final remark is in order: the action (3.1) still depends on the parameter b, so

by the same token as above one may introduce a second BF term eliminating it.

This second BF term corresponds to the one introduced by Cangemi and Jackiw,

while the one present in (3.1) may be considered as eliminating φ0 from the action,

as seen from the on-shell value of B in (3.7).

References

[1] R. Dijkgraaf, H. Verlinde, and E. Verlinde, String propagation in a black hole

geometry, Nucl. Phys. B371 (1992) 269–314.

[2] E. Witten, On string theory and black holes, Phys. Rev. D44 (1991) 314–324.

[3] G. Mandal, A. M. Sengupta, and S. R. Wadia, Classical solutions of two-dimensional

string theory, Mod. Phys. Lett. A6 (1991) 1685–1692.

[4] S. Elitzur, A. Forge, and E. Rabinovici, Some global aspects of string

compactifications, Nucl. Phys. B359 (1991) 581–610.

[5] C. G. Callan, Jr., S. B. Giddings, J. A. Harvey, and A. Strominger, Evanescent black

holes, Phys. Rev. D45 (1992) 1005–1009, [hep-th/9111056].

[6] A. A. Tseytlin, On the form of the black hole solution in d = 2 theory, Phys. Lett.

B268 (1991) 175–178.

– 264 –

[7] I. Jack, D. R. T. Jones, and J. Panvel, Exact bosonic and supersymmetric string

black hole solutions, Nucl. Phys. B393 (1993) 95–110,

[http://arXiv.org/abs/hep-th/9201039].

[8] C. Teitelboim, Gravitation and Hamiltonian structure in two space-time dimensions,

Phys. Lett. B126 (1983) 41;

R. Jackiw, Lower dimensional gravity, Nucl. Phys. B252 (1985) 343–356.

[9] A. A. Tseytlin, Effective action of gauged WZW model and exact string solutions,

Nucl. Phys. B399 (1993) 601–622, [hep-th/9301015].

[10] I. Bars and K. Sfetsos, Exact effective action and space-time geometry in gauged

WZW models, Phys. Rev. D48 (1993) 844–852, [hep-th/9301047].

[11] K. Becker, Strings, black holes and conformal field theory,

http://arXiv.org/abs/hep-th/9404157.

[12] V. A. Kazakov and A. A. Tseytlin, On free energy of 2-d black hole in bosonic string

theory, JHEP 06 (2001) 021, [http://arXiv.org/abs/hep-th/0104138].

[13] M. J. Perry and E. Teo, Nonsingularity of the exact two-dimensional string black

hole, Phys. Rev. Lett. 70 (1993) 2669–2672,

[http://arXiv.org/abs/hep-th/9302037].

[14] P. Yi, Nonsingular 2-D black holes and classical string backgrounds, Phys. Rev. D48

(1993) 2777–2788, [http://arXiv.org/abs/hep-th/9302070].

[15] D. Grumiller and D. V. Vassilevich, Non-existence of a dilaton gravity action for the

exact string black hole, JHEP 11 (2002) 018,

[http://arXiv.org/abs/hep-th/0210060].

[16] D. Grumiller, W. Kummer, and D. V. Vassilevich, Dilaton gravity in two dimensions,

Phys. Rept. 369 (2002) 327–429, [http://arXiv.org/abs/hep-th/0204253].

[17] P. Schaller and T. Strobl, Poisson structure induced (topological) field theories, Mod.

Phys. Lett. A9 (1994) 3129–3136, [http://arXiv.org/abs/hep-th/9405110].

[18] J. G. Russo and A. A. Tseytlin, Scalar tensor quantum gravity in two-dimensions,

Nucl. Phys. B382 (1992) 259–275, [arXiv:hep-th/9201021].

[19] T. Banks and M. O’Loughlin, Two-dimensional quantum gravity in Minkowski space,

Nucl. Phys. B362 (1991) 649–664.

[20] S. D. Odintsov and I. L. Shapiro, One loop renormalization of two-dimensional

induced quantum gravity, Phys. Lett. B263 (1991) 183–189.

[21] V. P. Frolov, Two-dimensional black hole physics, Phys. Rev. D46 (1992) 5383–5394.

[22] R. B. Mann, Conservation laws and 2-d black holes in dilaton gravity, Phys. Rev.

D47 (1993) 4438–4442, [hep-th/9206044].

– 265 –

[23] H. Grosse, W. Kummer, P. Presnajder, and D. J. Schwarz, Novel symmetry of

nonEinsteinian gravity in two- dimensions, J. Math. Phys. 33 (1992) 3892–3900,

[hep-th/9205071].

[24] T. Strobl, Gravity in two spacetime dimensions, hep-th/0011240. Habilitation

thesis.

[25] G. Guralnik, A. Iorio, R. Jackiw, and S. Y. Pi, Dimensionally reduced gravitational

Chern-Simons term and its kink, Ann. Phys. 308 (2003) 222–236, [hep-th/0305117].

[26] D. Grumiller and W. Kummer, The classical solutions of the dimensionally reduced

gravitational Chern-Simons theory, Ann. Phys. 308 (2003) 211–221,

[hep-th/0306036].

[27] L. Bergamin, D. Grumiller, A. Iorio, and C. Nunez, Chemistry of Chern-Simons

supergravity: Reduction to a BPS kink, oxidation to M-theory and thermodynamical

aspects, JHEP 11 (2004) 021, [hep-th/0409273].

[28] T. Klosch and T. Strobl, Classical and quantum gravity in 1+1 dimensions. Part II:

The universal coverings, Class. Quant. Grav. 13 (1996) 2395–2422,

[arXiv:gr-qc/9511081].

[29] H. Liebl, D. V. Vassilevich, and S. Alexandrov, Hawking radiation and masses in

generalized dilaton theories, Class. Quant. Grav. 14 (1997) 889–904,

[arXiv:gr-qc/9605044].

[30] D. Grumiller and D. Mayerhofer, On static solutions in 2d dilaton gravity with scalar

matter, Class. Quant. Grav. 21 (2004) 5893–5914, [gr-qc/0404013].

[31] M. Walker, Block diagrams and the extension of timelike two-surfaces, J. Math.

Phys. 11 (1970) 2280.

[32] A. A. Tseytlin, On field redefinitions and exact solutions in string theory, Phys. Lett.

B317 (1993) 559–564, [hep-th/9308042].

[33] M. O. Katanaev, W. Kummer, and H. Liebl, On the completeness of the black hole

singularity in 2d dilaton theories, Nucl. Phys. B486 (1997) 353–370,

[gr-qc/9602040].

[34] G. W. Gibbons and M. J. Perry, The Physics of 2-D stringy space-times, Int. J.

Mod. Phys. D1 (1992) 335–354, [http://arXiv.org/abs/hep-th/9204090].

[35] C. R. Nappi and A. Pasquinucci, Thermodynamics of two-dimensional black holes,

Mod. Phys. Lett. A7 (1992) 3337–3346, [http://arXiv.org/abs/gr-qc/9208002].

[36] G. W. Gibbons and S. W. Hawking, Action integrals and partition functions in

quantum gravity, Phys. Rev. D15 (1977) 2752–2756.

[37] S. M. Christensen and S. A. Fulling, Trace anomalies and the Hawking effect, Phys.

Rev. D15 (1977) 2088–2104.

– 266 –

[38] W. Kummer and D. V. Vassilevich, Hawking radiation from dilaton gravity in (1+1)

dimensions: A pedagogical review, Annalen Phys. 8 (1999) 801–827,

[http://arXiv.org/abs/gr-qc/9907041].

[39] J. Gegenberg, G. Kunstatter, and D. Louis-Martinez, Observables for

two-dimensional black holes, Phys. Rev. D51 (1995) 1781–1786,

[http://arXiv.org/abs/gr-qc/9408015].

[40] T. M. Fiola, J. Preskill, A. Strominger, and S. P. Trivedi, Black hole

thermodynamics and information loss in two- dimensions, Phys. Rev. D50 (1994)

3987–4014, [hep-th/9403137];

R. C. Myers, Black hole entropy in two-dimensions, Phys. Rev. D50 (1994)

6412–6421, [hep-th/9405162];

J. D. Hayward, Entropy in the RST model, Phys. Rev. D52 (1995) 2239–2244,

[gr-qc/9412065].

[41] O. B. Zaslavsky, Quantum corrections to temperature and mass of 1+1 dilatonic

black holes and the trace anomaly, Phys. Lett. B375 (1996) 43–46.

[42] A. Strominger, AdS(2) quantum gravity and string theory, JHEP 01 (1999) 007,

[hep-th/9809027];

M. Cadoni and S. Mignemi, Entropy of 2d black holes from counting microstates,

Phys. Rev. D59 (1999) 081501, [hep-th/9810251];

S. Carlip, Black hole entropy from conformal field theory in any dimension, Phys.

Rev. Lett. 82 (1999) 2828–2831, [hep-th/9812013];

S. N. Solodukhin, Conformal description of horizon’s states, Phys. Lett. B454 (1999)

213–222, [hep-th/9812056].

[43] S. Carlip, Horizon constraints and black hole entropy, hep-th/0408123.

[44] J. L. Cardy, Operator content of two-dimensional conformally invariant theories,

Nucl. Phys. B270 (1986) 186–204;

H. W. J. Bloete, J. L. Cardy, and M. P. Nightingale, Conformal invariance, the

central charge, and universal finite size amplitudes at criticality, Phys. Rev. Lett. 56

(1986) 742–745.

[45] D. Grumiller, W. Kummer, and D. V. Vassilevich, Positive specific heat of the

quantum corrected dilaton black hole, JHEP 07 (2003) 009, [hep-th/0305036].

[46] D. Grumiller, Long time black hole evaporation with bounded Hawking flux, JCAP 05

(2004) 005, [gr-qc/0307005].

[47] R. M. Wald, The thermodynamics of black holes, Living Rev. Rel. 4 (2001) 6,

[gr-qc/9912119];

– 267 –

T. Padmanabhan, Gravity and the thermodynamics of horizons, Phys. Rept. 406

(2005) 49–125, [gr-qc/0311036];

D. N. Page, Hawking radiation and black hole thermodynamics, hep-th/0409024.

[48] W. Kummer and P. Widerin, Conserved quasilocal quantities and general covariant

theories in two-dimensions, Phys. Rev. D52 (1995) 6965–6975, [gr-qc/9502031].

[49] C. G. Callan, Jr., E. J. Martinec, M. J. Perry, and D. Friedan, Strings in background

fields, Nucl. Phys. B262 (1985) 593.

[50] J. L. Karczmarek, J. Maldacena, and A. Strominger, Black hole non-formation in

the matrix model, hep-th/0411174.

[51] E. Martinec and K. Okuyama, Scattered results in 2D string theory, JHEP 10 (2004)

065, [hep-th/0407136].

[52] J. J. Friess and H. Verlinde, Hawking effect in 2-D string theory, hep-th/0411100.

[53] J. H. Horne and G. T. Horowitz, Exact black string solutions in three-dimensions,

Nucl. Phys. B368 (1992) 444–462, [hep-th/9108001].

[54] Y.-C. Park and A. Strominger, Supersymmetry and positive energy in classical and

quantum two-dimensional dilaton gravity, Phys. Rev. D47 (1993) 1569–1575,

[arXiv:hep-th/9210017].

[55] N. Ikeda, Gauge theory based on nonlinear Lie superalgebras and structure of 2-d

dilaton supergravity, Int. J. Mod. Phys. A9 (1994) 1137–1152.

[56] T. Strobl, Target-superspace in 2d dilatonic supergravity, Phys. Lett. B460 (1999)

87–93, [arXiv:hep-th/9906230];

M. Ertl, W. Kummer, and T. Strobl, General two-dimensional supergravity from

Poisson superalgebras, JHEP 01 (2001) 042, [arXiv:hep-th/0012219];

L. Bergamin and W. Kummer, Graded Poisson sigma models and dilaton-deformed

2d supergravity algebra, JHEP 05 (2003) 074, [hep-th/0209209]; The complete

solution of 2D superfield supergravity from graded Poisson-Sigma models and the

super pointparticle, Phys. Rev. D68 (2003) 104005, [hep-th/0306217].

[57] L. Bergamin, D. Grumiller, and W. Kummer, Supersymmetric black holes in 2d

dilaton supergravity: baldness and extremality, J. Phys. A37 (2004) 3881–3901,

[hep-th/0310006].

[58] M. W. Choptuik, Universality and scaling in gravitational collapse of a massless

scalar field, Phys. Rev. Lett. 70 (1993) 9–12.

[59] C. Gundlach, Critical phenomena in gravitational collapse, Adv. Theor. Math. Phys.

2 (1998) 1–49, [arXiv:gr-qc/9712084].

– 268 –

[60] T. Takayanagi and N. Toumbas, A matrix model dual of type 0B string theory in two

dimensions, JHEP 07 (2003) 064, [hep-th/0307083].

[61] M. R. Douglas et. al., A new hat for the c = 1 matrix model, hep-th/0307195.

[62] Y. Nakayama, Liouville field theory: A decade after the revolution, Int. J. Mod.

Phys. A19 (2004) 2771–2930, [hep-th/0402009].

[63] P. Ginsparg and G. W. Moore, Lectures on 2-d gravity and 2-d string theory,

http://arXiv.org/abs/hep-th/9304011;

P. Di Francesco, P. H. Ginsparg, and J. Zinn-Justin, 2-D Gravity and random

matrices, Phys. Rept. 254 (1995) 1–133, [hep-th/9306153];

S. Alexandrov, Matrix quantum mechanics and two-dimensional string theory in

non-trivial backgrounds, hep-th/0311273.

[64] V. Kazakov, I. K. Kostov, and D. Kutasov, A matrix model for the two-dimensional

black hole, Nucl. Phys. B622 (2002) 141–188, [hep-th/0101011].

[65] D. M. Thompson, AdS solutions of 2D type 0A, Phys. Rev. D70 (2004) 106001,

[hep-th/0312156];

A. Strominger, A matrix model for AdS(2), JHEP 03 (2004) 066, [hep-th/0312194];

S. Gukov, T. Takayanagi, and N. Toumbas, Flux backgrounds in 2D string theory,

JHEP 03 (2004) 017, [hep-th/0312208];

J. L. Davis, L. A. Pando Zayas, and D. Vaman, On black hole thermodynamics of

2-D type 0A, JHEP 03 (2004) 007, [hep-th/0402152];

U. H. Danielsson, J. P. Gregory, M. E. Olsson, P. Rajan, and M. Vonk, Type 0A 2D

black hole thermodynamics and the deformed matrix model, JHEP 04 (2004) 065,

[hep-th/0402192].

J. L. Davis and R. McNees, Boundary counterterms and the thermodynamics of 2-D

black holes, hep-th/0411121.

[66] W. Kummer, H. Liebl, and D. V. Vassilevich, Integrating geometry in general 2d

dilaton gravity with matter, Nucl. Phys. B544 (1999) 403–431, [hep-th/9809168];

D. Grumiller, W. Kummer, and D. V. Vassilevich, The virtual black hole in 2d

quantum gravity, Nucl. Phys. B580 (2000) 438–456, [gr-qc/0001038]; Virtual black

holes in generalized dilaton theories (and their special role in string gravity),

European Phys. J. C30 (2003) 135–143, [http://arXiv.org/abs/hep-th/0208052];

P. Fischer, D. Grumiller, W. Kummer, and D. V. Vassilevich, S-matrix for s-wave

gravitational scattering, Phys. Lett. B521 (2001) 357–363,

[http://arXiv.org/abs/gr-qc/0105034]. Erratum ibid. B532 (2002) 373;

D. Grumiller, Virtual Black Holes and the S-matrix, Int. J. Mod. Phys. D13 (2004)

1973–2002, [hep-th/0409231].

– 269 –

[67] W. Kummer, H. Liebl, and D. V. Vassilevich, Exact path integral quantization of

generic 2-d dilaton gravity, Nucl. Phys. B493 (1997) 491–502,

[http://arXiv.org/abs/gr-qc/9612012].

[68] L. Bergamin, D. Grumiller, and W. Kummer, Quantization of 2d dilaton supergravity

with matter, JHEP 05 (2004) 060, [hep-th/0404004].

[69] D. Cangemi and R. Jackiw, Gauge invariant formulations of lineal gravity, Phys.

Rev. Lett. 69 (1992) 233–236, [http://arXiv.org/abs/hep-th/9203056].

[70] A. Achucarro, Lineal gravity from planar gravity, Phys. Rev. Lett. 70 (1993)

1037–1040, [hep-th/9207108].

[71] H. Verlinde, Black holes and strings in two dimensions, in Trieste Spring School on

Strings and Quantum Gravity, pp. 178–207, April, 1991. the same lectures have been

given at MGVI in Japan, June, 1991.

– 270 –

Preprint typeset in JHEP style - HYPER VERSION LU-ITP-2006/004

Ramifications of Lineland

Daniel Grumiller∗ and Rene Meyer∗†

∗Institute for Theoretical Physics, University of Leipzig

Augustusplatz 10-11, D-04109 Leipzig, Germany

†Max Planck Institute for Mathematics in the Sciences

Inselstrasse 22, D-04103 Leipzig, Germany

E-mail: grumiller@itp.uni-leipzig.de, Rene.Meyer@itp.uni-leipzig.de.

Abstract: A non-technical overview on gravity in two dimensions is provided. Ap-

plications discussed in this work comprise 2D type 0A/0B string theory, Black Hole

evaporation/thermodynamics, toy models for quantum gravity, for numerical Gen-

eral Relativity in the context of critical collapse and for solid state analogues of

Black Holes. Mathematical relations to integrable models, non-linear gauge theories,

Poisson-sigma models, KdV surfaces and non-commutative geometry are presented.

Keywords: Black Holes in String Theory, 2D Gravity, Integrable Models.

Contents

1. Introduction 273

2. Gravity as non-linear gauge theory 274

2.1 First order formulation 274

2.2 Generic classical solutions 275

2.3 Constant dilaton vacua 277

2.4 Topological generalizations 278

2.5 Non-topological generalizations 278

3. Strings in 2D 280

3.1 Target space formulation of 2D type 0A/0B string theory 280

3.2 Exact string Black Hole 281

4. Black Holes 284

4.1 Classical analysis 284

4.2 Thermodynamics 284

4.3 Semi-classical analysis 286

4.4 Long time behavior 286

4.5 Killing horizons kill horizon degrees 287

4.6 Critical collapse 288

4.7 Quasinormal modes 290

4.8 Solid state analogues 291

5. Geometry from matter 291

5.1 Scalar matter 292

5.2 Fermionic matter 295

6. Mathematical issues 296

6.1 Remarks on the Einstein-Hilbert action in 2D 297

6.2 Relations to 3D: Chern-Simons and BTZ 297

6.3 Integrable systems, Poisson-sigma models and KdV surfaces 297

6.4 Torsion and non-metricity 298

6.5 Non-commutative gravity 299

– 272 –

1. Introduction

The study of gravity in 2D — boring to some, fascinating to others [1] — has the

undeniable disadvantage of eliminating a lot of structure that is present in higher

dimensions; for instance, the Riemann tensor is determined already by the Ricci

scalar, i.e., there is no Weyl curvature and no trace-free Ricci part. On the other

hand, it has the undeniable advantage of eliminating a lot of structure that is present

in higher dimensions; for instance, non-perturbative results may be obtained with

relative ease due to technical simplifications, thus allowing one to understand some

important conceptual issues arising in classical and quantum gravity which are uni-

versal and hence of relevance also for higher dimensions.

The scope of this non-technical overview is broad rather than focussed, since

there exist already various excellent reviews and textbooks presenting the techni-

cal pre-requisites in detail,1 and because the broadness envisaged here may lead to

a cross-fertilization between otherwise only loosely connected communities. Some

recent results are presented in more detail. It goes without saying that the topics

selected concur with the authors’ preferences; by no means it should be concluded

that an issue or a reference omitted here is devoid of interest.

The common link between all applications mentioned here is 2D dilaton gravity,2

S2DG =1

2

∫

d2x√−g

[

XR+ U(X) (∇X)2 − 2V (X)]

, (1.1)

the action of which depends functionally on the metric gµν and on the scalar field X.

Note that very often, in particular in the context of string theory, the field redefinition

X = e−2φ is employed; the field φ is the dilaton of string theory, hence the name

“dilaton gravity”. However, it is emphasized that the natural interpretation of X

need not be the one of a dilaton field — it may also play the role of surface area,

dual field strength, coordinate of a suitable target space or black hole (BH) entropy,

depending on the application. The curvature scalar R and covariant derivative ∇are associated with the Levi-Civita connection and Minkowskian signature is implied

unless stated otherwise. The potentials U , V define the model; several examples will

be provided below. A summary is contained in table 1.

This proceedings contribution is organized as follows: section 2 is devoted to a

reformulation of (1.1) as a non-linear gauge theory, which considerably simplifies the

construction of all classical solutions; section 3 discusses applications in 2D string

theory; section 4 summarizes applications in BH physics; section 5 demonstrates how

to reconstruct geometry from matter in a quantum approach; section 6 contains not

only mathematical issues but also some open problems.

1For instance, the status of the field in the late 1980ies is summarized in [2].2The 2D Einstein-Hilbert action will not be discussed except in section 6.1.

– 273 –

Model (cf. (1.1) or (2.2)) U(X) V (X) w(X) (cf. (2.3))

1. Schwarzschild [5] − 12X

−λ2 −2λ2√X

2. Jackiw-Teitelboim [6, 7] 0 −ΛX −12ΛX2

3. Witten BH/CGHS [8, 9] − 1X

−2b2X −2b2X

4. CT Witten BH [8,9] 0 −2b2 −2b2X

5. SRG (D > 3) − D−3(D−2)X

−λ2X(D−4)/(D−2) −λ2 D−2D−3

X(D−3)/(D−2)

6. (A)dS2 ground state [10] − aX

−B2X a 6= 2 : − B

2(2−a)X2−a

7. Rindler ground state [11] − aX

−B2Xa −B

2X

8. BH attractor [12] 0 −B2X−1 −B

2lnX

9. All above: ab-family [13] − aX

−B2Xa+b b 6= −1 : − B

2(b+1)Xb+1

10. Liouville gravity [14] a beαX a 6= −α : ba+α

e(a+α)X

11. Scattering trivial [15] generic 0 const.

12. Reissner-Nordstrom [16] − 12X

−λ2 + Q2

X−2λ2

√X − 2Q2/

√X

13. Schwarzschild-(A)dS [17] − 12X

−λ2 − ℓX −2λ2√X − 2

3ℓX3/2

14. Katanaev-Volovich [18] α βX2 − Λ∫ X

eαy(βy2 − Λ) dy

15. Achucarro-Ortiz [19] 0 Q2

X− J

4X3 − ΛX Q2 lnX + J8X2 − 1

2ΛX2

16. KK reduced CS [20, 21] 0 12X(c−X2) −1

8(c−X2)2

17. Symmetric kink [22] generic −XΠni=1(X

2 −X2i ) cf. [22]

18. 2D type 0A/0B [23,24] − 1X

−2b2X + b2q2

8π−2b2X + b2q2

8πlnX

19. exact string BH [25,26] (3.11) (3.11) (3.13)

Table 1: Selected list of models

2. Gravity as non-linear gauge theory

It has been known for a long time how to obtain all classical solutions of (1.1) not

only locally, but globally. Two ingredients turned out to be extremely useful: a

reformulation of (1.1) as a first order action and the imposition of a convenient

(axial or Eddington-Finkelstein type) gauge, rather than using conformal gauge.3

Subsequently we will briefly recall these methods. For a more comprehensive review

cf. [4].

2.1 First order formulation

The Jackiw-Teitelboim model (cf. the second model in table (1)) allows a gauge

theoretic formulation based upon (A)dS2,

[Pa, Pb] = ΛεabJ , [Pa, J ] = εabPb , (2.1)

with Lorentz generator J , translation generators Pa and Λ 6= 0. A corresponding

first order action, S =∫

XAFA, has been introduced in [27]. The field strength

3In string theory almost exclusively conformal gauge is used. A notable exception is [3].

– 274 –

F = dA + [A,A]/2 contains the SO(1, 2) connection A = eaPa + ωJ , and the La-

grange multipliers XA transform under the coadjoint representation. This example

is exceptional insofar as it allows a formulation in terms of a linear (Yang-Mills

type) gauge theory. Similarly, the fourth model in table 1 allows a gauge theoretic

formulation [28] based upon the centrally extended Poincare algebra [29]. The gen-

eralization to non-linear gauge theories [30] allowed a comprehensive treatment of all

models (1.1) with U = 0, which has been further generalized to U 6= 0 in [31]. The

corresponding first order gravity action

SFOG = −∫

[

XaTa +XR+ ǫ

(

X+X−U(X) + V (X))]

(2.2)

is equivalent to (1.1) (with the same potentials U, V ) upon elimination of the auxiliary

fields Xa and the torsion-dependent part of the spin-connection. Here is our notation:

ea = eaµdx

µ is the dyad 1-form. Latin indices refer to an anholonomic frame, Greek

indices to a holonomic one. The 1-form ω represents the spin-connection ωab =

εabω = εa

bωµ dxµ with the totally antisymmetric Levi-Civita symbol εab (ε01 = +1).

With the flat metric ηab in light-cone coordinates (η+− = 1 = η−+, η++ = 0 = η−−)

it reads ε±± = ±1. The torsion 2-form present in the first term of (2.2) is given by

T± = (d±ω) ∧ e±. The curvature 2-form Rab can be represented by the 2-form R

defined by Rab = εa

bR with R = dω. It appears in the second term in (2.2). Since no

confusion between 0-forms and 2-forms should arise the Ricci scalar is also denoted

by R. The volume 2-form is denoted by ǫ = e+∧e−. Signs and factors of the Hodge-∗operation are defined by ∗ǫ = 1. It should be noted that (2.2) is a specific Poisson-

sigma model [31] with a 3D target space, with target space coordinates X,X±, see

section 6.3 below. A second order action similar to (1.1) has been introduced in [32].

2.2 Generic classical solutions

It is useful to introduce the following combinations of the potentials U and V :

I(X) := exp

∫ X

U(y) dy , w(X) :=

∫ X

I(y)V (y) dy (2.3)

The integration constants may be absorbed, respectively, by rescalings and shifts of

the “mass”, see equation (2.9) below. Under dilaton dependent conformal transfor-

mations Xa → Xa/Ω, ea → eaΩ, ω → ω+Xaea d ln Ω/ dX the action (2.2) is mapped

to a new one of the same type with transformed potentials U , V . Hence, it is not

invariant. It turns out that only the combination w(X) as defined in (2.3) remains

invariant, so conformally invariant quantities may depend on w only. Note that I is

positive apart from eventual boundaries (typically, I may vanish in the asymptotic

region and/or at singularities). One may transform to a conformal frame with I = 1,

solve all equations of motion and then perform the inverse transformation. Thus, it

– 275 –

is sufficient to solve the classical equations of motion for U = 0,

dX + X−e+ − X+e− = 0 , (2.4)

(d±ω)X± ∓ e±V (X) = 0 , (2.5)

(d±ω) ∧ e± = 0 , (2.6)

which is what we are going to do now. Note that the equation containing dω is

redundant, whence it is not displayed.

Let us start with an assumption: X+ 6= 0 for a given patch. To get some

physical intuition as to what this condition could mean: the quantities Xa, which

are the Lagrange multipliers for torsion, can be expressed as directional derivatives of

the dilaton field by virtue of (2.4) (e.g. in the second order formulation a term of the

form XaXa corresponds to (∇X)2). For those who are familiar with the Newman-

Penrose formalism: for spherically reduced gravity the quantities Xa correspond to

the expansion spin coefficients ρ and ρ′ (both are real). If X+ vanishes a (Killing)

horizon is encountered and one can repeat the calculation below with indices + and

− swapped everywhere. If both vanish in an open region by virtue of (2.4) a constant

dilaton vacuum emerges, which will be addressed separately below. If both vanish on

isolated points the Killing horizon bifurcates there and a more elaborate discussion is

needed [33]. The patch implied by X+ 6= 0 is a “basic Eddington-Finkelstein patch”,

i.e., a patch with a conformal diagram which, roughly speaking, extends over half of

the bifurcate Killing horizon and exhibits a coordinate singularity on the other half.

In such a patch one may redefine e+ = X+Z with a new 1-form Z. Then (2.4) implies

e− = dX/X+ + X−Z and the volume form reads ǫ = e+ ∧ e− = Z ∧ dX. The +

component of (2.5) yields for the connection ω = − dX+/X+ + ZV (X). One of the

torsion conditions (2.6) then leads to dZ = 0, i.e., Z is closed. Locally (in fact, in the

whole patch) it is also exact: Z = du. It is emphasized that, besides the integration

of (2.8) below, this is the only integration needed! After these elementary steps one

obtains already the conformally transformed line element in Eddington-Finkelstein

(EF) gauge

ds2 = 2e+e− = 2 du dX + 2X+X− du2 , (2.7)

which nicely demonstrates the power of the first order formalism. In the final step

the combination X+X− has to be expressed as a function of X. This is possible by

noting that the linear combination X+×[(2.5) with − index] + X−×[(2.5) with +

index] together with (2.4) establishes a conservation equation,

d(X+X−) + V (X) dX = d(X+X− + w(X)) = 0 . (2.8)

Thus, there is always a conserved quantity (dM = 0), which in the original conformal

frame reads

M = −X+X−I(X)− w(X) , (2.9)

– 276 –

where the definitions (2.3) have been inserted. It should be noted that the two free

integration constants inherent to the definitions (2.3) may be absorbed by rescalings

and shifts of M , respectively. The classical solutions are labelled by M , which may

be interpreted as mass (see section 4.2). Finally, one has to transform back to the

original conformal frame (with conformal factor Ω = I(X)). The line element (2.7)

by virtue of (2.9) may be written as

ds2 = 2I(X) du dX − 2I(X)(w(X) +M) du2 . (2.10)

Evidently there is always a Killing vector K ·∂ = ∂/∂u with associated Killing norm

K2 = −2I(w +M). Since I 6= 0 Killing horizons are encountered at X = Xh where

Xh is a solution of

w(Xh) +M = 0 . (2.11)

It is recalled that (2.10) is valid in a basic EF patch, e.g., an outgoing one. By

redoing the derivation above, but starting from the assumption X− 6= 0 one may

obtain an ingoing EF patch, and by gluing together these patches appropriately one

may construct the Carter-Penrose diagram, cf. [4, 33, 34].

As pointed out in the introduction the full geometric information resides in the

Ricci scalar. The one related to the generic solution (2.10) reads

R =2

I(X)

d

dX

(

U(X)(M + w(X)) + I(X)V (X))

. (2.12)

There are two important special cases: for U = 0 the Ricci scalar simplifies to

R = 2V ′(X), while for w(X) ∝ 1/I(X) it scales proportional to the mass, R =

2MU ′(X)/I(X). The latter case comprises so-called Minkowskian ground state mod-

els (for examples cf. the first, third, fifth and last line in table 1). Note that for many

models in table 1 the potential U(X) has a singularity at X = 0 and consequently a

curvature singularity arises.

2.3 Constant dilaton vacua

For sake of completeness it should be mentioned that in addition to the family of

generic solutions (2.10), labelled by the mass M , isolated solutions may exist, so-

called constant dilaton vacua (cf. e.g. [22]), which have to obey4 X = XCDV = const.

with V (XCDV ) = 0. The corresponding geometry has constant curvature, i.e., only

Minkowski, Rindler or (A)dS2 are possible space-times for constant dilaton vacua.5

The Ricci scalar is determined by

RCDV = 2V ′(XCDV) = const. (2.13)

4Incidentally, for the generic case (2.10) the value of the dilaton on an extremal Killing horizon

is also subject to these two constraints.5In quintessence cosmology in 4D such solutions serve as late time dS4 attractor [35]. In 2D

dilaton supergravity solutions preserving both supersymmetries are necessarily constant dilaton

vacua [36].

– 277 –

Examples are provided by the last eighth entries in table 1. For instance, 2D type

0A strings with an equal number q of electric and magnetic D0 branes (cf. the

penultimate entry in table 1) allow for an AdS2 vacuum with XCDV = q2/(16π) and

RCDV = −4b2 [37].

2.4 Topological generalizations

In 2D there are neither gravitons nor photons, i.e. no propagating physical modes

exist [38]. This feature makes the inclusion of Yang-Mills fields in 2D dilaton gravity

or an extension to supergravity straightforward. Indeed, both generalizations can be

treated again in the first order formulation as a Poisson-sigma model, cf. e.g. [39]. In

addition to M (see (2.9)) more locally conserved quantities (Casimir functions) may

emerge and the integrability concept is extended.

As a simple example we include an abelian Maxwell field, i.e., instead of (2.2)

we take

SMDG = −∫

[

XaTa +XR+BF + ǫ

(

X+X−U(X,B) + V (X,B))]

, (2.14)

where B is an additional scalar field and F = dA is the field strength 2-form. Varia-

tion with respect to A immediately establishes a constant of motion, B = Q, where

Q is some real constant, the U(1) charge. Variation with respect to B may produce

a relation that allows to express B as a function of the dilaton and the dual field

strength ∗F . For example, suppose that V (X,B) = V (X) + 12B2. Then, variation

with respect to B gives B = −∗F . Inserting this back into the action yields a stan-

dard Maxwell term. The solution of the remaining equations of motion reduces to

the case without Maxwell field. One just has to replace B by its on-shell value Q in

the potentials U , V .

Concerning supergravity we just mention a couple of references for further ori-

entation [36, 40, 41].

2.5 Non-topological generalizations

To get a non-topological theory one can add scalar or fermionic matter. The action

for a real, self-interacting and non-minimally coupled scalar field T ,

ST =1

2

∫

[

F (X) dT ∧ ∗ dT + ǫf(X, T )]

, (2.15)

in our convention requires F < 0 for the kinetic term to have the correct sign;

e.g. F = −κ or F = −κX.

While scalar matter couples to the metric and the dilaton, fermions6 couple

directly to the Zweibein (A←→d B = A dB − (dA)B),

Sχ =

∫

[ i

2F (X) (∗ea) ∧ (χγa

←→d χ) + ǫH(X)g(χχ)

]

, (2.16)

6We use the same definition for the Dirac matrices as in [42].

– 278 –

but not — and this is a peculiar feature of 2D — to the spin connection. The

self-interaction is at most quartic (a constant term may be absorbed in V (X)),

g(χχ) = mχχ+ λ(χχ)2 . (2.17)

The quartic term (henceforth: Thirring term [43]) can also be recast into a classically

equivalent form by introducing an auxiliary vector potential,

λ

∫

ǫ(χχ)2 =λ

2

∫

[A ∧ ∗A+ 2A ∧ (∗ea)χγaχ] , (2.18)

which lacks a kinetic term and thus does not propagate by itself.

We speak of minimal coupling if the coupling functions F (X), f(X, T ), H(X) do

not depend on the dilaton X, and of nonminimal coupling otherwise.

As an illustration we present the spherically reduced Einstein-massless-Klein-

Gordon model (EMKG). It emerges from dimensional reduction of 4D Einstein-

Hilbert (EH) gravity (cf. the first model in table 1) with a minimally coupled scalar

field, with the choices f(X, τ) = 0 and

w(X) = −2λ2√X , F (X) = −κX , I(X) =

1√X, (2.19)

where λ is an irrelevant scale parameter and κ encodes the (also irrelevant) Newton

coupling. Minimally coupled Dirac fermions in four dimensions yield upon dimen-

sional reduction two 2-spinors coupled to each other through intertwinor terms, which

is not covered by (2.16) (see [44] for details on spherical reduction of fields of arbitrary

spin and the spherical reduced standard model).

With matter the equation of motion (2.5) and the conservation law (2.8) obtain

contributions W± = δ(ST + Sχ)/δe∓ and X−W+ +X+W−, respectively, destroying

integrability because Z is not closed anymore: dZ = W+ ∧ Z/X+. In special cases

exact solutions can be obtained:

1. For (anti-)chiral fermions and (anti-)selfdual scalars with W+ = 0 (W− = 0)

the geometric solution (2.7) is still valid [4] and the second equation of motion

(2.5) implies W− = W−u du. Such solutions have been studied e.g. in [45, 46].

They arise also in the Aichelburg-Sexl limit [47] of boosted BHs [48].

2. A one parameter family of static solutions of the EMKG has been discovered

in [49]. Studies of static solutions in generic dilaton gravity may be found

in [50, 51]. A static solution for the line-element with time-dependent scalar

field (linear in time) has been discussed for the first time in [52]. It has been

studied recently in more detail in [53].

3. A (continuously) self-similar solution of the EMKG has been discoverd in [54].

– 279 –

4. Specific models allow for exact solutions even in the presence of more general

matter sources; for instance, the conformally transformed CGHS model (fourth

in table 1), Rindler ground state models (seventh in table 1) and scattering

trivial models (eleventh in table 1).

3. Strings in 2D

Strings propagating in a 2D target space are comparatively simple to describe because

the only propagating degree of freedom is the tachyon (and if the latter is switched off

the theory becomes topological). Hence several powerful methods exist to describe

the theory efficiently, e.g. as matrix models. In particular, strings in non-trivial

backgrounds may be studied in great detail. Here are some references for further

orientation: For the matrix model description of 2D type 0A/0B string theory cf. [23,

55] (for an extensive review on Liouville theory and its relation to matrix models and

strings in 2D cf. [14]; some earlier reviews are refs. [56]; the matrix model for the 2D

Euclidean string BH has been constructed in [57]; a study of Liouville theory from

the 2D dilaton gravity point of view may be found in [58]). The low energy effective

action for 2D type 0A/0B string theory in the presence of RR fluxes has been studied

from various aspects e.g. in [23, 24, 37, 59].

3.1 Target space formulation of 2D type 0A/0B string theory

For sake of definiteness focus will be on 2D type 0A with an equal number q of electric

and magnetic D0 branes, but other cases may be studied as well. For vanishing

tachyon the corresponding target space action is given by (setting κ2 = 1)

S0A =1

2

∫

d2x√−g

[

e−2φ(

R− 4 (∇φ)2 + 4b2)

− b2q2

4π

]

, (3.1)

Obviously, this is a special case of the generic model (1.1), with U, V given by the

penultimate model in table 1, to which all subsequent considerations — in particular

thermodynamical issues — apply. Note that the dilaton fields X and φ are related

by X = exp (−2φ). The constant b2 = 2/α′ defines the physical scale. In the absence

of D0 branes, q = 0, the model simplifies to the Witten BH, cf. the third line in table

1.

The action defining the tachyon sector up to second order in T is given by

(cf. (2.15))

ST =1

2

∫

d2x√−g [F (X)gµν(∂µT )(∂νT ) + f(X, T )] , (3.2)

with

F (X) = X , f(T , X) = b2T 2

(

X − q2

2π

)

. (3.3)

The total action is S0A + ST .

– 280 –

3.2 Exact string Black Hole

The exact string black hole (ESBH) was discovered by Dijkgraaf, Verlinde and Ver-

linde more than a decade ago [25]. The construction of a target space action for it

which does not display non-localities or higher order derivatives had been an open

problem which could be solved only recently [26]. There are several advantages of

having such an action available: the main point of the ESBH is its non-perturbative

aspect, i.e., it is believed to be valid to all orders in the string-coupling α′. Thus, a

corresponding action captures non-perturbative features of string theory and allows,

among other things, a thorough discussion of ADM mass, Hawking temperature and

Bekenstein–Hawking entropy of the ESBH which otherwise requires some ad-hoc

assumption. Therefore, we will devote some space to its description. At the per-

turbative level actions approximating the ESBH are known: to lowest order in α′

one has (3.1) with q = 0. Pushing perturbative considerations further Tseytlin was

able to show that up to 3 loops the ESBH is consistent with sigma model conformal

invariance [60]. In the strong coupling regime the ESBH asymptotes to the Jackiw–

Teitelboim model [6]. The exact conformal field theory methods used in [25], based

upon the SL(2,R)/U(1) gauged Wess–Zumino–Witten model, imply the dependence

of the ESBH solutions on the level k. A different (somewhat more direct) deriva-

tion leading to the same results for dilaton and metric was presented in [61]. For a

comprehensive history and more references [62] may be consulted.

In the notation of [63] for Euclidean signature the line element of the ESBH is

given by

ds2 = f 2(x) dτ 2 + dx2 , (3.4)

with

f(x) =tanh(bx)

√

1− p tanh2(bx). (3.5)

Physical scales are adjusted by the parameter b ∈ R+ which has dimension of inverse

length. The corresponding expression for the dilaton,

φ = φ0 − ln cosh(bx)− 1

4ln(

1− p tanh2(bx))

, (3.6)

contains an integration constant φ0. Additionally, there are the following relations

between constants, string-coupling α′, level k and dimension D of string target space:

α′b2 =1

k − 2, p :=

2

k=

2α′b2

1 + 2α′b2, D − 26 + 6α′b2 = 0 . (3.7)

For D = 2 one obtains p = 89, but like in the original work [25] we will treat general

values of p ∈ (0; 1) and consider the limits p → 0 and p → 1 separately: for p = 0

one recovers the Witten BH geometry; for p = 1 the Jackiw–Teitelboim model is

obtained. Both limits exhibit singular features: for all p ∈ (0; 1) the solution is

– 281 –

regular globally, asymptotically flat and exactly one Killing horizon exists. However,

for p = 0 a curvature singularity (screened by a horizon) appears and for p = 1 space-

time fails to be asymptotically flat. In the present work exclusively the Minkowskian

version of (3.4)

ds2 = f 2(x) dτ 2 − dx2 , (3.8)

will be needed. The maximally extended space-time of this geometry has been stud-

ied in [64]. Winding/momentum mode duality implies the existence of a dual so-

lution, the Exact String Naked Singularity (ESNS), which can be acquired most

easily by replacing bx→ bx+ iπ/2, entailing in all formulas above the substitutions

sinh→ i cosh, cosh→ i sinh.

After it had been realized that the nogo result of [65] may be circumvented with-

out introducing superfluous physical degrees of freedom by adding an abelian BF -

term, a straightforward reverse-engineering procedure allowed to construct uniquely

a target space action of the form (1.1), supplemented by aforementioned BF -term,

SESBH = −∫

[

XaTa +XESBHR + ǫ

(

X+X−UESBH + VESBH

)]

−∫

BF , (3.9)

where B is a scalar field and F = dA an abelian field strength 2-form. Per construc-

tionem SESBH reproduces as classical solutions precisely (3.5)–(3.8) not only locally

but globally. A similar action has been constructed for the ESNS. The relation

(X − γ)2 = arcsinh 2γ in conjunction with the definition γ := exp (−2φ)/B may be

used to express the auxiliary dilaton field X entering the action (1.1) in terms of the

“true” dilaton field φ and the auxiliary field B. The two branches of the square root

function correspond to the ESBH (main branch) and the ESNS (second branch),

respectively:

XESBH = γ + arcsinh γ , XESNS = γ − arcsinh γ . (3.10)

The potentials read [26]

VESBH = −2b2γ , UESBH = − 1

γN+(γ), VESNS = −2b2γ , UESNS = − 1

γN−(γ),

(3.11)

with

N±(γ) = 1 +2

γ

(

1

γ±√

1 +1

γ2

)

. (3.12)

Note that N+N− = 1. The conformally invariant combination (2.3),

wESBH = −b(

1 +√

γ2 + 1)

, wESNS = −b(

1−√

γ2 + 1)

, (3.13)

of the potentials shows that the ESBH/ESNS is a Minkowskian ground state model,

w ∝ 1/I. In figure 1 the potential U is plotted as function of the auxiliary dilaton

– 282 –

1 2 3 4 5

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Figure 1: The potentials U(γ) for the ESNS, the Witten BH and the ESBH.

γ. The lowest branch is associated with the ESBH, the one on top with the ESNS

and the one in the middle with the Witten BH (i.e., the third entry in table 1). The

regularity of the ESBH is evident, as well as the convergence of all three branches

for γ → ∞, encoding (T-)self-duality of the Witten BH. For small values of the

dilaton the discrepancy between the ESBH, the ESNS and the Witten BH is very

pronounced. Note that U remains bounded globally only for the ESBH, concurring

with the absence of a curvature singularity.

The two constants of motion — mass and charge — may be parameterized by k

and φ0, respectively. Thus, the level k is not fixed a priori but rather emerges as a

constant of motion, namely essentially the ADM mass. A rough interpretation of this

— from the stringy point of view rather unexpected — result has been provided in [26]

and coincides with a similar one in [63]. There is actually a physical reason why k

defines the mass: in the presence of matter the conservation equation dM = 0 (with

M from (2.9)) acquires a matter contribution, dM = W (m), where W (m) = dC(m) is

an exact 1-form defined by the energy-momentum tensor (cf. section 5 of [4] or [66]).

In a nutshell, the addition of matter deforms the total mass which now consists of a

geometric and a matter part, M and C(m), respectively. Coming back to the ESBH,

the interpretation of k as mass according to the preceding discussion implies that the

addition of matter should “deform” k. But this is precisely what happens: adding

matter will in general change the central charge and hence the level k. Thus, from

an intrinsically 2D dilaton gravity point of view the interpretation of k as mass is

not only possible but favored.

It could be interesting to generalize the target space action of 2D type 0A/0B,

(3.1), as to include the non-perturbative corrections implicit in the ESBH by adding

(3.2) (not necessarily with the choice (3.3)) to the ESBH action (3.9). However, it

is not quite clear how to incorporate the term from the D0 branes — perturbatively

one should just add b2q2/8π to V in (3.11), but non-perturbatively this need not be

correct. More results and speculations concerning applications of the ESBH action

can be found in [26].

– 283 –

4. Black Holes

BHs are fascinating objects, both from a theoretical and an experimental point of

view [67]. Many of the features which are generic for BHs are already exhibited

by the simplest members of this species, the Schwarzschild and Reissner-Nordstrom

BHs (sometimes the Schwarzschild BH even is dubbed as “Hydrogen atom of General

Relativity”). Since both of them, after integrating out the angular part, belong to

the class of 2D dilaton gravity models (the first and twelfth model in table 1), the

study of (2.2) at the classical, semi-classical and quantum level is of considerable

importance for the physics of BHs.

4.1 Classical analysis

In section 2.2 it has been recalled briefly how to obtain all classical solutions in basic

EF patches, (2.10). By looking at the geodesics of test particles and completeness

properties it is straightforward to construct all Carter-Penrose diagrams for a generic

model (2.2) (or, equivalently, (1.1)). For a detailed description of this algorithm

cf. [4, 33, 34] and references therein.

4.2 Thermodynamics

Mass The question of how to define “the” mass in theories of gravity is notori-

ously cumbersome. A nice clarification for D = 4 is contained in [68]. The main

conceptual point is that any mass definition is meaningless without specifying 1. the

ground state space-time with respect to which mass is being measured and 2. the

physical scale in which mass units are being measured. Especially the first point is

emphasized here. In addition to being relevant on its own, a proper mass definition

is a pivotal ingredient for any thermodynamical study of BHs. Obviously, any mass-

to-temperature relation is meaningless without defining the former (and the latter).

For a large class of 2D dilaton gravities these issues have been resolved in [69]. One

of the key ingredients is the existence [70, 71] of a conserved quantity (2.9) which

has a deeper explanation in the context of first order gravity [72] and Poisson-sigma

models [31]. It establishes the necessary prerequisite for all mass definitions, but by

itself it does not yet constitute one. Ground state and scale still have to be defined.

Actually, one can take M from (2.9) provided the two ambiguities from integration

constants in (2.3) are fixed appropriately. This is described in detail in appendix

A of [51]. In those cases where this notion makes sense M then coincides with the

ADM mass.

Hawking temperature There are many ways to calculate the Hawking tempera-

ture, some of them involving the coupling to matter fields, some of them being purely

geometrical. Because of its simplicity we will restrict ourselves to a calculation of

the geometric Hawking temperature as derived from surface gravity (cf. e.g. [73]).

– 284 –

If defined in this way it turns out to be independent of the conformal frame. How-

ever, it should be noted that identifying Hawking temperature with surface gravity is

somewhat naive for space-times which are not asymptotically flat. But the difference

is just a redshift factor and for quantities like entropy or specific heat actually (4.1) is

the relevant quantity as it coincides with the period of Euclidean time (cf. e.g. [74]).

Surface gravity can be calculated by taking the normal derivative d/ dX of the Killing

norm (cf. (2.10)) evaluated on one of the Killing horizons X = Xh, where Xh is a

solution of (2.11), thus yielding

TH =1

2π

∣

∣

∣w′(X)

∣

∣

∣

X=Xh

. (4.1)

The numerical prefactor in (4.1) can be changed e.g. by a redefinition of the Boltz-

mann constant. It has been chosen in accordance with refs. [4, 75].

Entropy In 2D dilaton gravity there are various ways to calculate the Bekenstein-

Hawking entropy. Using two different methods (simple thermodynamical consid-

erations, i.e., dM = T dS, and Wald’s Noether charge technique [76]) Gegenberg,

Kunstatter and Louis-Martinez were able to calculate the entropy for rather generic

2D dilaton gravity [77]: entropy equals the dilaton field evaluated at the Killing

horizon,

S = 2πXh . (4.2)

There exist various ways to count the microstates by appealing to the Cardy formula

[78] and to recover the result (4.2). However, the true nature of these microstates

remains unknown in this approach, which is a challenging open problem. Many

different proposals have been made [79].

Specific heat By virtue of Cs = T dS/ dT the specific heat reads

Cs = 2πw′

w′′

∣

∣

∣

∣

X=Xh

= γS TH , (4.3)

with γS = 4π2 sign (w′(Xh))/w′′(Xh). Because it is determined solely by the con-

formally invariant combination of the potentials, w as defined in (2.3), specific heat

is independent of the conformal frame, too. On a curious sidenote it is mentioned

that (4.3) behaves like an electron gas at low temperature with Sommerfeld constant

γS (which in the present case may have any sign). If Cs is positive and CsT2 ≫ 1

one may calculate logarithmic corrections to the canonical entropy from thermal

fluctuations and finds [80]

Scan = 2πXh +3

2ln∣

∣

∣w′(Xh)

∣

∣

∣− 1

2ln∣

∣

∣w′′(Xh)

∣

∣

∣+ . . . . (4.4)

– 285 –

Hawking-Page like phase transition In their by now classic paper on thermo-

dynamics of BHs in AdS, Hawking and Page found a critical temperature signalling a

phase transition between a BH phase and a pure AdS phase [17]. This has engendered

much further research, mostly in the framework of the AdS/CFT correspondence (for

a review cf. [81]). This transition is displayed most clearly by a change of the specific

heat from positive to negative sign: for Schwarzschild-AdS (cf. the thirteenth entry

in table 1) the critical value of Xh is given by Xch = ℓ2/3. For Xh > Xc

h the specific

heat is positive, for Xh < Xch it is negative.7 By analogy, a similar phase transition

may be expected for other models with corresponding behavior of Cs. Interesting

speculations on a phase transition at the Hagedorn temperature Th = k/(2π) in-

duced by a tachyonic instability have been presented recently in the context of 2D

type 0A strings (cf. the penultimate model in table 1) by Olsson [83]. From equation

(22) of that work one can check easily that indeed the specific heat (at fixed q),

Cs = (q2/8)(T/Th)/(1− T/Th), changes sign at T = Th.

4.3 Semi-classical analysis

After the influential CGHS paper [9] there has been a lot of semi-classical activity in

2D, most of which is summarized in [4, 75, 84]. In many applications one considers

(1.1) coupled to a scalar field (2.15) with F = const. (minimal coupling). Tech-

nically, the crucial ingredient for 1-loop effects is the Weyl anomaly (cf. e.g. [85])

< T µµ >= R/(24π), which — together with the semi-classical conservation equation

∇µ < T µν >= 0 — allows to derive the flux component of the energy momentum

tensor after fixing some relevant integration constant related to the choice of vacuum

(e.g. Unruh, Hartle-Hawking or Boulware). This method goes back to Christensen

and Fulling [86]. For non-minimal coupling, e.g. F ∝ X, there are some important

modifications — for instance, the conservation equation no longer is valid but ac-

quires a right hand side proportional to F ′(X). The first calculation of the conformal

anomaly in that case has been performed by Mukhanov, Wipf and Zelnikov [87]. It

has been confirmed and extended e.g. in [88].

4.4 Long time behavior

The semi-classical analysis, while leading to interesting results, has the disadvan-

tage of becoming unreliable as the mass of the evaporating BH drops to zero. The

long time behavior of an evaporating BH presents a challenge to theoretical physics

and touches relevant conceptual issues of quantum gravity, such as the information

paradox. There are basically two strategies: top-down, i.e., to construct first a full

quantum theory of gravity and to discuss BH evaporation as a particular applica-

tion thereof, and bottom-up, i.e., to sidestep the difficulties inherent to the former

7Actually, in the original work [17] Hawking and Page did not invoke the specific heat directly.

The consideration of the specific heat as an indicator for a phase transition is in accordance with

the discussion in [82].

– 286 –

approach by invoking “reasonable” ad-hoc assumptions. The latter route has been

pursued in [12]. A crucial technical ingredient has been Izawa’s result [89] on con-

sistent deformations of 2D BF theory, while the most relevant physical assumption

has been boundedness of the asymptotic matter flux during the whole evaporation

process. Together with technical assumptions which can be relaxed, the dynamics of

the evaporating BH has been described by means of consistent deformations of the

underlying gauge symmetries with only one important deformation parameter. In

this manner an attractor solution, the endpoint of the evaporation process, has been

found (cf. the eighth model in table 1).

Ideologically, this resembles the exact renormalization group approach, cf. e.g.

[90, 91] and references therein, which is based upon Weinberg’s idea of “asymptotic

safety”.8 There are, however, several conceptual and technical differences, especially

regarding the truncation of “theory space”: in 4D a truncation to EH plus cosmo-

logical constant, undoubtedly a very convenient simplification, may appear to be

somewhat ad-hoc, whereas in 2D a truncation to (2.2) comprises not only infinitely

many different theories, but essentially9 all theories with the same field content as

(2.2) and the same kind of local symmetries (Lorentz transformations and diffeomor-

phisms).

The global structure of an evaporating BH can also be studied, and despite of

the differences between various approaches there seems to be partial agreement on

it, cf. e.g. [12,91,93–96]. The crucial insight might be that a BH in the mathematical

sense (i.e., an event horizon) actually never forms, but only some trapped region,

cf. figure 5 in [96].

4.5 Killing horizons kill horizon degrees

As pointed out by Carlip [97], the fact that very different approaches to explain

the entropy of BHs nevertheless agree on the result urgently asks for some deeper

explanation. Carlip’s suggestion was to consider an underlying symmetry, somehow

attached to the BH horizon, as the key ingredient, and he noted that requiring the

presence of a horizon imposes constraints on the physical phase space. Actually, the

change of the phase-space structure due to a constraint which imposes the existence

of a horizon in space-time is an issue which is of considerable interest by itself.

In a recent work [98] we could show that the classical physical phase space is

smaller as compared to the generic case if horizon constraints are imposed. Con-

versely, the number of gauge symmetries is larger for the horizon scenario. In agree-

ment with a conjecture by ’t Hooft [99], we found that physical degrees of freedom

are converted into gauge degrees of freedom at a horizon. We will now sketch the

derivation of this result briefly for the action (2.2) which differs from the one used

8In the present context also [92] should be mentioned.9Actually, one should replace in (2.2) the term X+X−U(X)+V (X) by V(X+X−, X). However,

only (2.2) allows for standard supergravity extensions [41].

– 287 –

in [98] by a (Gibbons-Hawking) boundary term. For sake of concreteness we will sup-

pose the boundary is located at x1 = const. Consistency of the variational principle

then requires

X+δe−0 +X−δe+0 +Xδω0 = 0 (4.5)

at the boundary. Note that one has to fix the parallel component of the spin-

connection at the boundary rather than the dilaton field, which is the main differ-

ence to [98]. The generic case imposes δe±0 = 0 = δω0, while a horizon allows the

alternative prescription δe−0 = X− = 0 = δω0. One can now proceed in the same

way as in [98], i.e., derive the constraints (the only boundary terms in the secondary

constraints are now X and X±, while the primary ones have none) and calculate the

constraint algebra. All primary constraints and the Lorentz constraint turn out to

be first class, even at the boundary, whereas the Poisson bracket between the two

diffeomorphism constraints (G2, G3 in the notation of [98]) acquires a boundary term

of the form

X(U ′X+X− + V ′) + U(X)X+X− − V (X) . (4.6)

Notably, it vanishes only for V ∝ X and U ∝ 1/X, e.g. for the second, third and

sixth model in table 1, i.e., (A)dS2 ground state models. The boundary constraints

for the generic case convert all primary constraints into second class constraints. The

construction of the reduced phase space works in the same way as in section 6 of [98],

thus establishing again one physical degree of freedom “living on the boundary”. Ac-

tually, this had been known already before [100]. The horizon constraints, however,

lead to more residual gauge symmetries and to a stronger fixing of free functions —

in fact, no free function remains and the reduced phase space is empty. Thus, the

physical degree of freedom living on a generic boundary is killed by a Killing horizon.

It would be interesting to generalize this physics-to-gauge conversion at a horizon

to the case with matter. Obviously, it will no longer be a Killing horizon, but one

can still employ the (trapping) horizon condition X− = 0.

4.6 Critical collapse

Critical phenomena in gravitational collapse have been discovered in the pioneering

numerical investigations of Choptuik [101]. He studied a free massless scalar field

coupled to spherically symmetric EH gravity in 4D (the EMKG) with sophisticated

numerical techniques that allowed him to analyze the transition in the space of initial

data between dispersion to infinity and the formation of a BH. Thereby the famous

scaling law

MBH ∝ (p− p∗)γ , (4.7)

has been established, where p ∈ [0, 1] is a free parameter characterizing a one-

parameter family of initial data with the property that for p < p∗ a BH never

forms while for p > p∗ a BH always forms with mass MBH determined by (4.7) for

– 288 –

p sufficiently close to p∗. The critical parameter p∗ ∈ (0, 1) may be found by elabo-

rate numerical analysis and depends on the specific family under consideration; but

the critical exponent γ ≈ 0.37 is universal, albeit model dependent. Other systems

may display a different critical behavior, cf. the review [102]. The critical solution

p = p∗, called the “Choptuon”, in general exhibits remarkable features, e.g. discrete

or continuous self-similarity and a naked singularity.

Since the original system studied by Choptuik, (2.19), is a special case of (1.1)

(with U, V as given by the first line in table 1) coupled to (2.15), it is natural to

inquire about generalizations of critical phenomena to arbitrary 2D dilaton gravity

with scalar matter. Indeed, in [103] a critical exponent γ = 1/2 has been derived

analytically for the RST model [104], a semi-classical generalization of the CGHS

model (cf. the third line in table 1). Later, in [105] critical collapse within the

CGHS model has been considered and γ ≈ 1/2 has been found numerically. More

recently the generalization of the original Choptuik system to D dimensions has

been considered [106–108]. For 3.5 ≤ D ≤ 14 the approximation γ(D) ≈ 0.47(1 −exp (−0.41D)) shows that γ increases monotonically10 with D. Since formally the

CGHS corresponds to the limit D → ∞ one may expect that γ(D) asymptotes to

the value γ ≈ 1/2.

In the remainder of this subsection we will establish evolution equations for

generic 2D dilaton gravity with scalar matter, to be implemented numerically anal-

ogously to [109,110]. In these works for various reasons Sachs-Bondi gauge has been

used. Thus we employ

e+0 = 0 , e−0 = 1 , x0 = X , (4.8)

while the remaining Zweibein components are parameterized as

e−1 = α(u,X) , e+1 = I(X)e2β(u,X) . (4.9)

In the gauge (4.8) with the parameterization (4.9) the line element reads

ds2 = 2I(X)e2β(u,X) du (dX + α(u,X) du) . (4.10)

A trapping horizon emerges either if α = 0 or β →∞. The equations of motion may

be reduced to the following set:

Slicing condition : ∂Xα(u,X) = −e2β(u,X)w′(X) (4.11)

Hamiltonian constraint : ∂Xβ(u,X) = −F (X)(∂XT (u,X))2 (4.12)

Klein−Gordon equation : T (u,X) = 0 (4.13)

with

= 2∂X∂u − 2∂X(α(u,X)∂X)− F ′(X)

F (X)(2α(u,X)∂X − ∂u) . (4.14)

10In [107] a maximum in γ near D=11 has been found. The most recent study suggests it is an

artifact of numerics [108]. Another open question concerns the limit D→ 3: does γ remain finite?

– 289 –

These equations should be compared with (2.12a), (2.12b) in [109] or with (2.4) (and

for the Klein-Gordon equation also (2.3)) in [110], where they have been derived for

spherically symmetric EH gravity in 4D. In the present case they are valid for generic

2D dilaton gravity coupled non-minimally to a free massless scalar field. Thus, the

set of equations (4.11)-(4.14) is a suitable starting point for numerical simulations in

generic 2D dilaton gravity. The Misner-Sharp mass function

m(u,X) = −X+X−I(X)− w(X) = −α(u,X)e−2β(u,X) − w(X) (4.15)

allows to rewrite the condition for a trapped surface as αe−2β = 0 (cf. (2.11) with

(2.9)). Thus, as noted before, either α has to vanish or β →∞; it is the latter type

of horizon that is of relevance for numerical simulations of critical collapse. One may

use the Misner-Sharp function instead of α and thus obtains instead of (4.11)

∂Xm(u,X) = (m(u,X) + w(X))2F (X)(∂XT (u,X))2 . (4.16)

To monitor the emergence of a trapped surface numerically one has to check whether

m(u0, Xh) + w(Xh) ≈ 0 (4.17)

is fulfilled to a certain accuracy at a given retarded time u0; the quantity Xh corre-

sponds to the value of the dilaton field at the horizon. By analogy to (2.16) of [110]

one may now introduce a compactified “radial” coordinate, e.g. X/(1+X), although

there may be more convenient choices.

As a consistency check the original Choptuik system in the current notation will

be reproduced. We recall that (2.19) describes the EMKG. Using dr = I(X) dX the

evolution equations for geometry read:

∂rβ =κ

2r(∂rT )2 (4.18)

∂rα = λ2e2β (4.19)

They look almost the same as (2.4) in [110]. The coupling constant κ just has to

be fixed appropriately in (4.18) (i.e. κ = 4π). Also, the scaling constant λ must be

fixed. Note that the line element reads

ds2 = 22

re2β(u,X(r)) du

(

drr

2+ α(u,X(r)) du

)

= 2e2β du

(

dr +2α

rdu

)

(4.20)

This shows that β here really coincides with β in [110] and α here coincides, up to a

numerical factor, with V there (and there are some signs due to different conventions).

4.7 Quasinormal modes

The term “quasinormal modes” refers to some set of modes with a complex frequency,

associated with small perturbations of a BH. For U = 0 and monomial V in [111]

– 290 –

quasinormal modes arising from a scalar field, (2.15) with f = 0 and F ∝ Xp, have

been studied in the limit of high damping by virtue of the “monodromy approach”,

and the relation

eω/TH = −(1 + 2 cos (π(1− p))) (4.21)

for the frequency ω has been found (TH is Hawking temperature as defined in (4.1)).

Minimally coupled scalar fields (p = 0) lead to the trivial result ω/TH = 2πin. High

damping implies that the integer n has to be large. For the important case of p = 1

(relevant for the first and fifth entry in table 1) one obtains from (4.21)

ω

TH

= 2πi

(

n+1

2

)

+ ln 3 . (4.22)

The result (4.22) coincides with the one obtained for the Schwarzschild BH with 4D

methods, both numerically [112] and analytically [113]. Moreover, consistency with

D > 4 is found as well [114]. This shows that the 2D description of BHs is reliable

also with respect to highly damped quasinormal modes.

4.8 Solid state analogues

BH analogues in condensed matter systems go back to the seminal paper by Unruh

[115]. Due to the amazing progress in experimental condensed matter physics, in

particular Bose-Einstein condensates, in the past decade the subject of BH analogues

has flourished, cf. e.g. [116] and references therein.

In some cases the problem effectively reduces to 2D. It is thus perhaps not

surprising that an analogue system for the Jackiw-Teitelboim model has been found

[117] for a cigar shaped Bose-Einstein condensate. More recently this has led to

some analogue 2D activity [118]. Note, however, that some issues, like the one of

backreaction, might not be modelled very well by an effective action method [119].

Indeed, 2D dilaton gravity with matter could be of interest in this context, because

these systems might allow not just kinematical but dynamical equivalence, i.e., not

only the fluctuations (e.g. phonons) behave as the corresponding gravitational ones

(e.g. Hawking quanta), but also the background dynamics does (e.g. the flow of the

fluid or the metric, respectively). Such a system would be a necessary pre-requisite

to study issues of mass and entropy in an analogue context. At least for static

solutions this is possible [120], but of course the non-static case would be much more

interesting. Alas, it is not only more interesting but also considerably more difficult,

and a priori there is no reason why one should succeed in finding a fully fledged

analogue model of 2D dilaton gravity with matter. Still, one can hope and try.

5. Geometry from matter

In first order gravity (2.2) coupled to scalar (2.15) or fermionic (2.16) matter the

– 291 –

geometry can be quantized exactly: after analyzing the constraints, fixing EF gauge

(ω0, e−0 , e

+0 ) = (0, 1, 0) (5.1)

and constructing a BRST invariant Hamiltonian, the path integral can be evaluated

exactly and a (nonlocal) effective action is obtained [121]. Subsequently the matter

fields can be quantized by means of ordinary perturbation theory. To each order all

backreactions are included automatically by this procedure.

i0

i-

i+

ℑ -

ℑ +

y

Figure 2: VBH

Although geometry has been integrated out exactly, it can

be recovered off-shell in the form of interaction vertices of the

matter fields, some of which resemble virtual black holes (VBHs)

[15, 122, 123]. This metamorphosis of geometry however does

not take place in the matterless case [124], where the quantum

effective action coincides with the classical action in EF gauge.

We hasten to add that one should not take this off-shell geometry

at face value — this would be like over-interpreting the role

of virtual particles in a loop diagram. But the simplicity of

such geometries and the fact that all possible configurations are

summed over are both nice qualitative features of this picture.

A Carter-Penrose diagram of a typical VBH configuration

is depicted in figure 2. The curvature scalar of such effective

geometries is discontinuous and even has a δ-peak. A typical

effective line element (for the EMKG) reads

(ds)2 = 2 dr du+

(

1− θ(ry − r)δ(u− uy)

(

2m

r+ ar − d

))

(du)2 , (5.2)

It obviously has a Schwarzschild part with ry-dependent “mass” m and a Rindler

part with ry-dependent “acceleration” a, both localized on a lightlike cut. This

geometry is nonlocal in the sense that it depends not just on the coordinates r, u

but additionally on a second point ry, uy. While the off-shell geometry (5.2) is highly

gauge dependent, the ensuing S-matrix — the only physical observable in this context

[125] — appears to be gauge independent, although a formal proof of this statement,

e.g. analogously to [126], is lacking.

5.1 Scalar matter

After integrating out geometry and the ghost sector (for f(X, T ) = 0), the effective

Lagrangian (w is defined in (2.3))

LeffT = F (X)(∂0T )(∂1T )− w′(X) + sources (5.3)

contains the quantum version of the dilaton field X = X(∇−20 (∂0T )2), depending

non-locally on T . The quantity X solves the equation of motion of the classical

– 292 –

V(4)(x,y)a

x y

∂0 S

q’

∂0 S

q

∂0 S

k’

∂0 S

k

+

V(4)(x,y)b

x y

∂0 S

q’

∂0 S

q

∂1 S

k’

∂0 S

k

Figure 3: Non-local 4-point vertices

dilaton field, with matter terms and external sources for the geometric variables in

EF gauge. The simplicity of (5.3) is in part due to the gauge choice (5.1) and in

part due to the linearity of the gauge fixed Lagrangian in the remaining gauge field

components, thus producing delta-functionals upon path integration.

In principle, the interaction vertices can be extracted by expanding the nonlocal

effective action in a power series of the scalar field T . However, this becomes cum-

bersome already at the T 4 level. Fortunately, the localization technique introduced

in [121] simplifies the calculations considerably. It relies on two observations: First,

instead of dealing with complicated nonlocal kernels one may solve corresponding

differential equations after imposing asymptotic conditions on the solutions. Second,

instead of taking the n-th functional derivative of the action with respect to bilin-

ear combinations of T , the matter fields may be localized at n different space-time

points, which mimics the effect of functional differentiation. For tree-level calcula-

tions it is then sufficient to solve the classical equations of motion in the presence of

these sources, which is achieved most easily via appropriate matching conditions.

It turns out (as anticipated from (5.2)) that the conserved quantity (2.9) is

discontinuous for a VBH. This phenomenon is generic [15].11 The corresponding

Feynman diagrams are contained in figure 3.12 For free, massless, non-minimally

coupled scalars (F 6= const.) both the symmetric and the non-symmetric 4-point

vertex

V (4) =

∫

d2x d2y(∂0T )2x

[

Va(x, y)(∂0T )2y + Vb(x, y)(∂0T )y(∂1T )y

]

(5.4)

are given in [15], and have the following properties:

1. They are local in one coordinate (e.g. containing δ(x1 − y1)) and nonlocal in

the other.

2. They vanish in the local limit (x0 → y0). Additionally, Vb vanishes for minimal

coupling F = const.

11With the exception of scattering trivial models, cf. the eleventh entry in table 1.12The scalar field T is denoted by S in these graphs.

– 293 –

3. The symmetric vertex depends only on the conformal invariant combination

w(X) and the asymptotic value M∞ of (2.9). The non-symmetric one is inde-

pendent of U , V and M∞. Thus if M∞ is fixed in all conformal frames, both

vertices are conformally invariant.

4. They respect the Z2 symmetry F (X) 7→ −F (X).

It should be noted that the class of models with UV +V ′ = 0 and F = const (contain-

ing the CGHS model, the seventh and eleventh entry in table 1) shows “scattering

triviality”, i.e., the classical vertices vanish, and scattering can only arise from higher

order quantum backreactions. For these models the VBH has no classically observ-

able consequences, but at 1-loop level physical observables like the specific heat are

modified appreciably [127].

The 2D Klein-Gordon equation relevant for the construction of asymptotic states

is also conformally invariant. For minimal coupling it simplifies considerably, and a

complete set of asymptotic states can be obtained explicitly. Since both, asymptotic

states and vertices, only depend on w(X) and M∞, at tree level conformal invariance

holds nonperturbatively (to all orders in T ), but it is broken at 1-loop level due

to the conformal anomaly. Because asymptotically geometry does not fluctuate, a

standard Fock space may be built with creation/annihilation operators a±(k) obeying

the standard commutation relations. The S-matrix for two ingoing (q, q′) into two

outgoing (k, k′) asymptotic modes is determined by (cf. (5.4))

T (q, q′; k, k′) =1

2〈0∣

∣a−(k)a−(k′)V (4)a+(q)a+(q′)∣

∣ 0〉 . (5.5)

The simple choice M∞ = 0 yields a “standard QFT vacuum” |0〉, provided the model

under consideration has a Minkowskian ground state (e.g. the first, third, fifth and

last model in table 1).

For the physically interesting case of the EMKG model such an S-matrix was

obtained in [123,128]. Both the symmetric and the non-symmetric vertex contribute,

each giving a divergent contribution to the S-matrix, but the sum of both turned

out to be finite! The whole calculation is highly nontrivial, involving cancellations

of polylogarithmic terms, but at the end giving the surprisingly simple result

T (q, q′; k, k′) = −iκδ (k + k′ − q − q′)2(4π)4|kk′qq′|3/2

E3T , (5.6)

with ingoing (q, q′) and outgoing (k, k′) spatial momenta, total energy E = q + q′,

T :=1

E3

[

Π lnΠ2

E6+

1

Π

∑

p∈k,k′,q,q′

p2 lnp2

E2·(

3kk′qq′ − 1

2

∑

r 6=p

∑

s 6=r,p

(

r2s2)

)]

, (5.7)

and the momentum transfer function Π = (k + k′)(k − q)(k′ − q). The factor T is

invariant under rescaling of the momenta p 7→ ap, and the whole amplitude trans-

forms monomial like T 7→ a−4T . It should be noted that due to the non-locality of

– 294 –

the vertices there is just one δ-function of momentum conservation (but no separate

energy conservation) present in (5.6). This is advantageous because it eliminates the

problem of “squared δ-functions” that is otherwise present in 2D theories of massless

scalar fields (cf. e.g. [129]). In this sense gravity acts as a regulator of the theory.

The corresponding differential cross section also reveals interesting features [123]:

1. For vanishing Π forward scattering poles are present.

2. There is an approximate self-similarity close to the forward scattering peaks.

Far away from them it is broken, however.

3. It is CPT invariant.

4. An ingoing s-wave can decay into three outgoing ones. Although this may be

expected on general grounds, within the present formalism it is possible to

provide explicit results for the decay rate.

Although it seems straightforward to generalize (5.5) to arbitrary n-point vertices,

no such calculation has been attempted so far. This is related to the fact that

the derivation of (5.6) has been somewhat tedious and lengthy. Thus, it could be

worthwhile to find a more efficient way to obtain this interesting S-matrix element.

5.2 Fermionic matter

Recently we considered 2D dilaton gravity (2.2) coupled to fermions (2.16) along the

lines of the previous subsection. The results will be published elsewhere, but we give

a short summary with emphasis on differences to the scalar case.

The constraint analysis for the general case (2.16) has been worked out first

in [42]. Three first class constraints generating the two diffeomorphisms and the

local Lorentz symmetry and four well-known second class constraints relating the four

real components of the Dirac spinor to their canonical momenta are present in the

system. As anticipated the Hamiltonian is fully constrained. After introducing the

Dirac bracket the constructions of the BRST charge and the gauge fixed Hamiltonian

are straightforward. Path integration over geometry is even simpler than in the

scalar case, because the second class constraints are implemented in the path integral

through delta functionals, allowing to integrate out the fermion momenta. The

effective Lagrangian

Leffχ =

i√2F (X)(χ∗

1

←→∂1 χ1)

+I(X)

(

i√2F (X)(χ∗

0

←→∂0 χ0) +H(X)g(χχ)− V (X)

)

+ sources (5.8)

again depends on the quantum version X = X(∇−20 (χ∗

1

←→∂0 χ1)) of the dilaton field

and exhibits non-locality in the matter field.

– 295 –

Some properties remain the same as compared to previous studies with scalar

matter. For instance, the VBH phenomenon is still present, now even for the eleventh

model in table 1. In fact, the conserved quantity (2.9) now becomes continuous only

for the trivial case F (X) = 0. But there are also some notable differences. For

example, the non-selfinteracting system already has three 4-point vertices, two of

them being the symmetric and asymmetric vertices of the scalar case and a new

third one, arising from the first term in the second line of (5.8). All vertices show

the first two properties listed above, and the symmetric and non-symmetric ones also

the third one.

The new vertex however does not vanish for minimal coupling, and thus in

contrast to the scalar case there are two vertices present even for this simple case. It is

not conformally invariant, but rather transforms additively because it contains a term

proportional to U(X). However, since also the external legs have a conformal weight,

conformal invariance of the tree-level S-matrix still is expected to hold, despite of

the non-invariance of some of the vertices and some of the asymptotic modes.

At 1-loop level and for minimal coupling conformal symmetry is broken and,

exactly as in the case of scalar matter, the conformal anomaly can be integrated to

the non-local Polyakov action [130]. This has been applied e.g. in [131]. A possible

Thirring term can be reformulated using (2.18) and integrated by use of the chiral

anomaly, giving a Wess-Zumino [132] contribution to the effective action. In this

case, a path integral over the auxiliary vector potential remains, with a highly non-

local self-interaction. Whether this treatment is favourable over treating the Thirring

term directly as an interaction vertex has to be decided by application.

Another peculiar feature of 2D field theories is bosonization, e.g. the quantum

equivalence of the Thirring model and the Sine-Gordon model, both in flat 1+1 di-

mensions [133]. This issue has been addressed recently on a curved background by

Frolov, Kristjansson and Thorlacius [134] to investigate the effect of pair-production

on BH space times in regions of small curvature (as compared to the microscopic

length scale of quantum theory). In the framework of first order gravity it may be

possible to investigate the question of bosonization even outside this simple frame-

work, since one is able to integrate out geometry non-perturbatively.

6. Mathematical issues

In the absence of matter many of the interesting features discussed in the previous

three sections are absent: there is no tachyon dynamics, no Hawking radiation, no

interesting semi-classical behavior, no critical collapse, no quasinormal modes, no

relevant solid state analogue, no scattering processes and no reconstruction of geom-

etry from matter. Nevertheless, some basic features remain, like the global structure

of the classical solutions or the physics-to-gauge conversion mentioned in section 4.5.

Mathematically, however, the absence of matter bears some attractiveness and re-

– 296 –

veals beautiful structures responsible for the classical integrability of (2.2). They may

allow some relevant generalizations of (2.2), e.g. in the context of non-commutative

gravity.

6.1 Remarks on the Einstein-Hilbert action in 2D

In 2D the Einstein tensor vanishes identically for any 2D metric and thus conveys

no useful information. Similarly, the 2D EH action, supplemented appropriately by

boundary and corner terms, just counts the number of holes of a compact Riemannian

manifold, cf. e.g. [135]. Thus, as compared to (1.1) or (2.2) the study of “pure” 2D

gravity, i.e., without coupling to a dilaton field, is of rather limited interest. If one

adds a cosmological constant term one may study quantum gravity in 2D by means

of dynamical triangulations, cf. e.g. [136] and references therein. The EH part of the

action plays no essential role, however.

It is possible to consider EH gravity in 2 + ε dimensions, an idea which seems

to go back to [137]. After taking the limit ε→ 0 in a specific way [138] one obtains

again a dilaton gravity model (1.1) with V = 0 and U = const. (cf. the eleventh

model in table 1). That such a limit can be very subtle has been shown recently

by Jackiw [139] in the context of Weyl invariant scalar field dynamics: if one simply

drops the EH term in equation (3.5) of that work the Liouville model is obtained

(cf. the tenth model in table 1), but Weyl invariance is lost.

6.2 Relations to 3D: Chern-Simons and BTZ

The gravitational Chern-Simons term [140] and the 3D BTZ BH [141] have inspired

a lot of further research. Here we will focus on relations to (1.1) and (2.2): di-

mensional reduction of the BTZ to 2D has been performed in [19], cf. the fifteenth

model in table 1. A reduction of the gravitational Chern-Simons term from 3D to 2D

has been performed in [20], cf. the sixteenth model in table 1. Recently [142], such

reductions have been exploited to calculate the entropy of a BTZ BH in the pres-

ence of gravitational Chern-Simons terms, something which is difficult to achieve in

3D because there is no manifestly covariant formulation of the Chern-Simons term,

whereas the reduced theory is manifestly covariant. It is not unlikely that also other

open problems of 3D gravity may be tackled with 2D methods.

6.3 Integrable systems, Poisson-sigma models and KdV surfaces

Some of the pioneering work has been mentioned already in section 2.1 and in table

1. In two seminal papers by Kummer and Schwarz [143] the usefulness of light-

cone gauge for the Lorentz frame and EF gauge for the curved metric has been

demonstrated for the fourteenth model in table 1, which is a rather generic one as

it has non-vanishing U and non-monomial V . A Hamiltonian analysis [72] revealed

an interesting (W-)algebraic structure of the secondary constraints together with

– 297 –

the fields X,X± as generators. The center of this algebra consists of the conserved

quantity (2.9) and its first derivative, ∂1M (which, of course, vanishes on the surface

of constraints). Consequently, it has been shown by Schaller and Strobl [31] that

(2.2) is a special case of a Poisson-sigma model,13

SPSM = −∫

M

[

XI dAI −1

2P IJAJ ∧ AI

]

, (6.1)

with a 3D target space, the coordinates of which are XI = X,X+, X−. The gauge

fields comprise the Cartan variables, AI = ω, e−, e+. Because the dimension of the

Poisson manifold is odd the Poisson tensor (I, J ∈ X,±)

PX± = ±X± , P+− = X+X−U(X) + V (X) , P IJ = −P JI , (6.2)

cannot have full rank. Therefore, always a Casimir function, (2.9), exists, which

may be interpreted as “mass”. Note that (6.2) indeed fulfills the required Jacobi-

identities, P IL∂LPJK +perm (IJK) = 0. For a generic (graded) Poisson-sigma model

(6.1) the commutator of two symmetry transformations

δXI = P IJεJ , δAI = − dεI −(

∂IPJK)

εK AJ , (6.3)

is a (non-linear) symmetry modulo the equations of motion. Only for P IJ linear in

XI a Lie algebra is obtained; cf. the second model in table 1. For (6.2) the symme-

tries (6.3) on-shell correspond to local Lorentz transformations and diffeomorphisms.

Generalizations discussed in section 2.4 are particularly transparent in this approach;

essentially, one has to add more target space coordinates to the Poisson manifold,

some of which will be fermionic in supergravity extensions, cf. e.g. [39].

Actually, there exist various approaches to integrability of gravity models in 2D,

cf. e.g. [145], and we can hardly do them justice here. We will just point out a

relation to Korteweg-de Vries (KdV) surfaces as discussed recently in [146]. These

are 2D surfaces embedded in 3D Minkowski space arising from the KdV equation

∂tw = ∂3xw + 6w∂xw, with line element (cf. (11) in [146]; u there coincides with

w here) ds2 = 2 dX du − (4λ − w(X, u)) du2, where X ∝ x, u ∝ t and λ is some

constant. For static KdV solutions, ∂uw = 0, this line element is also a solution of

(2.2) as can bee seen from (2.10), with λ playing the role of the mass M . In the

non-static case it describes a solution of (2.2) coupled to some energy-momentum

tensor. It could be of interest to pursue this relation in more depth.

6.4 Torsion and non-metricity

For U = 0 the equation of motion R = 2V ′(X), if invertible, allows to rewrite

the action (1.1) as SR =∫

d2x√−gf(R), cf. e.g. [147] and references therein. As

13Dirac-sigma models [144] are a recent generalization thereof.

– 298 –

compared to such theories, the literature on models with torsion τa = ∗T a,

SRT =

∫

d2x√−gf(R, τaτa) , (6.4)

is relatively scarce and consists mainly of elaborations based upon the fourteenth

model in table 1, where f = Aτaτa +BR2 +CR+ Λ, also known as “Poincare gauge

theory”, cf. [148] and references therein. This model in particular (and a large class

of models of type (6.4)) allows an equivalent reformulation as (2.2). Thus, they need

not be discussed separately.

A generalization which includes also effects from non-metricity has been studied

in [149]. Elimination of non-metricity leads again to models of type (1.1), (2.2),

but one has to be careful with such reformulations as test-particles moving along

geodesics or, alternatively, along auto-parallels, may “feel” the difference. Thus, it

could be of interest to generalize (2.2) (which already contains torsion if U 6= 0) as

to include non-metricity, thus dropping the requirement that the connection ωab is

proportional to εab. However, a formulation as Poisson-sigma model (6.1) (with 6D

target space) seems to be impossible as there are only trivial solutions to the Jacobi

identities.

6.5 Non-commutative gravity

In the 1970ies/1980ies theories have been supersymmetrized, in the 1990ies/2000s

theories have been “non-commutativized”, for reviews cf. e.g. [150]. The latter pro-

cedure still has not stopped as the original idea, namely to obtain a fully satisfactory

non-commutative version of gravity, has not been achieved so far. In order to get

around the main conceptual obstacles it is tempting to consider the simplified frame-

work of 2D.

There it is possible to construct non-commutative dilaton gravity models with a

usual (non-twisted) realization of gauge symmetries.14 A non-commutative version

of the Jackiw-Teitelboim model (cf. the second entry in table 1),

SNCJT = −1

2

∫

d2x εµν[

Xa ⋆ Taµν +Xab ⋆

(

Rabµν − Λea

µ ⋆ ebν

)]

, (6.5)

has been constructed in [152] and then quantized in [153]. A non-commutative

version of the fourth model in table 1 was suggested in [154]. For a definition of the

Moyal-⋆ and further notation cf. these two references. A crucial change as compared

to (2.2), besides the ⋆, is the appearance of a second dilaton field ψ in 2Xab =

Xεab − iψηab. However, interesting as these results may be, there seems to be no

way to generalize them to generic 2D dilaton gravity without twisting the gauge

symmetries [155]. Moreover, the fact that the metric can be changed by “Lorentz

14Another approach has been pursued in [151].

– 299 –

transformations” seems questionable from a physical point of view, cf. [156] for a

similar problem.

An important step towards constructing a satisfactory non-commutative gravity

was recently made by Wess and collaborators [157], who understood how one can

construct diffeomorphism invariants, including the EH action, on non-commutative

spaces (see also [158] for a real formulation). There is, however, a price to pay. The

diffeomorphism group becomes twisted, i.e., there is a non-trivial coproduct [159].

Recently it could be shown [160] that twisted gauge symmetries close for arbitrary

gauge groups and thus a construction of twisted-invariant actions is straightforward.

The main element in that construction (cf. also [157–159,161] and [162]) is the twist

operator

F = expP, P =i

2θµν∂µ ⊗ ∂ν , (6.6)

which acts on the tensor products of functions φ1⊗φ2. With the multiplication map

µ(φ1 ⊗ φ2) = φ1 · φ2 and (6.6) the Moyal-Weyl representation of the star product,

φ1 ⋆ φ2 = µ F(φ1 ⊗ φ2) = µ⋆(φ1 ⊗ φ2) , (6.7)

can be constructed. Consider now generators u of some symmetry transformations

which form a Lie algebra. If one knows the action of these transformations on

primary fields, δuφ = uφ, the action on tensor products is defined by the coproduct

∆. In the undeformed case the coproduct is primitive, ∆0(u) = u ⊗ 1 + 1 ⊗ u and

δu(φ1 ⊗ φ2) = ∆0(u)(φ1 ⊗ φ2) = uφ1 ⊗ φ2 + φ1 ⊗ uφ2 satisfies the usual Leibniz rule.

The action of symmetry generators on elementary fields is left undeformed, but the

coproduct is twisted,

∆(u) = exp(−P)∆0(u) exp(P) . (6.8)

Obviously, twisting preserves the commutation relations. Therefore, the commuta-

tors of gauge transformations for an arbitrary gauge group close.

It seems plausible that a corresponding generalization to twisted non-linear gauge

symmetries will be a crucial technical pre-requisite to a successful construction of

generic non-commutative 2D dilaton gravity.15 It would allow, among other things,

a thorough discussion of non-commutative BHs, along the lines of sections 2-5.

Acknowledgments

DG and RM would like to thank cordially L. Bergamin, W. Kummer and D. Vassile-

vich for a long-time collaboration and helpful discussions, respectively, on most of the

topics reviewed in this work. Moreover, DG is grateful to M. Adak, P. Aichelburg,

S. Alexandrov, H. Balasin, M. Bojowald, M. Cadoni, S. Carlip, T. Dereli, M. Gurses,

A. Iorio, R. Jackiw, M. Katanaev, C. Lechner, F. Meyer, S. Mignemi, C. Nunez,

15The relation of (6.1) to a specific Lie algebroid [163] could be helpful in this context.

– 300 –

Y. Obukhov, M.-I. Park, M. Purrer, R. Schutzhold, T. Strobl, W. Unruh, P. van

Nieuwenhuizen and S. Weinfurtner for helpful discussions and/or correspondence. In

addition, DG would like to thank the organizers of the Fifth Workshop on QUAN-

TIZATION, DUALITIES AND INTEGRABLE SYSTEMS in Denizli, Turkey, in

particular M. Adak for the kind invitation.

DG has been supported by project GR-3157/1-1 of the German Research Foun-

dation (DFG). Additional financial support due to Pamukkale University is acknowl-

edged gratefully. RM has been supported financially by the MPI and expresses his

gratitude to J. Jost in this regard.

References

[1] Such a life, with all vision limited to a Point, and all motion to a Straight Line,

seemed to me inexpressibly dreary; and I was surprised to note the vivacity and

cheerfulness of the King. [Edwin A. Abbot, “Flatland — A Romance of Many

Dimensions.” Dover Publications 1992, New York. (first published under the

pseudonym A. Square in 1884, Seeley & Co., London)].

[2] J. Brown, Lower Dimensional Gravity. World Scientific, 1988.

[3] A. M. Polyakov, “Quantum gravity in two-dimensions,” Mod. Phys. Lett. A2

(1987) 893.

[4] D. Grumiller, W. Kummer, and D. V. Vassilevich, “Dilaton gravity in two

dimensions,” Phys. Rept. 369 (2002) 327–429, hep-th/0204253.

[5] P. Thomi, B. Isaak, and P. Hajıcek, “Spherically symmetric systems of fields and

black holes. 1. Definition and properties of apparent horizon,” Phys. Rev. D30

(1984) 1168.

P. Hajıcek, “Spherically symmetric systems of fields and black holes. 2. Apparent

horizon in canonical formalism,” Phys. Rev. D30 (1984) 1178.

[6] C. Teitelboim, “Gravitation and Hamiltonian structure in two space-time

dimensions,” Phys. Lett. B126 (1983) 41.

[7] R. Jackiw, “Lower dimensional gravity,” Nucl. Phys. B252 (1985) 343–356.

[8] E. Witten, “On string theory and black holes,” Phys. Rev. D44 (1991) 314–324.

G. Mandal, A. M. Sengupta, and S. R. Wadia, “Classical solutions of

two-dimensional string theory,” Mod. Phys. Lett. A6 (1991) 1685–1692.

S. Elitzur, A. Forge, and E. Rabinovici, “Some global aspects of string

compactifications,” Nucl. Phys. B359 (1991) 581–610.

[9] C. G. Callan, Jr., S. B. Giddings, J. A. Harvey, and A. Strominger, “Evanescent

black holes,” Phys. Rev. D45 (1992) 1005–1009, hep-th/9111056.

– 301 –

[10] J. P. S. Lemos and P. M. Sa, “The black holes of a general two-dimensional dilaton

gravity theory,” Phys. Rev. D49 (1994) 2897–2908, arXiv:gr-qc/9311008.

[11] A. Fabbri and J. G. Russo, “Soluble models in 2d dilaton gravity,” Phys. Rev. D53

(1996) 6995–7002, hep-th/9510109.

[12] D. Grumiller, “Long time black hole evaporation with bounded Hawking flux,”

JCAP 05 (2004) 005, gr-qc/0307005.

[13] M. O. Katanaev, W. Kummer, and H. Liebl, “On the completeness of the black

hole singularity in 2d dilaton theories,” Nucl. Phys. B486 (1997) 353–370,

gr-qc/9602040.

[14] Y. Nakayama, “Liouville field theory: A decade after the revolution,” Int. J. Mod.

Phys. A19 (2004) 2771–2930, hep-th/0402009.

[15] D. Grumiller, W. Kummer, and D. V. Vassilevich, “Virtual black holes in

generalized dilaton theories (and their special role in string gravity),” European

Phys. J. C30 (2003) 135–143, hep-th/0208052.

[16] H. Reissner, “Uber die Eigengravitation des elektrischen Feldes nach der

Einsteinschen Theorie,” Ann. Phys. 50 (1916) 106.

G. Nordstrom, “On the energy of the gravitation field in Einstein’s theory,” Proc.

Kon. Ned. Akad. Wet. 20 (1916) 1238.

[17] S. W. Hawking and D. N. Page, “Thermodynamics of black holes in anti-de Sitter

space,” Commun. Math. Phys. 87 (1983) 577.

[18] M. O. Katanaev and I. V. Volovich, “String model with dynamical geometry and

torsion,” Phys. Lett. B175 (1986) 413–416; “Two-dimensional gravity with

dynamical torsion and strings,” Ann. Phys. 197 (1990) 1.

[19] A. Achucarro and M. E. Ortiz, “Relating black holes in two-dimensions and three-

dimensions,” Phys. Rev. D48 (1993) 3600–3605, hep-th/9304068.

[20] G. Guralnik, A. Iorio, R. Jackiw, and S. Y. Pi, “Dimensionally reduced

gravitational Chern-Simons term and its kink,” Ann. Phys. 308 (2003) 222–236,

hep-th/0305117.

[21] D. Grumiller and W. Kummer, “The classical solutions of the dimensionally

reduced gravitational Chern-Simons theory,” Ann. Phys. 308 (2003) 211–221,

hep-th/0306036.

L. Bergamin, D. Grumiller, A. Iorio, and C. Nunez, “Chemistry of Chern-Simons

supergravity: Reduction to a BPS kink, oxidation to M-theory and

thermodynamical aspects,” JHEP 11 (2004) 021, hep-th/0409273.

[22] L. Bergamin, “Constant dilaton vacua and kinks in 2d (super-)gravity,”

hep-th/0509183.

– 302 –

[23] M. R. Douglas et al., “A new hat for the c = 1 matrix model,” hep-th/0307195.

[24] S. Gukov, T. Takayanagi, and N. Toumbas, “Flux backgrounds in 2D string

theory,” JHEP 03 (2004) 017, hep-th/0312208.

[25] R. Dijkgraaf, H. Verlinde, and E. Verlinde, “String propagation in a black hole

geometry,” Nucl. Phys. B371 (1992) 269–314.

[26] D. Grumiller, “An action for the exact string black hole,” JHEP 05 (2005) 028,

hep-th/0501208.

[27] K. Isler and C. A. Trugenberger, “A gauge theory of two-dimensional quantum

gravity,” Phys. Rev. Lett. 63 (1989) 834.

A. H. Chamseddine and D. Wyler, “Gauge theory of topological gravity in

(1+1)-dimensions,” Phys. Lett. B228 (1989) 75.

[28] H. Verlinde, “Black holes and strings in two dimensions,” in Trieste Spring School

on Strings and Quantum Gravity, pp. 178–207. April, 1991. the same lectures have

been given at MGVI in Japan, June, 1991.

[29] D. Cangemi and R. Jackiw, “Gauge invariant formulations of lineal gravity,” Phys.

Rev. Lett. 69 (1992) 233–236, hep-th/9203056.

A. Achucarro, “Lineal gravity from planar gravity,” Phys. Rev. Lett. 70 (1993)

1037–1040, hep-th/9207108.

[30] N. Ikeda and K. I. Izawa, “General form of dilaton gravity and nonlinear gauge

theory,” Prog. Theor. Phys. 90 (1993) 237–246, hep-th/9304012.

[31] P. Schaller and T. Strobl, “Poisson structure induced (topological) field theories,”

Mod. Phys. Lett. A9 (1994) 3129–3136, hep-th/9405110.

[32] J. G. Russo and A. A. Tseytlin, “Scalar tensor quantum gravity in

two-dimensions,” Nucl. Phys. B382 (1992) 259–275, arXiv:hep-th/9201021.

S. D. Odintsov and I. L. Shapiro, “One loop renormalization of two-dimensional

induced quantum gravity,” Phys. Lett. B263 (1991) 183–189.

T. Banks and M. O’Loughlin, “Two-dimensional quantum gravity in Minkowski

space,” Nucl. Phys. B362 (1991) 649–664.

R. B. Mann, A. Shiekh, and L. Tarasov, “Classical and quantum properties of

two-dimensional black holes,” Nucl. Phys. B341 (1990) 134–154.

[33] T. Klosch and T. Strobl, “Classical and quantum gravity in 1+1 dimensions. Part

II: The universal coverings,” Class. Quant. Grav. 13 (1996) 2395–2422,

arXiv:gr-qc/9511081.

[34] T. Klosch and T. Strobl, “Classical and quantum gravity in (1+1)-dimensions. Part

I: A unifying approach,” Class. Quant. Grav. 13 (1996) 965–984,

arXiv:gr-qc/9508020.

– 303 –

[35] J.-G. Hao and X.-Z. Li, “Constructing dark energy models with late time de Sitter

attractor,” Phys. Rev. D68 (2003) 083514, hep-th/0306033.

[36] L. Bergamin, D. Grumiller, and W. Kummer, “Supersymmetric black holes in 2d

dilaton supergravity: baldness and extremality,” J. Phys. A37 (2004) 3881–3901,

hep-th/0310006.

[37] D. M. Thompson, “AdS solutions of 2D type 0A,” Phys. Rev. D70 (2004) 106001,

hep-th/0312156.

[38] D. Birmingham, M. Blau, M. Rakowski, and G. Thompson, “Topological field

theory,” Phys. Rept. 209 (1991) 129–340.

[39] T. Strobl, “Gravity in two spacetime dimensions,” hep-th/0011240. Habilitation

thesis.

[40] Y.-C. Park and A. Strominger, “Supersymmetry and positive energy in classical

and quantum two-dimensional dilaton gravity,” Phys. Rev. D47 (1993) 1569–1575,

arXiv:hep-th/9210017.

J. M. Izquierdo, “Free differential algebras and generic 2d dilatonic

(super)gravities,” Phys. Rev. D59 (1999) 084017, arXiv:hep-th/9807007.

T. Strobl, “Target-superspace in 2d dilatonic supergravity,” Phys. Lett. B460

(1999) 87–93, arXiv:hep-th/9906230.

M. Ertl, W. Kummer, and T. Strobl, “General two-dimensional supergravity from

Poisson superalgebras,” JHEP 01 (2001) 042, arXiv:hep-th/0012219.

M. Ertl, Supergravity in two spacetime dimensions. PhD thesis, Technische

Universitat Wien, 2001. arXiv:hep-th/0102140.

L. Bergamin and W. Kummer, “The complete solution of 2D superfield supergravity

from graded Poisson-Sigma models and the super pointparticle,” Phys. Rev. D68

(2003) 104005, hep-th/0306217; “Two-dimensional N=(2,2) dilaton supergravity

from graded Poisson-Sigma models I: Complete actions and their symmetries.,”

Eur. Phys. J. C39 (2005) S41–S52, hep-th/0402138; “Two-dimensional N = (2,2)

dilaton supergravity from graded Poisson-Sigma models. II: Analytic solution and

BPS states,” Eur. Phys. J. C39 (2005) S53–S63, hep-th/0411204.

L. Bergamin, D. Grumiller, and W. Kummer, “Quantization of 2d dilaton

supergravity with matter,” JHEP 05 (2004) 060, hep-th/0404004.

[41] L. Bergamin and W. Kummer, “Graded Poisson sigma models and

dilaton-deformed 2d supergravity algebra,” JHEP 05 (2003) 074, hep-th/0209209.

[42] R. Meyer, “Constraints in two-dimensional dilaton gravity with fermions,”

hep-th/0512267.

[43] W. E. Thirring, “A soluble relativistic field theory,” Annals Phys. 3 (1958) 91–112.

– 304 –

[44] H. Balasin, C. G. Boehmer, and D. Grumiller, “The spherically symmetric standard

model with gravity,” Gen. Rel. Grav. 37 (2005) 1435–1482, gr-qc/0412098.

[45] W. Kummer, “Deformed ISO(2,1) symmetry and non-Einsteinian 2d-gravity with

matter,” in HADRON STRUCTURE ’92, D. Bruncko and J. Urban, eds.

September, 1992. Stara Lesna, Czechoslovakia.

[46] H. Pelzer and T. Strobl, “Generalized 2d dilaton gravity with matter fields,” Class.

Quant. Grav. 15 (1998) 3803–3825, arXiv:gr-qc/9805059.

[47] P. C. Aichelburg and R. U. Sexl, “On the gravitational field of a massless particle,”

Gen. Rel. Grav. 2 (1971) 303–312.

[48] H. Balasin and D. Grumiller, “The ultrarelativistic limit of 2d dilaton gravity and

its energy momentum tensor,” Class. Quant. Grav. 21 (2004) 2859–2872,

gr-qc/0312086.

[49] I. Z. Fisher, “Scalar mesostatic field with regard for gravitational effects,” Zh. Eksp.

Teor. Fiz. 18 (1948) 636–640, gr-qc/9911008.

[50] A. T. Filippov and D. Maison, “Horizons in 1+1 dimensional dilaton gravity

coupled to matter,” Class. Quant. Grav. 20 (2003) 1779–1786, gr-qc/0210081.

[51] D. Grumiller and D. Mayerhofer, “On static solutions in 2d dilaton gravity with

scalar matter,” Class. Quant. Grav. 21 (2004) 5893–5914, gr-qc/0404013.

[52] M. Wyman, “Static spherically symmetric scalar fields in general relativity,” Phys.

Rev. D24 (1981) 839–841.

[53] A. H. Bilge and D. Daghan, “Partial decoupling and exact static solutions for

Choptuik’s spacetime,” gr-qc/0508020.

[54] M. D. Roberts, “Scalar field counterexamples to the cosmic censorship hypothesis,”

Gen. Rel. Grav. 21 (1989) 907–939.

[55] T. Takayanagi and N. Toumbas, “A matrix model dual of type 0B string theory in

two dimensions,” JHEP 07 (2003) 064, hep-th/0307083.

[56] P. Ginsparg and G. W. Moore, “Lectures on 2-d gravity and 2-d string theory,”

hep-th/9304011.

P. Di Francesco, P. H. Ginsparg, and J. Zinn-Justin, “2-D Gravity and random

matrices,” Phys. Rept. 254 (1995) 1–133, hep-th/9306153.

S. Alexandrov, “Matrix quantum mechanics and two-dimensional string theory in

non-trivial backgrounds,” hep-th/0311273.

[57] V. Kazakov, I. K. Kostov, and D. Kutasov, “A matrix model for the

two-dimensional black hole,” Nucl. Phys. B622 (2002) 141–188, hep-th/0101011.

– 305 –

[58] L. Bergamin, D. Grumiller, W. Kummer, and D. V. Vassilevich, “Classical and

quantum integrability of 2D dilaton gravities in Euclidean space,” Class. Quant.

Grav. 22 (2005) 1361–1382, hep-th/0412007.

[59] I. R. Klebanov and A. A. Tseytlin, “D-branes and dual gauge theories in type 0

strings,” Nucl. Phys. B546 (1999) 155–181, hep-th/9811035.

A. Strominger, “A matrix model for AdS(2),” JHEP 03 (2004) 066,

hep-th/0312194.

J. L. Davis, L. A. Pando Zayas, and D. Vaman, “On black hole thermodynamics of

2-D type 0A,” JHEP 03 (2004) 007, hep-th/0402152.

U. H. Danielsson, J. P. Gregory, M. E. Olsson, P. Rajan, and M. Vonk, “Type 0A

2D black hole thermodynamics and the deformed matrix model,” JHEP 04 (2004)

065, hep-th/0402192.

J. L. Davis and R. McNees, “Boundary counterterms and the thermodynamics of

2-D black holes,” JHEP 09 (2005) 072, hep-th/0411121.

[60] A. A. Tseytlin, “On the form of the black hole solution in d = 2 theory,” Phys.

Lett. B268 (1991) 175–178.

I. Jack, D. R. T. Jones, and J. Panvel, “Exact bosonic and supersymmetric string

black hole solutions,” Nucl. Phys. B393 (1993) 95–110, hep-th/9201039.

[61] A. A. Tseytlin, “Effective action of gauged WZW model and exact string

solutions,” Nucl. Phys. B399 (1993) 601–622, hep-th/9301015.

I. Bars and K. Sfetsos, “Exact effective action and space-time geometry in gauged

WZW models,” Phys. Rev. D48 (1993) 844–852, hep-th/9301047.

[62] K. Becker, “Strings, black holes and conformal field theory,” hep-th/9404157.

[63] V. A. Kazakov and A. A. Tseytlin, “On free energy of 2-d black hole in bosonic

string theory,” JHEP 06 (2001) 021, hep-th/0104138.

[64] M. J. Perry and E. Teo, “Nonsingularity of the exact two-dimensional string black

hole,” Phys. Rev. Lett. 70 (1993) 2669–2672, hep-th/9302037.

P. Yi, “Nonsingular 2-D black holes and classical string backgrounds,” Phys. Rev.

D48 (1993) 2777–2788, hep-th/9302070.

[65] D. Grumiller and D. V. Vassilevich, “Non-existence of a dilaton gravity action for

the exact string black hole,” JHEP 11 (2002) 018, hep-th/0210060.

[66] W. Kummer and P. Widerin, “Conserved quasilocal quantities and general

covariant theories in two-dimensions,” Phys. Rev. D52 (1995) 6965–6975,

arXiv:gr-qc/9502031.

[67] V. Frolov and I. Novikov, Black Hole Physics. Kluwer Academic Publishers, 1998.

– 306 –

[68] L. D. Faddeev, “The energy problem in Einstein’s theory of gravitation,” Sov.

Phys. Usp. 25 (1982) 130–142.

[69] H. Liebl, D. V. Vassilevich, and S. Alexandrov, “Hawking radiation and masses in

generalized dilaton theories,” Class. Quant. Grav. 14 (1997) 889–904,

arXiv:gr-qc/9605044.

[70] V. P. Frolov, “Two-dimensional black hole physics,” Phys. Rev. D46 (1992)

5383–5394.

[71] R. B. Mann, “Conservation laws and 2-d black holes in dilaton gravity,” Phys. Rev.

D47 (1993) 4438–4442, hep-th/9206044.

[72] H. Grosse, W. Kummer, P. Presnajder, and D. J. Schwarz, “Novel symmetry of

nonEinsteinian gravity in two- dimensions,” J. Math. Phys. 33 (1992) 3892–3900,

hep-th/9205071.

[73] R. M. Wald, “The thermodynamics of black holes,” Living Rev. Rel. 4 (2001) 6,

gr-qc/9912119.

[74] G. W. Gibbons and S. W. Hawking, eds., Euclidean quantum gravity. Singapore:

World Scientific, 1993.

[75] W. Kummer and D. V. Vassilevich, “Hawking radiation from dilaton gravity in

(1+1) dimensions: A pedagogical review,” Annalen Phys. 8 (1999) 801–827,

gr-qc/9907041.

[76] R. M. Wald, “Black hole entropy is the Nother charge,” Phys. Rev. D48 (1993)

3427–3431, gr-qc/9307038.

V. Iyer and R. M. Wald, “Some properties of Nother charge and a proposal for

dynamical black hole entropy,” Phys. Rev. D50 (1994) 846–864, gr-qc/9403028.

[77] J. Gegenberg, G. Kunstatter, and D. Louis-Martinez, “Observables for

two-dimensional black holes,” Phys. Rev. D51 (1995) 1781–1786, gr-qc/9408015.

[78] H. W. J. Bloete, J. L. Cardy, and M. P. Nightingale, “Conformal invariance, the

central charge, and universal finite size amplitudes at criticality,” Phys. Rev. Lett.

56 (1986) 742–745.

J. L. Cardy, “Operator content of two-dimensional conformally invariant theories,”

Nucl. Phys. B270 (1986) 186–204.

[79] A. Strominger and C. Vafa, “Microscopic Origin of the Bekenstein-Hawking

Entropy,” Phys. Lett. B379 (1996) 99–104, hep-th/9601029.

K. V. Krasnov, “On statistical mechanics of gravitational systems,” Gen. Rel.

Grav. 30 (1998) 53–68, gr-qc/9605047.

A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, “Quantum geometry and black

hole entropy,” Phys. Rev. Lett. 80 (1998) 904–907, gr-qc/9710007.

– 307 –

M. Cadoni and S. Mignemi, “Entropy of 2d black holes from counting microstates,”

Phys. Rev. D59 (1999) 081501, hep-th/9810251.

S. Carlip, “Black hole entropy from conformal field theory in any dimension,” Phys.

Rev. Lett. 82 (1999) 2828–2831, hep-th/9812013.

S. N. Solodukhin, “Conformal description of horizon’s states,” Phys. Lett. B454

(1999) 213–222, hep-th/9812056.

M.-I. Park and J. H. Yee, “Comments on ’Entropy of 2D black holes from counting

microstates’,” Phys. Rev. D61 (2000) 088501, hep-th/9910213.

S. Carlip, “Reply to the comment by Park and Ho on ’Black hole entropy from

conformal field theory in any dimension’,” Phys. Rev. Lett. 83 (1999) 5596,

hep-th/9910247.

M.-I. Park and J. Ho, “Comments on ’Black hole entropy from conformal field

theory in any dimension’,” Phys. Rev. Lett. 83 (1999) 5595, hep-th/9910158.

I. Sachs and S. N. Solodukhin, “Horizon holography,” Phys. Rev. D64 (2001)

124023, hep-th/0107173.

M.-I. Park, “Hamiltonian dynamics of bounded spacetime and black hole entropy:

Canonical method,” Nucl. Phys. B634 (2002) 339–369, hep-th/0111224.

N. Pinamonti and L. Vanzo, “Central charges and boundary fields for two

dimensional dilatonic black holes,” Phys. Rev. D69 (2004) 084012,

hep-th/0312065.

G. Kang, J.-I. Koga, and M.-I. Park, “Near-horizon conformal symmetry and black

hole entropy in any dimension,” Phys. Rev. D70 (2004) 024005, hep-th/0402113.

[80] D. Grumiller, “Logarithmic corrections to the entropy of the exact string black

hole,” in Path Integrals from Quantum Information to Cosmology, C. Burdik,

N. Navratil, and S. Posta, eds. JINR Publishing Department, Prague, June, 2005.

hep-th/0506175.

[81] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, “Large N field

theories, string theory and gravity,” Phys. Rept. 323 (2000) 183–386,

hep-th/9905111.

[82] J. D. Brown, J. Creighton, and R. B. Mann, “Temperature, energy and heat

capacity of asymptotically anti-de Sitter black holes,” Phys. Rev. D50 (1994)

6394–6403, gr-qc/9405007.

[83] M. E. Olsson, “The stringy nature of the 2d type-0A black hole,” hep-th/0511106.

[84] J. A. Harvey and A. Strominger, “Quantum aspects of black holes,” in Recent

directions in particle theory: from superstrings and black holes to the standard

model (TASI - 92). 1992. hep-th/9209055.

A. Strominger, “Les Houches lectures on black holes,” arXiv:hep-th/9501071.

Talk given at NATO Advanced Study Institute.

– 308 –

L. Thorlacius, “Black hole evolution,” Nucl. Phys. Proc. Suppl. 41 (1995) 245–275,

hep-th/9411020.

S. B. Giddings, “Quantum mechanics of black holes,” Trieste HEP Cosmology

(1994) 0530–574, arXiv:hep-th/9412138.

[85] M. J. Duff, “Twenty years of the weyl anomaly,” Class. Quant. Grav. 11 (1994)

1387–1404, hep-th/9308075.

[86] S. M. Christensen and S. A. Fulling, “Trace anomalies and the Hawking effect,”

Phys. Rev. D15 (1977) 2088–2104.

[87] V. Mukhanov, A. Wipf, and A. Zelnikov, “On 4-D Hawking radiation from effective

action,” Phys. Lett. B332 (1994) 283–291, hep-th/9403018.

[88] T. Chiba and M. Siino, “Disappearance of black hole criticality in semiclassical

general relativity,” Mod. Phys. Lett. A12 (1997) 709–718.

S. Ichinose, “Weyl anomaly of 2D dilaton-scalar gravity and hermiticity of system

operator,” Phys. Rev. D57 (1998) 6224–6229, hep-th/9707025.

W. Kummer, H. Liebl, and D. V. Vassilevich, “Hawking radiation for

non-minimally coupled matter from generalized 2d black hole models,” Mod. Phys.

Lett. A12 (1997) 2683–2690, hep-th/9707041.

[89] K. I. Izawa, “On nonlinear gauge theory from a deformation theory perspective,”

Prog. Theor. Phys. 103 (2000) 225–228, hep-th/9910133.

[90] O. Lauscher and M. Reuter, “Ultraviolet fixed point and generalized flow equation

of quantum gravity,” Phys. Rev. D65 (2002) 025013, hep-th/0108040.

[91] A. Bonanno and M. Reuter, “Spacetime structure of an evaporating black hole in

quantum gravity,” hep-th/0602159.

[92] M. Niedermaier, “Dimensionally reduced gravity theories are asymptotically safe,”

Nucl. Phys. B673 (2003) 131–169, hep-th/0304117.

[93] V. P. Frolov and G. A. Vilkovisky, “Spherically symmetric collapse in quantum

gravity,” Phys. Lett. B106 (1981) 307–313.

[94] M. K. Parikh and F. Wilczek, “Global structure of evaporating black holes,” Phys.

Lett. B449 (1999) 24–29, gr-qc/9807031.

[95] A. Ashtekar and M. Bojowald, “Black hole evaporation: A paradigm,” Class.

Quant. Grav. 22 (2005) 3349–3362, gr-qc/0504029.

[96] S. A. Hayward, “Formation and evaporation of regular black holes,” Phys. Rev.

Lett. 96 (2006) 031103, gr-qc/0506126.

[97] S. Carlip, “Horizon constraints and black hole entropy,” Class. Quant. Grav. 22

(2005) 1303–1312, hep-th/0408123; “Horizon constraints and black hole entropy,”

gr-qc/0508071; “Horizons, constraints, and black hole entropy,” gr-qc/0601041.

– 309 –

[98] L. Bergamin, D. Grumiller, W. Kummer, and D. V. Vassilevich, “Physics-to-gauge

conversion at black hole horizons,” Class. Quant. Grav. 23 (2006) 3075–3101,

hep-th/0512230.

[99] G. ’t Hooft, “Horizons,” gr-qc/0401027.

[100] K. V. Kuchar, “Geometrodynamics of Schwarzschild black holes,” Phys. Rev. D50

(1994) 3961–3981, gr-qc/9403003.

[101] M. W. Choptuik, “Universality and scaling in gravitational collapse of a massless

scalar field,” Phys. Rev. Lett. 70 (1993) 9–12.

[102] C. Gundlach, “Critical phenomena in gravitational collapse,” Adv. Theor. Math.

Phys. 2 (1998) 1–49, arXiv:gr-qc/9712084.

[103] A. Strominger and L. Thorlacius, “Universality and scaling at the onset of quantum

black hole formation,” Phys. Rev. Lett. 72 (1994) 1584–1587, hep-th/9312017.

[104] J. G. Russo, L. Susskind, and L. Thorlacius, “Cosmic censorship in two-dimensional

gravity,” Phys. Rev. D47 (1993) 533–539, hep-th/9209012.

[105] Y. Peleg, S. Bose, and L. Parker, “Choptuik scaling and quantum effects in 2D

dilaton gravity,” Phys. Rev. D55 (1997) 4525–4528, gr-qc/9608040.

[106] M. Birukou, V. Husain, G. Kunstatter, E. Vaz, and M. Olivier, “Spherically

symmetric scalar field collapse in any dimension,” Phys. Rev. D65 (2002) 104036.

[107] E. Sorkin and Y. Oren, “On Choptuik’s scaling in higher dimensions,” Phys. Rev.

D71 (2005) 124005, hep-th/0502034.

[108] J. Bland, B. Preston, M. Becker, G. Kunstatter, and V. Husain, “Dimension

dependence of the critical exponent in spherically symmetric gravitational

collapse,” Class. Quant. Grav. 22 (2005) 5355–5364.

[109] S. Husa, C. Lechner, M. Purrer, J. Thornburg, and P. C. Aichelburg, “Type II

critical collapse of a self-gravitating nonlinear sigma-model,” Phys. Rev. D62

(2000) 104007, gr-qc/0002067.

[110] M. Purrer, S. Husa, and P. C. Aichelburg, “News from critical collapse: Bondi mass,

tails and quasinormal modes,” Phys. Rev. D71 (2005) 104005, gr-qc/0411078.

[111] J. Kettner, G. Kunstatter, and A. J. M. Medved, “Quasinormal modes for single

horizon black holes in generic 2-d dilaton gravity,” Class. Quant. Grav. 21 (2004)

5317–5332, gr-qc/0408042.

[112] H.-P. Nollert, “Quasinormal modes of Schwarzschild black holes: The determination

of quasinormal frequencies with very large imaginary parts,” Phys. Rev. D47

(1993) 5253–5258.

N. Andersson, “On the asymptotic distribution of quasinormal-mode frequencies for

Schwarzschild black holes,” Class. Quant. Grav. L10 (1993) 61–67.

– 310 –

[113] L. Motl, “An analytical computation of asymptotic Schwarzschild quasinormal

frequencies,” Adv. Theor. Math. Phys. 6 (2003) 1135–1162, gr-qc/0212096.

[114] L. Motl and A. Neitzke, “Asymptotic black hole quasinormal frequencies,” Adv.

Theor. Math. Phys. 7 (2003) 307–330, hep-th/0301173.

[115] W. G. Unruh, “Experimental black hole evaporation,” Phys. Rev. Lett. 46 (1981)

1351–1353.

[116] M. Novello, M. Visser, and G. Volovik, eds., Artificial black holes. World Scientific,

River Edge, USA, 2002.

G. E. Volovik, The universe in a helium droplet. Clarendon, Oxford, UK, 2003.

C. Barcelo, S. Liberati, and M. Visser, “Analogue gravity,” Living Rev. Rel. 8

(2005) 12, gr-qc/0505065.

[117] P. O. Fedichev and U. R. Fischer, “Hawking radiation from sonic de Sitter horizons

in expanding Bose-Einstein-condensed gases,” Phys. Rev. Lett. 91 (2003) 240407,

cond-mat/0304342.

[118] R. Balbinot, S. Fagnocchi, A. Fabbri, and G. P. Procopio, “Backreaction in acoustic

black holes,” Phys. Rev. Lett. 94 (2005) 161302, gr-qc/0405096.

M. Cadoni, “Acoustic analogs of two-dimensional black holes,” Class. Quant. Grav.

22 (2005) 409–420, gr-qc/0410138.

M. Cadoni and S. Mignemi, “Acoustic analogues of black hole singularities,” Phys.

Rev. D72 (2005) 084012, gr-qc/0504143.

[119] R. Schutzhold, M. Uhlmann, Y. Xu, and U. R. Fischer, “Quantum back-reaction in

dilute Bose-Einstein condensates,” Phys. Rev. D72 (2005) 105005,

cond-mat/0503581.

[120] D. Grumiller, “Black holes and analogues in two dimensions.” talk presented at

QUASIM05 in Dresden, July, 2005.

[121] W. Kummer, H. Liebl, and D. V. Vassilevich, “Integrating geometry in general 2d

dilaton gravity with matter,” Nucl. Phys. B544 (1999) 403–431, hep-th/9809168.

[122] D. Grumiller, W. Kummer, and D. V. Vassilevich, “The virtual black hole in 2d

quantum gravity,” Nucl. Phys. B580 (2000) 438–456, gr-qc/0001038.

[123] D. Grumiller, “Virtual black hole phenomenology from 2d dilaton theories,” Class.

Quant. Grav. 19 (2002) 997–1009, gr-qc/0111097; “Virtual Black Holes and the

S-matrix,” Int. J. Mod. Phys. D13 (2004) 1973–2002, hep-th/0409231.

[124] W. Kummer, H. Liebl, and D. V. Vassilevich, “Exact path integral quantization of

generic 2-d dilaton gravity,” Nucl. Phys. B493 (1997) 491–502, gr-qc/9612012;

“Exact path integral quantization of 2-d dilaton gravity,” gr-qc/9710033.

– 311 –

[125] W. Kummer, “Progress and problems in quantum gravity,” gr-qc/0512010.

D. Grumiller and W. Kummer, “How to approach quantum gravity: Background

independence in 1+1 dimensions,” in What comes beyond the Standard Model?

Symmetries beyond the standard model, N. M. Borstnik, H. B. Nielsen, C. D.

Froggatt, and D. Lukman, eds., vol. 4 of Bled Workshops in Physics, pp. 184–196,

EURESCO. Portoroz, Slovenia, July, 2003. gr-qc/0310068. based upon two talks.

[126] W. Kummer, “On the gauge-independence of the S-matrix,” Eur. Phys. J. C21

(2001) 175–179, hep-th/0104123.

[127] D. Grumiller, W. Kummer, and D. V. Vassilevich, “Positive specific heat of the

quantum corrected dilaton black hole,” JHEP 07 (2003) 009, hep-th/0305036.

[128] P. Fischer, D. Grumiller, W. Kummer, and D. V. Vassilevich, “S-matrix for s-wave

gravitational scattering,” Phys. Lett. B521 (2001) 357–363, gr-qc/0105034.

Erratum ibid. B532 (2002) 373.

D. Grumiller, Quantum dilaton gravity in two dimensions with matter. PhD thesis,

Technische Universitat Wien, 2001. gr-qc/0105078.

[129] H. Balasin, W. Kummer, O. Piguet, and M. Schweda, “On the regularization of the

mass zero 2-D propagator,” Phys. Lett. B287 (1992) 138–144.

[130] A. M. Polyakov, “Quantum geometry of bosonic strings,” Phys. Lett. B103 (1981)

207–210.

[131] S. Nojiri and I. Oda, “Charged dilatonic black hole and Hawking radiation in two-

dimensions,” Phys. Lett. B294 (1992) 317–324, hep-th/9206087.

A. Ori, “Evaporation of a two-dimensional charged black hole,” Phys. Rev. D63

(2001) 104016, gr-qc/0102067.

[132] J. Wess and B. Zumino, “Consequences of anomalous Ward identities,” Phys. Lett.

B37 (1971) 95.

[133] S. R. Coleman,“Quantum Sine-Gordon Equation As The Massive Thirring Model,”

Phys. Rev. D11 (1975) 2088.

S. Mandelstam, “Soliton Operators For The Quantized Sine-Gordon Equation,”

Phys. Rev. D11 (1975) 3026.

[134] A. V. Frolov, K. R. Kristjansson, and L. Thorlacius, “Semi-classical geometry of

charged black holes,” Phys. Rev. D72 (2005) 021501, hep-th/0504073.

[135] V. Guillemin and A. Pollack, Differential Topology. Englewood Cliffs, NJ:

Prentice-Hall, 1974.

[136] J. Ambjorn, R. Loll, J. L. Nielsen, and J. Rolf, “Euclidean and Lorentzian quantum

gravity: Lessons from two dimensions,” Chaos Solitons Fractals 10 (1999) 177–195,

hep-th/9806241.

– 312 –

[137] R. Gastmans, R. Kallosh, and C. Truffin, “Quantum gravity near two-dimensions,”

Nucl. Phys. B133 (1978) 417.

S. M. Christensen and M. J. Duff, “Quantum gravity in two + epsilon dimensions,”

Phys. Lett. B79 (1978) 213.

S. Weinberg in General Relativity, an Einstein Centenary Survey, S. Hawking and

W. Israel, eds. Cambridge University Press, 1979.

[138] R. B. Mann and S. F. Ross, “The D → 2 limit of general relativity,” Class. Quant.

Grav. 10 (1993) 345–351, gr-qc/9208004.

[139] R. Jackiw, “Weyl symmetry and the Liouville theory,” hep-th/0511065.

[140] S. Deser, R. Jackiw, and S. Templeton, “Topologically massive gauge theories,”

Ann. Phys. 140 (1982) 372–411; Erratum-ibid. 185 (1988) 406; “Three-dimensional

massive gauge theories,” Phys. Rev. Lett. 48 (1982) 975–978.

[141] M. Banados, C. Teitelboim, and J. Zanelli, “The black hole in three-dimensional

space-time,” Phys. Rev. Lett. 69 (1992) 1849–1851, hep-th/9204099.

M. Banados, M. Henneaux, C. Teitelboim, and J. Zanelli, “Geometry of the (2+1)

black hole,” Phys. Rev. D48 (1993) 1506–1525, gr-qc/9302012.

[142] B. Sahoo and A. Sen, “BTZ black hole with Chern-Simons and higher derivative

terms,” hep-th/0601228.

[143] W. Kummer and D. J. Schwarz, “Renormalization of R**2 gravity with dynamical

torsion in d = 2,” Nucl. Phys. B382 (1992) 171–186; “General analytic solution of

R**2 gravity with dynamical torsion in two-dimensions,” Phys. Rev. D45 (1992)

3628–3635.

[144] A. Kotov, P. Schaller, and T. Strobl, “Dirac sigma models,” Commun. Math. Phys.

260 (2005) 455–480, hep-th/0411112.

[145] H. Nicolai, D. Korotkin, and H. Samtleben, “Integrable classical and quantum

gravity,” hep-th/9612065.

D. Korotkin and H. Samtleben, “Canonical quantization of cylindrical gravitational

waves with two polarizations,” Phys. Rev. Lett. 80 (1998) 14–17, gr-qc/9705013;

“Yangian symmetry in integrable quantum gravity,” Nucl. Phys. B527 (1998)

657–689, hep-th/9710210.

D. Bernard and N. Regnault, “Vertex operator solutions of 2-d dimensionally

reduced gravity,” Commun. Math. Phys. 210 (2000) 177–201, solv-int/9902017.

L. D. Faddeev, R. M. Kashaev, and A. Y. Volkov, “Strongly coupled quantum

discrete Liouville theory. I: Algebraic approach and duality,” Commun. Math.

Phys. 219 (2001) 199–219, hep-th/0006156.

J. Teschner, “Liouville theory revisited,” Class. Quant. Grav. 18 (2001)

R153–R222, hep-th/0104158.

– 313 –

[146] M. Gurses and S. Tek, “KdV Surfaces,” nlin.si/0511049.

[147] H.-J. Schmidt, “The classical solutions of two-dimensional gravity,” Gen. Rel. Grav.

31 (1999) 1187–1210, gr-qc/9905051.

[148] Y. N. Obukhov and F. W. Hehl, “Black holes in two dimensions,” hep-th/9807101.

[149] T. Dereli and R. W. Tucker, “Nonmetricity induced by dilaton gravity in

two-dimensions,” Class. Quant. Grav. 11 (1994) 2575–2583.

Y. N. Obukhov, “Two-dimensional metric-affine gravity,” Phys. Rev. D69 (2004)

064009, gr-qc/0311091.

M. Adak, “Nonmetricity and torsion induced by dilaton gravity in two dimension,”

gr-qc/0509010.

[150] M. R. Douglas and N. A. Nekrasov, “Noncommutative field theory,” Rev. Mod.

Phys. 73 (2001) 977–1029, hep-th/0106048.

R. J. Szabo, “Quantum field theory on noncommutative spaces,” Phys. Rept. 378

(2003) 207–299, hep-th/0109162.

[151] M. Buric and J. Madore, “Noncommutative 2-dimensional models of gravity,”

hep-th/0406232; “A dynamical 2-dimensional fuzzy space,” Phys. Lett. B622

(2005) 183–191, hep-th/0507064.

[152] S. Cacciatori et al., “Noncommutative gravity in two dimensions,” Class. Quant.

Grav. 19 (2002) 4029–4042, hep-th/0203038.

[153] D. V. Vassilevich, “Quantum noncommutative gravity in two dimensions,” Nucl.

Phys. B715 (2005) 695–712, hep-th/0406163.

[154] D. V. Vassilevich, “Constraints, gauge symmetries, and noncommutative gravity in

two dimensions,” hep-th/0502120.

[155] D. V. Vassilevich, R. Fresneda, and D. M. Gitman, “Stability of a noncommutative

Jackiw-Teitelboim gravity,” hep-th/0602095.

[156] D. Grumiller, W. Kummer, and D. V. Vassilevich, “A note on the triviality of

kappa-deformations of gravity,” Ukr. J. Phys. 48 (2003) 329–333, hep-th/0301061.

[157] P. Aschieri et al., “A gravity theory on noncommutative spaces,” Class. Quant.

Grav. 22 (2005) 3511–3532, hep-th/0504183.

[158] B. M. Zupnik, “Reality in noncommutative gravity,” hep-th/0512231.

[159] M. Chaichian, P. P. Kulish, K. Nishijima, and A. Tureanu, “On a Lorentz-invariant

interpretation of noncommutative space-time and its implications on

noncommutative QFT,” Phys. Lett. B604 (2004) 98–102, hep-th/0408069.

J. Wess, “Deformed coordinate spaces: Derivatives,” hep-th/0408080.

– 314 –

[160] D. V. Vassilevich, “Twist to close,” hep-th/0602185.

P. Aschieri, M. Dimitrijevic, F. Meyer, S. Schraml, and J. Wess, “Twisted gauge

theories,” hep-th/0603024.

[161] M. Chaichian, P. Presnajder, and A. Tureanu, “New concept of relativistic

invariance in NC space-time: Twisted Poincare symmetry and its implications,”

Phys. Rev. Lett. 94 (2005) 151602, hep-th/0409096.

P. Aschieri, M. Dimitrijevic, F. Meyer, and J. Wess, “Noncommutative geometry

and gravity,” Class. Quant. Grav. 23 (2006) 1883–1912, hep-th/0510059.

A. Kobakhidze, “Theta-twisted gravity,” hep-th/0603132.

[162] R. Oeckl, “Untwisting noncommutative R**d and the equivalence of quantum field

theories,” Nucl. Phys. B581 (2000) 559–574, hep-th/0003018.

[163] M. Bojowald, A. Kotov, and T. Strobl, “Lie algebroid morphisms, Poisson Sigma

Models, and off- shell closed gauge symmetries,” J. Geom. Phys. 54 (2005)

400–426, math.dg/0406445.

– 315 –

MIT-CTP 3772hep-th/0609197

Duality in 2-dimensional dilaton gravity

D. Grumiller1 and R. Jackiw2

Center for Theoretical Physics, Massachusetts Institute of Technology,

77 Massachusetts Ave., Cambridge, MA 02139

Abstract

We descry and discuss a duality in 2-dimensional dilaton gravity.

1e-mail: grumil@lns.mit.edu2e-mail: jackiw@lns.mit.edu

1 Introduction

A large class of 2-dimensional (2D) gravity models is described by a gen-eral dilaton gravity action, which determines the line element in terms of aparameter in the action and an additional integration constant. We demon-strate that a well-defined transformation constructs another, inequivalentaction, which belongs to the same class of models, leads to the same line ele-ment, but with the action parameter and integration constant interchanged.We call this transformation a duality: when carried out a second time, itreproduces the initial action.

In Section 2, we recall some results and present a specific example of theduality transformation. This example suggests the general procedure, whichis described in Section 3. Section 4 provides applications to various modelsand Section 5 addresses coupling to matter.

2 Recapitulation of some results

In a recent investigation [1] concerning geometry, we produced the 2D lineelement (X ∈ [0, π])

ds2 = 2 du dX + du2 (λ cos (X/2) + M) , (1)

or equivalently

ds2 =1

cosh (X/2)

[

2 du dX + du2(

M cosh (X/2) + λ)]

, (2)

withtanh (X/2) = sin (X/2) . (3)

We observed that (1) or (2) are solutions to the equations of motion thatfollow from the 2D gravity action (we use the same sign conventions as in[1])

I =1

8π2

∫

d2x√

−g(x)

[

X(x)R(x) +λ

2sin (X(x)/2)

]

. (4)

The parameter M in (1) and (2) arises as an integration constant for theequations of motion implied by (4). These equations are solved in terms oftwo functions X(x) and u(x), which must allow expressing uniquely xα =xα(u, X), but otherwise are arbitrary. They are used as coordinates in (1)

318

or, after the redefinition (3), in (2). The parameter λ appears in the action asa coupling constant with dimension inverse length squared, while the scalar(dilaton) field X is dimensionless and R is the Ricci scalar. The observationthat we made is that the line element, directly in its representation (2) or,after the redefinition (3), as (1), also arises as the solution to the equationsof motion descending from the action

I =1

8π2

∫

d2x√−g

[

XR +1

2tanh (X/2)(∇X)2 − M

4sinh X

]

. (5)

While the two line elements (1), (2) are related through the definition (3), Iis not similarly related to I. Their functional forms differ and especially with(5) M is a parameter of the action and λ occurs as an integration constant,which is opposite to the situation with (4).

In this note we show that the phenomenon observed in our example aboveis generally true for 2D dilaton gravity theories with the action

I =1

8π2

∫

d2x√−g

[

XR + U(X)(∇X)2 − λV (X)]

. (6)

The functions U, V define the model, and many examples will be providedbelow. As before, the dilaton field X is dimensionless, λ is a parameter withdimension inverse length squared and V is dimensionless. We prove thatfor each action (6) there is another one which, although inequivalent to (6),leads to the same 2-parameter family of 2D geometries in a sense made precisebelow. We call the two actions duals of each other and we demonstrate howthis works.

In order to proceed we state some well-known results. For detailed expla-nations and references the review article [2] may be consulted. The generalsolution for the line element derived from the action (6) may be presented inEddington-Finkelstein gauge as

ds2 = eQ[

2 du dX + (λw + M) du2]

, (7)

with the constant of motion M and the definitions

Q′(X) := −U(X) , w′(X) := eQ(X)V (X) , (8)

where prime means differentiation with respect to the argument. Two inte-gration constants arise in the integrated versions of (8). The first one (present

319

in Q) is called “scaling ambiguity”, the second one “shift ambiguity”. Wediscuss later how to fix them appropriately; let us now just assume that theyhave been fixed in some way. Once the functions Q and w are known allother quantities may be derived without ambiguity.

Let us collect some properties of the solutions (7). Evidently, there isalways (at least) one Killing vector ∂u. The square of its norm is given byeQ(λw + M), and therefore Killing horizons emerge for X = Xh, where Xh

is a solution ofλw(Xh) + M = 0 . (9)

Like in the example (1) the dilaton field is used as one of the coordinatesin (7). This is possible globally, except at points where the Killing horizonbifurcates. The Ricci scalar is given by

R(X) = λV ′(X) − 2λV (X)U(X) − U ′(X)e−Q(X)(M + λw(X)) , (10)

and becomes −U ′e−Q M if the condition

eQw = constant (11)

holds. This implies Minkowski spacetime for M = 0, and hence a model withthe property (11) is called a “Minkowski ground state model” (MGS).

In addition to the family of line elements (7) there are isolated solutionswith constant dilaton vacuum (CDV) and maximally symmetric line elementfor each solution X = XCDV of the equation V (XCDV ) = 0. The correspond-ing Ricci scalar is given by

R = λV ′(X)|X=XCDV= constant . (12)

3 Duality in generic 2D dilaton gravity

To obtain the dual action we consider the definitions

dX :=dX

w(X), eQ(X) dX := eQ(X) dX , w(X) :=

1

w(X). (13)

We assume that w is strictly positive3 and therefore the definitions (13) arewell-defined (except for boundary values of X, typically either X = 0 or

3If it is strictly negative similar considerations apply, but for sake of clarity we disregardthis case. If w has zeros but is bounded either from above or from below one may exploitthe shift-ambiguity inherent to the definition of w to give it a definite sign. If w isunbounded from below and from above the definitions (13) necessarily have singularities.

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|X| = ∞). Differentiating (8) and inserting the definitions (13) leads to thedual potentials

U(X) = w(X)U(X) − eQ(X)V (X) , V (X) = − V (X)

w2(X), (14)

which can be used to define a new action of the form

I =1

8π2

∫

d2x√−g

[

XR + U(X)(∇X)2 − MV (X)]

. (15)

The line element (7), following from the original action (6), is identical tothe line element following from the dual action (15) [with the dual potentials(14)]. This can be shown as follows. We start with the action (6) and theensuing line element (7). Factoring w in the latter allows presenting

ds2 = eQw

[

2 dudX

w+ (

M

w+ λ) du2

]

. (16)

Inserting the definitions (13) yields

ds2 = eQ[

2 du dX + (Mw + λ) du2]

, (17)

But according to the general result (7) this is the line element following fromthe dual action (15) with dual potentials (14). The quantity λ appears asa constant of motion in the dual formulation, so the roles of M, λ are inter-changed. Evidently (13) defines a line element-preserving diffeomorphism ofX, with Jacobian w and eQ transforming as a density.

We call the relation (13) [together with (14)] between the actions (6)and (15) a “duality”. One reason for this nomenclature is that both ofthem generate the same set of geometries, as we have just shown, but withinterchanged roles of parameter of the action and constant of motion. Theother relevant observation is that the dual of the dual is always the originalquantity (for Q, w, U , V and X). So there is always a pair of actions relatedto each other in the way presented above, which justifies the use of the name“duality”.

3.1 Properties

Here we list some general properties of the dual theories (6) and (15) [with(13) and (14).]

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• The respective families of line elements are identical by construction;in particular, the number and types of Killing horizons are the samebecause every solution of (9) is also a solution of Mw +λ = 0 and viceversa. Moreover, also the type of horizon remains the same: if it isextremal, i.e., if in addition to (9) also the relation V (Xh) = 0 is ful-filled, then also the corresponding dual relation holds. Similarly, non-extremal horizons are also non-extremal horizons of the dual theory.Therefore, the respective global structures also coincide, except possi-bly in the asymptotic region and at the singularity, where w and/or eQ

may vanish or diverge.

• The respective line elements are not only the same, but also the numberof CDVs where V vanishes coincides for both theories, because V = 0implies with (14) also V = 0 and vice versa. Moreover, if in additionto w > 0 also the inequalities λ > 0 > M hold, then (A)dS CDVsare dual to (A)dS CDVs, because signs coincide of the respective Ricciscalars, given by (12) and its dual version. Flat CDVs are always dualto flat CDVs.

• MGS models, i.e., models with the property (11), are dual to modelswith U = 0.

• Within the first order approach to 2D dilaton gravity the constant ofmotion M has an interpretation as a Casimir function of a certainPoisson-sigma model [3]. If one replaces λ in the action (6) by a scalarfield B and adds a BF term to the action, where F is an Abelianfield strength, then B turns out to be a second Casimir function, theon-shell value of which is given by λ. The duality then acts by swap-ping these two Casimir functions. This mechanism can be extended tomodels which have even more Casimirs, for instance 2D dilaton gravitywith additional gauge fields or with potentials V containing couplingconstants additional to λ.

• Physically the duality exchanges the respective roles of reference mass[λ in (6) and M in (15)] and spacetime mass (as emerging from theconstant of motion).

It should be noted that our duality, which relates different actions but leavesinvariant the space of solutions for the metric, is quite different from the

322

target space duality in [4, 5], which acts on the space of solutions and leavesinvariant the gauge fixed action.

3.2 Fixing the ambiguities

The functions w and eQ are defined (8) up to two integration constantsonly, corresponding to a shift ambiguity w → w + β and a scale ambiguityeQ → αeQ and w → αw, where α, β are some real numbers. The shiftambiguity can be fixed for a large class of models as follows: The duality (13)is meaningful only if w does not have any zeros. If it has zeros but is boundedfrom above or from below (in the range of definition of the dilaton), one mayexploit the shift ambiguity to eliminate all zeros. If one does this in such away that the only zero of w lies at either of the boundaries (typically X = 0 or|X| = ∞) then the shift ambiguity is fixed uniquely. We shall exhibit how thisworks for concrete examples below. Physically, this ambiguity correspondsto a choice of the ground state solution, M = 0. Often there is a preferredchoice, such as a maximally symmetric spacetime, but if there is none thenone just has to choose any particular solution as the ground state. The scalingambiguity is harmless for classical considerations and may be absorbed by arescaling of the coordinate u together with an appropriate rescaling of themass M . Additionally, the representation of the potential V in the action (6)is ambiguous because one may multiply λ by some dimensionless constantand divide V through the same constant. This ambiguity is not essential,because it can be absorbed into the scaling ambiguity discussed above.

Some definitions involve sign ambiguities. Without loss of generality werequire λ > 0 and, as mentioned before, for sake of definiteness also w > 0.For solutions which have a Killing horizon from (9) we deduce M < 0. Thusminus M is directly related to the physical mass of black hole (BH) solutions.In the dual formulation the situation is reversed, i.e., plus λ is directly relatedto the physical mass. As an illustration we consider the model w = e−Q =√

X. Introducing dr = eQ dX and inserting the functions Q, w into the lineelement (7) yields ds2 = 2 du dr + (λ + 2M/r) du2, which is the 2D partof the Schwarzschild (S) BH in Eddington-Finkelstein gauge. However, ris dimensionless, u has dimension length squared and λ, M have dimensioninverse length squared. Therefore, we redefine u :=

√λu, r := r/

√λ and

obtain

ds2 = 2 du dr +

(

1 − 2MADM

r

)

du2 , (18)

323

with MADM = −Mλ−3/2 in units of length. For fixed positive λ the constantof motion −M determines the ADM mass, while in the dual formulation forfixed negative M the constant of motion λ−3/2 determines the ADM mass.

4 Examples and applications

As a demonstration let us now apply the definitions and results from theprevious Section to the example presented before, (1)-(5), with particularemphasis on various ambiguities. Starting point is the action (4). The func-tion U vanishes in that case, while V ∝ sin (X/2), and w ∝ cos (X/2) + c.We may fix the shift ambiguity by setting c = 0 so that w > 0 for X ∈ [0, π)and w → 0 for X → π. The scaling ambiguity is fixed by identifying λ in (4)with λ in (6) in order to reproduce (3) with the same numerical factors. Thisgives V = −1

2sin (X/2) and w = cos (X/2). The formulas (13), (14) lead to

U = 12tanh (X/2), V = 1

4sinh X, Q = − ln cosh (X/2) and w = cosh (X/2),

which after insertion into (15) correctly reproduces (5).Next we apply the general procedure to several models. Among the best-

known 2D dilaton gravity models are the S BH (spherically reduced to 2D),the Jackiw-Teitelboim (JT) model [6] and the Witten (W) BH [7]. It wasrealized in [8] that all of them can be summarized in a 2-parameter familyof 2D dilaton gravity models of the form

eQ = X−a , w = Xb+1 → U =a

X, V = (b + 1)Xa+b . (19)

The shift ambiguity inherent to w has been fixed by requiring w(0) = 0for b > −1 and w(∞) = 0 for b < −1 (for b = −1 there is no preferredway to fix this ambiguity, so we choose w = 1). The scale ambiguity hasbeen fixed conveniently. The line element (7) may be re-parameterized as(X−a dX = dr)

ds2 = 2 du dr +(

λXb+1−a(r) + MX−a(r))

du2 . (20)

The Ricci scalar (10) reads

R = λb(b + 1 − a)Xa+b−1 + MaXa−2 . (21)

Notably R is constant for all λ, M if and only if (a−1)2 = 1 and b(b2−1) = 0.When a− b = 1 the MGS property (11) holds. If a + b = 1 the ground state

324

solution (M = 0) is (A)dS. For a = 0 R becomes independent from M .Models with a 6= 1, b = 0 are called Rindler ground state models, becauseR vanishes and the Killing norm is linear in the coordinate r for M = 0.We shall assume b 6= 0 and discuss this special case separately at the end.Applying our rules (13) to the models given by (19) yields

eQ = (−bX)−1+(a−1)/b , w = (−bX)1+1/b . (22)

After the field redefinition −bX → X the functions Q and w are again in theform (19), so our duality maps one model of the ab family to another one ofthe same family with new parameters a and b given by

a = 1 − a − 1

b, b =

1

b. (23)

The fixed points under duality transformations are b = a = 1 and b = −1, aarbitrary.

It should be mentioned that the combination

ρ :=(a − 1)√

|b|= sign (−b)

(a − 1)√

|b|(24)

is invariant under the duality for b < 0 and goes to ρ = −ρ for b > 0.This leads to a useful representation of the “phase space” of Carter-Penrosediagrams. In Fig. 6 of [8] that phase space is depicted as function of a and b.We present the same graph as a function of ρ and ξ = ln

√

|b|, discriminatingbetween positive b (Fig. 1) and negative b (Fig. 2). In the former case dualityacts by reflection at the origin, in the latter case duality acts by reflectionat the ρ axis. So in Fig. 1 the white region is mapped onto itself (with theorigin as fixed point), whereas the light and dark gray regions are mappedonto each other by duality. By contrast, in Fig. 2 each of the three differentlyshaded regions is mapped onto itself (with the ρ axis as line of fixed points).Therefore, in Fig. 1 the singularities and asymptotic regions are exchanged bythe duality, while in Fig. 2 they remain the same. The four exponential curvesin both graphs correspond to the line of MGS models (a = 1 + b), modelswhich have an (A)dS ground state (a = 1 − b), models with no kinetic termfor the dilaton (a = 0) and models with a = 2. At the intersection points ofthese curves lie models which have maximally symmetric spacetimes for anyvalue of λ, M . The JT model appears in Fig. 1, the S BH in Fig. 2. The WBH emerges as an asymptotic limit (ξ → −∞ on the ξ-axis) in both graphs.

325

Figure 1: b > 0

Figure 2: b < 0

326

In particular, for spherically reduced models from D dimensions we haveb = −1/(D− 2) and a = (D− 3)/(D− 2). Since X has a higher dimensionalinterpretation as surface area it is fair to ask for a physical interpretationof X. It may be checked easily that X ∝ X1/(D−2) is the surface radius.The dual model is conformally related4 to spherically reduced gravity fromD dimensions, with D = (2D − 3)/(D − 2). Obviously, only D = 3 is self-dual, while the S BH (D = 4) is related to D = 5/2. We mention that analternative representation of the dual action is obtained by eliminating thedual dilaton field by means of its equation of motion, X ∝ R1/(1−D):

I = ND

∫

d2x√−g R1+1/(1−D) , (25)

where ND is a constant following from the normalization chosen in (15). Itgrows with D for D ≫ 1. This class of theories has been studied in [9].

The JT model (a = 0, b = 1) is very special as it is not only dual, butalso conformally related to the model a = 2, b = 1.

Finally we discuss the models with b = 0. We assume first a 6= 1, so thatthe ground state solution is Rindler spacetime. The dual dilaton is givenby X = ln X. Therefore, the dual model does not belong to the ab-family;rather it belongs to the class of Liouville gravities [10, 11]: Q = (1 − a)X

and w = e−X . In particular, for a = 0 the CGHS model emerges (cf. the lastRef. [7]). Its dual is given by the “almost Weyl invariant” Liouville model[11]. In [12] it was observed that the Ricci scalar obtained from Liouvillegravity is independent from the constant of motion. This feature is simplya consequence of the duality of Liouville gravity to Rindler ground statemodels. The W BH (a = 1, b = 0) has the MGS property (11). Its dual is

given by Q = 0 and w = e−X , thus leading to the dual action

IWBH =1

8π2

∫

d2x√−g

(

XR + Me−X)

. (26)

Elimination of the dual dilaton field by means of its equation of motion,X = − ln (R/M), and re-insertion into (26) allows to represent the dual WBH action (up to Einstein-Hilbert terms) as

IWBH = − 1

8π2

∫

d2x√−g R ln |R| . (27)

4By “conformally related” we mean that the difference between the corresponding lineelements is a conformal factor which is regular globally, except possibly at boundary pointswhere e

Q may vanish or diverge.

327

This dual action for the W BH was presented for the first time in [13] andit arises also as the D → ∞ limit of (25), concurrent with the fact that theD → ∞ limit of spherically reduced gravity yields the W BH. This limitingprocedure resembles the one discussed in [11].

5 Outlook

The duality (13) [together with (14)] between the actions (6) and (15) leavesintact the line element (7) but changes the dilaton field. This has someconsequences for coupling to matter as well as for thermodynamical andsemi-classical considerations, which we shall outline briefly. Generally anyphenomenon that is based upon a fixed background geometry and that isnot sensitive to the dilaton field will be invariant under the duality, but evenquantities that are sensitive to the dilaton field (like quasi-normal modes ofa scalar field) may be duality-invariant.

An example of a duality-invariant observable is the Hawking temperature,as derived either naively from surface gravity or from the Hawking flux ofa Klein-Gordon field propagating on the (fixed) BH background. An exam-ple of a duality-non-invariant observable is the Bekenstein-Hawking entropy,which is proportional to the dilaton field [14] and thus changes under theduality.

It would be interesting to find observables that are duality-invariant inthe full dynamical and self-consistent system of geometry plus matter, i.e.,not based upon some fixed background approximation.

Acknowledgments

DG is grateful to Max Kreuzer, Wolfgang Kummer, Alessandro Torrielli,Dimitri Vassilevich for discussions and to Ralf Lehnert for help with theFigures. We thank Mariano Cadoni for calling our attention to [5].

This work is supported in part by funds provided by the U.S. Depart-ment of Energy (DOE) under the cooperative research agreement DEFG02-05ER41360. DG has been supported by the Marie Curie Fellowship MC-OIF021421 of the European Commission under the Sixth EU Framework Pro-gramme for Research and Technological Development (FP6).

328

References

[1] D. Grumiller and R. Jackiw, “Kaluza-Klein reduction of conformallyflat spaces,” math-ph/0609025.

[2] D. Grumiller, W. Kummer, and D. V. Vassilevich, “Dilaton gravity intwo dimensions,” Phys. Rept. 369 (2002) 327–429, hep-th/0204253.

[3] P. Schaller and T. Strobl, “Poisson structure induced (topological) fieldtheories,” Mod. Phys. Lett. A9 (1994) 3129–3136, hep-th/9405110.

[4] A. Giveon, “Target space duality and stringy black holes,” Mod. Phys.

Lett. A6 (1991) 2843.

[5] M. Cadoni and S. Mignemi, “Dualities of Lorentzian and Euclideanblack holes in two-dimensional string generated models,” Mod. Phys.

Lett. A10 (1995) 367 hep-th/9403113.

[6] R. Jackiw, C. Teitelboim, in Quantum Theory Of Gravity,S. Christensen, ed., Adam Hilger, Bristol, 1984.

[7] E. Witten, “On string theory and black holes,” Phys. Rev. D44 (1991)314–324.

G. Mandal, A. M. Sengupta, and S. R. Wadia, “Classical solutions oftwo-dimensional string theory,” Mod. Phys. Lett. A6 (1991) 1685–1692.

S. Elitzur, A. Forge, and E. Rabinovici, “Some global aspects of stringcompactifications,” Nucl. Phys. B359 (1991) 581–610.

C. G. Callan, Jr., S. B. Giddings, J. A. Harvey, and A. Strominger,“Evanescent black holes,” Phys. Rev. D45 (1992) 1005–1009,hep-th/9111056.

[8] M. O. Katanaev, W. Kummer, and H. Liebl, “On the completeness ofthe black hole singularity in 2d dilaton theories,” Nucl. Phys. B486

(1997) 353–370, gr-qc/9602040.

[9] H.-J. Schmidt, “Scale invariant gravity in two-dimensions,” J. Math.

Phys. 32 (1991) 1562–1566.

[10] Y. Nakayama, “Liouville field theory: A decade after the revolution,”Int. J. Mod. Phys. A19 (2004) 2771–2930, hep-th/0402009.

329

[11] R. Jackiw, “Weyl symmetry and the Liouville theory,”hep-th/0511065.

[12] L. Bergamin, D. Grumiller, W. Kummer, and D. V. Vassilevich,“Classical and quantum integrability of 2D dilaton gravities inEuclidean space,” Class. Quant. Grav. 22 (2005) 1361–1382,hep-th/0412007.

[13] V. P. Frolov, “Two-dimensional black hole physics,” Phys. Rev. D46

(1992) 5383–5394.

[14] J. Gegenberg, G. Kunstatter, and D. Louis-Martinez, “Observables fortwo-dimensional black holes,” Phys. Rev. D51 (1995) 1781–1786,gr-qc/9408015.

330

Preprint typeset in JHEP style - HYPER VERSION Brown-HET-1478MIT-CTP-3825

hep-th/0703230

Thermodynamics of Black Holes in Two (and Higher)

Dimensions

Daniel Grumiller

Center for Theoretical Physics, Massachusetts Institute of Technology,77 Massachusetts Ave., Cambridge, MA 02139Email: grumil@lns.mit.edu

Robert McNees

High Energy Theory Group, Department of Physics, Brown University,Providence, RI 02912Email: mcnees@het.brown.edu

Abstract: A comprehensive treatment of black hole thermodynamics in two-dimensionaldilaton gravity is presented. We derive an improved action for these theories and constructthe Euclidean path integral. An essentially unique boundary counterterm renders the im-proved action finite on-shell, and its variational properties guarantee that the path integralhas a well-defined semi-classical limit. We give a detailed discussion of the canonical en-semble described by the Euclidean partition function, and examine various issues related tostability. Numerous examples are provided, including black hole backgrounds that appearin two dimensional solutions of string theory. We show that the Exact String Black Holeis one of the rare cases that admits a consistent thermodynamics without the need for anexternal thermal reservoir. Our approach can also be applied to certain higher-dimensionalblack holes, such as Schwarzschild-AdS, Reissner-Nordstrom, and BTZ.

Keywords: Black hole thermodynamics, two-dimensional dilaton gravity, string theoryin two dimensions, spherically symmetric black holes, BTZ black hole.

Contents

1. Introduction 333

2. Semi-Classical Approximation of the Dilaton Gravity Path Integral 3352.1 Equations of Motion 3352.2 Black Holes 3362.3 Variational Properties of the Action 3382.4 The Hamilton-Jacobi Boundary Term 3402.5 Path Integral Based on the Improved Action 341

3. Thermodynamics in the Canonical Ensemble 3433.1 Dilaton Charge and Free Energy 3433.2 Entropy and Area 3453.3 Dilaton Chemical Potential and Surface Pressure 3463.4 Internal Energy and First Law 3473.5 Specific Heat at Constant Dilaton Charge and Fluctuations 3483.6 Free Enthalpy and Gibbs-Duhem Violation 3493.7 Miscellaneous Thermodynamical Quantities 3503.8 Charged Black Holes 3513.9 Conformal Properties and Microcanonical Thermodynamics 3543.10 Stability and Phase Transitions 3553.11 Tunneling and Constant Dilaton Vacua 357

4. Two Dimensional Examples 3594.1 Minkowski Ground State Models 3594.2 Models in the a-b Family 360

4.2.1 Thermodynamics of the a-b Family 3614.2.2 Equation of State and Scaling Properties 362

5. Examples from String Theory 3655.1 Witten Black Hole 3675.2 String Theory Is Its Own Reservoir: Exact String Black Hole 368

5.2.1 Thermodynamics of the Exact String Black Hole 3695.2.2 Thermodynamical Derivation of the Exact String Black Hole 371

6. Spherically Symmetric Black Holes in Higher Dimensions 3726.1 Schwarzschild Black Hole in d+ 1 ≥ 4 Dimensions 3746.2 Asymptotically AdS Space-Times 3756.3 Reduction of BTZ 376

7. Discussion 380

– 1 –

A. Conventions, Dimensions, and a Dilaton Gravity Bestiary 382

B. Weyl Rescaled Metric 385

C. Proof of Gauge-Independence 387

D. Formulas 388

1. Introduction

Two-dimensional gravity provides a useful expedient for testing ideas about quantum grav-ity in higher dimensions. Technical simplifications in two dimensions often lead to exactresults, and it is hoped that this might help to address some of the conceptual problemsposed by quantum gravity. At the same time, these simplifications remove many of the fea-tures that make gravity interesting. Striking a balance between models that are tractableand models that seem relevant is an art in its own right.

In this paper we study black hole (BH) thermodynamics in two-dimensional modelsof dilaton gravity. In the absence of matter these models can be solved exactly, but theycontain no dynamical degrees of freedom. This seems too restrictive, so we sacrifice inte-grability and consider models that are coupled to matter. A detailed description of thematter is not necessary for our purposes. We only require the existence of some propagat-ing degrees of freedom, analogous to the radiation emitted by a BH in higher dimensions.We then perform a semi-classical analysis and obtain a comprehensive description of BHthermodynamics for arbitrary models. Despite the fact that we work in two dimensions,these systems exhibit a wide range of interesting thermodynamic phenomena. To someextent this is due to our consideration of ‘quasilocal’ thermodynamic potentials describingBHs in a finite cavity. Such a treatment reveals aspects of BH thermodynamics that arenot visible in an asymptotic analysis. Several universal properties of dilaton gravity BHsare identified, as well as structures that are common to families of models. Our approachallows us to study the thermodynamics of backgrounds, both approximate [1] and exact [2],associated with two-dimensional solutions of string theory. We are also able to extend ouranalysis to spherically symmetric BHs in higher dimensions. In particular, we obtain anexpression that describes the free energy of both asymptotically flat and asymptoticallyAnti-de Sitter (AdS) BHs.

Dilaton gravity in two dimensions is conventionally described by the (Euclidean) action

I = − 116πG2

∫Md 2x√g[X R− U(X) (∇X)2 − 2V (X)

]− 1

8πG2

∫∂Mdx√γ X K . (1.1)

We define the dilaton X in terms of its coupling to the Ricci scalar by the expression X R.Different models are distinguished by the kinetic and potential functions U(X) and V (X),cf. e.g. [3] for a review. The remaining part of the action is the analog of the Gibbons-Hawking-York boundary term [4,5], where γab is the induced metric on the boundary and

– 333 –

K is the trace of the extrinsic curvature. Our conventions for curvature tensors and othergeometric quantities can be found in appendix A. To study the thermodynamics of thesetheories we employ the Euclidean path integral,

Z =∫

DgDX exp(−1

~I[g,X]

). (1.2)

The path integral is evaluated by imposing boundary conditions on the fields and thenperforming the weighted sum over all relevant space-times (M, g) and dilaton configurationsX. In the semi-classical limit it is dominated by contributions from stationary points ofthe action. This can be verified by expanding it around a classical solution

I[gcl + δg,Xcl + δX] = I[gcl, Xcl] + δI[gcl, Xcl; δg, δX] +12δ2I[gcl, Xcl; δg, δX] + . . . (1.3)

where δI and δ2I are the linear and quadratic terms in the Taylor expansion. The linearterm is assumed to vanish on-shell. Then, if the leading term is finite and the quadraticterm is positive definite, the saddle point approximation of the path integral is given by

Z ∼ exp(−1

~I[gcl, Xcl]

) ∫DδgDδX exp

(− 1

2~δ2I[gcl, Xcl; δg, δX]

). (1.4)

This is interpreted as the partition function in the canonical ensemble, with the temperatureset by the path integral boundary conditions.

The transition from (1.2) to (1.4) is complicated by the fact that the action (1.1)does not possess any of the properties listed above. First, the on-shell action diverges.This is familiar from calculations in higher dimensions, and is commonly addressed usinga technique known as ‘background subtraction’ [5, 6]. Second, the linear term in (1.3)does not vanish for all field configurations that contribute to the path integral [7, 8]. Thisis due to boundary terms whose contribution to δI are usually not considered in detail.Schematically, these boundary terms take the form

δI∣∣∣on−shell

∼∫∂Mdx√γ[πab δγab + πX δX

]6= 0 . (1.5)

Even when we consider models where the boundary conditions imply δγab → 0 and δX → 0at ∂M, the coefficients of these terms tend to diverge so rapidly that δI does not vanish.Finally, the Gaussian integral in (1.4) is often divergent. In that case the canonical partitionfunction is not well-defined. Instead of describing the thermodynamics of a stable system,(1.4) yields information about decay rates between various field configurations with thespecified boundary conditions [9].

We solve the first two problems by replacing (1.1) with an ‘improved’ action Γ. Theimproved action is related to (1.1) by a boundary counterterm. The counterterm, whichis essentially unique, is obtained by extending the results of [10] to arbitrary models ofdilaton gravity. The improved action is finite on-shell, and δΓ vanishes for any variationof the fields that preserve the path integral boundary conditions. However, it does notaddress the remaining problem. The quadratic term δ2Γ may result in a Gaussian integral,

– 334 –

as in (1.4), that is divergent. This is directly related to the thermodynamic stability of thesystem. Essentially, the density of states grows so rapidly that the canonical ensemble isnot defined. We deal with this in the same manner as York [11]. The BH is placed insidea cavity that couples the system to a thermal reservoir, with boundary conditions fixed atthe wall of the cavity. Following Gibbons and Perry [12], we identify the location of thewall with the value of a conserved dilaton charge. The canonical ensemble that emergesfrom this procedure is well defined if and only if the specific heat of the system is positive.Typically this requires a finite cavity, but there are some notable exceptions. Havingresolved the three problems described above, the approximation (1.4), with Γ replacing I,gives the canonical partition function for the system.

We proceed as follows: in section 2 we review BH solutions of two-dimensional dilatongravity [13, 14], and examine the action (1.1) and its variation evaluated on these back-grounds. We then derive the improved action Γ and verify its properties. The Euclideanpath integral is used to define the canonical partition function in section 3, from which allof our results on thermodynamics follow. We obtain expressions for thermodynamic quan-tities in an arbitrary model of two-dimensional dilaton gravity, establish universal resultslike the first law of BH thermodynamics, and examine several issues related to stability. Insection 4 these results are applied to two significant families of models. Section 5 addressesBHs that appear in models obtained from string theory. This includes a treatment of theExact String BH [2]. The results obtained in section 3 can also be applied, in certaincases, to BH solutions of higher dimensional theories. This is discussed in section 6, wherewe study spherically symmetric BHs with or without a cosmological constant, and thetoroidal Kaluza-Klein reduction of the BTZ BH [15]. Finally, in section 7 we summarizeour results, discuss further applications, and suggest possible generalizations. A collectionof appendices contain our conventions, a review of various models, and additional technicalresults.

Before proceeding to section 2, we point out a few important conventions. Euclideansignature is used throughout this paper. Nevertheless, we frequently use terms such as‘Minkowski’ and ‘horizon’ as if we were working in Lorentzian signature. The dimensionlessNewton’s constant that appears in the action is set to 8πG2 = 1. It is restored in certainexpressions, when necessary.

2. Semi-Classical Approximation of the Dilaton Gravity Path Integral

In this section we consider the Euclidean path integral formulation of dilaton gravity intwo dimensions. Our analysis shows that the standard form of the action is not stationaryunder the full class of field variations that preserve the boundary conditions. We constructan improved action that solves this problem, and obtain the semi-classical approximationof the corresponding path integral.

2.1 Equations of Motion

The equations of motion are obtained by extremizing the action (1.1) with respect to small

– 335 –

variations of the fields that vanish at the boundary. This yields

U(X)∇µX∇νX −12gµνU(X)(∇X)2 − gµνV (X) +∇µ∇νX − gµν∇2X = 0 (2.1)

R+ ∂XU(X)(∇X)2 + 2U(X)∇2X − 2 ∂XV (X) = 0 . (2.2)

Solutions of these equations always possess at least one Killing vector whose orbits arecurves of constant X [16, 17]. We fix the gauge so that the metric is diagonal. Thesolutions

X = X(r) , ds2 = ξ(r) dτ2 +1ξ(r)

dr2 , (2.3)

with

∂rX = e−Q(X) (2.4)

ξ(X) = w(X) eQ(X)

(1− 2M

w(X)

)(2.5)

are expressed in terms of two model-dependent functions,

Q(X) := Q0 +∫ X

dX U(X) , (2.6)

w(X) := w0 − 2∫ X

dX V (X) eQ(X) . (2.7)

Here Q0 and w0 are constants, and the integrals are evaluated at X. Notice that w0

and the integration constant M contribute to ξ(X) in the same manner. Together, theyrepresent a single parameter that has been partially incorporated into the definition ofw(X). By definition they transform as w0 → e∆Q0w0 and M → e∆Q0M under the shiftQ0 → Q0 +∆Q0. This ensures that the functions (2.4) and (2.5) transform homogeneously,allowing Q0 to be absorbed into a rescaling of the coordinates. Therefore, the solution isparameterized by a single constant of integration. With an appropriate choice of w0 wecan restrict M to take values in the range M ≥ 0 for physical solutions. As evident from(2.3) the Killing vector described above is ∂τ . For brevity we refer to

√ξ(X) from (2.5)

as the “Killing norm”. If it vanishes we encounter a Killing horizon. The square of theKilling norm for the solution with M = 0 is

ξ0(X) := w(X) eQ(X) . (2.8)

This solution, which we often refer to as the ‘ground state’, plays an important role through-out this paper 1.

2.2 Black Holes

The constant of integration that labels solutions of the equations of motion is analogousto the mass parameter that appears in the Schwarzschild metric. This constant has been

1If the function V (X) has one or more zeroes at finite values of X there is a second, inequivalent family

of solutions with constant dilaton. In that case, the identification of the actual ground state of the model

becomes more involved. This issue will be addressed in subsection 3.11.

– 336 –

partially absorbed into the definition of w(X) in such a way that M = 0 represents theground state of a particular model. We refer to solutions with M > 0, which may exhibithorizons, as BHs.

In the models we consider the metric function ξ(X) is non-negative over a semi-infiniteinterval

Xh ≤ X <∞ . (2.9)

The upper end of this interval corresponds to the asymptotic region of the space-time. Thefunction w(X) generally diverges in this limit

limX→∞

w(X)→∞ (2.10)

so that the asymptotic behavior of the metric is typified by the ground state solution (2.8).The lower end of the interval (2.9) is either the origin, or a Killing horizon. Assuming thatthe function eQ(X) is non-zero for finite values of X, the location of the horizon is relatedto the parameter M by

w(Xh) = 2M . (2.11)

If this condition admits multiple solutions then Xh is taken to be the outermost horizon,so that w(X) > 2M for X > Xh.

Regularity of the metric at the horizon fixes the periodicity τ ∼ τ +β of the Euclideantime, which is given by

β =4π∂rξ

∣∣∣∣rh

=4π

w′(X)

∣∣∣∣Xh

. (2.12)

The inverse periodicity is related to the surface gravity of the BH by 2πβ−1 = κ. In anasymptotically flat space-time, β−1 is also the temperature measured by an observer atinfinity. In a slight abuse of notation, we denote this quantity by T

T = β−1 =w′(X)

4π

∣∣∣∣Xh

. (2.13)

This should not be confused with the proper local temperature, which is related to β−1 bya position dependent red-shift (the ‘Tolman factor’ [18])

T (X) =1√ξ(X)

β−1 . (2.14)

The proper temperature at infinity coincides with (2.13) only if ξ(X)→ 1 as X →∞.At this point it is instructive to evaluate the action for the BH solution, which we

expect is related to the semi-classical approximation of the BH partition function. Thebulk terms in (1.1) are straightforward, but we need to specify how the boundary is treatedin order to calculate the Gibbons-Hawking-York term. It is apparent from (2.3) that theonly component of ∂M that contributes to this calculation is a surface orthogonal to thecoordinate r. Depending on the choice of coordinate system this may occur either at somefinite value of r, or in a limit such as r → ∞. In either case, the boundary is assumedto correspond to X → ∞, which means that the boundary terms in the action should be

– 337 –

calculated via a suitable limiting procedure. We implement this by placing a regulatorX ≤ Xreg on the dilaton, and treating the isosurface X = Xreg as the boundary. Then theon-shell action is

Ireg = β(

2M − w(Xreg)− 2πXh T). (2.15)

The regulator should be removed by taking the limit Xreg → ∞, which recovers the fullBH space-time. At this point we encounter a problem: the asymptotic behavior (2.10) ofw(Xreg) causes the action to diverge. How should this term be addressed? One possibilityis that it should be discarded, and the finite remainder then interpreted as the action for thespace-time. This strategy is unsatisfactory for two reasons. First, it fails to reproduce thecorrect thermodynamics of the BH. Second, as we shall demonstrate in the next subsection,the variational properties of the action (1.1) are not consistent with our assumptions aboutthe path integral. Neither issue is resolved by the sort of ad-hoc subtraction describedabove.

2.3 Variational Properties of the Action

The equations of motion were described in section 2.1 as the conditions for extremizingthe action. In this section we show that this is true for ‘localized’ variations of the fields,but not for generic variations that preserve the boundary conditions. This has importantconsequences for the path integral formulation of the theory. We begin by consideringsmall, independent variations of the metric and dilaton. The corresponding change in theaction is

δI = −12

∫Md 2x√g[Eµν δgµν + EX δX

]+∫∂Mdx√γ[πab δγab + πX δX

]. (2.16)

The terms Eµν and EX in the bulk integral are the left-hand-sides of (2.1) and (2.2),respectively. In addition to the bulk term, there is also a boundary term that depends onmomenta conjugate to γab and X

πab = −12γab nµ∇µX , πX = U(X)nµ∇µX − K . (2.17)

These momenta are defined with respect to evolution of the fields along the direction normalto the boundary, indicated here by the vector nµ. When evaluated on a solution of theclassical equations of motion, the change in the action (2.16) should vanish for any variationof the fields that preserves the boundary conditions. A direct calculation, which we shallcarry out below, shows that this is not the case. Instead, we find that δI 6= 0 unless werestrict our attention to localized variations of the fields. In this context a field variationis ‘localized’ if its asymptotic behavior satisfies some condition that is more restrictivethan simply preserving the boundary conditions. Essentially, the change in the action onlyvanishes for variations that fall off more rapidly at infinity than required by the boundaryconditions.

The claim that δI does not vanish for physically reasonable variations of the fieldsmay seem odd. The Gibbons-Hawking-York term in (1.1) is usually motivated by statingthat it leads to an action with a “well defined” variational principle. But the Gibbons-Hawking-York term only ensures that fields obey Dirichlet conditions at ∂M. It does

– 338 –

not guarantee that the boundary term in δI vanishes for arbitrary δγab and δX thatpreserve these boundary conditions. This issue has been studied in higher dimensions forgravitational theories with both asymptotically flat [7, 8] and asymptotically AdS [19, 20]boundary conditions. It is especially important for the path integral formulation of thetheory. Given a solution (ξcl, Xcl) of the equations of motion, the path integral also receivescontributions from field configurations whose asymptotic behavior is (ξcl + δξ,Xcl + δX),where δξ and δX are generic variations consistent with the boundary conditions. The~ → 0 limit of the path integral is dominated by solutions of the classical equations ofmotion if, and only if, δI vanishes for all such variations.

We show now that the action (1.1) does not satisfy this condition. Evaluating (2.16)for a solution of the form (2.3) gives

δIreg =∫dτ

[−1

2∂rX δξ +

(U(X) ξ(X) ∂rX −

12∂rξ

)δX

]. (2.18)

The subscript on δI indicates that the boundary term must be evaluated using a regulator,as in the previous subsection. The boundary term should vanish as the regulator is removedto infinity. To show that this is not the case, it is enough to set δX = 0 and consider thevariation δξ. Using (2.4), the coefficient of δξ is proportional to e−Q(X). Depending onthe model, this term may either diverge, approach a constant, or vanish at ∂M. Now weneed to specify how a generic variation δξ is allowed to behave at ∂M. The models weconsider allow for a wide range of asymptotic behavior. In particular, ξ might diverge inthe X → ∞ limit. Thus, we cannot assume that δξ vanishes at ∂M, as we would if ξapproached a constant at ∂M. Instead we must appeal to the general solution (2.5), fromwhich we infer the asymptotic behavior of a δξ that preserves the boundary conditions.Recall that the leading asymptotic behavior of ξ is determined by ξ0 = weQ, since w →∞at X → ∞. We assume that the boundary conditions are preserved by variations whoseasymptotic behavior is similar to the sub-leading term in (2.5). This means that we mustconsider the change in the action due to

δξ = δM eQ(X) (2.19)

where δM is an arbitrary infinitesimal constant. The corresponding change in the actionis non-zero and independent of the regulator

δI =∫dτ δM 6= 0 . (2.20)

This shows that solutions of (2.1) and (2.2) do not extremize the action (1.1) againstgeneric variations δξ that preserve the boundary conditions on ξ. In appendix B we repeatthis analysis in terms of a new metric variable ξ that obeys the standard Dirichlet conditionξ = 1 at ∂M. Then, in appendix C, we show that the result (2.20) does not depend onthe diagonal gauge used in (2.3).

This explains why one cannot drop the regulator-dependent term in (2.15) and assigna meaning to the part that remains. The semi-classical approximation of the path integralis only well-defined if δI = 0 for the full class of field variations encountered in the path

– 339 –

integral. With this in mind, we look for an improved action that is both finite and stationaryfor solutions of the classical equations of motion.

2.4 The Hamilton-Jacobi Boundary Term

In the previous section we found that the variational properties of the action are notcompatible with the semi-classical approximation of the path integral. The same problemwas encountered in [10], where the authors studied BH solutions of the tree-level equationsfor type 0A string theory in two dimensions. In this section we generalize the techniquesused in [10] to arbitrary theories of dilaton gravity. The result is a new action Γ that isrelated to I by

I = Γ + ICT . (2.21)

The term ICT is a boundary integral of the form

ICT =∫∂MdxLCT (γ,X) . (2.22)

The integrand may depend on the fields at the boundary, and their derivatives along theboundary, but not their normal derivatives. This ensures that Γ and I yield the sameequations of motion. The term ICT is then defined so that solutions of these equationsare stationary points of Γ, with respect to any variation of the fields that preserves theboundary conditions. This construction is similar to the boundary counterterm techniqueused in the AdS/CFT correspondence. Although this approach was originally devisedto cancel divergences in the action [21–24], it was later shown that the boundary coun-terterms follow from the requirement that the action admits a well-defined variationalproblem with asymptotically AdS boundary conditions [19]. A similar modification of theEinstein-Hilbert action was proposed in [8, 25], where the authors considered space-timeswith asymptotically flat boundary conditions.

Following the derivation in [10], the boundary counterterm is identified with a solutionof the Hamilton-Jacobi equation for the on-shell action. This equation can be derived bystarting with the Hamiltonian constraint that follows from the action (1.1)

H = 2πX γab πab + 2U(X)(γab π

ab)2

+ V (X) = 0 . (2.23)

Recall that the momenta, defined in (2.17), appear as boundary terms in the variationof the action. When δI is evaluated for a solution of the equations of motion the bulkterm vanishes, leaving only the boundary term. Thus, the momenta can be expressed asfunctional derivatives of the on-shell action

πab =1√γ

δ

δγab

(I∣∣∣E=0

)πX =

1√γ

δ

δX

(I∣∣∣E=0

). (2.24)

Rewriting the Hamiltonian constraint (2.23) in terms of these expressions gives a non-linearfunctional differential equation for the on-shell action which is difficult to solve. But in ourcase symmetry arguments reduce the problem to a linear differential equation for a functionof X, as we now demonstrate. The solution ICT should be invariant under diffeomorphisms

– 340 –

of ∂M, so we express its integrand as LCT (γ,X) =√γ LCT (γ,X). Because there are no

other intrinsic, diffeomorphism covariant scalars constructed from γ in one dimension, itfollows that LCT can depend only on X and its derivatives along the boundary. But termsinvolving derivatives can be ignored, because the boundary is an isosurface of X. Thus,the scalar density LCT depends only on X and the boundary counterterm takes the form

ICT =∫∂Mdτ√γLCT (X) . (2.25)

The functional derivatives (2.24) applied to this integral give ‘momenta’ that we denote

pab =12γab LCT (X) pX = ∂XLCT (X) (2.26)

to distinguish them from πab and πX . Applying these expressions in (2.23), the Hamiltonianconstraint becomes a linear differential equation for L2

CT ,

12∂X(LCT (X)2

)+

12U(X)LCT (X)2 + V (X) = 0 . (2.27)

The general solution of this equation is

LCT (X) = −√e−Q(X) (w(X) + c) (2.28)

where Q(X) and w(X) were defined in (2.6) and (2.7), and c is an arbitrary constant.Note that the Hamilton-Jacobi equation determines (LCT ) 2, and the negative root hasbeen chosen in (2.28). With the definition (2.21), this root leads to an action Γ with thecorrect properties.

In many models there are natural arguments for setting the constant c to zero. Forinstance, in models associated with two-dimensional string theories the action I is invariantunder Buscher duality [26]. The action Γ is expected to possess the same invariances asI, and this is only the case if c = 0, as pointed out in [10]. More generally, setting c = 0guarantees that LCT is independent of the scaling ambiguity associated with the constantQ0 that appears in (2.6). Therefore, we set c = 0 and obtain the boundary counterterm

ICT = −∫∂Mdx√γ√w(X)e−Q(X) . (2.29)

This leads to the ‘improved’ action

Γ = I +∫∂Mdx√γ√w(X)e−Q(X) . (2.30)

2.5 Path Integral Based on the Improved Action

Based on the analysis of the previous subsections, we conclude that the Euclidean pathintegral for two-dimensional dilaton gravity should weight field configurations using theexponential of the improved action,

Z =∫

DgDX exp(−1

~Γ[ g,X]

), (2.31)

– 341 –

rather than (1.2). In this section we justify this claim by establishing two importantproperties of Γ. First we show that solutions of (2.1) and (2.2) are stationary points of(2.30) with respect to variations of the fields that preserve the boundary conditions. Thismeans that the semi-classical approximation of (2.31) is dominated by solutions of theclassical equations of motion (2.1) and (2.2). Second, we evaluate the action for the BHspace-times discussed in section 2.2 and show that the result is finite.

The variation of the action Γ is

δ Γ =− 12

∫Md 2x√g[Eµν δgµν + EX δX

]+∫∂Mdx√γ[

(πab − pab) δγab + (πX − pX) δX].

(2.32)

This differs from (2.16) by boundary terms coming from the variation of (2.29). We canrepeat the analysis of section 2.3, evaluating δΓ on-shell and checking that the boundaryterm vanishes for appropriate variations of the metric and dilaton. The terms involvingπab and πX were evaluated in (2.18). Using (2.26), the contributions due to ICT are

pab = −12γab e−Q(X)

√ξ0(X) , pX =

12e−Q(X) U(X)

√ξ0(X)− 1

2w′(X)√ξ0(X)

. (2.33)

As before, the boundary term is evaluated at the cut-off X = Xreg

δ Γreg =12

∫dτ

[e−Q(X)

(√ξ0(X)ξ(X)

− 1

)δξ

+

(ξ(X)U(X) e−Q(X)

(1−

√ξ0(X)ξ(X)

)− w′(X)

(1−

√ξ(X)ξ0(X)

))δX

]. (2.34)

This result simplifies a great deal when the regulator is moved far into the asymptoticregion of the space-time. The function w(X) becomes very large in this limit, and thecoefficients of δξ and δX can be expanded in powers of 1/w. The leading order terms inthis expansion are

δ Γreg =M

2

∫dτ

[1

ξ0(X)δξ −

(U(X) +

w′(X)w(X)

)δX

]+ . . . (2.35)

The variation of the metric δξ behaves as (2.19), so the first term clearly vanishes in theXreg → ∞ limit. However, the analysis of (2.35) is much simpler if we use the changeof variables gµν = e2σ(X) gµν defined in (B.1), with σ = w(X)eQ(X) as in (B.10). Thenthe variation of Γ is given by (2.34) with all un-hatted quantities replaced by their hattedequivalents. Using (B.12), the leading terms in the large-w expansion of δΓ are

limXreg→∞

δΓreg =M

2lim

Xreg→∞

∫dτ

[δξ +M δ

( 1w

)]. (2.36)

The new variable ξ obeys the standard Dirichlet condition ξ = 1 at the boundary, whichmeans that δξ vanishes as Xreg →∞. The second term must also vanish in this limit since

– 342 –

we assume that w →∞ asymptotically. Thus we have established

limXreg→∞

δ Γreg = 0 . (2.37)

We conclude that solutions of the classical equations of motion are stationary points of Γwith respect to all variations of the fields that preserve the boundary conditions.

Now we evaluate the action for the BH solution. In section 2.2 we found that theregulated on-shell action Ireg contains a term that diverges in theXreg →∞ limit. Includingthe contribution from the boundary counterterm, the improved action is

Γreg = β

(w(Xreg)

√1− 2M

w(Xreg)− w(Xreg) + 2M − 2πXH T

). (2.38)

The regulator can be safely removed to obtain a finite result

limXreg→∞

Γreg = β(M − 2πXH T

). (2.39)

Notably, this differs by an amount βM from the part of (2.15) that was independent ofthe regulator.

3. Thermodynamics in the Canonical Ensemble

Motivated by York’s path integral formulation of gravitational thermodynamics [11], wenow introduce an upper bound Xc on the dilaton. This constrains the dilaton to a ‘cavity’X ≤ Xc whose wall is the dilaton isosurfaceX = Xc. The cavity is in contact with a thermalreservoir that maintains boundary conditions at Xc

2. The semi-classical approximationof the corresponding path integral yields the partition function in the canonical ensemble.The consistency of this interpretation rests upon the condition of thermodynamic stability,which we discuss in subsections 3.5 and 3.10.

3.1 Dilaton Charge and Free Energy

In standard thermodynamics the canonical ensemble fixes the temperature and volume ofa system. The local temperature Tc at the cavity wall is fixed by requiring the properperiodicity of the Euclidean time to be βc. But the volume of the cavity at constantEuclidean time is not a well-defined quantity, due to the possibility of a horizon Xh < Xc.In higher dimensions this issue is addressed by holding the area of the cavity wall fixed,instead [11]. We essentially do the same, but in an indirect way. In two-dimensional dilatongravity any sufficiently regular function of the dilaton can be used to construct a conservedcurrent. As a result, there are an infinite number of conserved charges associated with the

2Note that Xc has a clear physical interpretation, unlike the regulator Xreg introduced in the previous

section. The regulator appeared as an ingredient of the limiting procedure used to evaluate boundary

quantities. As such, the limit Xreg → ∞ was always assumed to be the final step of any calculation. This

is not the case with Xc, which may remain finite.

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dilaton. Following Gibbons and Perry [12], we choose a particular conserved charge thatwe denote by D(X) and hold this quantity fixed at Xc. One convenient choice is

Dc = Xc , (3.1)

which plays a role similar to the area of the cavity wall in higher dimensions. This choicecan be regarded as “natural” as it is precisely the quantity that couples to the Ricci scalarand the extrinsic curvature in the action. Another “natural” possibility for the dilatoncharge,

Dc = e−Qc√ξ0(Xc) , (3.2)

coincides with the solution of the Hamilton-Jacobi equation (2.29). We shall discuss indetail how various thermodynamical properties depend on the choice of dilaton charge.

Fixing Dc and βc specifies the boundary conditions for the path integral. The classicalsolutions consistent with these boundary conditions correspond to real, non-negative valuesof M that satisfy the condition

βc =√ξ(Dc,M)β(M) , (3.3)

where β(M) is the periodicity of the Euclidean time given in (2.12). In general, there maybe zero, one, or multiple classical solutions that satisfy (3.3). Issues related to the existenceof multiple solutions will be discussed in subsection 3.10. For the next several subsections,we assume that we are working with a single solution. In that case the Helmholtz freeenergy of the solution is related to the on-shell action Γc by

Fc(Tc, Dc) = Tc Γc(Tc, Dc) . (3.4)

From (2.14), (2.38) and (3.4) we obtain

Fc = −2πXh Tc + e−Qc(√

ξ0 −√ξc

). (3.5)

It is worth mentioning that the free energy for the ground state solution vanishes if eitherT = 0 or Xh = 0. This statement does not depend on the location of the cavity wall.

The interpretation of (3.4) as the Helmholtz free energy assumes thermodynamic sta-bility. The consistency of this assumption will be addressed in subsections 3.5 and 3.10. Inmost models thermodynamic stability requires a finite cavity. However, we are often inter-ested in the properties of an isolated BH without an enclosing cavity. This is accomplishedby taking the limit Xc → ∞, while simultaneously changing βc according to (3.3). Thislimit, if it exists, can be thought of as removing the cavity wall to infinity while adiabati-cally relaxing the temperature to preserve M . Motivated by curiosity, we shall sometimesapply this limit to the result of a calculation without worrying about whether or not it isconsistent with thermodynamic stability (usually it is not). We emphasize that, becausethe on-shell action is finite in this limit, any singular behavior in Fc as Xc →∞ is due tothe Tolman factor in Tc, as evident from (3.4).

Another issue related to stability concerns the possibility of decay into a field configu-ration with a smaller free energy. The BH solution is unstable against decay into any field

– 344 –

configuration (that obeys the boundary conditions) with a free energy Fg smaller than Fcfrom (3.5). We shall discuss this issue in more detail later on. For now, we simply pointout that there are usually relevant field configurations with a non-positive free energy, soa minimal condition for stability of the BH solution is Fc ≤ 0. It implies

2M − 2πXhT ≤ wc

(1−

√1− 2M

wc

). (3.6)

This condition is always fulfilled for boundary conditions that admit solutions with horizonsclose to the cavity wall: wc = 2M + ε. As we shall see in later parts of this section, theinequality (3.6) is usually not fulfilled for solutions (3.3) with 2M wc.

Keeping in mind the caveats described above, the collection of formulas in appendixD can be applied to the Helmholtz free energy (3.5) to calculate all thermodynamicalquantities. We start with the most prominent one.

3.2 Entropy and Area

The entropy is given by

Sc = − ∂Fc∂Tc

∣∣∣∣Dc

. (3.7)

The variation of Fc at fixed Dc can be expressed as

dFc

∣∣∣Dc

= −2πXh dTc +(

1√ξc− TcT

)dM . (3.8)

The last term, which was obtained using (D.1) and (D.3), vanishes by the defining relation(2.14). Thus, we find the well-known result [10,12,27–29]

S = 2πXh . (3.9)

The entropy is independent of Xc, and depends only on the value of the dilaton at the hori-zon. As expected entropy is exclusively a property of the horizon. Furthermore, this resultis universal and applies to any model (1.1) regardless of the potentials U and V . Restoringthe two-dimensional Newton’s constant G2 the result (3.9) becomes S = Xh/(4G2). Itis useful to introduce the notion of area at this point in order to compare with the stan-dard Bekenstein-Hawking formula. The area of a sphere of radius r embedded in d spatialdimensions is given by Ad = 2πd/2rd−1/Γ(d/2). In the limit d→ 1 the area becomes inde-pendent of the radius and the “sphere” consists of two disjoint points. It is suggestive torestrict to a connected component and therefore to associate only one of these two pointswith the horizon. Thus we define Ah = A1/2 = 1. The effective Newton coupling in ascalar-tensor theory is given by Geff = G2/X, where X is either approximately constantor evaluated at some characteristic scale. The only scale available here is provided by thehorizon, so we define Geff = G2/Xh. Then (3.9) is equivalent to the Bekenstein-Hawkingrelation

S =Ah

4Geff, (3.10)

– 345 –

so that the entropy is one quarter of the horizon area in ‘effective Planck units’. In section6, when we consider the reduction of a spherically symmetric BH from d+1 > 2 dimensionsdown to two dimensions, we shall find that the conserved charge Dc always corresponds tothe higher-dimensional area, in d+ 1 dimensional Planck units, of the cavity surroundingthe BH. This supports our earlier statement about “naturalness” of the choice (3.1).

3.3 Dilaton Chemical Potential and Surface Pressure

The chemical potential associated with the dilaton charge (3.1) is given by

ψc = − ∂Fc∂Dc

∣∣∣∣Tc

. (3.11)

The variation of Fc at fixed Tc can be expressed as

dFc

∣∣∣Tc

= −2π Tc dXh +12e−Qc

(dξ0√ξ0− dξc√

ξc

)− Uce−Qc

(√ξ0 −

√ξc

)dDc . (3.12)

Using (D.1), (D.2) and (D.5) we find

ψc = −12Uc e

−Qc(√

ξc −√ξ0

)+

12w′c

(1√ξc− 1√

ξ0

). (3.13)

We have defined the dilaton chemical potential with a minus sign in (3.11) so that it followsthe same conventions as a pressure in standard thermodynamics. This analogy becomesprecise in several examples where Dc plays the role of the area of a cavity surrounding thehorizon.

The dilaton chemical potential (3.13) exhibits some interesting properties: It divergeswith the Tolman factor as the horizon is approached. For the ground state solution M = 0it vanishes. Finally, it transforms covariantly with the dilaton charge Dc(Dc), viz., ψc =ψc dDc/dDc. With the choice (3.2) the dilaton chemical potential simplifies to

ψc = η

√ξ0

ξc+ (1− η)

√ξcξ0− 1 (3.14)

whereη :=

11− Ucwc/w′c

. (3.15)

This definition is particularly useful if η is constant, such as for models with U = 0,Minkowskian ground state models and the so-called ab-family. We shall discuss them indetail in section 4.

It is sometimes of interest to determine the values of the dilaton charge for which thedilaton chemical potential (3.13) vanishes. This can be thought of as extremizing the freeenergy with respect to the dilaton charge. The dilaton chemical potential vanishes if eitherof the following conditions is satisfied√

ξc =√ξ0 ,

√ξc = − w′c

wcUc

√ξ0 . (3.16)

– 346 –

The first possibility can only be realized asymptotically. Thus, one natural value of thecut-off is the asymptotic region, which might have been anticipated. The second possibilityarises only if −1 < w′c/(wcUc) < 0. In the latter case, the system is only stable againstchanges in Dc if the free energy is a minimum. This is equivalent to positivity of theisothermal compressibility (3.38) discussed below.

3.4 Internal Energy and First Law

All calculations so far have referred to the canonical ensemble, with the temperature Tcand dilaton charge Dc at the cavity wall held fixed. We can now perform a Legendretransformation to obtain the internal energy Ec as a function of the entropy and thedilaton charge

Ec(S,Dc) = Fc(Tc, Dc) + Tc S . (3.17)

From (3.5) and (3.9) we obtain

Ec = e−Q(Dc)(√

ξ0 −√ξc

)≥ 0 . (3.18)

The last inequality holds because we assume M,wc ≥ 0. The internal energy Ec agreeswith the proper energy obtained from the quasi-local stress tensor of Brown and York [30].This stress tensor is defined as

T ab :=2√γ

δΓδγab

. (3.19)

Using (2.32) obtainsT ab = 2

(πab − pab

). (3.20)

Contracting T ab with two copies of the unit-norm Killing vector ua = δ τa gives the proper(Euclidean) energy density. This can be read off from (2.17) and (2.33)

T abuaub = e−Q(Dc)(√

ξ0 −√ξc

). (3.21)

This is precisely the internal energy given in (3.18). Notice that it is not the same as theconserved charge associated with the Killing vector ∂τ ,

Q∂τ = limXc→∞

√ξc T

abuaub = M , (3.22)

where√ξ is the lapse function3 in the 1+1 decomposition of the metric (2.3). The definition

of the conserved charge Q∂τ involves the limit Xc →∞, which should be thought of as thetwo-dimensional analog of the limiting procedure used to integrate over a sphere at spatialinfinity in higher dimensions. The asymptotic limit of the internal energy is proportionalto the conserved charge M , rescaled by the Tolman factor

limwc→∞

Ec = limwc→∞

M√ξc. (3.23)

3Here ’lapse function’ refers to the standard temporal evolution, to be distinguished from the lapse

function of the radial evolution introduced in appendix C.

– 347 –

At finite values of Xc the result (3.18) can be used to express the conserved quantity M asa function of the internal energy

M =√ξ0Ec −

12wc

(√ξ0Ec)2 . (3.24)

This has a simple interpretation in space-times where the Killing norm√ξ approaches a

constant, which we may set to unity without loss of generality. In that case M coincideswith the ADM mass, which is related to the internal energy in the region X ≤ Xc by

M = Ec −E2c

2wc. (3.25)

The ADM mass contains two contributions: the first one is just the internal energy Ec andthe second one is the gravitational binding energy associated with collecting the energy Ecin a region determined by Xc. Consistently, the second term is absent asymptotically.

Using (D.2) and the results for entropy (3.9) and dilaton chemical potential (3.13) onecan show that the internal energy obeys the first law of thermodynamics

dEc = TcdS − ψcdDc . (3.26)

This quasilocal form of the first law holds regardless of the particular model, the locationXc of the cavity wall, or choice of the dilaton charge. In this regard it contains a greatdeal more information about the system than the microcanonical expression dM = TdS.

3.5 Specific Heat at Constant Dilaton Charge and Fluctuations

Specific heat at constant dilaton charge is given by

CD =∂Ec∂Tc

∣∣∣∣Dc

= Tc∂S

∂Tc

∣∣∣∣Dc

. (3.27)

Using (3.18), (D.1) and (D.3) we find

CD =4π w′h(wc − 2M)

2w′′h(wc − 2M) + w′h2 . (3.28)

If w′′h 6= 0 then the asymptotic limit of DD reduces to the microcanonical specific heat

limwc→∞

CD = 2πw′hw′′h

= −(dSdM

)2d2SdM2

. (3.29)

We have implicitly assumed in the previous subsections that the specific heat is positive.This assumption will be examined in more detail in subsection 3.10, but we can offer apartial justification by considering the near horizon limit of (3.28). If the classical solutiondetermined by (3.3) satisfies wc = wh + ε, with 0 < ε 1, then the specific heat simplifiesto

CD =ε

T. (3.30)

– 348 –

Remarkably, it is always positive and vanishes as the cut-off approaches the horizon. Thisstatement is model independent.

If the specific heat is positive and the temperature is large enough then one can takeinto account the effect of thermal fluctuations of the internal energy. The temperature is‘large enough’ if quantum fluctuations are considerably smaller than thermal fluctuations.Quantum fluctuations are typically of the order ∆qEc ≈ 1, whereas thermal fluctuations,∆tEc =

√〈E2

c 〉 − 〈Ec〉2 =√CDTc, depend on specific heat at constant dilaton charge and

on temperature. Thermal fluctuations dominate over quantum fluctuations if CDT 2c 1.

The canonical entropy, which takes into account thermal fluctuations, is given by the well-known result

Sfluct = S +12

lnCDT 2c + . . . (3.31)

Despite the fact that Tc →∞ at the horizon, the fluctuations remain finite there

limwc→2M

CDT2c = e−QhT . (3.32)

For some models (including the Schwarzschild BH) (3.32) does not satisfy CDT 2c 1, even

for large BH masses. In these cases thermal fluctuations do not dominate over quantumfluctuations near the horizon.

3.6 Free Enthalpy and Gibbs-Duhem Violation

With the exception of the dilaton chemical potential, the thermodynamical quantities ob-tained in the previous subsections do not depend on the specific choice of the dilatoncharge. This is not the case for quantities calculated in this and the following subsection.Quantities without a bar on top either refer to the dilaton charge (3.1) or are independentfrom it; quantities with a bar refer to the dilaton charge (3.2).

We start with the free enthalpy (Gibbs function),

Gc(Tc, ψc) = F (Tc, Dc) + ψcDc = Ec(S,Dc)− TcS + ψcDc . (3.33)

Even though we have Legendre-transformed the internal energy with respect to the quanti-ties that usually are considered extensive (entropy and “volume”, viz., dilaton charge) theresult is, in general, not zero. This can be thought of as a ‘violation’ of the standard form ofthe Gibbs-Duhem relation 4. There are, however, a few notable exceptions which we shalldiscuss in sections 4 and 5. This shows that extensitivity properties are not necessarilyas in standard thermodynamics. The physical reason underlying these observations is theexistence of gravitational binding energy, as in (3.25). Notably, free enthalpy is finite atthe horizon, despite the fact that free energy diverges there:

Gh =1√ξ0

(2M − TS +

12π

MSUc

)∣∣∣∣Dc=Xh

. (3.34)

For comparison it is instructive to calculate

Gc =1√ξc

(2ηM − TS) , (3.35)

4For a discussion of alternative forms of this relation relevant for BHs, see [31].

– 349 –

with η as defined in (3.15). For constant η the quantity√ξcGc exclusively depends on

properties of the horizon and the Gibbs-Duhem relation holds if 2ηM = TS. The result(3.35) is not only considerably simpler than the expression for Gc (which we omitted topresent) but it is also divergent at the horizon because of the Tolman factor. This is a cleardemonstration of the fact that free enthalpy crucially depends on the choice of the dilatoncharge.

The enthalpy

Hc(S, ψc) = Ec(S,Dc) + ψcDc = Gc(Tc, ψc) + TcS (3.36)

can be calculated from free enthalpy (3.35) with the result

Hc = 2ηM√ξc. (3.37)

It is positive if η > 0 and also scales with the Tolman factor. The quantity Hc is muchmore complicated so we do not present it here.

3.7 Miscellaneous Thermodynamical Quantities

The isothermal compressibility

κT = − 1Dc

∂Dc

∂ψc

∣∣∣∣Tc

(3.38)

requires the calculation of ∂ψc/∂Dc|Tc performed in appendix D, cf. (D.10). Althoughthe dilaton chemical potential diverges at the horizon isothermal compressibility in generalremains finite there,

limwc→2M

κT =2πS· (2M)3/2eQh/2

w′′hM − 4π2T 2 − 4πTMUh − 2M2U ′h +M2U2h

. (3.39)

We do not include the result for κT here, but it may be deduced from the results for CDand Cψ.

Specific heat at constant dilaton chemical potential is defined by

Cψ =∂Hc

∂Tc

∣∣∣∣ψc

= Tc∂S

∂Tc

∣∣∣∣ψc

. (3.40)

Using (D.11) the difference between the specific heats,

Cψ − CD = κT DcTc

(∂ψc∂Tc

∣∣∣∣Dc

)2

, (3.41)

turns out to be positive if and only if isothermal compressibility is positive. The explicitcalculation of Cψ is somewhat lengthy. A simpler result is obtained starting with thedilaton charge (3.2):

Cψ = 2πw′hw′′h

. (3.42)

– 350 –

This coincides with the asymptotic limit (3.29) and with the microcanonical specific heat.The difference between the specific heats is

Cψ − CD = 2πw′hw′′h·

w′h2

w′h2 + 2w′′h(wc − 2M)

. (3.43)

For generic thermodynamical systems mechanical stability requires positive isothermalcompressibility or, equivalently [cf. (3.41)], Cψ > CD > 0. However, as we have demon-strated the quantity Cψ depends crucially on the definition of the dilaton charge. Thus,unless there is a good physical reason to prefer one particular dilaton charge over all otherpossible choices, we cannot provide an unambiguous answer to the question of mechanicalstability. It is intriguing that, for the dilaton charge (3.2), thermodynamic stability impliesmechanical stability: κT > 0. This is because in a region where both CD and Cψ arepositive their difference Cψ − CD is also positive, as evident from (3.43).

Additional thermodynamical quantities can be deduced from the ones we have pre-sented here. One can basically exploit the results of standard thermodynamics by identify-ing the dilaton charge with “volume”, and the dilaton chemical potential with “pressure”.A few examples will be provided in section 4. We conclude with a supplementary observa-tion regarding an interesting geometric description of the thermodynamics of equilibriumsystems due to Ruppeiner [32]. The main idea is to introduce a metric on the thermody-namic state space,

ds2R := GRIJdX

IdXJ , GIJ = −∂I∂JS(X) , XI = (Ec, Dc, . . . ) , (3.44)

and to relate its geometric properties to thermodynamic quantities. The entropy S hereis expressed as a function of extensive variables XI , which in our case comprise internalenergy (3.18), the dilaton charge (3.1), and any additional extensive quantities associatedwith generalizations of the metric-dilaton system. This formalism was applied to variousBH systems and led to some intriguing results [33]. It is worth mentioning that even inthe absence of U(1)-charges the Ruppeiner metric is two-dimensional, which differs fromthe microcanonical analysis in [34], where the Ruppeiner metric was two-dimensional forcharged BHs only.

3.8 Charged Black Holes

Gauge fields in two dimensions have no propagating physical degrees of freedom [35], whichmakes the generalization of our results to charged BHs straightforward. We consider theaddition of a single abelian gauge field, but the extension to several abelian or non-abeliangauge fields is straightforward. The improved action for dilaton gravity coupled to a gaugefield is

Γ =− 12

∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)

]+∫Md2x√g f(X)FµνFµν

−∫∂Mdx√γ X K +

∫∂Mdx√γ e−Q(X)

√ξ0(X) . (3.45)

– 351 –

The abelian field strength Fµν = ∇µAν−∇νAµ is coupled to the dilaton field via the func-tion f(X). If this function is constant then we say that the gauge field is minimally cou-pled, and non-minimally coupled otherwise. We emphasize that the gauge field Aµ cannotcontribute to the boundary counterterm because we require gauge- and diffeomorphism-invariance. The only gauge-invariant scalar that is linear in Aµ or its derivatives comesfrom the contraction of the field strength with the ε-tensor: Fµνεµν . However, this termcontains a derivative normal to the boundary and is not allowed in a counter-term of theform (2.22). All other gauge-invariant scalars constructed from Aµ, like FµνFµν , necessarilycontain such derivatives and are also ruled out.

The solution of the Maxwell equation ∇µ(f(X)Fµν) = 0 is

Fµν =q

4f(X)εµν , (3.46)

where q is the electric charge, and the factor of 14 has been introduced for convenience.

When working in the diagonal gauge (2.3) we use the convention ετr = −εrτ = +1. Theequations of motion for the metric and dilaton can be solved in a manner similar to un-charged BHs. The metric and dilaton are again given by (2.3) and (2.4), but the square ofthe Killing norm is now

ξ(X) = eQ(X)

(w(X)− 2M +

14q2 h(X)

)(3.47)

The ‘ground state’ is defined as the solution with vanishing mass and charge, M = q = 0,leading to the same Killing norm

√ξ0 as in (2.8). The function h(X) is defined as

h(X) :=∫ X

dXeQ(X)

f(X), (3.48)

where the integral is evaluated at X and the integration constant is absorbed into thedefinition of w. The result (3.47) could also have been obtained from our earlier solutionfor the uncharged BH, by replacing V (X) in (2.1) and (2.2) with an effective potential ofthe form

V eff(X) = V (X)− q2

8f(X). (3.49)

We shall return to this point at the end of this subsection.To obtain the on-shell action we use∫

Md2x√g fFµνFµν = 2

∫Md2x√g∇µ(fFµνAν)− 2

∫Md2x√g Aν∇µ(fFµν) (3.50)

and note that the second term vanishes on-shell. The first term, a total derivative, can alsobe interpreted as one-dimensional Chern-Simons terms with support at both the boundaryand the horizon

2∫Md 2x√g∇µ(fFµνAν) = −q

2

∫Aτdτ

∣∣∣∣XcXh

= −q2

(CS1(Xc)− CS1(Xh)

)= −q

2β(Aτ (Xc)−Aτ (Xh)

). (3.51)

– 352 –

It is convenient to fix the gauge so that Ar = 0. In that case the τ component of the gaugefield is given by

Aτ (X) = −q4

(h(X)− h(Xh)) +Aτ (Xh) . (3.52)

This is just the potential difference between the cavity wall and the horizon, plus anarbitrary constant. The proper electrostatic potential between the cavity wall and thehorizon is related to Aτ by

Φ(X) :=Aτ (X)−Aτ (Xh)√

ξ(X). (3.53)

Note that it contains a Tolman factor, like the proper temperature.We are now ready to present the generalization of (2.38) to the case of charged BHs,

Γc = βc

(e−Qc

(√ξ0 −

√ξc

)− 2πXhTc − qΦc

). (3.54)

As pointed out in [10] the logarithm of the partition function,

Yc(Tc, Dc,Φc) = Tc Γc = −2πXh Tc + e−Qc(√

ξ0 −√ξc

)− qΦc , (3.55)

is not the Helmholtz free energy. Rather, Yc is the Legendre transform of Fc with respectto the proper electrostatic potential Φc. This is because the Maxwell equation is obtainedby varying the action with respect to the gauge field, so it is the gauge field (and not thefield strength) that obeys Dirichlet conditions at ∂M. The on-shell action is therefore afunction of Aµ evaluated at the boundary. The derivative of Yc with respect to Φc, holdingTc and Dc fixed, gives the conserved electric charge

− ∂Yc∂Φc

∣∣∣∣Dc,Tc

= q . (3.56)

The Legendre transformation

Fc(Tc, Dc, q) = Yc(Tc, Dc,Φc) + qΦc (3.57)

leads to the same expressions for the Helmholtz free energy (3.5) and consequently also forentropy (3.9) and internal energy (3.18).

For minimally coupled Maxwell fields the electrostatic potential Φ ∝ (Xc − Xh) isproportional to the separation between the cavity wall and the horizon. If the Killing normasymptotes to unity then the proper electrostatic potential Φc diverges in the Xc → ∞limit. As a result, the improved action for these models diverges in the limit where thecavity wall is removed from the system. We emphasize that this divergence reflects thephysics of the linear electrostatic potential and therefore cannot be removed from theaction. The same rule that was mentioned at the end of subsection 2.2 applies here, aswell. One does not obtain the correct improved action by simply dropping divergent terms.In the present case, the action depends on the potential difference between the wall andthe horizon, which is gauge invariant. Dropping the divergent term from the action leadsto a violation of gauge invariance. An example of a charged BH with minimal coupling

– 353 –

is provided by 2D type 0A string theory [10], where e−Q = w = λX and f = πα′. Asan example of non-minimal coupling we mention the Reissner-Nordstrom BH, e−Q = w =2√X and f(X) = X (with convenient normalizations for eQ and f). Our formula (3.48)

gives h = −1/√X = −1/r in this case, which is the correct Coulomb-potential. Thus, the

typical linear (confining) behavior of the electrostatic potential in two dimensions can bemodified by a non-minimal coupling f(X).

Maxwell fields open up an interesting possibility. Consider a theory without Maxwellfields whose potential V takes the form

V (X) = V(X)− q2

8f(X). (3.58)

We can also think of this potential as the result of integrating out a gauge field withcoupling function f(X), in a theory with dilaton potential V(X). We simply re-interpret(3.58) as (3.49). These two points of view will generally lead to different conclusions aboutthe thermodynamics. In particular, the “natural” choice of ground state is different in thetwo formulations. If we treat the system as dilaton gravity with a Maxwell field then thenatural ground state is given by M = q = 0. But in the original description, with noMaxwell fields, the ground state is associated with M = 0. For instance, the ground stateM = q = 0 is Minkowski space for the Reissner-Nordstrom BH, whereas in the alternativedescription without Maxwell field the natural ground state M = 0 is a BPS solution. It isstraightforward to generalize these considerations to potentials of the form

V =∞∑

n=−∞λnX

n/a , (3.59)

where a is some natural number. This applies to almost all models in the literature. Now wemay introduce Maxwell fields to convert all but one of the terms in the generalized Laurentseries into Maxwell terms. This leads to a rather simple thermodynamical description andmay be useful in various contexts. We shall provide an example in 6.3 when we considerthe toroidal reduction of the BTZ BH.

3.9 Conformal Properties and Microcanonical Thermodynamics

In appendix B we discuss some consequences of dilaton dependent Weyl rescalings (B.1).If interpreted as a conformal transformation,5 rather than as a change of variables, a newaction (1.1) is obtained with potentials given by (B.3). We say that these models areconformally related to each other and sometimes refer to them as representing different‘conformal frames’ within an equivalence class of models. We consider now the transfor-mation behavior of thermodynamical quantities. To this end it is very helpful to note(B.4) that w is conformally invariant. Therefore, quantities which depend only on w andits derivatives are conformally invariant, but quantities that depend on Q are not.

5We use the term ‘conformal transformation’ in this section, as commonly done in the literature, though

(B.1) should really be referred to as a Weyl transformation.

– 354 –

The conformally invariant thermodynamical quantities comprise the on-shell action Γc(2.38), surface gravity T (2.13), mass parameter M (2.11) and entropy S (3.9). It is inter-esting to note that the specific heat at constant dilaton charge CD (3.28) is conformallyinvariant, despite the fact that both Ec and Tc depend on the conformal frame. This isbecause the temperature Tc (2.14), free energy Fc (3.5) and internal energy Ec (3.18) areconformally covariant, with a conformal weight of −1 due to the Tolman factor 1/

√ξc.

With this convention the metric is defined to have weight +2. Quantities that are neitherinvariant nor covariant are the dilaton chemical potential ψc (3.14), free enthalpy Gc (3.35),enthalpy Hc (3.37) and isothermal compressibility κT (3.38). These statements are inde-pendent from the choice of the dilaton charge. An interesting issue arises for the specificheat at constant dilaton chemical potential: while the result for Cψ (3.42) is conformallyinvariant, this is not the case for Cψ with a generic choice of dilaton charge Dc. Thisprovides another rationale for working with the charge Dc, as defined in (3.2).

As a simplification one can consider microcanonical thermodynamics (or “horizon ther-modynamics”) without referring to an actual observer [28]. If one is interested only in themicrocanonical entropy S, surface gravity T , and their relation to the mass parameter M ,then one can use a conformal transformation to set U = 0. In this scenario the two relevantformulas, besides the result for entropy (3.9), are the Smarr formula for two-dimensionaldilaton gravity,

M =w(S/2π)

2(3.60)

and the microcanonical first lawdM = TdS . (3.61)

It is important to realize that the microcanonical thermodynamics is not equivalent tocanonical thermodynamics in a limit where the dilaton charge approaches some specificvalue, with rare exceptions. The two pictures are equivalent only if there is a value ofthe dilaton charge where the dilaton chemical potential vanishes, ψc = 0, the propertemperature coincides the with Hawking temperature, T = Tc, and the canonical ensembleis well-defined. In that case the canonical first law (3.26) coincides with the microcanonicalone (3.61), and the mass M coincides with the internal energy Ec. The first two conditionsare satisfied in the asymptotic region of models with a Minkowskian ground state. But thethird condition is typically violated: in these models the canonical ensemble quite oftenis not defined in the Xc → ∞ limit. We show now why this is the case and under whichconditions the cut-off can be removed.

3.10 Stability and Phase Transitions

The most important issue related to stability in the canonical ensemble is the effect of ther-mal fluctuations. If the specific heat at constant dilaton charge is negative, CD < 0, thenthermal fluctuations in the internal energy have an imaginary component and the systemis unstable. From the point of view of the path integral, CD < 0 implies that the Gaus-sian integral around a particular extremum of the action does not converge. Fortunately,the assumptions wc > wh and w′h > 0 guarantee that there is an open region around thehorizon where the specific heat (3.30) is positive. This result is model-independent. Thus,

– 355 –

if the horizon is located sufficiently close to the cavity wall then the canonical ensemble isalways well-defined. Of course, a quantitative definition of ‘sufficiently close’ will dependon the features of the particular model.

In most models a classical solution is only stable when the cavity wall is located at anXc that is less than some finite, critical value of the dilaton. Consider a BH associated witha solution M of the condition (3.3). We assume that this system has a positive specific heatat constant dilaton charge. Now we want to change the boundary conditions Xc and βcwhile holding M fixed. As described at the beginning of section 3, this is accomplished bymoving Xc away from the horizon while simultaneously relaxing βc according to (3.3). Theequation (3.28) allows us to make a quantitative statement about the point at which thisprocess destabilizes the system. Suppose that w is a monotonic function. Then a solutionXcrit > Xh of

w(Xcrit) = 2M −w′h

2

2w′′h(3.62)

represents a critical value of the dilaton where the specific heat diverges and changes sign.A well-defined canonical ensemble containing the classical solution exists for any valueXh < Xc < Xcrit, but not for Xc > Xcrit. A similar argument can be made for w that arenot monotonic. Notice that in either case the specific heat approaches zero at Xc = Xh.This reflects the fact that the system is only defined for a cavity wall that is external tothe horizon.

In general, the condition (3.3) identifies multiple classical solutions consistent with theboundary conditions [11]. We refer to each solution as a ‘branch’ of (3.3), denoted byMi(Xc, βc). The thermodynamic stability of each branch can be determined by looking fora solution Xi

crit > Xh of (3.62). There are three possibilities:

1. There is no solution Xicrit > Xh. This implies that CD is positive, and the branch is

thermodynamically stable.

2. There is a solution Xicrit > Xc > Xh. Again, the specific heat is positive and the

branch is thermodynamically stable.

3. There is a solution such that Xc > Xicrit > Xh. In this case the specific heat is

negative, and the branch is not thermodynamically stable.

In the third case the solution Mi(Xc, βc) may decay into another field configuration withlower free energy via a Hawking-Page phase transition [36]. Because these considerationsinvolve only w the emergence of a phase transition happens in any conformal frame.

We are now able to address the issue of the Xc → ∞ limit in more detail. Given abranch Mi of (3.3) that is thermodynamically stable for some initial values of the boundaryconditions, we are interested in the process described by

Xc →∞ βc → limXc→∞

√ξ(Dc,Mi)β(Mi) (3.63)

while holding Mi fixed. This removes the cavity wall, leaving an isolated BH. The processis only defined if βc remains real and non-negative for all values of Xc. Furthermore, it may

– 356 –

be defined on some branches but not others. Finally, even if the process (3.63) is definedits endpoint may not be thermodynamically stable. The existence of a thermodynamicallystable limit (3.63) is far from being a generic feature in the models we consider. Forinstance, the Schwarzschild BH has limXc→∞CD < 0. In models where the endpointmight be stable, the equation (3.62) can be used to group solutions into two sectors. If(3.62) admits a finite solution Xcrit > Xh for a particular value of M , then that solution isunstable. But if there is no Xcrit > Xh for that value of M then the solution is stable. Inhigher dimensions the best known example of this phenomenon is the Schwarzschild-AdSsolution, where the presence of a cosmological constant stabilizes very massive BHs withoutthe need for an external reservoir. An analogous two-dimensional dilaton gravity modelhas wSAdS = 2

√X(1 +X/`2) with X = r2. For large enough masses, such that Xh > `2/3,

the quantity w′′h is positive and the system is stable, while for Xh < `2/3 the quantityw′′h becomes negative and the system is unstable. This can be translated into a criticaltemperature, where the specific heat diverges. It signals a phase transition [36] between aBH phase for large masses (small temperatures) and a thermal AdS phase for small masses(large temperatures). We shall encounter similar examples in subsection 6.2.

3.11 Tunneling and Constant Dilaton Vacua

In addition to the generic family of solutions (2.3), certain models may admit isolatedsolutions known as constant dilaton vacua (CDV). Namely, for each solution X = X0 ofV (X) = 0 there is a CDV with a line element (2.3) whose Killing norm

√ξ is given by

ξ = c+ ar − 12λr2 . (3.64)

Here c and a are constants of motion. The constant λ is determined by

RCDV = −∂2r ξ = λ = 2V ′(X0) . (3.65)

Because they have a different number of constants of motion CDVs and the generic solutionsbelong to different ’superselection sectors’. Therefore there is no perturbative way inwhich a generic solution could decay into a CDV or vice versa. This is consistent withthe result [37] that fully supersymmetric solutions necessarily are CDVs, while half BPSsolutions can never be CDVs. If a supersymmetric CDV exists then the natural groundstate solution in the generic sector is a half BPS solution, and both kinds of solutions arestable. However, once matter is switched on there could be non-perturbative interactionswhich allow tunneling from one sector into the other. We would like to determine whethersuch a tunneling between a generic solution and a CDV is possible, and if so, which of thetwo configurations is favored.

We immediately encounter an obstacle: CDVs are isolated in the sense that for anopen interval of boundary conditions, Xc ∈ (X<, X>), there is only a countable numberof values of the dilaton consistent with the equations of motion. Typically this number iseither zero or of order unity. Thus, at first glance CDVs are irrelevant because they seemto be a set of measure zero in the partition function (1.2). Only for infinite finetuning,Xc = X0, there may be a contribution from CDVs. In order to decide whether or not CDVs

– 357 –

may be disregarded we consider an open interval of boundary conditions for the dilaton andthe metric. This interval may be arbitrarily small, but it should include exactly one valueXc = X0 that leads to a CDV.6 In order to get the number of classical solutions determinedby this set of boundary data we take the product of the subset of initial conditions consistentwith the classical equations of motion times the space of continuous constants of motionthat may be adjusted freely. We call the dimension of this space “weight” and want tocompare the weight of generic solutions with the weight of CDVs. Let us work in fixedphysical units. Then, for generic solutions (2.3) there is only a discrete set of values forM consistent with the boundary conditions on Xc and ξc. So the weight is 2 (from theset of boundary conditions consistent with the equations of motion) plus 0 (from the spaceof continuous constants of motion that may be adjusted freely) and thus equals to 2. ForCDVs the space of allowed boundary conditions is just one-dimensional since the dilaton isfixed to a certain value, but now there are two constants of motion, c and a. Therefore, thespace of constants of motion that may be adjusted freely is one-dimensional. We concludethat also for CDVs the weight is 2 and they should not be regarded as a set of measurezero. Below we use again sharp boundary conditions and assume Xc = X0 for simplicity.

Now that we have convinced ourselves that CDVs are not a set of measure zero weaddress the question whether tunneling from a generic solution into a CDV is a favorableprocess or not. To this end we compare the generic result for the ’improved’ on-shell action(2.38) with the on-shell action for a CDV,

ΓCDV = −2πX0 χ+ β√ξe−Q(X0)/2

√w(X0) . (3.66)

The first term in (3.66) contains the Euler-characteristic χ of the manifold M. The lastterm is the universal counterterm (2.29) evaluated at X = X0. It has the same structureand, for the same set of boundary data, also the same magnitude as the correspondingterm in the ’improved’ on-shell action for a generic solution. The periodicity in Euclideantime β is well-defined and positive provided the inequality a2 + 2cλ > 0 holds, which weassume henceforth. Minkowski or AdS vacua require a non-vanishing Rindler accelerationa while de Sitter solutions (cλ > 0) are compatible with a = 0. We conclude that tunnelingfrom a generic solution into a CDV is favored if the inequality

X0 χ−Xh >2w(X0)− 2w(Xh)

w′h(3.67)

holds. Here Xh < X0 is the dilaton evaluated at the horizon of the generic solution.Since the right hand side is non-negative the Euler characteristic must be positive for thisinequality to hold. Therefore we restrict the remaining discussion to the topologies of eithera sphere (χ = 2) or a disk (χ = 1). The latter case is consistent with a BH interpretation,while the former case may arise for de Sitter. As an example for a model with enhancedtunneling to its CDV solution we mention w(X) = c3 + (X − c)3. The CDV solutionsolution for this model is X0 = c > 0, and the inequality (3.67) holds for any value of M

6In fact, in any experimental situation this smearing of boundary data is physically more reasonable

than an infinite finetuning.

– 358 –

consistent with X0 > Xh. For generic models there are two interesting limits. If X0 isclose to the horizon, X0 = Xh + ε, then for χ = 2 the inequality (3.67) holds and tunnelingto a CDV is enhanced, whereas for χ = 1 tunneling is suppressed. If X0 1 then (3.67)simplifies to X0χ > 2w(X0)/w′h + . . . Since we demand that w → ∞ as X → ∞ thisinequality establishes a dichotomy in the space of dilaton gravity theories: if the functionw diverges asymptotically faster (slower) than linearly tunneling into a CDV is suppressed(enhanced), regardless of whether χ = 1 or χ = 2.

In summary, a transition between generic solutions and CDVs is possible only throughtunneling. If the inequality (3.67) holds tunneling from a generic solution into a CDV isenhanced, otherwise it is suppressed. We remark that such a tunneling is far from beinga generic feature: first of all, many models do not exhibit CDVs, and second, many of themodels that do allow for CDV solutions violate the inequality (3.67) for all possible valuesof Xh < X0. In fact, none of the models that we are going to discuss explicitly in sections4.2, 5 and 6 allows for an enhanced tunneling into a CDV.

4. Two Dimensional Examples

4.1 Minkowski Ground State Models

If the relation

eQw = 1 (4.1)

holds then ξ0 = 1 and the ground state solution (2.8) is (upon Wick rotation back toLorentzian signature) Minkowski space-time. These models are of particular interest asthey contain asymptotically flat solutions like the Schwarzschild and Witten BHs. As weshall see (4.1) leads to considerable simplifications.

The free energy

FMGSc = EMGS

c − ST√1− 2M

wc

(4.2)

and internal energy

EMGSc = wc

(1−

√1− 2M

wc

)(4.3)

have a finite limit if the cut-off is removed, limwc→∞ FMGSc = M−TS and limwc→∞E

MGSc =

M . The asymptotic version of the first law (3.26) simplifies to the microcanonical dM =TdS. The chemical potential conjugate to the dilaton charge DMGS

c = wc (3.2),

ψMGSc =

12

(1− 2M

wc

)1/2

+12

(1− 2M

wc

)−1/2

− 1 ≥ 0 , (4.4)

is always non-negative and vanishes for the ground state solution only. It is useful tonotice that the quantity defined in (3.15) becomes a universal constant, η = 1/2. For theformulation based upon the dilaton charge (3.1) we get ψMGS

c = ψMGSc w′c, which is also

– 359 –

non-negative provided w′c ≥ 0. Asymptotically (4.4) implies limwc→∞ ψMGSc = M2/(2w2

c ).From (3.35) we obtain the free enthalpy

GMGSc =

M − TS√1− 2M

wc

. (4.5)

The relevant quantities for isothermal compressibility simplify to fMGSc = w′′c , gMGS

c =−w′′c /2 − w′cNc/2, hMGS

c = −w′′c /2 + w′cNc/2 and Nc = (2Mw′cw′′h)/[wc(w′h

2 + 2w′′h(wc −2M))]. Evaluation at the horizon yields limwc→2M κMGS

T = 2π/(Sw′′h). In the alternativeformulation the result is

κTMGS = wc

√1− 2M

wc·

2wc − 4M + w′h2/w′′h

4M2. (4.6)

The zerosXc > Xh of κTMGS coincide with the poles of CMGSD (3.28). We do not investigate

these models in more detail here but shall encounter them again in this and the next section.

4.2 Models in the a-b Family

The so-called ab-family of models [38],

U = − a

X, V = −B

2Xa+b . (4.7)

includes the dimensional reduction of spherically symmetric BHs in d + 1 space-time di-mensions [a = (d − 2)/(d − 1), b = −1/(d − 1)], the Witten BH (a = 1, b = 0) [1], theJackiw-Teitelboim (JT) model (a = 0, b = 1) [39], and many others. In particular, theSchwarzschild BH arises for a = −b = 1/2. Minkowskian ground state models obey therelation a = 1 + b. Models with a = 1 − b have an (A)dS ground state and models withb = 0 a Rindler ground state. Even more general models often are approximated quite wellby the monomial potentials of the ab family. Therefore our discussion here will reveal somegeneric features of the thermodynamics of dilaton gravity in two dimensions. For the sakeof definiteness, and to comply with our working assumption (2.10), we assume b > −1 andB > 0. In addition, we reduce clutter by imposing B = b + 1, without loss of generality.We fix the constants in (2.6) and (2.7) so that

w = Xb+1 , eQ = X−a . (4.8)

With these assumptions the BH horizon is located at

Xh = (2M)1/(b+1) . (4.9)

Note that a negative value of M would result in a naked singularity at X = 0 if a < 2.Surface gravity yields

T =b+ 14π

(2M)b/(b+1) , (4.10)

which leads to a mass-to-temperature relation M ∝ T 1+1/b. The Killing norm squared andasymptotic Killing norm squared are given by

ξc = ξ0 − 2MX−ac , ξ0 = Xb−a+1c ≥ ξc . (4.11)

If we demand ξ0 6= 0 for Xc →∞ the inequality b− a+ 1 ≥ 0 must be met.

– 360 –

4.2.1 Thermodynamics of the a-b Family

Entropy (3.9) readsS = 2π(2M)1/(b+1) . (4.12)

The free energy (3.5) simplifies to

Fc = Xac (√ξ0 −

√ξc)−

(b+ 1)M√ξc

. (4.13)

For b < 0 free energy changes its sign at a particular value of the cut-off,√ξc =

1 + b

1− b√ξ0 , (4.14)

and the asymptotic free energy is positive. This equation always has exactly one solutionunder the assumptions −1 < b < 0 and M > 0. For Schwarzschild this happens at√X = 9M/4. With the dilaton charge Dc = X

(a+b+1)/2c the dilaton chemical potential

(3.14) simplifies to

ψc =b+ 1

a+ b+ 1

√ξ0

ξc+

a

a+ b+ 1

√ξcξ0− 1 . (4.15)

It vanishes at a particular value of the dilaton given by√ξc =

1 + b

a

√ξ0 . (4.16)

This equation has a solution if 0 < b + 1 < a. For Minkowskian ground state modelsthe dilaton chemical potential vanishes for Xc →∞ only. For (A)dS ground state modelsthe dilaton chemical potential vanishes precisely at the value of the cut-off that leads tovanishing free energy. Internal energy (3.18) is given by

Ec = X(a+b+1)/2c

(1−

√1− 2MX−1−b

c

). (4.17)

The specific heat at constant dilaton charge (3.28) simplifies to

CD = SXb+1c − 2M

bXb+1c − (b− 1)M

. (4.18)

It changes sign at Xcrit if the inequality

Xcrit =(Mb− 1b

)1/(1+b)

> Xh (4.19)

holds, which is possible for −1 < b < 0 only. Thermal fluctuations near the horizon aredetermined by

limwc→2M

CDT2c ∝M (a+b)/(b+1) , (4.20)

with a proportionality constant of order of unity. If b > −a (b < −a) thermal (quantum)fluctuations dominate for large masses. If a+ b = 0 both kinds of fluctuations are relevant.

– 361 –

The only Minkowskian ground state model with this property is the Schwarzschild BH.Specific heat at constant dilaton chemical potential (3.42),

Cψ =S

b, (4.21)

is positive for b > 0 and coincides with the Xc →∞ limit of (4.18). The free enthalpy

Gc = (1− a− b) b+ 1a+ b+ 1

· M√ξc

(4.22)

obeys the Gibbs-Duehm relation Gc = 0 for (A)dS ground state models only. To get theisothermal compressibility (3.38) we calculate (D.6): fc = 1

4(a+b+1)(a+b−1)X(a+b−3)/2c ,

gc = a2X

a−2c (1− a−XcNc), and hc = 1

2(b+ 1)Xb−1c (−b+XcNc), with

Nc = − b

2Xc

2aM − (a− b− 1)X1+bc

(b− 1)M − bX1+bc

. (4.23)

For any model without (A)dS ground state (a 6= 1 − b) the poles of Nc and CD coincidewith each other. For (A)dS ground state models Nc = b/Xc and therefore κT vanishes.

For Minkowskian ground state models we can discuss the life-time of a BH withoutspecifying explicitly the degrees of freedom that carry the Hawking quanta. Let us assumethat the two-dimensional Stefan-Boltzmann law, asymptotic flux∝ T 2, is valid. Then themass loss per time is determined by dM/dt ∝ −T 2. The proportionality constant dependson the matter content and on the physical units. With (4.10) it is straightforward tocalculate the time scale ∆t for the evaporation process

∆t ∝M (1−b)/(1+b)[= Md/(d−2)

]. (4.24)

The second expression in (4.24) refers to the Schwarzschild BH in d+ 1 space-time dimen-sions. The result (4.24) provides a good estimate of the time-scale associated with BHevaporation, as long as the semi-classical approximation remains valid throughout most ofthe process. Typically, this requires that the BH initially have large M .

4.2.2 Equation of State and Scaling Properties

For some purposes it is useful to extract an equation of state ψc(Tc, Dc) from the formulasabove. It can be derived from (4.15) upon expressing the right hand side as a function ofDc and Tc. To this end one has to find M or T as a function of Dc and Tc using (4.10),(4.11) and the defining relation (2.14). As we shall demonstrate, for given Tc and Dc thereis in general an ambiguity, i.e., the BH mass M or, equivalently, its Hawking temperatureT does not follow uniquely from specifying these quantities [cf. the discussion around (3.3)].A simple exception is provided by Rindler ground state models, where T from (4.10) notonly is unique but actually is constant, thus establishing

Rindler : ψc(Tc, Dc) =1

a+ 14πTc D(1−a)/(1+a)

c +a

a+ 11

4πTcD(a−1)/(a+1)c − 1 . (4.25)

– 362 –

The dilaton chemical potential becomes minimal at Tc =√aD

(a−1)/(a+1)c /(4π). For the

Witten BH a = 1 the dilaton chemical potential depends on Tc only and attains its mini-mum in the asymptotic region. Also models with b = 1 lead to a unique T and to a simpleequation of state,

ψc(Tc, Dc) =2

a+ 2

(1 + 4π2T 2

c D−2a/(a+2)c

)1/2+

a

a+ 2

(1 + 4π2T 2

c D−2a/(a+2)c

)−1/2− 1 .

(4.26)For the JT model a = 0 the dilaton chemical potential becomes independent from thedilaton charge. The same happens for all (A)dS ground state models

(A)dS : ψc(Tc) =1 + b

2Tc

k(Tc)+

1− b2

k(Tc)Tc− 1 . (4.27)

However, the function k(Tc) in general has more than one branch and is known implicitlyonly. For all other models one has to solve

T = Tc D(1+b−a)/(1+b+a)c

√1− D−2(1+b)/(1+b+a)

c

(4πT1 + b

)1+1/b

(4.28)

for T in terms of Tc and Dc. The solution of (4.28) is not unique in general. For instance,the Schwarzschild BH in 5 space-time dimensions, a = −2b = 2/3, leads to two branchesT± = Tc

(1 ±

√1− 144π2/(T 4

c Dc))1/2

/√

2. If we restrict the discussion to an equation ofstate for the asymptotic region Dc →∞ it suffices to solve (4.28) perturbatively to leadingorder, in which case there is only one branch. For Minkowski ground state models theresult is

Minkowski ground state : limDc→∞

ψc(Tc, Dc) ∝T

2+2/bc

D2c

(4.29)

while for all other models it reads

Others : limDc→∞

ψc(Tc, Dc) ∝ T 1+1/bc D−(1+b)(a+b−1)/(b(a+b+1))

c . (4.30)

The proportionality constants in these equations are numerical coefficients depending ona, b. Asymptotically the equation of state for an ideal gas, ψc ∝ Tc/Dc, arises in the limitb→∞, only. As a cautionary remark we recall that the asymptotic limit is not accessiblefor b < 0 because the asymptotic specific heat (4.21) is negative. By contrast, the nearhorizon approximation is always accessible. It corresponds to the limit Tc →∞ and allowsto derive a simple equation of state for the branch where T from (4.28) is finite at thehorizon,

Near horizon : ψc(Tc, Dc) ≈4π

a+ b+ 1Tc D

(1−a−b)/(a+b+1)c . (4.31)

Extensitivity properties of BHs are quite unusual; generically, the temperature is notintensive and depending on the choice of the dilaton charge, which is extensive by defini-tion, either entropy is non-extensive or internal energy or both. For (3.1) the dilaton fieldDc = Xc is extensive by definition and thus also entropy is extensive. However, genericallyno other quantity is extensive or intensive. For (3.2) all thermodynamical potentials are

– 363 –

Quantity Witten JT (A)dS MGS Rindler a = 0 genericDc, Ec, Fc, Gc, Hc ext. ext. ext. ext. ext. ext. ext.ψc, κT int. int. int. int. int. int. int.S,CD, Cψ ext. ext. ext. — — — —Tc int. int. int. — — — —√ξc int. ext. — int. — ext. —

T int. ext. — — int. — —M ext. — — ext. — — —

Table 1: Extensitivity properties based upon the dilaton charge (3.2)

extensive and the dilaton chemical potential becomes intensive. However, generically en-tropy is not extensive. We summarize the extensitivity properties with respect to (3.2) ofall ab-models in table 1. Extensive (intensive) quantities are denoted by “ext.” (“int.”),while quantities which are neither are denoted by “—”.

Conf. Scale Shift Ext. Tol. G−12 Sign SignGS Dc-dep.

eQ 2 1 0 — 0 0 + + now 0 1 -1 — 0 0 + + noDc -1 0 — 1 0 1 + + yesξc 2 2 0 — -2 0 + + noΓc 0 0 — — 0 1 ? 0 noM 0 1 1 — 0 0 + 0 noT 0 1 0 — 0 0 + 0 noTc -1 0 0 — 1 0 + 0 noFc -1 0 — 1 1 1 ? 0 noS 0 0 0 — 0 1 + 0 noEc -1 0 — 1 0 1 + 0 noCD 0 0 0 — -2 1 ? 0 noTcS -1 0 0 1 1 1 + 0 noM/w 0 0 — 0 0 0 + 0 noψc ? 0 — 0 1 0 ? 0 yesκT ? 0 — 0 -1 0 ? ? yesGc ? 0 — 1 1 1 ? 0 yesHc ? 0 — 1 1 1 ? 0 yesCψ 0 0 0 — 0 1 ? 0 yesCψ − CD 0 0 0 — 0 1 ? 0 yes

Table 2: Summary of scaling properties

Besides extensitivity properties there are additional interesting scaling properties ofthermodynamical quantities, which are summarized for generic models in table 2. Here isan explanation of all abbreviations: “Conf.” refers to the conformal weight (0 means thequantity is conformally invariant and “?” means it transforms inhomogeneously); “Scale”

– 364 –

refers to the scale ambiguity eQ → ceQ, w → cw inherent to the definitions of thesefunctions (0 means again that the quantity is independent from such rescalings); similarly,“Shift” refers to the shift ambiguity w → w+w0 (quantities which have no well-defined shift-weight are denoted by “—”); “Ext.” denotes extensitivity properties (1 means extensiveand 0 means intensive); “Tol.” refers to the scaling with powers of the Tolman factor 1/

√ξc

close to the horizon (if it is positive the corresponding quantity diverges at the horizon andif the entry is underlined the whole cut-off dependence comes from a single Tolman factorwith corresponding weight); “G−1

2 ” determines the scaling with the gravitational couplingconstant in front of the action; “Sign” is + if the quantity is positive everywhere outsidethe horizon and “?” if the quantity may change its sign; “SignGS” is the same as “Sign”but for the ground state solution (therefore many quantities vanish); the last entry justkeeps track of quantities which depend in an essential way on the definition of the dilatoncharge. All scalings act multiplicatively [s(A) · s(B) = s(AB)] except for “Shift” whichacts additively [s(A) + s(B) = s(A + B)]. The bold entries in this table are definitionsrather than results; e.g. the quantity eQ is the conformal factor in front of the metric andthus by definition has a conformal weight of +2.

Finally, we collect various restrictions on the parameters from thermodynamical con-siderations. We recall that we have assumed B > 0 for positive w, M ≥ 0 for non-negativemass and b > −1 for a well-defined (positive) temperature. If we want space-time to beregular in the asymptotic region we encounter the inequalities a ≤ 2 and a + b ≤ 1. Ifwe demand that the dilaton chemical potential be non-negative everywhere then we need0 ≤ a ≤ 1 + b. Finally, if we wish the specific heat to be asymptotically positive theinequality b ≥ 0 must hold. These results are summarized in figure 1. The red (dark gray)region is excluded because b < −1 there, thus violating (2.10). The orange (medium gray)region exhibits a curvature singularity in the “asymptotic” region Xc → ∞. In the green(light gray) region the dilaton chemical potential can become negative. In the region belowthe a-axis the specific heat is asymptotically negative. The white region above the a-axis isfree from thermodynamical pathologies regardless of the location of the cavity. The pointsS (W) [JT] correspond to the Schwarzschild BH (Witten BH) [JT model]. The dashed linedenotes (A)dS ground state models and the dotted line Minkowskian ground state models.The a-axis contains all Rindler ground state models.

Two particular models of this class deserve further study. The Schwarzschild BH willbe addressed in more detail in section 6 and the Witten BH will be studied in the nextsection.

5. Examples from String Theory

In this section we consider two-dimensional BHs that emerge as solutions, either approxi-mate or exact, of string theory. In [1] it was shown that the Euclidean BHs studied in [40]admit an exact CFT description in terms of an SL(2,R)/U(1) gauged WZW model. Inthe large k limit, where k is the level of the SL(2) current algebra, the correspondingbackground takes the form

ds2 = α′ k(dy2 + tanh2 y dθ2

)X = X0 cosh2 y . (5.1)

– 365 –

Figure 1: Summary of the ab-family

This solution is commonly referred to as the ‘Witten BH’. It is related7 to the exactbackground obtained in [2]

ds2 = dx2 +tanh2(b x)

1− p tanh2(b x)dτ2 X = X0 cosh2(b x)

√1− p tanh2(b x) (5.2)

where the parameters b and p are

α′ b2 =1

k − 2p :=

2α′ b2

1 + 2α′ b2=

2k. (5.3)

The background (5.2) can be expressed in the form (2.3) by means of coordinate transfor-mations described in [41].

In the following subsections we wish to consider BHs for the full range k ∈ (2,∞)allowed by the CFT. However, in order for the background (5.2) to be a solution of stringtheory it must satisfy the condition

Dim− 26 + 6α′ b2 = 0 . (5.4)

Because the target space here is two-dimensional, Dim = 2, requiring the correct centralcharge fixes the level at the critical value kcrit = 9/4. Following [42], we vary k by allow-ing for additional matter fields that contribute to the total central charge, modifying thecondition (5.4) so that k 6= 9/4.

7The precise relationship is as follows: in the limit k → ∞ (or, equivalently, p → 0) upon identifying

y = bx and τ =√α′k θ the line-element and dilaton (5.1) emerge from (5.2).

– 366 –

5.1 Witten Black Hole

The Witten BH has the most intriguing thermodynamics of all the models we have exam-ined so far [10,12,27]. We focus on the background (5.1) as a solution of the string theoryβ-functions at lowest order in α′. The β-functions can be derived from a target-space actionof the form (1.1) with

U = − 1X

V = −λ2

2X (5.5)

where λ = 4/√α′. With these potentials and convenient choices for the integrations con-

stants in (2.6) and (2.7) the functions Q and w are given by

eQw = 1 , w = λX . (5.6)

Because w is linear in X, the surface gravity and asymptotic temperature T = λ/(4π) donot depend on the mass M . As a result, the condition (3.3) identifies a single solutionconsistent with the boundary conditions βc and Dc. The relation 2M = λXh then impliesa simple proportionality between mass and entropy

M = T S . (5.7)

An immediate consequence is that the Witten BH satisfies the Gibbs-Duhem relationship:GWc = GWc = 0.

The two definitions for the dilaton charge (3.1) and (3.2) are both linear in Xc forthe Witten BH. They are related by Dc = λDc. Therefore, the dilaton chemical potentialassociated with Dc,

ψWc =12

(1− Xh

Xc

)1/2

+12

(1− Xh

Xc

)−1/2

− 1 , (5.8)

is related to ψWc from (3.13) by the constant rescaling ψWc = λ ψWc . Both the internalenergy and the Helmholtz free energy

EWc = λXc

(1−

√1− Xh

Xc

)FWc = EWc − TWc S (5.9)

have finite asymptotic limits, EW → M and FW → 0, respectively. The extensitivityproperties of the Witten BH are comparable to those of the standard thermodynamics ofideal gases, as can be seen from table 1 on page 364.

There are various reasons to expect that a semi-classical analysis of the Witten BHmight encounter problems. For instance, the curvature is of order 1/α′ near the horizon,indicating that corrections have to be taken into account. It is not surprising that thissystem also exhibits some peculiar thermodynamic properties. The most obvious is thebehavior of the the specific heat, which is given by

CWD = 4π(Xc −Xh) . (5.10)

The specific heat is positive for all Xc > Xh, but it diverges linearly if we try to removethe cavity wall to infinity. In addition, the quantities Cψ and κT are not well-defined.

– 367 –

However, since we know that the Witten BH is related to the exact solution (5.2), it isreasonable to expect that these problems might be absent in a more complete treatment.As we shall demonstrate in the next subsection, α′ corrections have a significant impact onthe thermodynamics.

5.2 String Theory Is Its Own Reservoir: Exact String Black Hole

The discussion of the canonical ensemble in section 3 required the addition of a cavitywall at Xc that couples the system to a thermal reservoir. In most cases thermodynamicstability is only possible for finite values of Xc. We have not questioned the nature ofthis reservoir, or the form of the coupling between the two systems. In previous exampleswe simply assumed that it was possible to couple a given model to some other degreesof freedom that functioned as the reservoir. String theory is fundamentally different inthis regard. It is not possible to couple arbitrary degrees of freedom to the theory, orplace an abrupt cut-off on the space-time fields. Therefore, if we apply our methods to anon-compact solution of string theory any cut-offs introduced during a calculation must beremoved.

As shown in the previous subsection, the specific heat of the Witten BH diverges asthe cut-off is removed to infinity. Remarkably, this and other issues are resolved when α′

corrections are taken into account. The metric and dilaton for the ESBH were given in(5.2). A target space action of the form (1.1) was constructed for the Exact String BHin [41]. In our current notation the relevant functions determining the model are given by8

eQw = 1 , w(X) = 2b(√

ρ2 + 1 + 1), (5.11)

where b was defined in (5.3), and the canonical dilaton X is related to a new field ρ by

X = ρ+ arcsinh ρ . (5.12)

It will generally be more convenient to work with the field ρ, rather than X. The relation(5.12) between the two fields is globally invertible, and the limits X → ∞ and X → 0coincide with the corresponding limits for ρ. Therefore, ρ can be employed not only locallybut globally. Expressed in terms of ρ, the square of the Killing norm is given by

ξ = 1− k√ρ2 + 1 + 1

. (5.13)

This vanishes atρh =

√k(k − 2) . (5.14)

which establishes the location of the horizon in terms of the level k.As in [41], the integration constant in w(X) has been chosen assuming a Minkowski

ground state, i.e. eQw = 1. Comparing the general form of the Killing norm (2.5) with

8Winding/momentum mode duality relates the Exact String BHs to a different set of target space

geometries, namely the Exact String Naked Singularities. They are also described by (5.11) and (5.12), but

with sign changes in front of the square root and in front of the arcsinh .

– 368 –

(5.13) shows that the mass parameter is then proportional to the level: MMGS = bk. Thismeans that the choice of w0 in (5.11) identifies the ground state with k = 0. However, theCFT description requires k > 2. This can also be seen from the Hawking temperature [42],

TESBH =b

2π

√1− 2

k, (5.15)

which is real and positive only if k > 2. Therefore, we must shift the constant w0 so thatthe ground state corresponds to level k = 2. This is accomplished by replacing w withw−4b in (5.11). Notice that this is the largest shift by a negative constant that is consistentwith w > 0. The result is

eQESBH(X) =

12b

1√ρ2 + 1 + 1

, wESBH(X) = 2b(√

ρ2 + 1− 1). (5.16)

Comparing the resulting expression for the Killing norm with (5.13) gives the mass param-eter in terms of b and k

MESBH = b(k − 2) . (5.17)

With (5.16) the dilaton charge (3.2) reads

DESBHc = 2b ρc . (5.18)

For given βc and Dc the equation (3.3) has a unique solution with level k > 2. In theasymptotic region Dc → ∞ the temperature is bounded from above and below for k ∈(2,∞) according to (5.15). As we shall show in the next subsection, the thermal partitionfunction diverges in the k → ∞ limit, which identifies the maximal temperature TH =b/(2π) as the Hagedorn temperature. As might have been anticipated on general groundsthe asymptotic temperature has to be positive and below the Hagedorn temperature forthe existence of the ESBH solution.

5.2.1 Thermodynamics of the Exact String Black Hole

With (5.16) as a starting point we can calculate all thermodynamical quantities of interestfor the ESBH by applying the general results of section 3. The cut-off is always removed atthe end of a calculation, so we drop the subscript ‘c’ from the following expressions. Theresult for the entropy,

SESBH = 2π(√

k(k − 2) + arcsinh√k(k − 2)

), (5.19)

agrees with equation (4.17) found in [41] (In that reference M = k/2.). For k = 9/4 weobtain SESBH = 2π(3/4 + ln 2). In the large k limit the entropy SESBH = 2πk +O(ln k)asymptotes to the result for the Witten BH with logarithmic α′ corrections [43]. The freeenergy,

FESBH = −b√

1− 2k

arcsinh√k(k − 2) (5.20)

– 369 –

is manifestly non-positive 9. This would not have been the case with the original choice ofw0 that lead to (5.11). The result (5.20) coincides with (5.7) in [41]. The internal energyis given by the mass, EESBH = MESBH = b(k − 2). The specific heat

CD = Cψ = 2π k√k(k − 2) (5.21)

is positive and finite. Thus the problem encountered for the Witten BH is absent: removingthe cut-off is possible for the Exact String BH and leads to a positive and finite specificheat. It is worth mentioning that the specific heat and entropy both vanish in the limitT → 0, so the Exact String BH is one of the few examples for which the third law ofthermodynamics holds.

Even though the Minkowski ground state condition is not satisfied by (5.16), the Killingnorm still approaches unity as the cut-off is removed. Therefore the asymptotic limit of(3.15) is constant, ηESBH = 1/2. As a consequence the dilaton-chemical potential vanishesas X−2

c in the limit Xc →∞. The Gibbs-Duhem relation

GESBH = FESBH = MESBH − TESBHSESBH 6= 0 (5.22)

is violated even in the large k limit,

limk→∞

GESBH = limk→∞

FESBH = −b ln 2k +O(

1k

ln k). (5.23)

Thus, comparing the thermodynamic behavior of the exact String BH with that of theWitten BH demonstrates the importance of α′ corrections, even in the large k limit. Inthe limit k → 2 the Gibbs-Duhem relation,

limk→2

GESBH = limk→2

FESBH = −b(k − 2) +O(k − 2)2 , (5.24)

is restored.It is interesting to observe how the free energy scales with the level k in various

limits. For large values of k the free energy scales logarithmically, while for k close to theground state value it is linear in k− 2. This implies corresponding behavior of the thermalpartition function (2.31). Suppose that we fix the factor in front of the action as σ/(2π),where σ is some positive parameter, so that the curvature term in the action 1.1 is givenby −σXR/(4π). Then the thermal partition function in the large k limit is given by

ZESBH∣∣∣∣k2

= (2k)σ + · · · ∼(SESBH

)σ+ . . . (5.25)

In this limit the partition function grows as a positive power of k. It diverges in the k →∞limit, as the temperature approaches the asymptotic value TH = b/(2π). In this sense, THrepresents the Hagedorn temperature. Similarly, for k → 2 we obtain

ZESBH∣∣∣∣k→2

= eσ√

2k−4 + · · · ∼ eσ4πSESBH + . . . (5.26)

9It is worth mentioning that there is also a CDV solution with X = 0 and AdS geometry with curvature

proportional to b2. It has vanishing free energy. Therefore the Exact String BH is stable against tunneling

(cf. subsection 3.11).

– 370 –

Thus, in the weak coupling limit the partition function grows only with a power of entropy,while in the strong coupling limit it grows exponentially with entropy. This is consistentwith fermionization in the strong coupling regime.10 For arbitrary values of k we obtain

ZESBH(k) = eσ arcsinh√k(k−2) . (5.27)

Because exp [arcsinh (3/4)] = 2 we obtain the interesting result

ZESBH(9/4) = 2σ . (5.28)

As a caveat, we remind the reader that these results apply to the semi-classical approxima-tion of the thermal partition function, based on the action for the ESBH given in [41]. Theydo not take into account contributions from the tachyon that appears in the bosonic stringspectrum, or from any matter that must be coupled to the ESBH to maintain the correctcentral charge (5.4). Furthermore, while we have focused on the effects of α′ corrections,we have not taken into account contributions from world-sheet topologies other than theleading term.

5.2.2 Thermodynamical Derivation of the Exact String Black Hole

It is interesting to note that the potentials (5.11) can be derived purely from thermody-namical considerations. Of course, this requires some ad-hoc assumption: we look for amodel which may be shifted to a Minkowski ground state model and whose BH solutionshave a specific heat whose asymptotic behavior is

CD = Cψ ∝M2T (5.29)

for all values of M and T . The rationale behind this assumption comes from the con-sideration of the weak (k → ∞) and strong (k → 2) coupling limit. In the former casewe have seen that the ESBH is approximated by the Witten BH which has a divergentasymptotic specific heat. Taking into account corrections from fluctuations at 1-loop levelleads to a specific heat CD ∝M2T [45]. This is a non-stringy calculation, but as long as weare not interested in the precise proportionality factor it should not matter which kind offluctuations we consider perturbatively: thermal fluctuations, α′ corrections and quantumfluctuations of some additional massless matter fields are expected to yield the same qual-itative behavior. In the strong coupling limit the system can be fermionized (cf. footnote10). A free Fermi gas has a specific heat of the form C ∝ T . The proportionality constantcan be rescaled such that it includes a factor M2. Then the weak and strong coupling

10 In the WZW formulation the stress tensor is given by

T (z) =1

k − 2ηabJ

aJb +k

4(∂φ)2 + b∂c .

Here ηab is the metric on the algebra sl(2,R) and the Ja are given by equation (2.11) in [2]. For k → 2

the first term dominates and the system behaves like a free Fermi gas, cf. e.g. [44]. This suggests that an

appropriate interpretation of the Exact String BH in that limit is not in terms of some effective geometry

but rather in terms of free fermions.

– 371 –

limits qualitatively both yield the same result for the specific heat, which motivates ourAnsatz (5.29).

With CD ∝ w′h/w′′h, T ∝ w′H and 2M = wh the most general model (1.1) obeying

(5.29) has to fulfill the autonomous differential equation

w′′ ∝ 1w2

. (5.30)

Without loss of generality we fix the proportionality constant in (5.30) to unity. Its generalsolution is ∫

dw√c1 − 2/w

= c0 ±X . (5.31)

The integration constant c1 may be adjusted by rescaling w, c0 and X, so we fix it to unity.The integration constant c0 may be adjusted by shifting the origin of X, so we fix it tozero. The sign ambiguity is resolved by requiring X to be positive. Thus we obtain√

w(w − 2) + ln[w − 1 +

√w(w − 2)

]= X . (5.32)

Defining ρ =√w(w − 2) reveals that (5.32) actually is equivalent to (5.12). Solving w in

terms of ρ and taking the branch of positive w leads to

w(X) =√ρ2 + 1 + 1 , X = ρ+ arcsinh ρ . (5.33)

This is the same as w(X) in (5.11) for b = 1/2. If we had chosen a different proportionalityconstant in (5.30) we would obtain a different value for b but otherwise the same result forw. The function Q is fixed by the Minkowski ground state property (4.1). Thus, the ExactString BH in the Minkowski ground state representation (5.11) is the most general two-dimensional dilaton gravity model with a Minkowskian ground state that has an asymptoticspecific heat CD = Cψ ∝M2T .

6. Spherically Symmetric Black Holes in Higher Dimensions

Up to this point we have focused on theories in two dimensions, but the results of section3 can also be applied to solutions of gravitational theories in higher dimensions, as longas the on-shell action reduces to the form (2.38) or (3.54). With a few caveats, this leadsto a simple description of the thermodynamics of a class of spherically symmetric BHs ind+ 1 > 2 dimensions, with or without a cosmological constant.

Pure gravity with a cosmological constant in d+ 1 > 2 dimensions is described by theaction

Id+1 = − 116πGd+1

∫Md d+1x

√gd+1 (Rd+1 − 2 Λ)− 1

8πGd+1

∫∂Md dx√γd (Kd + . . .) . (6.1)

Subscripts ‘d+ 1’ and ‘d’ have been used to distinguish quantities from their two- and one-dimensional analogs. The ‘. . .’ in the boundary integral indicates boundary counterterms,whose precise definition depends on the space-times asymptotics [8,19]. We are interested

– 372 –

in solutions of this theory that can be expressed as a direct product of the two-dimensionalmetric (2.3) and the metric on a round d− 1 sphere

ds2 = ξ(r) dτ2 +1ξ(r)

dr2 + (Gd+1)2d−1 ϕ(r)2 dΩ 2

d−1 . (6.2)

Explicit factors of the Newton’s constant have been included in this expression so that thefield ϕ(r) is dimensionless. Ignoring for a moment the contributions from the boundarycounterterms, the on-shell reduction of (6.1) on the d−1 sphere leads to a two-dimensionalexpression of the form (1.1). The dilaton X in this action is related to the scalar field ϕ by

X(r) = ΥGd+1 ϕ(r)d−1 Υ :=Ad−1

8πGd+1(6.3)

where Ad−1 is the solid angle subtended by the d − 1 sphere, and the constant Υ hasbeen defined for convenience. Thus, the dilaton X(r) is the proper area of a sphere withcoordinate radius r, in d+ 1 dimensional Planck units. The kinetic and potential functionsfor the dilaton are

U(X) = −(d− 2d− 1

)1X

(6.4)

V (X) = −12

(d− 1)(d− 2) Υ2d−1 X

d−3d−1 + e

d(d− 1)2 `2

X . (6.5)

The last term in V comes from the cosmological constant, which we parameterize in termsof a length scale ` by

Λ := ed(d− 1)

2 `2e = ±1, 0 . (6.6)

The coefficient of the reduced action explicitly sets the two-dimensional Newton’s constantto 8πG2 = 1. From (6.4) and (6.5) we obtain

eQ(X) =Υ

11−d

d− 1X

2−dd−1 , (6.7)

w(X) = (d− 1) Υ1d−1 X

d−2d−1

(1− e

`2Υ−

2d−1 X

2d−1

), (6.8)

where we have fixed the free constants in the definitions (2.6) and (2.7) conveniently.The results of section 3 can be applied to these solutions of the d + 1 dimensional

theory. This is because the on-shell action, after integrating over the sphere, takes theform (2.38). Of course, the contributions from the boundary counterterms, alluded to bythe ‘. . .’ in (6.1), must be properly accounted for. Otherwise the reduced on-shell actionwould diverge, as in (2.15). This raises an interesting technical problem. The formulationof the boundary counterterms in the d+ 1 dimensional theory depends on the asymptoticsof the space-time [8, 46]. This would seem to imply that the reduction of theories withΛ = 0 and Λ 6= 0 must be studied separately. We take a different approach. Ratherthan working out the reduction of the appropriate boundary counterterms in (6.1), weassume that the two-dimensional boundary counterterm derived in subsection 2.4 gives a

– 373 –

sensible renormalization of the reduced on-shell action. For the solution described above,the boundary counterterm is

ICT = −β w(X)

√1− 2M

w(X). (6.9)

This counterterm gives the ‘correct’ on-shell action in the sense that the analysis of section3 recovers the standard results associated with the thermodynamics of BHs whose metricstake the form (6.2). An immediate consequence, due to (6.3), is

S = 2πXh =Ah

4Gd+1. (6.10)

The entropy of these BHs is always given by one-quarter of the proper area of the horizon,in d+ 1 dimensional Planck units.

There is no reason to believe that the counterterm (6.9) can be lifted to higher di-mension, to recover the counterterms of the original theory. The combination of the two-dimensional metric and scalar field are simply not sufficient to reproduce the appropriatefunctional dependence on the higher-dimensional metric. But despite this caveat, it seemsremarkable that the quasi-local thermodynamics of spherically symmetric BHs with differ-ent space-time asymptotics can be recovered from the reduced on-shell action, renormalizedby the two-dimensional counterterm (2.29).

6.1 Schwarzschild Black Hole in d+ 1 ≥ 4 Dimensions

As a first example, consider the on-shell reduction of (6.1) with Λ = 0. Setting ϕ ∼ r, upto factors of the Newton’s constant, gives the standard form of the Schwarzschild solution

ξ(r) = 1− 2Mrd−2

(6.11)

X(r) = Υ rd−1 (6.12)

where the mass parameter M is related to the constant M by

M :=M

(d− 1)Υ. (6.13)

In section 3 the dilaton charge Dc = Xc and proper periodicity βc were held fixed at r = rc.This is precisely the canonical ensemble studied in [30,47], where the BH is enclosed insidea cavity of radius rc whose area and proper temperature are held fixed by means of anexternal thermal reservoir. The relation (6.3) identifies the dilaton charge Dc = Xc as thearea of the cavity, so the dilaton chemical potential (3.13) is the corresponding pressure

pc =12

(d− 2) r−1c

(√ξ(rc) +

1√ξ(rc)

− 2

). (6.14)

The internal energy of the system can be expressed as

Ec = (d− 1) Υ r d−2c

1−

√1− 2M

rd−2c

. (6.15)

– 374 –

The asymptotic limit of this quantity gives the ADM mass M . Furthermore, the internalenergy satisfies the quasi-local form of the first law (3.26),

dEc(S,Ac) = Tc dS − pc dAc , (6.16)

where Ac is the area of the cavity in Planck units (8πGd+1 = 1). Finally, the expression(3.28) for the heat capacity recovers the well-known result for the thermodynamic stabilityof the Schwarzschild BH enclosed by an isothermal cavity in d+1 dimensions

CD = −4πM rhr d−2c − 2M

r d−2c − d M

. (6.17)

This quantity is positive for rh < rc < rmax, as indicated in figure 2.

Figure 2: Heat capacity for the Schwarzschild BH in dimension d+ 1 ≥ 4.

6.2 Asymptotically AdS Space-Times

Now we turn our attention to the case of non-zero cosmological constant. For definiteness,we take the cosmological constant to be negative (e = −1) and consider asymptoticallyAdS space-times11. Setting ϕ ∼ r leads to the Schwarzschild-AdS solution

ξ(r) = 1 +r2

`2− 2 Mrd−2

X = Υ rd−1 . (6.18)

As in the previous case, holding Dc = Xc and βc fixed at rc corresponds to an AdSBH enclosed by an isothermal cavity. The thermodynamics of this system was studiedin [48,49].

Rather than reproduce the full catalog of results regarding the Schwarzschild-AdSsolution, we focus on a few interesting properties which are qualitatively different than theprevious example. First, the internal energy (3.18) is given by

Ec =(d− 1)`

Υ rd−1c

√1 +`2

r2c

−

√1 +

`2

r2c

− 2M`2

rdc

. (6.19)

11Asymptotically de Sitter space-times with Λ > 0 can also be addressed in this framework, but many

of the working assumptions made in sections 2.1 and 3 must be changed. We shall discuss this briefly in

section 7.

– 375 –

Unlike the result (6.15), the asymptotic limit of the internal energy vanishes due to red-shifting

limrc→∞

Ec = 0 . (6.20)

However, as in section 3.4, the conserved charge is [30]

Q∂τ := limrc→∞

√ξ(rc)Ec = M . (6.21)

This is the same result obtained in higher dimensions using any one of a number of methods[50]. But, notably, it does not agree with the standard AdS/CFT result when the space-time dimension d+ 1 is odd. In that case, global AdS (the ground state) has Q∂τ = Mcas,where Mcas is a Casimir energy associated with the dual field theory on S1×Sd−1 [22,23].Reproducing the Casimir term is possible in the present context: an appropriate shift in w0

can be used to shift the value of (6.21) by an amount Mcas, for all solutions. But from thetwo-dimensional point of view the shift is arbitrary. The counterterm (6.9) is completelyignorant of the physics attached to this issue in the higher-dimensional theory.

Another interesting result is the structure of the heat capacity for an asymptoticallyAdS BH. The result is, in general, very complicated for finite rc [49]. But the presenceof a cosmological constant can stabilize the BH even when the cavity wall is removed toinfinity. Using (3.29), the rc →∞ limit of the heat capacity is

limrc→∞

CD = (d− 1) 2πXh

dX 2d−1

h + (d− 2) `2 Υ2d−1

dX2d−1

h − (d− 2) `2 Υ2d−1

(6.22)

Unlike the asymptotically flat solutions studied in the previous subsection, the heat capacitymay be positive for a large enough BH. The relation between the heat capacity and thesize of the BH is shown in figure 3. The condition CD > 0 is satisfied if the horizon radiusis larger than a critical size set by the dimension and the AdS length scale

rcrit =

√d− 2d

` . (6.23)

This corresponds to a critical periodicity for the Euclidean time given by

βcrit =

√d(d− 2)2π`

. (6.24)

The possibility of a phase transition, as studied by Hawking and Page [36], is encounteredwhen rh drops below the critical value given by (6.23). This is consistent with our generaldiscussion in subsection 3.10.

6.3 Reduction of BTZ

Gravity with a negative cosmological constant in 3 dimensions admits a solution known asthe BTZ BH [15]. If Λ = −1/`2 then (Euclidean) line element of the BTZ BH is

ds2BTZ = ξ(r)dτ2 +

1ξ(r)

dr2 + r2(dφ− 4G3J

r2dt)2 (6.25)

– 376 –

Figure 3: Heat capacity for the Schwarzschild-AdS BH in d + 1 dimensions, with the cavity wallsent to infinity. The horizontal axis is the horizon radius rh, and the vertical axis is the Xc → ∞limit of CD.

where

ξ(r) =r2

`2− 8G3M +

16G23J

2

r2. (6.26)

If the inequality |J | ≤M` holds then two horizons exist,

r± = `√

4G3M

√1±

√1− J2

M2`2. (6.27)

The normalization chosen in the rest of this work would require G3 = 1/4, but we setG3 = 1/8 to obtain the same normalization of the action used in [15, 51]. Some usefulformulas are

r2+ + r2

−`2

= M ,2r+r−`

= J ,r2

+ − r2−

`2= M

√1− J2

M2`2. (6.28)

Achucarro and Ortiz performed a Kaluza-Klein reduction

ds2BTZ = gαβdx

αdxβ +X2 (dφ+Aαdxα)2 (6.29)

to two dimensions. The dilaton field is the surface radius, X = r, and the two-dimensionalline element gαβdxαdxβ corresponds to the τr-part of (6.25). They integrated out thegauge field Aα and obtained an action of the type (1.1) with [51]

UAO = 0 , V AO = −X`2

+J2

4X3. (6.30)

We point out an important deficiency of (6.30): the angular momentum J enters here as aparameter in the action rather than emerging as a constant of motion. Consequently, thethermodynamics is not reproduced correctly from the two-dimensional model (1.1) withthe potentials (6.30). But there is a simple solution to this problem: we have demonstrated

– 377 –

in subsection 3.8 how to treat charged BHs and noted that upon introducing Maxwell fieldsa parameter in the action may be converted into a constant of motion, namely a conservedcharge. In fact, the Kaluza-Klein Ansatz (6.29) already contains such a Maxwell field andall one has to do is to refrain from integrating it out. Therefore, we study an action (3.45)[multiplied by 2 to get the normalizations above] with the functions

UBTZ(X) = 0 , (6.31)

V BTZ(X) = −X`2, (6.32)

fBTZ(X) = −18X3 . (6.33)

The potentials (6.31) and (6.32) are consistent with (6.4) and (6.5), respectively, for d+1 =3. The Maxwell field with coupling function (6.33) arises because we are treating a spinningBH rather than a spherically symmetric one. The conserved U(1) charge therefore has tobe identified with the angular momentum, q ∝ J . The proportionality factor depends onthe conventions. With our choices the correct relation is q = J/2. For further calculationswe need the functions

QBTZ(X) = 0 , wBTZ(X) =X2

`2, hBTZ(X) =

4X2

(6.34)

and note that the ground state solution M = J = 0 is pure AdS3, ξ0 = X2/`2.We show now that this effective two-dimensional description leads to the correct ther-

modynamics for the BTZ BH. To this end we apply the general results of subsection 3.8to the specific choice (6.34) recalling that q = J/2. The square of the Killing norm

ξ(X) =X2

`2−M +

J2

4X2(6.35)

indeed is equivalent to (6.26) for G3 = 1/8 and X = r. This together with the gaugepotential

Ar = 0 , Aτ = − J

2X2, (6.36)

plugged into (6.29) allows to recover the BTZ line-element (6.25). With the proper angularmomentum [cf. (3.53)]

Ωc := Φc =J

2√ξc

(1X2h

− 1X2c

)(6.37)

the thermodynamic potential (3.55) is given by

Y (Tc, Xc,Ωc) = −TcS + Ec − ΩcJ . (6.38)

Entropy is determined by the value of the dilaton at the outer horizon, Xh = r+. Takinginto account that the normalization of Newton’s constant differs in this subsection fromthe one used in the rest of this work by a factor of 2 we get from (3.9)

S = 4πr+ . (6.39)

– 378 –

The result (6.39) coincides with the entropy of the BTZ calculated by counting of mi-crostates, cf. e.g. [52]. The inverse Euclidean periodicity determines temperature,

T =r2

+ − r2−

2πr+`2=

2MS

√1− J2

M2`2. (6.40)

This coincides with the result for the temperature of the BTZ BH [15]. Obviously, inthe limit |J |/(M`) → 1 temperature vanishes, which is consistent with the fact that for|J | = M` the BTZ BH becomes extremal. If we define the quantity Ω as

Ω := limXc→∞

√ξcΩc =

J

2r2+

(6.41)

then we obtain the correct first law relating the conserved charges,

dM = TdS + ΩdJ . (6.42)

The integrated version

2M = TS + 2ΩJ (6.43)

shows that extensitivity properties differ slightly from standard thermodynamics. However,they are as expected for a BH in AdS: if the dilaton (and thus entropy) is extensive thenthe square-root of the mass M is extensive, consistently with the extensitivity propertiesin table 1. This explains the factor of 2 on the left hand side of (6.43). The factor 2 onthe right hand side is the standard result for spinning BHs. With the results above, theexpression for internal energy [note again the factor of 2 as compared to (3.18) because ofthe different normalization of the action],

Ec(Tc, Xc, J) = 2√ξ0 − 2

√ξc =

2Xc

`

(1−

√1− M`2

X2c

+J2`2

4X4c

), (6.44)

and surface pressure

ψc =Ec

`√ξc− J2

2X3c

√ξc

(6.45)

lead to the quasilocal form of the first law

dEc = TcdS + ΩcdJ − ψcdXc . (6.46)

All our results are compatible with [48] (cf. also [53]). This shows that the BTZ BH thermo-dynamics is compatible with a Kaluza-Klein reduction to two dimensions. A generalizationto a charged BTZ BH is straightforward and requires the addition of another Maxwell fieldwith a linear coupling to the dilaton. A different generalization involves the coupling toa gravitational Chern-Simons term [54], which yields upon dimensional reduction [55] amodel of type (3.45) [56] whose entropy recently has been derived in [57] using Wald’sNoether charge technique [58].

– 379 –

7. Discussion

In this paper we have studied black hole thermodynamics for arbitrary models of dilatongravity in two dimensions. The analysis of these models was complicated by the factthat the path integral defined with respect to the action (1.1) does not have a sensiblesemi-classical limit. We resolved this problem by constructing an improved action,

Γ = − 116πG2

∫Md 2x√g[X R− U(X) (∇X)2 − 2V (X)

]− 1

8πG2

∫∂Mdx√γ X K +

18πG2

∫∂Mdx√γ√w(X) e−Q(X) , (7.1)

generalizing the techniques first used in [10] to obtain the boundary counterterm (2.29).The counterterm is unique up to a shift ambiguity12 which we fixed by demanding that itbe independent under changes of scale associated with constant shifts in Q(X). Thus, oneshould consider (7.1), instead of (1.1), as ‘the generic dilaton gravity action’. Followingthe approach of York [11], we proceeded to define the canonical partition function for ablack hole enclosed in the ‘cavity’ X ≤ Xc. The Helmholtz free energy (3.5) for this systemis related to the improved action by Fc = β−1

c Γc, where the proper inverse temperatureβ−1c is fixed at the cavity wall Xc by an external thermal reservoir. All of the results

on thermodynamics in section 3 follow from this identification. After deriving variousthermodynamical quantities, we generalized our results to include two dimensional Maxwellfields and their associated conserved charges. We then examined the conditions under whichthe system is thermodynamically and mechanically stable, and described processes whichmight destabilize the system. Our results were then applied to a wide range of models insections 4-6. This includes backgrounds that appear in two-dimensional solutions of stringtheory (section 5). Finally, we discovered that our results can be used to describe thethermodynamics of certain higher dimensional black holes (section 6).

An important aspect of our analysis is universality. For instance, we confirmed thatthe entropy (3.9) for a dilaton gravity black hole in two dimensions is essentially given byXh, the dilaton evaluated at the horizon. This result is independent of the details thatdefine a particular model. If we define the ‘area’ of the horizon via the d→ 1 limit of a ddimensional sphere and restore Newton’s constant, then this result can be expressed as

S =Ah

4Geff. (7.2)

The effective Newton’s constant is defined as Geff = G2/Xh. Since G2 is dimensionlessthere is no notion of Planck length in two dimensions. However, we can think of Geff asdefining an effective resolution at the horizon, in the sense that decreasing Geff increases thenumber of bits that can be stored by the black hole. Then (7.2) can be interpreted in muchthe same way as the usual relationship between entropy and area in higher dimensions. Insection 3.4 we obtained a simple expression for the (proper) internal energy of the system

Ec = e−Q(Dc)(√

ξ0 −√ξc

). (7.3)

12In addition there is a sign ambiguity which is trivially fixed by demanding finiteness of the improved

on-shell action.

– 380 –

This agrees with the calculation of the energy using the quasilocal stress tensor defined byBrown and York [30]. We were able to show that every dilaton gravity black hole satisfiesa quasilocal form of the first law of black hole thermodynamics

dEc = Tc dS − ψc dDc + Φc dq . (7.4)

This expression holds at arbitrary values of Xc, incorporates the dilaton charge Dc and itschemical potential ψc, allows for Maxwell fields with proper electrostatic potentials Φc andconserved charges q, and accounts for the (non-linear) effects of gravitational binding energypresent in Ec. The canonical first law (7.4) contains considerably more information thanthe microcanonical dM = T dS, which becomes a triviality if expressed as dwh = w′hdXh.

Despite our best attempts to provide a comprehensive study, there are several direc-tions that we have not had a chance to explore. A number of ‘working assumptions’ weremade throughout the paper and it is important to understand which ones can be relaxed.An example is our decision to focus on models where w → ∞ as X → ∞. This is a rea-sonable assumption, as it seems to be the most common behavior among dilaton gravitymodels. But there are certainly models for which it does not apply. For instance, onemight consider models where w approaches a constant, or w → −∞, as X → ∞. Ourapproach should still be applicable, though many of the considerations in section 3 mustbe modified. In the first case, where w approaches a constant, we can make a constantshift so that w → 0 as X →∞. Then one can exploit the duality described in [59], whichpreserves the classical solution for the metric but replaces w with w = 1/w. This impliesw →∞ as X →∞, which can be analyzed under our initial assumptions. The second case,w → −∞ as X →∞, is perhaps more interesting as it includes the dimensional reductionof the static patch of de Sitter space. We now have to consider the region X ≤ XdS ,where XdS corresponds to the observer-dependent cosmological horizon in the static patch.Although we have not performed this analysis, we expect that our approach should repro-duce the standard results for the thermodynamics of cosmological horizons. An additionalcomplication emerges when one considers a second horizon, Xh < XdS , associated with ablack hole. The analysis of that system is beyond the scope of this paper.

It would be interesting to study the impact of the boundary counterterm in (7.1)on the exact path integral quantization procedure in the first order formulation of dilatongravity, cf. e.g. [3,60]. While the bulk term and the Gibbons-Hawking-York boundary termlook quite different from (1.1) in the first order formulation, we expect that the boundarycounterterm (2.29) can be translated easily to its first order counterpart, because it dependssolely on the dilaton field and the induced volume form at the boundary.

Another idea that deserves to be studied in more detail was mentioned at the end ofsubsection 3.8. Consider a dilaton gravity model with a potential of the form (3.59). Thesame model can be obtained by integrating out a number of Maxwell fields whose couplingfunctions f(X) are related to terms in the potential. This means that various couplingconstants in the potential V (X) can be converted to conserved U(1) charges associated withMaxwell fields. This technique allowed us to study the thermodynamics of the BTZ BH insubsection 6.3. The first law for the rotating, three dimensional solution was recovered by‘integrating in’ a gauge field that Achucarro and Ortiz had integrated out in their toroidal

– 381 –

reduction [51]. It would be interesting to study the extent to which the thermodynamicsof higher dimensional solutions can be reproduced in this manner.

The most pressing generalization of our work involves coupling two dimensional dila-ton gravity to matter, such as a scalar field. This is important for several reasons. Asmentioned in the introduction, we have assumed that these models will be coupled tosome form of propagating matter, and scalar fields are a prime candidate. We have alsoconsidered backgrounds that arise in two dimensional solutions of string theory, and anyfurther analysis of these models should try to incorporate the tachyon. In general, couplingadditional fields to the dilaton gravity model will introduce complications at the level ofthe Hamilton-Jacobi equation. However, even if this equation can no longer be solvedexactly, it can still be addressed perturbatively. This approach has been used successfullyfor gravity coupled to a scalar field in higher dimensions [61–64]. We hope to address thisin a future publication.

Acknowledgments

The authors would like to thank Josh Davis, Roman Jackiw, Antal Jevicki, Mark Spradlin,Alessandro Torrielli and Dima Vassilevich for useful discussions. In addition DG wouldlike to thank Jan Aman, Ingemar Bengtsson and Narit Pidokrajt for the hospitality atStockholm University and for their interest. The work of DG is supported in part byfunds provided by the U.S. Department of Energy (DoE) under the cooperative researchagreement DEFG02-05ER41360. DG has been supported by the Marie Curie FellowshipMC-OIF 021421 of the European Commission under the Sixth EU Framework Programmefor Research and Technological Development (FP6). The research of RM was supportedin part by the DoE through Grant DE-FG02-91ER 40688, Task A (Brown University).

A. Conventions, Dimensions, and a Dilaton Gravity Bestiary

Let M be a two-dimensional manifold with boundary ∂M. We consider a metric gµν onM that induces a metric γab on ∂M. The (one-dimensional) boundary indices are usefulfor keeping track of factors of γ and its inverse.

The Riemann tensor on (M, g) and its contractions are defined so that spheres havepositive curvature and hyperbolic spaces have negative curvature

Rλµρν = ∂ρΓλµν + ΓκµνΓλκρ − (ν ↔ ρ) (A.1)

Rµν = Rλµλν . (A.2)

The extrinsic curvature associated with the embedding of (∂M, γ) in (M, g) is given by

Kab =12nµ∂µγab (A.3)

where nµ is an outward-pointing unit vector normal to ∂M. The trace of the extrinsiccurvature is K = γabKab. Because the boundary is one-dimensional, it follows that

Kab − γabK = 0 . (A.4)

– 382 –

This implies that πab would vanish if not for the coupling between X and the Ricci scalarin (1.1).

Throughout the paper we work with Euclidean metrics of the form

ds2 = ξ(r) dτ2 +1ξ(r)

dr2 . (A.5)

Calculating the action and its variation requires the Ricci scalar and the extrinsic curvatureof the boundary. The Ricci scalar is given by

R = − ∂2ξ

∂ r 2= −e−Q

[w′′ + Uw′ + U ′(w − 2M)

]. (A.6)

The last expression, which follows from (2.4) and (2.5), is coordinate independent andrelates the Ricci scalar to the dilaton field. For models obeying the Minkowskian groundstate condition (4.1) it simplifies to R = 2M(w′2/w − w′′). Boundary quantities are ob-tained from the X →∞ limit of a constant X surface, as described in subsection 2.2. Theinduced metric on the surface is γab = ξ(X), and the trace of its extrinsic curvature is

K =1

2√ξ

∂ξ

∂r= e−Q/2

(w − 2M)U + w′

2√w − 2M

. (A.7)

For models obeying the Minkowskian ground state condition (4.1) it simplifies to K =Mw′/(w

√ξ). The Euler characteristic for the space-time is related to (A.6) and (A.7) by

4π χ =∫Md 2x√g R+ 2

∫∂Mdx√γ K . (A.8)

One can verify that the BH solutions of discussed in 2.2 have the topology of a disk, χ = 1.The dimensions of various quantities are assigned conveniently, but at times differently,

in the examples we consider. Here we clarify the role of physical dimensions, how to fixthem, and how to rescale them. Unless stated otherwise we always work in natural units~ = c = kB = 1 and choose the dilaton field X to be dimensionless. This means that thepotential U(X) is dimensionless, as well. If the coordinates are taken to have the naturaldimension of length then the relation (2.4) implies that eQ must also have dimensionsof length. This can be accomplished by defining eQ = λ exp

∫ XdXU(X), where λ is a

dimensionful constant with units of length that includes the factor of eQ0 . In that case,both w and M must have units of inverse-length, so that (2.5) is dimensionless. Similarly,coordinates with units of length imply that the function V has dimensions of one overlength squared. Units of length can then be changed by appropriate rescalings of thedimensionful constants in V , as well as λ and M . Alternatively, if the coordinates aretaken to be dimensionless then U , eQ, V , w, and M are all dimensionless, as well.

After these general observations we focus on the choices of relevance for our work. Withone exception (mentioned below) we demand that the dilaton field X be dimensionless.In some of our examples we do not exhibit the dimensionful constants, for simplicity.Instead we treat all quantities as dimensionless, including the line-element, and restoreappropriate physical dimensions when necessary. This applies to the discussion of the

– 383 –

ab-family in subsection (4.2), and to the discussion of the BTZ BH in subsection (6.3)if X, `, J,M are considered to be dimensionless. In other examples we keep track of thephysical dimensions for easier comparison with the literature. This applies to sections 5,6.1, and 6.2. Alternatively, in subsection 6.3 the quantities X, `, J can be thought of ashaving dimensions of length, with M dimensionless.

Model U(X) V (X) eQ(X) w(X) Reference

Schwarzschild − 12X − 1

2G4

√G42X

√2XG4

[65]

Jackiw-Teitelboim 0 −ΛX 1 ΛX2 [39]

Witten BH − 1X −λ2

2 X 1λX λX [1]

CGHS 0 −λ2 1 λX [66]

(A)dS2 Ground State − aX −1

2 BX1Xa

B2−a X

2−a [67]

Rindler Ground State − aX −1

2 BXa 1

Xa BX [68]

BH Attractor 0 − B2X 1 B lnX [69]

ab-Family − aX −B

2 Xa+b 1

XaBb+1 X

b+1 [38]

Liouville Gravity a beαX eaX − 2ba+α e

(a+α)X [70]

Exact String BH [41] [41] (5.16) (5.16), (5.12) [2]

Schwarzschild-(A)dS − 12X − 1

2G4− 3

`2X

√G42X

√2XG4

(1 + 2G4

`2X)

[36]

Katanaev-Volovich α βX2 − λ eαX 2α e

αX(λ− βX2 + [71]

+2βα X − 2 βα2

)KK Reduced CS 0 1

2 X(c−X2

)1 1

4

(c−X2

)2 − 14 c

2 [55, 56]

Table 3: Summary of Dilaton Gravity models, adapted from [29].

Model U(X) V (X) f(X) eQ(X) w(X) h(X) Reference

Reissner-Nordstrom − 12X − 1

2G4X

√G42X

√2XG4

−√

2G4X [72]

Achucarro-Ortiz 0 −X`2

−X3

8 1 X2

`24X2 [51]

2D Type 0A/0B − 1X −λ2

2 X πα′ 1λX λX lnX

λπα′ [73]

Table 4: Dilaton Gravity models with Maxwell fields.

The analysis of sections 2 and 3 is carried out for a general model of two-dimensionaldilaton gravity. As a convenience, the properties of several models are summarized in tables

– 384 –

3 and 4. Table 3 lists models that contain only a metric and a dilaton, while table 4 listsmodels that also contain Maxwell fields. The last three models in table 3 can be convertedinto models belonging to table 4 as explained at the end of subsection 3.8. In most entriesin these tables we have chosen the scaling and shift ambiguities inherent to Q and w asto achieve coincidence with the choices in the main text. In some brief examples in ourpaper the four-dimensional Newton constant is fixed as G4 = 1/2, thus simplifying the firstentries in both tables, as well as the Schwarzschild-(A)dS entry in table 3. In subsection4.2 we set B = b+ 1, which simplifies the corresponding entry ’ab-Family’ in table 3.

B. Weyl Rescaled Metric

In subsections 2.3 and 2.5 we employ a change of variables that involves a Weyl rescalingof the metric. Then, in subsection 3.9, we consider the effect of a conformal transformationon a model. In this appendix we work out some of the relevant details.

Consider a new metric related to gµν by

gµν = e2σ(X) gµν (B.1)

where σ(X) is an arbitrary function of the dilaton. If we express the action (1.1) in termsof this new metric we find

I[X, gµν ] = −12

∫Md 2x

√g[XR− U(X)(∇X)2 − 2 V (X)

]−∫∂Mdx√γXK . (B.2)

This has the same functional form as (1.1), with dilaton potentials U and V that are relatedto U and V by

U(X) = U(X)− 2σ′(X) V (X) = e2σ(X) V (X) . (B.3)

The functions Q(X) and w(X) transform according to

Q(X) = Q(X)− 2σ(X) w(X) = w(X) . (B.4)

A change of variables with 2σ(X) = Q(X) leads to vanishing U . This transformation iscommon in the dilaton gravity literature because it simplifies many calculations. Noticethat w(X) is invariant under (B.1). Referring to (2.15), which depends only on w(X), X,and the parameter M , we confirm that the on-shell action is invariant. This is as it shouldbe: the value of the action cannot depend on the variables we use to evaluate it.

Now we want to examine the variation of the action using the new metric variable. Interms of gµν and σ a general variation of gµν takes the form

δgµν = e2σ δgµν + 2 e2σ gµνσ′δX . (B.5)

The change in the action (2.16) due to a small variation of the fields can be written as

δI[X, gµν ] =∫Md 2x

√g[Eµνδgµν + EX δX

]+∫∂Mdx√γ[πabδγab + πXδX

]. (B.6)

– 385 –

Both the equations of motion Eµν and EX , and the momenta πab and πX have the samefunctional form as in the main text, with all un-hatted quantities being replaced by theirhatted counterparts. The solution of the equations of motion is

X = X(r) ds2 = ξ(r) dτ2 +1

ξ(r)dr2 (B.7)

∂rX = e−Q(X) ξ(X) = eQ(X) (w(X)− 2M) . (B.8)

This is just the solution from section 2.1, expressed in terms of the new metric and writtenin a coordinate system (τ, r) with ∂r = e2σ(X)∂r. Evaluating (B.6) for this solution gives

δI[X, gµν ]∣∣∣E=0

=∫∂Mdx

[−1

2e−Q δξ + e−Q

(U ξ − 1

2ξ′)δX

]. (B.9)

We can now choose the function σ(X) in (B.1) to simplify the analysis of this expression.If we define

e2σ(X) = w(X) eQ(X) , (B.10)

then the (square of the) Killing norm for the new metric is given by

ξ(X) = 1− 2Mw(X)

. (B.11)

This makes the meaning of Dirichlet boundary conditions unambiguous: ξ takes the con-stant value ξ = 1 at ∂M, and any variation δξ that preserves this condition must vanishat ∂M. The functions Q and U are determined as

eQ(X) =1

w(X)U = −w

′(X)w(X)

(B.12)

so that (B.9) yields

δIreg =∫dτ

[−1

2w δξ −

(1− M

w

)w′ δX

]. (B.13)

The subscript indicates that this quantity is evaluated using a regulator, as in section 2.3.Following the arguments in the main text we ignore the δX term and concentrate on avariation of the metric δξ = δMeQ = δM/w. The corresponding change in the action is

limXreg→∞

δIreg =∫dτ δM 6= 0 (B.14)

which of course agrees with (2.20). Changing variables to a metric that obeys standardDirichlet conditions at ∂M simplifies the analysis of the improved action, as well, and isuseful in establishing the result (2.37).

In subsection 3.9 we treat (B.1) as a Weyl transformation, as opposed to a change ofvariable. Then the action (B.2) is interpreted as a functional of X and gµν . This gives anew model of type (1.1) with potentials given by (B.3).

– 386 –

C. Proof of Gauge-Independence

In this appendix we show that the main results of section 2 are independent of the choiceof gauge (2.3). We begin by establishing two important properties of solutions of (2.1) and(2.2). Consider the vector kµ defined by

kµ = εµν eQ(X)∇νX (C.1)

where eQ(X) was given in (2.6). The Lie derivative of the dilaton along kµ vanishes byconstruction

LkX = eQ(X)εµν ∂µX∂νX = 0 . (C.2)

The same result also applies to the metric. It follows from the equation of motion (2.1)that the vector kµ satisfies Killing’s equation

Lkgµν = ∇µkν +∇νkµ = 0 . (C.3)

Therefore, solutions of (2.1) and (2.2) always possess at least one Killing vector whoseorbits are isosurfaces of the dilaton.

The properties of kµ allow us to construct a coordinate system (τ, r) where the Killingvector is kµ∂µ = ∂τ , and the metric and dilaton take the form

ds2 = N(r)2 dr2 + ξ(r) (dτ +N τ (r) dr)2 X = X(r) . (C.4)

The functions N and N τ that appear in the metric are the lapse function and shift ‘vector’,respectively, associated with evolution in the direction orthogonal to kµ. Normally, thelapse and shift both appear in the action as Lagrange multipliers. In this case, the factthat all fields are constant along the direction singled out by the Killing vector implies thatthe action is independent of N τ (r). Furthermore, the coefficient of δN τ in the variation ofthe action vanishes identically. This means that we can freely set N τ = 0 in (C.4) with noloss of generality. The metric then simplifies to

ds2 = N(r)2 dr2 + ξ(r) dτ2 . (C.5)

One can substitute this form of the metric into the action and recover all of the results of afully covariant analysis. From the point of view of the action, the functions ξ(r) and N(r)must be treated as independent fields. However, we are always free to present a solutionin a coordinate system where N(r)2 = ξ(r)−1, as in (2.3).

Now we want to check whether or not the conclusions of subsection 2.3 are sensitiveto the choice of gauge. Integrating the equations of motion for an arbitrary lapse function,we find

∂rX = N(r)√ξ(X) e−Q(X)

ξ(X) = w(X) eQ(X)

(1− 2M

w(X)

).

(C.6)

The only difference between these expressions and the solution (2.4) and (2.5) is an overallfactor that appears in ∂rX. Notably, we obtain the same expression for ξ(X), concurrent

– 387 –

with its coordinate independent interpretation as square of the Killing norm. The changein the action due to a variation in the metric, with the bulk terms set to zero, is now givenby

δIreg = −12

∫dτ

∂rX

N(r)√ξ(r)

δξ(r) . (C.7)

The subscript is a reminder that the integrand is evaluated using a regulator. This resultcan be obtained either by evaluating the covariant expression (2.16) for the solution (C.6),or by expressing the action (1.1) in terms of ξ(r) and N(r) and then varying the fields.It is clear from (C.6) that the explicit factor of N(r) in (C.7) cancels against the factorin ∂rX, and we recover the result (2.20). Therefore, the conclusions of subsection 2.3 areindependent of the choice of gauge. This exercise can be repeated for the variation of theaction Γ from (2.30). Regardless of the choice of gauge, δΓ = 0 for all variations of thefields that preserve the boundary conditions.

D. Formulas

The variations of Xh and T can be expressed entirely in terms of dM , using the definitionsof the horizon (2.11) and T (2.13)

dXh =1

2πTdM dT =

12πT

∂T

∂XhdM . (D.1)

Similarly, the definitions of ξ (2.5), Q (2.6), and w (2.7) give

dξc = −2 eQc dM +(Uc ξc + eQc w′c

)dDc . (D.2)

When Dc is held fixed the variation of ξc reduces to

dξc

∣∣∣Dc

= −2 eQc dM . (D.3)

A useful consequence is∂ξc∂T

∣∣∣∣Dc

= −4πeQcw′hw′′h

. (D.4)

Holding the temperature Tc fixed implies a relationship between dM and dDc, which canbe obtained using the definition (2.14), and the relations (D.1) and (D.2):(

12π

∂T

∂Xh+ eQc T 2

c

)dM

∣∣∣Tc

=12T 2c

(Uc ξc + eQc w′c

)dDc

∣∣∣Tc. (D.5)

Using (3.5) and the formulas above we obtain

∂ψc∂Dc

∣∣∣∣Tc

= −(fc + gc

√ξc + hc

1√ξc

)(D.6)

with

f =e−Q/2

2√w

[w′′ − w′2

2w− (wU)′ +

12wU2

]g =

12e−Q

(U ′ − U2 + UN

)h = −1

2w′′ +

12w′N

(D.7)

– 388 –

where

N = −w′′h(2M − w)U − w′

w′h2 − 2w′′h(2M − w)

. (D.8)

The relations (D.4) and

∂ξc∂Tc

∣∣∣∣Dc

=∂ξc∂T

∣∣∣∣Dc

√ξc

(1− T

2ξc∂ξc∂T

∣∣∣∣Dc

)−1

(D.9)

allow to determine

∂ψc∂Tc

∣∣∣∣Dc

= −14ξ−3/2c

(w′c + ξcUce

−Qc) ∂ξc∂Tc

∣∣∣∣Dc

. (D.10)

Another useful formula is

Tc∂S

∂Tc

∣∣∣∣ψc

= −Tc∂ψc∂Tc

∣∣∣S

∂ψc∂S

∣∣∣Tc

. (D.11)

References

[1] E. Witten, “On string theory and black holes,” Phys. Rev. D44 (1991) 314–324.

[2] R. Dijkgraaf, H. Verlinde, and E. Verlinde, “String propagation in a black hole geometry,”Nucl. Phys. B371 (1992) 269–314.

[3] D. Grumiller, W. Kummer, and D. V. Vassilevich, “Dilaton gravity in two dimensions,”Phys. Rept. 369 (2002) 327–429, hep-th/0204253.

[4] J. W. York, Jr., “Role of conformal three geometry in the dynamics of gravitation,” Phys.Rev. Lett. 28 (1972) 1082–1085.

[5] G. W. Gibbons and S. W. Hawking, “Action integrals and partition functions in quantumgravity,” Phys. Rev. D15 (1977) 2752–2756.

[6] H. Liebl, D. V. Vassilevich, and S. Alexandrov, “Hawking radiation and masses in generalizeddilaton theories,” Class. Quant. Grav. 14 (1997) 889–904, arXiv:gr-qc/9605044.

[7] T. Regge and C. Teitelboim, “Role of surface integrals in the Hamiltonian formulation ofgeneral relativity,” Ann. Phys. 88 (1974) 286.

[8] R. B. Mann and D. Marolf, “Holographic renormalization of asymptotically flat spacetimes,”Class. Quant. Grav. 23 (2006) 2927–2950, hep-th/0511096.

[9] D. J. Gross, M. J. Perry, and L. G. Yaffe, “Instability of flat space at finite temperature,”Phys. Rev. D25 (1982) 330–355.

[10] J. L. Davis and R. McNees, “Boundary counterterms and the thermodynamics of 2-D blackholes,” JHEP 09 (2005) 072, hep-th/0411121.

[11] J. W. York, Jr., “Black hole thermodynamics and the Euclidean Einstein action,” Phys. Rev.D33 (1986) 2092–2099.

[12] G. W. Gibbons and M. J. Perry, “The Physics of 2-D stringy space-times,” Int. J. Mod.Phys. D1 (1992) 335–354, hep-th/9204090.

– 389 –

[13] V. P. Frolov, “Two-dimensional black hole physics,” Phys. Rev. D46 (1992) 5383–5394.

[14] T. Klosch and T. Strobl, “Classical and quantum gravity in (1+1)-dimensions. Part I: Aunifying approach,” Class. Quant. Grav. 13 (1996) 965–984, arXiv:gr-qc/9508020.

[15] M. Banados, C. Teitelboim, and J. Zanelli, “The black hole in three-dimensional space-time,”Phys. Rev. Lett. 69 (1992) 1849–1851, hep-th/9204099.

M. Banados, M. Henneaux, C. Teitelboim, and J. Zanelli, “Geometry of the (2+1) blackhole,” Phys. Rev. D48 (1993) 1506–1525, gr-qc/9302012.

[16] H.-J. Schmidt, “Scale invariant gravity in two-dimensions,” J. Math. Phys. 32 (1991)1562–1566.

[17] T. Banks and M. O’Loughlin, “Two-dimensional quantum gravity in Minkowski space,” Nucl.Phys. B362 (1991) 649–664.

[18] R. C. Tolman, Relativity, Thermodynamics, and Cosmology. Oxford University Press,Oxford, 1934.

[19] I. Papadimitriou and K. Skenderis, “Thermodynamics of asymptotically locally AdSspacetimes,” JHEP 08 (2005) 004, hep-th/0505190.

[20] R. McNees, “A new boundary counterterm for asymptotically AdS spacetimes,”hep-th/0512297.

[21] M. Henningson and K. Skenderis, “The holographic Weyl anomaly,” JHEP 07 (1998) 023,hep-th/9806087.

[22] V. Balasubramanian and P. Kraus, “A stress tensor for anti-de Sitter gravity,” Commun.Math. Phys. 208 (1999) 413–428, hep-th/9902121.

[23] R. Emparan, C. V. Johnson, and R. C. Myers, “Surface terms as counterterms in theAdS/CFT correspondence,” Phys. Rev. D60 (1999) 104001, hep-th/9903238.

[24] S. de Haro, S. N. Solodukhin, and K. Skenderis, “Holographic reconstruction of spacetimeand renormalization in the AdS/CFT correspondence,” Commun. Math. Phys. 217 (2001)595–622, hep-th/0002230.

[25] R. B. Mann, D. Marolf, and A. Virmani, “Covariant counterterms and conserved charges inasymptotically flat spacetimes,” Class. Quant. Grav. 23 (2006) 6357–6378, gr-qc/0607041.

[26] T. H. Buscher, “A symmetry of the string background field equations,” Phys. Lett. B194(1987) 59; “Path integral derivation of quantum duality in nonlinear sigma models,” Phys.Lett. B201 (1988) 466.

[27] C. R. Nappi and A. Pasquinucci, “Thermodynamics of two-dimensional black holes,” Mod.Phys. Lett. A7 (1992) 3337–3346, gr-qc/9208002.

[28] J. Gegenberg, G. Kunstatter, and D. Louis-Martinez, “Observables for two-dimensional blackholes,” Phys. Rev. D51 (1995) 1781–1786, gr-qc/9408015.

[29] D. Grumiller and R. Meyer, “Ramifications of lineland,” Turk. J. Phys. 30 (2006) 349,hep-th/0604049.

[30] J. D. Brown and J. W. York, Jr., “Quasilocal energy in general relativity,” gr-qc/9209012;“Quasilocal energy and conserved charges derived from the gravitational action,” Phys. Rev.D47 (1993) 1407–1419.

– 390 –

[31] G. W. Gibbons, M. J. Perry, and C. N. Pope, “The first law of thermodynamics for kerr -anti-de sitter black holes,” Class. Quant. Grav. 22 (2005) 1503–1526, hep-th/0408217.

[32] G. Ruppeiner, “Riemannian geometry in thermodynamic fluctuation theory,” Rev. Mod.Phys. 67 (1995) 605–659; Erratum-ibid. 68 (1996) 313.

[33] G. Arcioni and E. Lozano-Tellechea, “Stability and critical phenomena of black holes andblack rings,” Phys. Rev. D72 (2005) 104021, hep-th/0412118.

J.-Y. Shen, R.-G. Cai, B. Wang, and R.-K. Su, “Thermodynamic geometry and criticalbehavior of black holes,” Int. J. Mod. Phys. A22 (2007) 11–27, gr-qc/0512035.

J. E. Aman, I. Bengtsson, and N. Pidokrajt, “Flat information geometries in black holethermodynamics,” Gen. Rel. Grav. 38 (2006) 1305–1315, gr-qc/0601119.

T. Sarkar, G. Sengupta, and B. Nath Tiwari, “On the thermodynamic geometry of BTZblack holes,” JHEP 11 (2006) 015, hep-th/0606084.

[34] J. E. Aman, J. Bedford, D. Grumiller, N. Pidokrajt, and J. Ward, “Ruppeiner theory ofblack hole thermodynamics,” gr-qc/0611119.

[35] D. Birmingham, M. Blau, M. Rakowski, and G. Thompson, “Topological field theory,” Phys.Rept. 209 (1991) 129–340.

[36] S. W. Hawking and D. N. Page, “Thermodynamics of black holes in anti-de Sitter space,”Commun. Math. Phys. 87 (1983) 577.

[37] L. Bergamin, D. Grumiller, and W. Kummer, “Supersymmetric black holes in 2d dilatonsupergravity: baldness and extremality,” J. Phys. A37 (2004) 3881–3901, hep-th/0310006.

[38] M. O. Katanaev, W. Kummer, and H. Liebl, “On the completeness of the black holesingularity in 2d dilaton theories,” Nucl. Phys. B486 (1997) 353–370, gr-qc/9602040.

[39] R. Jackiw and C. Teitelboim in Quantum Theory Of Gravity, S. Christensen, ed. AdamHilger, Bristol, 1984.

[40] S. Elitzur, A. Forge, and E. Rabinovici, “Some global aspects of string compactifications,”Nucl. Phys. B359 (1991) 581–610.

G. Mandal, A. M. Sengupta, and S. R. Wadia, “Classical solutions of two-dimensional stringtheory,” Mod. Phys. Lett. A6 (1991) 1685–1692.

[41] D. Grumiller, “An action for the exact string black hole,” JHEP 05 (2005) 028,hep-th/0501208.

[42] V. A. Kazakov and A. A. Tseytlin, “On free energy of 2-d black hole in bosonic stringtheory,” JHEP 06 (2001) 021, hep-th/0104138.

[43] D. Grumiller, “Logarithmic corrections to the entropy of the exact string black hole,” in PathIntegrals from Quantum Information to Cosmology, C. Burdik, N. Navratil, and S. Posta,eds. JINR Publishing Department, Prague, June, 2005. hep-th/0506175.

[44] C. Ford, G. Jorjadze, and G. Weigt, “Integration of the SL(2,R)/U(1) gauged WZNW theoryby reduction and quantum parafermions,” Theor. Math. Phys. 128 (2001) 1046–1060,hep-th/0003246.

C. Kruger, “Exact operator quantization of the Euclidean black hole CFT,” hep-th/0411275.

[45] D. Grumiller, W. Kummer, and D. V. Vassilevich, “Positive specific heat of the quantumcorrected dilaton black hole,” JHEP 07 (2003) 009, hep-th/0305036.

– 391 –

[46] P. Kraus, F. Larsen, and R. Siebelink, “The gravitational action in asymptotically AdS andflat spacetimes,” Nucl. Phys. B563 (1999) 259–278, hep-th/9906127.

[47] J. D. Brown et al., “Thermodynamic ensembles and gravitation,” Class. Quant. Grav. 7(1990) 1433–1444.

J. D. Brown and J. W. York, Jr., “The path integral formulation of gravitationalthermodynamics,” gr-qc/9405024.

J. D. Brown, S. R. Lau, and J. W. York, “Action and energy of the gravitational field,”gr-qc/0010024.

[48] J. D. Brown, J. Creighton, and R. B. Mann, “Temperature, energy and heat capacity ofasymptotically anti-de Sitter black holes,” Phys. Rev. D50 (1994) 6394–6403,gr-qc/9405007.

[49] M. M. Akbar, “Schwarzschild-anti de Sitter within an isothermal cavity: Thermodynamics,phase transitions and the Dirichlet problem,” hep-th/0401228.

[50] S. Hollands, A. Ishibashi, and D. Marolf, “Comparison between various notions of conservedcharges in asymptotically AdS-spacetimes,” Class. Quant. Grav. 22 (2005) 2881–2920,hep-th/0503045.

[51] A. Achucarro and M. E. Ortiz, “Relating black holes in two-dimensions and three-dimensions,” Phys. Rev. D48 (1993) 3600–3605, hep-th/9304068.

[52] S. Carlip, “What we don’t know about BTZ black hole entropy,” Class. Quant. Grav. 15(1998) 3609–3625, hep-th/9806026.

[53] O. B. Zaslavsky, “Geometry of nonextreme black holes near the extreme state,” Phys. Rev.D56 (1997) 2188–2191, gr-qc/9707015; Erratum-ibid. D59 (1999) 069901.

A. J. M. Medved and G. Kunstatter, “One-loop corrected thermodynamics of the extremaland non- extremal spinning BTZ black hole,” Phys. Rev. D63 (2001) 104005,hep-th/0009050.

J. M. Izquierdo, J. Navarro-Salas, and P. Navarro, “Kaluza-Klein theory, AdS/CFTcorrespondence and black hole entropy,” Class. Quant. Grav. 19 (2002) 563–570,hep-th/0107132.

[54] S. Deser, R. Jackiw, and S. Templeton, “Three-dimensional massive gauge theories,” Phys.Rev. Lett. 48 (1982) 975–978; “Topologically massive gauge theories,” Ann. Phys. 281 (2000)409–449; Erratum-ibid. 185 (1988) 406.

[55] G. Guralnik, A. Iorio, R. Jackiw, and S. Y. Pi, “Dimensionally reduced gravitationalChern-Simons term and its kink,” Ann. Phys. 308 (2003) 222–236, hep-th/0305117.

[56] D. Grumiller and W. Kummer, “The classical solutions of the dimensionally reducedgravitational Chern-Simons theory,” Ann. Phys. 308 (2003) 211–221, hep-th/0306036.

[57] B. Sahoo and A. Sen, “BTZ black hole with Chern-Simons and higher derivative terms,”JHEP 07 (2006) 008, hep-th/0601228.

[58] R. M. Wald, “Black hole entropy is the Nother charge,” Phys. Rev. D48 (1993) 3427–3431,gr-qc/9307038.

[59] D. Grumiller and R. Jackiw, “Duality in 2-dimensional dilaton gravity,” Phys. Lett. B642(2006) 530, hep-th/0609197.

– 392 –

[60] W. Kummer, H. Liebl, and D. V. Vassilevich, “Exact path integral quantization of generic2-d dilaton gravity,” Nucl. Phys. B493 (1997) 491–502, gr-qc/9612012.

L. Bergamin, D. Grumiller, W. Kummer, and D. V. Vassilevich, “Classical and quantumintegrability of 2D dilaton gravities in Euclidean space,” Class. Quant. Grav. 22 (2005)1361–1382, hep-th/0412007.

[61] J. de Boer, E. P. Verlinde, and H. L. Verlinde, “On the holographic renormalization group,”JHEP 08 (2000) 003, hep-th/9912012.

J. de Boer, “The holographic renormalization group,” Fortsch. Phys. 49 (2001) 339–358,hep-th/0101026.

[62] D. Martelli and W. Muck, “Holographic renormalization and Ward identities with theHamilton-Jacobi method,” Nucl. Phys. B654 (2003) 248–276, hep-th/0205061.

[63] F. Larsen and R. McNees, “Holography, diffeomorphisms, and scaling violations in theCMB,” JHEP 07 (2004) 062, hep-th/0402050.

[64] A. Batrachenko, J. T. Liu, R. McNees, W. A. Sabra, and W. Y. Wen, “Black hole mass andHamilton-Jacobi counterterms,” JHEP 05 (2005) 034, hep-th/0408205.

[65] K. Schwarzschild, “On the gravitational field of a mass point according to Einstein’s theory,”Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) (1916) 189–196,arXiv:physics/9905030.

[66] C. G. Callan, Jr., S. B. Giddings, J. A. Harvey, and A. Strominger, “Evanescent black holes,”Phys. Rev. D45 (1992) 1005–1009, hep-th/9111056.

[67] J. P. S. Lemos and P. M. Sa, “The black holes of a general two-dimensional dilaton gravitytheory,” Phys. Rev. D49 (1994) 2897–2908, arXiv:gr-qc/9311008.

[68] A. Fabbri and J. G. Russo, “Soluble models in 2d dilaton gravity,” Phys. Rev. D53 (1996)6995–7002, hep-th/9510109.

[69] D. Grumiller, “Long time black hole evaporation with bounded Hawking flux,” JCAP 05(2004) 005, gr-qc/0307005.

[70] Y. Nakayama, “Liouville field theory: A decade after the revolution,” Int. J. Mod. Phys. A19(2004) 2771–2930, hep-th/0402009.

[71] M. O. Katanaev and I. V. Volovich, “String model with dynamical geometry and torsion,”Phys. Lett. B175 (1986) 413–416.

[72] H. Reissner, “Uber die Eigengravitation des elektrischen Feldes nach der EinsteinschenTheorie,” Ann. Phys. 50 (1916) 106.

G. Nordstrom, “On the energy of the gravitation field in Einstein’s theory,” Proc. Kon. Ned.Akad. Wet. 20 (1916) 1238.

[73] N. Berkovits, S. Gukov, and B. C. Vallilo, “Superstrings in 2D backgrounds with R-R fluxand new extremal black holes,” Nucl. Phys. B614 (2001) 195–232, hep-th/0107140.

– 393 –

Part III

Papers on 3D Gravity

Preprint typeset in JHEP style - HYPER VERSION MIT-CTP 3949

UUITP-08/08

Instability in cosmological topologically

massive gravity at the chiral point

Daniel Grumiller

Center for Theoretical Physics, Massachusetts Institute of Technology,

77 Massachusetts Ave., Cambridge, MA 02139

Email: grumil@lns.mit.edu

Niklas Johansson

Institutionen for Fysik och Astronomi, Uppsala Universitet

Box 803, S-751 08 Uppsala, Sweden

Email: Niklas.Johansson@fysast.uu.se

Abstract: We consider cosmological topologically massive gravity at the chiral

point with positive sign of the Einstein–Hilbert term. We demonstrate the presence

of a negative energy bulk mode that grows linearly in time. Unless there are physical

reasons to discard this mode, this theory is unstable. To address this issue we prove

that the mode is not pure gauge and that its negative energy is time-independent

and finite. The isometry generators L0 and L0 have non-unitary matrix represen-

tations like in logarithmic CFT. While the new mode obeys boundary conditions

that are slightly weaker than the ones by Brown and Henneaux, its fall-off behavior

is compatible with spacetime being asymptotically AdS3. We employ holographic

renormalization to show that the variational principle is well-defined. The corre-

sponding Brown–York stress tensor is finite, traceless and conserved. Finally we

address possibilities to eliminate the instability and prospects for chiral gravity.

Keywords: Cosmological topologically massive gravity, chiral gravity, gravity in

three dimensions, logarithmic CFT, holographic renormalization, AdS/CFT.

Contents

1. Introduction 398

2. CTMG and CCTMG 400

3. Logarithmic mode with negative energy 403

4. Variational principle and boundary stress tensor 405

5. Conclusions 408

A. Fefferman-Graham expansion 410

1. Introduction

Gravity in three dimensions is simple enough to be studied in great depth and com-

plicated enough to make such studies interesting. Pure Einstein–Hilbert gravity

exhibits no propagating physical degrees of freedom [1–3]. If the theory is deformed

by a negative cosmological constant it has black hole solutions [4]. Another possi-

ble deformation is to add a gravitational Chern–Simons term. The resulting theory

is called topologically massive gravity (TMG) and, remarkably, contains a massive

graviton [5]. Including both terms yields cosmological topologically massive grav-

ity [6] (CTMG), a theory that exhibits both gravitons and black holes.

Recently, Li, Song and Strominger [7] considered CTMG with the following ac-

tion

ICTMG =1

16πG

∫

M

d3x√−g

[

R +2

ℓ2+

1

2µελµνΓρ

λσ

(

∂µΓσνρ +

2

3Γσ

µτΓτνρ

)

]

, (1.1)

where the negative cosmological constant is parameterized by Λ = −1/ℓ2. Notably,

the sign of the Einstein–Hilbert action in (1.1) differs from the choice in [5] that

is required to make the graviton energy positive. The chosen sign in (1.1) has the

advantage of making the BTZ black hole energy positive in the limit of large µ, which

is not the case otherwise.

– 398 –

Exploiting the properties of the underlying SL(2,R)L × SL(2,R)R isometry al-

gebra, [7] argued that the would-be-negative energy of the massive graviton mode

actually is zero if the constants µ and ℓ satisfy the chiral condition1

µℓ = 1 . (1.2)

Thus, the sign choice in (1.1) would be admissible as long as (1.2) holds. For this

tuning the massive graviton mode ψM(µℓ) becomes identical to a mode that exists

already in cosmological Einstein gravity. This ‘left-moving’ mode ψL is not a physical

bulk degree of freedom, and thus the theory appears to lose one physical degree of

freedom at the chiral point.

A recent work by Carlip, Deser, Waldron and Wise disputes the claim that no

negative energy bulk mode arises for cosmological topologically massive gravity at

the chiral point (CCTMG) [8]: They find no loss of degree of freedom at the chiral

point. The approach of Carlip et al. is quite different though, which makes a direct

comparison cumbersome.

We clarify here this discrepancy by constructing a negative energy bulk mode

that was not considered in [7], employing their approach. The reason for its existence

is the very reason why CCTMG seemingly loses a degree of freedom: When two

linearly independent solutions to a differential equation degenerate, a logarithmic

solution appears. In the present case, the wave function ψM of the massive mode

degenerates with the left-moving mode ψL. Therefore, a new solution appears, whose

wave-function is given by

ψnew = limµℓ→1

ψM(µℓ) − ψL

µℓ− 1. (1.3)

In this work we study this mode and reveal several intriguing features. In particular it

grows linearly in time and the radial coordinate of AdS3. We compute its energy and

show that it is finite, negative and time-independent. We also demonstrate that the

variational principle is well-defined, including boundary issues. The new mode (1.3)

turns out not to contribute to the boundary stress tensor, which is finite, traceless

and conserved. To achieve these results we have to extend the analysis of Kraus and

Larsen [9], who dropped a term in the Fefferman–Graham expansion that becomes

relevant here. Furthermore, we demonstrate that the L0 and L0 isometry generators

have matrix representations identical to those in logarithmic CFT (LCFT), and

therefore the theory is not unitary.

1The point µℓ = 1 is special because one of the central charges of the dual boundary CFT

vanishes, cL = 0, cR 6= 0, and the mass M and angular momentum J of the BTZ black hole

solutions satisfy J = Mℓ. In [7] the theory (1.1) with (1.2) was dubbed “chiral gravity”, assuming

that all solutions obey the Brown–Henneaux boundary conditions [10]. We slightly relax the latter

assumption in our discussion, so to avoid confusion we stick to the name “cosmological topologically

massive gravity at the chiral point” and abbreviate it by CCTMG, where the first C stands for

“chiral”.

– 399 –

With well-defined variational principle and finite energy, we see no reason to

dismiss this mode a priori. Its negative energy renders CCTMG unstable, con-

current with [8]. We find it noteworthy, however, that the destabilizing mode of

CCTMG has characteristics quite different from the corresponding modes for gen-

eral µℓ. For instance, the new mode does not obey the original Brown–Henneaux

boundary conditions [10] (for a very recent treatment of CTMG imposing Brown–

Henneaux boundary conditions, see [11]), but a slightly weaker version thereof that

is still consistent with spacetime being asymptotically AdS3. Namely, our Fefferman-

Graham expansion for the metric in Gaussian coordinates is of the form

ds2 = ℓ2 dρ2 +(

e2ρ γ(0)ij + ρ γ

(1)ij + γ

(2)ij + . . .

)

dxidxj , (1.4)

which reduces to the Brown–Henneaux case for vanishing γ(1) only. Moreover, the

new mode is not periodic in time and therefore does not contribute to a finite tem-

perature partition function. This could mean that it is nevertheless possible to make

sense of CTMG exactly at the chiral point, as conjectured by Li, Song and Stro-

minger [7]. This would have to involve a consistent truncation of the new mode. We

shall argue in the Conclusions that even without such a truncation CCTMG and its

related LCFT provide interesting subjects for further studies.

This paper is organized as follows. We begin in Section 2 by recalling basic

features of CTMG and CCTMG. We construct the new physical mode and calculate

its energy in Section 3. In Section 4 we show that this mode is a valid classical

solution (including boundary issues) and we calculate the boundary stress tensor.

We conclude with a brief summary and discussion of future prospects for CCTMG

and chiral gravity in Section 5.

Before starting, we mention some of our conventions. We set 16πG = 1 and oth-

erwise use the same conventions for signature and sign definitions2 as in [7], including

Riemann tensor Rµνσλ = ∂σΓµ

νλ + . . . , Ricci tensor Rµν = Rλµλν and epsilon symbol

ǫ012 = ǫ01 = +1. The epsilon-tensor is denoted by ελµν = ǫλµν/√−g. For sake of

specificity we consider exclusively ℓ > 0. We use Greek indices for 3-dimensional

tensors and Latin indices for 2-dimensional ones. For adapted coordinates we take

x0 = τ , x1 = φ and x2 = ρ. Our conventions for boundary quantities and the

Fefferman–Graham expansion are summarized in Appendix A.

2. CTMG and CCTMG

In this section we review the powerful formulation of linearized CTMG developed

in [7]. We put particular emphasis on the behavior at the chiral point µℓ = 1.

2The Chern-Simons term in (1.1) has a sign different from [7], thus correcting a typo in that

work.

– 400 –

The background metric gµν is that of global AdS3,

ds2 = gµν dxµdxν = ℓ2

(

− cosh2ρ dτ 2 + sinh2ρ dφ2 + dρ2)

, (2.1)

whose isometry group is SL(2,R)L×SL(2,R)R. In light-cone coordinates u = τ +φ,

v = τ − φ the SL(2,R)L generators read

L0 = i∂u (2.2)

L−1 = ie−iu[cosh 2ρ

sinh 2ρ∂u − 1

sinh 2ρ∂v +

i

2∂ρ

]

(2.3)

L1 = ieiu[cosh 2ρ

sinh 2ρ∂u − 1

sinh 2ρ∂v −

i

2∂ρ

]

(2.4)

with algebra[

L0, L±1

]

= ∓L±1 ,[

L1, L−1

]

= 2L0 (2.5)

and quadratic Casimir

L2 =1

2

(

L1L−1 + L−1L1

)

− L20 . (2.6)

The SL(2,R)R generators L0, L−1, L1 satisfy the same algebra and are given by

(2.2)-(2.4) with u↔ v and L↔ L.

The full non-linear equations of motion of CTMG read

Gµν +1

µCµν = 0 , (2.7)

where

Gµν = Rµν −1

2gµνR− 1

ℓ2gµν (2.8)

is the Einstein tensor (including cosmological constant) and

Cµν =1

2εµ

αβ ∇αRβν + (µ ↔ ν) (2.9)

is essentially the Cotton tensor. To look for perturbative solutions to (2.7), we write

the metric as the sum of the AdS3 background (2.1) and fluctuations hµν .

gµν = gµν + hµν . (2.10)

Expanding in hµν produces the linearized equations of motion

Glinµν +

1

µC lin

µν = 0 , (2.11)

where

Glinµν = Rlin

µν −1

2gµνR

lin +2

ℓ2hµν (2.12)

– 401 –

and

C linµν =

1

2εµ

αβ∇αGlinβν + (µ↔ ν) (2.13)

are the linear versions of the Einstein and Cotton tensors, respectively. Expressions

for the linearized Ricci tensor Rlinµν and Ricci scalar Rlin can be found in [12].

By choosing the transverse and traceless gauge

∇µhµν = 0 , gµνhµν = 0 (2.14)

the linearized equations of motion (2.11) take the form

(

DRDLDMh)

µν= 0 . (2.15)

The mutually commuting differential operators DL/R/M are given by

(DL/R)µν = δν

µ ± ℓεµαν ∇α , (DM)µ

ν = δνµ +

1

µεµ

αν ∇α . (2.16)

Notice that for CCTMG DM = DL, and that the equations of motion for this case

read(

DRDLDLh)

µν= 0 . (2.17)

For generic values of µ and ℓ the three linearly independent solutions to (2.15) can

be taken to satisfy

(

DLhL)

µν= 0 ,

(

DRhR)

µν= 0 ,

(

DMhM)

µν= 0 . (2.18)

These branches of solutions are referred to as left-moving, right-moving and massive

gravitons, respectively. Solely the latter entails physical bulk degrees of freedom.

The basis of solutions (2.18) becomes inadequate at the chiral point µℓ = 1, since,

at that point, the L and M branches coincide. In the next Section we remedy this

deficiency by explicitly constructing a mode hnewµν satisfying3

(

DLDLhnew)

µν= 0 ,

(

DLhnew)

µν6= 0 . (2.19)

Using the SL(2,R) algebra, [7] finds all solutions to (2.18). These sets of solutions

consist of primaries satisfying L1ψ = L1ψ = 0, and descendants obtained by acting

with L−1 and L−1. The explicit form of the wave functions for the massive and

left-moving primaries will be of importance to us, so we recall them here.

ψMµν = e−(3/2+µℓ/2)iu−(−1/2+µℓ/2)iv sinh2ρ

(cosh ρ)1+µℓ

1 1 ia

1 1 ia

ia ia −a2

µν

(2.20)

3We are grateful to Roman Jackiw for suggesting to perform such a construction.

– 402 –

where

a :=1

sinh ρ cosh ρ(2.21)

The left-mover ψLµν is obtained from (2.20) by setting µℓ = 1. The real and imaginary

parts of ψµν separately solve the equations of motion. We take

hµν = ℜψµν . (2.22)

This concludes our recapitulation of CTMG.

3. Logarithmic mode with negative energy

In this Section we construct and discuss the new mode of CCTMG. Using the explicit

form (2.20) of ψM it is straight-forward to perform the standard construction:

ψnewµν := lim

µℓ→1

ψMµν(µℓ) − ψL

µν

µℓ− 1= y(τ, ρ)ψL

µν (3.1)

where we define the function y by

y(τ, ρ) := −iτ − ln cosh ρ . (3.2)

When analyzing the asymptotics of the new mode it will be convenient to have an

explicit expression for hnewµν . Using (2.20)-(2.22) we obtain

hnewµν =

sinh ρ

cosh3ρ

(

cos (2u) τ − sin (2u) ln cosh ρ)

0 0 1

0 0 1

1 1 0

µν

− tanh2ρ(

sin (2u) τ + cos (2u) ln cosh ρ)

1 1 0

1 1 0

0 0 − sinh−2ρ cosh−2ρ

µν

. (3.3)

We see that the new mode grows linearly in time, and also (asymptotically) in the

radial coordinate ρ.

To show that ψnewµν solves the bulk equations of motion, let us determine the

action of the isometry algebra. Acting on y we obtain

L0 y = L0 y =1

2, L1 y = L1 y = 0 . (3.4)

Correspondingly, on ψnew the action is

L0 ψnewµν = 2ψnew

µν +1

2ψL

µν , L0 ψnewµν =

1

2ψL

µν , L1 ψnew = L1 ψ

new = 0 . (3.5)

– 403 –

Note that ψnew is not an eigenstate of L0 or L0, but only of L0 − L0. Because of

the relations (3.5) it is impossible to decompose ψnew as a linear combination of

eigenstates to L0 and L0. The representation of L0 and L0 as matrices,

L0

(

ψnew

ψL

)

=

(

2 12

0 2

) (

ψnew

ψL

)

, L0

(

ψnew

ψL

)

=

(

0 12

0 0

) (

ψnew

ψL

)

, (3.6)

shows that their Jordan normal form is the same as in LCFT [13]. In the parlance

of LCFT literature ψnew is the logarithmic partner of ψL. (For reviews see [14, 15];

for some applications to AdS/LCFT see [16–20].4)

From the equations (3.5) we deduce

(DRDLψnew)µν = −ℓ2(

∇2 +2

ℓ2)

ψnewµν = 2

(

L2 + L2 + 2)

ψnewµν = −2ψL

µν (3.7)

and consequently(

DLDRDLψnew)

µν= 0 . (3.8)

The identity (3.8) shows that ψnew solves the classical equations of motion. Acting

on ψnew with L−1 and L−1 produces a tower of descendants.

As expected on general grounds, the new mode ψnew is indeed a physical mode

and not just pure gauge. To prove this it is sufficient to demonstrate that there is

no gauge preserving coordinate transformation ξµ that annihilates ψnewµν ,

∇(µξν) + ψnewµν = 0 . (3.9)

The quickest way to show that (3.9) has no solution for ξµ is as follows: for any ξµpreserving the gauge conditions ∇(µξν) solves the linearized Einstein equations,5 while

ψnewµν does not. We also mention that despite of the linear divergence of ψnew in the

radial coordinate ρ the linearized approximation does not break down asymptotically,

i.e., (3.3) really is a small perturbation of the AdS3 background.

Let us now compute the energy of the new mode. We do this by the procedure

described in [7, 21]. The Hamiltonian is given by

H =

∫

dx2(

hµνΠ(1)µν + (∇0hµν)Π

(2)µν − L)

, (3.10)

where L is the Lagrange density expanded to quadratic order in h, and the canonical

momenta Π(1)µν and Π(2)µν are given by

Π(1)µν = −√−g

4

(

∇0(2hµν + ℓ εµαβ∇αh

βν) − ℓ εβ0µ(∇2 +

2

ℓ2)hβν

)

(3.11)

4The relation to LCFTs was pointed out by John McGreevy during a talk by Andy Strominger

at MIT. We thank John McGreevy for discussions on LCFTs.5We thank Wei Li and Wei Song for providing this argument.

– 404 –

Π(2)µν = −√−g g00

4ℓ εβ

λµ∇λhβν . (3.12)

It is slightly lengthy, but straightforward to evaluate (3.10) on the solution (3.3). By

virtue of the on-shell relations (3.7) and L = 0 the on-shell Hamiltonian reduces to

H∣

∣

EOM=

1

2

∫

d2x√−g

[

(

(∇0hnewµν )(hµν

new + ℓ εµαβ∇αh

βνnew) +

1

ℓhnew

µν ε0µβ h

βνL

)

− g00∂0

(

hnewµν (hµν

new +ℓ

2εµα

β∇αhβνnew)

)

]

=: Enew (3.13)

Note the appearance of both hnew and hL in the integrand. Evaluating the integral

(3.13) leads to the result (with 16πG reinserted)

Enew =2π

16πG ℓ3

∞∫

1

dx( 8

x9log x− 9

2x9− 2

x7log x+

1

x7

)

= − 47

1152Gℓ3. (3.14)

We see explicitly that the energy is finite, negative and time-independent. While the

finiteness of (3.14) may seem surprising considering that hnew diverges, we recall that

it is not unusual for a mode to grow linear in time and still have time independent

finite energy. Comparable precedents are free motion in Newtonian mechanics and

static spherically symmetric solutions of the Einstein-massless-Klein-Gordon model

with a scalar field that grows linearly in time [22].

We conclude that the new mode (3.3) for CCTMG cannot be dismissed on phys-

ical grounds, since it is not merely pure gauge and its energy remains bounded.

Moreover, its energy is negative and thus CCTMG is unstable. The boundary issues

considered in the next Section do not alter this conclusion.

4. Variational principle and boundary stress tensor

We pose now the relevant question whether the new mode (3.3) is actually a classical

solution of CCTMG. To this end not only the bulk equations of motion (2.17) must

hold, as they do indeed, but also all boundary terms must be canceled so that the

first variation of the on-shell action

δICG

∣

∣

EOM= −

∫

∂M

d2x√−γ

(

Kij −(

K − 1

ℓ

)

γij)

δγij

+ ℓ

∫

∂M

d2x ǫij(

− Rkρjρ δγik +Ki

k δKkj −1

2Γk

li δΓlkj

)

(4.1)

vanishes for all variations preserving the boundary conditions. While answering this

question in the affirmative, we shall obtain as a byproduct the result for the boundary

stress tensor T ij, which follows also from the variation of the on-shell action

δICCTMG

∣

∣

EOM=

1

2

∫

∂M

d2x√

−γ(0) T ij δγ(0)ij . (4.2)

– 405 –

Here γ(0) is the metric on the conformal boundary as defined in Appendix A. In order

to proceed we must supplement the bulk action (1.1) with appropriate boundary

terms.

CCTMG requires two kinds of boundary terms, as most other gravitational theo-

ries do: a Gibbons–Hawking–York boundary term for making the Dirichlet boundary

value problem well-defined, and a boundary counterterm for making the variational

principle well-defined. It was shown by Kraus and Larsen [9] (for related considera-

tions see also [23]) that the fully supplemented CCTMG action is given by

ICCTMG =

∫

M

d3x√−g

(

R +2

ℓ2

)

+ 2

∫

∂M

d2x√−γ

(

K − 1

ℓ

)

+ℓ

2

∫

M

d3x√−g ελµνΓρ

λσ

(

∂µΓσνρ +

2

3Γσ

µτΓτνρ

)

(4.3)

Its first variation leads to (4.1) above. Remarkably, the boundary terms are just the

ones that are present already in cosmological Einstein-Hilbert gravity, i.e., the terms

in the first line of (4.3). However, the result (4.3) was derived assuming a restricted

Fefferman-Graham expansion of the boundary metric, i.e., one that does not involve

the term linear in ρ in (4.4) below. This is not sufficient to encompass the new mode

described in Section 3. Rather, we get the expansion announced in (1.4), viz.

γij = e2ργ(0)ij + ρ γ

(1)ij + γ

(2)ij + . . . (4.4)

for the boundary metric, which coincides with [9] for γ(1)ij = 0 only. Here γ

(0)ij is the

conformal metric at the boundary, γ(1)ij describes the linearly growing contribution

and γ(2)ij the constant contribution.

Let us comment briefly on the linear term in (4.4). Such a term is always

present in pure gravity for odd-dimensional AdS spacetimes with dimension D ≥ 5.

In D = 3 the coefficient in front of this term is set to zero by the Einstein equations

[24], and it is not included in the boundary conditions of Brown and Henneaux

[10]. However, it is also well-known that violations of the original Brown–Henneaux

boundary conditions can arise even in three dimensions if gravity couples to matter

[25], and that the linear term in (4.4) does not spoil the property of spacetime being

asymptotically AdS [26]. Interestingly, the coupling to a Chern-Simons term leads

to such a linear term, as we demonstrate here explicitly.

To identify the coefficients γ(i), we recall that the full metric is given by (2.10),

where gµν is the background metric (2.1) and hµν = hnewµν is the new mode (3.3). The

boundary metric

γ(0)ij =

ℓ2

4

(

−1 0

0 1

)

ij

(4.5)

– 406 –

is (trivially) conformal to the Minkowski metric, the linearly growing contribution

reads

γ(1)ij = − cos (2u)

(

1 1

1 1

)

ij

(4.6)

and the constant contribution is given by

γ(2)ij = −

(

sin (2u) τ − cos (2u) ln 2)

(

1 1

1 1

)

ij

− ℓ2

2

(

1 0

0 1

)

ij

. (4.7)

The first term in (4.7) comes from hnew and the second one from the next-to-leading

order term of the AdS3 background. As explained in Appendix A we use (4.4) to

expand relevant quantities like extrinsic curvature for large ρ.

Variations that preserve the boundary conditions are those where δγ(0) vanishes,

but δγ(1) and δγ(2) may be finite. Thus, a well-defined variational principle requires

that only δγ(0) remains in (4.1) after taking the limit ρ→ ∞. We have checked that

all the terms appearing in (4.1) indeed contain exclusively δγ(0)-terms, see (A.11)-

(A.16) in Appendix A. Therefore, we have generalized the conclusions of Kraus and

Larsen that CTMG has a well-defined variational principle to the case where the

Fefferman-Graham expansion (4.4) has a non-vanishing contribution from (4.6).

From the result (A.16) in Appendix A we can now read off the boundary stress

tensor as defined in (4.2).

T ij = limρ→∞

1

ℓ

[

ρ(

γij(1)−γil

(1)γ(0)lk ε

kj)

− 1

2

(

γij(1)−3γil

(1)γ(0)lk ε

kj)

+γij(2)−γil

(2)γ(0)lk ε

kj]

+(i↔ j)

(4.8)

For vanishing γ(1) this coincides with the result6 (5.14) of [9] if we take into account

the tracelessness of γ(2). For non-vanishing γ(1) apparently the boundary stress tensor

(4.8) diverges. However, with (4.6) we see that the expression

γij(1) − γil

(1)γ(0)lk ε

kj = 0 (4.9)

actually vanishes identically. Therefore, the linear divergence in ρ is not present in

the boundary stress tensor for the mode (3.3). The equations (4.5)-(4.9) establish

our result for the boundary stress tensor (with 16πG reinserted)

T ij = − 1

πG ℓ3

(

1 1

1 1

)ij

− 2

πG ℓ5cos (2u)

(

1 −1

−1 1

)ij

. (4.10)

The AdS stress tensor is interpreted as the Casimir energy of the dual field theory

[27,28]. The boundary stress tensor (4.10) is finite, traceless and conserved. Except

for the crucial first property of finiteness, these features might have been anticipated

on general grounds. The finiteness confirms our conclusion of the previous Section:

The new mode (3.3) cannot be dismissed on physical grounds.

6We note that in [9] there is a sign change between appendix and body of the paper.

– 407 –

5. Conclusions

To summarize, we have investigated CCTMG (1.1), (1.2) at the linearized level along

the lines of [7] and found a new mode (3.3), concurrent with the analysis of [8]. We

checked that this mode is physical, i.e., not pure gauge, and that it has finite, time-

independent negative energy (3.14). We showed also that this mode is a valid classical

solution in the sense that the variational principle is well-defined. Furthermore we

demonstrated that it has a Fefferman-Graham expansion (4.4) and therefore does not

spoil the property of spacetime being asymptotically AdS3. Thus, we may conclude

that CCTMG is unstable, because the new mode is physically acceptable, but has

negative energy. As a byproduct we calculated the boundary stress tensor (4.10)

and found that it is finite, traceless and conserved. By analyzing the action of the

isometry algebra on the new mode, we concluded that a dual CFT describing this

mode must be a logarithmic CFT and therefore is not unitary.

While the analysis in the current work used the linearized approximation, the

new mode is present also non-perturbatively. This can be checked easily by a canon-

ical analysis, which reveals that nothing special happens with the dimension of the

physical phase space as the chiral point (1.2) is approached.7

It is conceivable that nonperturbative effects stabilize CCTMG, i.e., that the

instability is an artifact of perturbation theory, but we have found no evidence for

this suggestion. Since very few exact solutions of CTMG are known [32–37] and

because the new mode (3.3) exhibits two commuting Killing vectors, a reasonable

strategy to find relevant nonperturbative solutions would be the consideration of

exact solutions with two commuting Killing vectors. To this end a 2-dimensional

dilaton gravity [38] approach extending the analysis of [39, 40] could be helpful (see

also [41]). We also recall that the gravitational modes have positive energy — not

just for CCTMG, but generically — if the sign of the Einstein-Hilbert term in (1.1)

is reversed. This sign change, however, leads to negative energy for BTZ black hole

solutions8 as emphasized in [7].

CCTMG can exist as a meaningful theory, which one might call chiral gravity,

if the new mode is absent. Thus, it is of interest to point out applications where the

new mode is eliminated. If one imposes boundary conditions that are stricter than

required by the variational procedure then the new mode can be discarded. This

is the case if one imposes the original Brown–Henneaux boundary conditions [10].

However, we reiterate that the expansion (4.4) is consistent with spacetime being

7We thank Steve Carlip for conveying this information to us. We have convinced ourselves

independently that this statement is true, but we do not include the corresponding analysis here.

We just mention that a simple way to derive this statement is to exploit the first order formulation

of [29] and count the number of first- and second-class constraints. See also Ref. [30] for a recent

canonical analysis along these lines and Ref. [31] for a corresponding analysis at ℓ = 0.8In [8] it was pointed out that this issue is resolved if one finds a superselection sector in which

BTZ black holes are excluded.

– 408 –

asymptotically AdS3 [26], so a priori there is no reason to impose stronger conditions.

Indeed, insisting on this stronger set of boundary conditions would also eliminate

physically interesting solutions in similar theories of gravity [25]. Whether such a

truncation of CCTMG to chiral gravity is quantum mechanically consistent remains

as a pivotal open issue.9

Alternatively, if one imposes periodic boundary conditions τ = τ + β on the

metric the new mode (3.3) is eliminated since it is linear in τ . Therefore, at finite

(but arbitrarily small) temperature the new mode appears to become irrelevant. This

conclusion applies as well to the descendants, which are obtained by acting with L−1

and L−1 on (3.1) and therefore have a contribution linear in τ .

The considerations in the previous paragraphs might be of interest for the Eu-

clidean approach/CFT approach pursued in [42–50]. We conclude with three options,

all of which are worthwhile pursuing:

1. A consistent quantum theory of Euclidean chiral gravity with a chiral CFT dual

may exist if the truncation of CCTMG can be shown to be admissible. At the

boundary this would involve a truncation of a LCFT to a unitary CFT. One

can check the viability of this option by studying correlators like 〈ψnew ψL ψL〉.If they are non-vanishing no truncation is possible.

2. If a truncation turns out to be impossible then an alternative option is to find

a unitary completion of the theory.

3. Even without truncation or completion CCTMG and its related LCFT provide

interesting subjects for further studies.10 LCFTs are not unitary, but still useful

as physical models [14, 15]. One could learn something about 3-dimensional

gravity in general and about the instability described here in particular, by

studying the dual LCFT. On the other hand, studying the bulk theory along

the lines of the present work may also shed some light on properties of the dual

LCFT, via the dictionary of AdS/LCFT [16–20].

Acknowledgments

We thank Steve Carlip, Stanley Deser, Henriette Elvang, Matthias Gaberdiel, Thomas

Hartman, Alfredo Iorio, Roman Jackiw, Per Kraus, Wei Li, Alexander Maloney, John

McGreevy, Robert McNees, Wei Song, Andy Strominger, Alessandro Torrielli, An-

drew Waldron, Derek Wise and Xi Yin for discussions. NJ thanks the CTP at MIT

for its kind hospitality during the main part of this work.

This work is supported in part by funds provided by the U.S. Department of

Energy (DoE) under the cooperative research agreement DEFG02-05ER41360. DG

9We thank Andy Strominger for helpful discussions on these issues.10In this case the attribute “chiral” in CCTMG is slightly misleading since the dual LCFT is not

chiral even for cL = 0. We thank Matthias Gaberdiel for pointing this out. See also Refs. [51, 52].

– 409 –

is supported by the project MC-OIF 021421 of the European Commission under

the Sixth EU Framework Programme for Research and Technological Development

(FP6). The research of NJ was supported in part by the STINT CTP-Uppsala

exchange program.

Note added: Previous versions of this paper contained a sign error in (4.1) with

relevant consequences for the finite part of the Brown–York stress tensor (4.8) and

(4.10). This sign error was corrected in an erratum [53], prepared together with

Sabine Ertl.

A. Fefferman-Graham expansion

While the conclusions of the analysis below are gauge independent, it is convenient

to use an adapted coordinate system. Even though the bulk metric (2.10) is not in

Gaussian coordinates

ds2 = gµν dxµdxν = dρ2 + γij dx

idxj (A.1)

its shift vector and lapse function do not contribute to the relevant order in a large

ρ expansion. Thus, for boundary purposes the bulk metric (2.10) actually is of the

form (A.1), up to a factor of ℓ2, which we shall take into account in the very end.

Therefore, we can exploit the standard features of Gaussian coordinates, e.g. that

the outward pointing unit normal vector nµ has only a ρ-component, nρ = 1, ni = 0,

and that the first fundamental form hµν = gµν − nµnν has only non-vanishing ij-

components given by hij = γij. Thus, γij is the boundary metric. Similarly, the

second fundamental form Kµν = hµαhν

β∇αnβ has only non-vanishing ij-components

given by Kij = −Γρij .

We expand the boundary metric in the limit of large ρ

γij = e2ρ/ℓ γ(0)ij +

ρ

ℓγ

(1)ij + γ

(2)ij + . . . (A.2)

as well as its inverse,

γij = e−2ρ/ℓ γij(0) − e−4ρ/ℓ ρ

ℓγij

(1) − e−4ρ/ℓ γij(2) + . . . (A.3)

and its determinant √−γ = e2ρ/ℓ√

−γ(0) + . . . (A.4)

In all expressions above and below we display the leading and next-to-leading order

terms (if they are non-vanishing) in powers of e2ρ/ℓ. The extrinsic curvature

Kij =1

2∂ργij = e2ρ/ℓ 1

ℓγ

(0)ij +

1

2ℓγ

(1)ij + . . . (A.5)

– 410 –

and its inverse

Kij = e−2ρ/ℓ 1

ℓγij

(0) − e−4ρ/ℓ 2ρ

ℓ2γij

(1) − e−4ρ/ℓ 2

ℓγij

(2) + e−4ρ/ℓ 1

2ℓγij

(1) + . . . (A.6)

in our case have a very simple trace

K =2

ℓ+ . . . (A.7)

because of the tracelessness gauge conditions [cf. (2.14)]

γij(0)γ

(1)ij = γij

(0)γ(2)ij = 0 . (A.8)

The Gauss-Codazzi equations

Riρjρ = −∂ρK

ij −Ki

kKkj (A.9)

yield

Rkρjρ = − 1

ℓ2δkj + e−2ρ/ℓ 1

ℓ2γkl

(1)γ(0)lj + . . . (A.10)

Analogous formulas are valid for the variations of these quantities. We use them

to derive

εijRkρjρ δγik =

1

ℓ2εij γkl

(1)γ(0)lj δγ

(0)ik + . . . (A.11)

and

εij Kik δKkj = −1

ℓεij

( ρ

ℓ2γlk

(1)γ(0)li +

1

ℓγlk

(2)γ(0)li − 1

2ℓγlk

(0)γ(1)li

)

δγ(0)kj + . . . (A.12)

In these expressions

εij =ǫij

√

−γ(0)(A.13)

denotes the ε-tensor with respect to the conformal boundary metric γ(0). For the

Einstein-Hilbert part of the action we need the quantity

√−γ(

Kij−(

K−1

ℓ

)

γij)

δγij = −√

−γ(0)( ρ

ℓ2γij

(1)+1

ℓγij

(2)−1

2ℓγij

(1)

)

δγ(0)ij +. . . (A.14)

The explicit form of the expression Γkli δΓ

lkj ∼ γ(0) δγ(0) is not needed in the present

work since it vanishes for flat γ(0). Dropping this term in the first variation of the

on-shell action (4.1) and using

ρ = ℓρ (A.15)

establishes

δICG

∣

∣

EOM= lim

ρ→∞

∫

∂M

d2x√

−γ(0) δγ(0)ij

[ρ

ℓ

(

γij(1) − γil

(1)γ(0)lk ε

kj)

− 1

2ℓ

(

γij(1) − 3γil

(1)γ(0)lk ε

kj)

+1

ℓ

(

γij(2) − γil

(2)γ(0)lk ε

kj)

]

. (A.16)

The terms in the first line of (A.16) diverge linearly with ρ, while the terms in the

second line are finite. We see explicitly from (A.16) that no δγ(1) or δγ(2) dependence

remains for large ρ. Thus, the variational principle is well-defined.

– 411 –

References

[1] S. Weinberg, Gravitation and cosmology: principles and applications of the general

theory of relativity. Wiley, New York, 1972.

[2] S. Deser, R. Jackiw, and G. ’t Hooft, “Three-dimensional einstein gravity: Dynamics

of flat space,” Ann. Phys. 152 (1984) 220.

[3] S. Deser and R. Jackiw, “Three-dimensional cosmological gravity: Dynamics of

constant curvature,” Annals Phys. 153 (1984) 405–416.

[4] M. Banados, C. Teitelboim, and J. Zanelli, “The black hole in three-dimensional

space-time,” Phys. Rev. Lett. 69 (1992) 1849–1851, hep-th/9204099.

[5] S. Deser, R. Jackiw, and S. Templeton, “Three-dimensional massive gauge theories,”

Phys. Rev. Lett. 48 (1982) 975–978. “Topologically massive gauge theories,” Ann.

Phys. 140 (1982) 372–411. Erratum-ibid. 185 (1988) 406.

[6] S. Deser, “Cosmological Topological Supergravity,” Print-82-0692 (Brandeis).

[7] W. Li, W. Song, and A. Strominger, “Chiral Gravity in Three Dimensions,”

arXiv:0801.4566 [hep-th].

[8] S. Carlip, S. Deser, A. Waldron, and D. K. Wise, “Cosmological Topologically

Massive Gravitons and Photons,” 0803.3998.

[9] P. Kraus and F. Larsen, “Holographic gravitational anomalies,” JHEP 01 (2006)

022, hep-th/0508218.

[10] J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization of

Asymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun.

Math. Phys. 104 (1986) 207–226.

[11] K. Hotta, Y. Hyakutake, T. Kubota and H. Tanida, “Brown–Henneaux’s Canonical

Approach to Topologically Massive Gravity,” arXiv:0805.2005 [hep-th].

[12] S. Deser and B. Tekin, “Energy in topologically massive gravity,” Class. Quant.

Grav. 20 (2003) L259, gr-qc/0307073. S. Olmez, O. Sarioglu and B. Tekin, “Mass

and angular momentum of asymptotically AdS or flat solutions in the topologically

massive gravity,” Class. Quant. Grav. 22 (2005) 4355, gr-qc/0507003.

[13] V. Gurarie, “Logarithmic operators in conformal field theory,” Nucl. Phys. B410

(1993) 535–549, hep-th/9303160.

[14] M. Flohr, “Bits and pieces in logarithmic conformal field theory,” Int. J. Mod. Phys.

A18 (2003) 4497–4592, hep-th/0111228.

[15] M. R. Gaberdiel, “An algebraic approach to logarithmic conformal field theory,” Int.

J. Mod. Phys. A18 (2003) 4593–4638, hep-th/0111260.

– 412 –

[16] A. M. Ghezelbash, M. Khorrami, and A. Aghamohammadi, “Logarithmic conformal

field theories and AdS correspondence,” Int. J. Mod. Phys. A14 (1999) 2581–2592,

hep-th/9807034.

[17] Y. S. Myung and H. W. Lee, “Gauge bosons and the AdS(3)/LCFT(2)

correspondence,” JHEP 10 (1999) 009, hep-th/9904056.

[18] I. I. Kogan, “Singletons and logarithmic CFT in AdS/CFT correspondence,” Phys.

Lett. B458 (1999) 66–72, hep-th/9903162.

[19] A. Lewis, “Logarithmic operators in AdS(3)/CFT(2),” Phys. Lett. B 480, 348

(2000) [arXiv:hep-th/9911163].

[20] S. Moghimi-Araghi, S. Rouhani and M. Saadat, “Correlation functions and

AdS/LCFT correspondence,” Nucl. Phys. B 696 (2004) 492 [arXiv:hep-th/0403150].

[21] I. L. Buchbinder, S. L. Lyahovich, and V. A. Krychtin, “Canonical quantization of

topologically massive gravity,” Class. Quant. Grav. 10 (1993) 2083–2090.

[22] M. Wyman, “Static spherically symmetric scalar fields in general relativity,” Phys.

Rev. D24 (1981) 839–841.

[23] S. N. Solodukhin, “Holography with gravitational Chern-Simons,” Phys. Rev. D74

(2006) 024015 hep-th/0509148. G. Compere and D. Marolf, “Setting the boundary

free in AdS/CFT,” arXiv:0805.1902 [hep-th].

[24] S. de Haro, S. N. Solodukhin, and K. Skenderis, “Holographic reconstruction of

spacetime and renormalization in the AdS/CFT correspondence,” Commun. Math.

Phys. 217 (2001) 595–622, hep-th/0002230.

[25] M. Henneaux, C. Martinez, R. Troncoso, and J. Zanelli, “Black holes and

asymptotics of 2+1 gravity coupled to a scalar field,” Phys. Rev. D65 (2002) 104007,

hep-th/0201170. “Asymptotically anti-de Sitter spacetimes and scalar fields with a

logarithmic branch,” Phys. Rev. D70 (2004) 044034, hep-th/0404236. M. I. Park,

“Fate of three-dimensional black holes coupled to a scalar field and the

Bekenstein-Hawking entropy,” Phys. Lett. B597 (2004) 237, hep-th/0403089.

[26] K. Skenderis, “Lecture notes on holographic renormalization,” Class. Quant. Grav.

19 (2002) 5849–5876, hep-th/0209067.

[27] V. Balasubramanian and P. Kraus, “A stress tensor for anti-de Sitter gravity,”

Commun. Math. Phys. 208 (1999) 413–428, hep-th/9902121.

[28] R. Emparan, C. V. Johnson, and R. C. Myers, “Surface terms as counterterms in the

AdS/CFT correspondence,” Phys. Rev. D60 (1999) 104001, hep-th/9903238.

[29] P. Baekler, E. W. Mielke, and F. W. Hehl, “Dynamical symmetries in topological 3d

gravity with torsion,” Nuovo Cim. 107B (1991) 91–110.

– 413 –

[30] M. I. Park, “Constraint Dynamics and Gravitons in Three Dimensions,”

arXiv:0805.4328 [hep-th].

[31] S. Deser and X. Xiang, “Canonical Formulations Of Full Nonlinear Topologically

Massive Gravity,” Phys. Lett. B263 (1991) 39.

[32] R. Percacci, P. Sodano, and I. Vuorio, “Topologically massive planar universes with

constant twist,” Ann. Phys. 176 (1987) 344.

[33] G. S. Hall, T. Morgan, and Z. Perjes, “Three-dimensional space-times,”.

KFKI-1986-95/B.

[34] Y. Nutku, “Exact solutions of topologically massive gravity with a cosmological

constant,” Class. Quant. Grav. 10 (1993) 2657–2661.

[35] A. N. Aliev and Y. Nutku, “A theorem on topologically massive gravity,” Class.

Quant. Grav. 13 (1996) L29–L32, gr-qc/9812089.

[36] T. Dereli and O. Sarioglu, “Topologically massive gravity and black holes in three

dimensions,” gr-qc/0009082.

[37] A. Bouchareb and G. Clement, “Black hole mass and angular momentum in

topologically massive gravity,” Class. Quant. Grav. 24 (2007) 5581–5594,

arXiv:0706.0263 [gr-qc].

[38] D. Grumiller, W. Kummer, and D. V. Vassilevich, “Dilaton gravity in two

dimensions,” Phys. Rept. 369 (2002) 327–429, hep-th/0204253. D. Grumiller and

R. Meyer, “Ramifications of lineland,” Turk. J. Phys. 30 (2006) 349–378,

hep-th/0604049.

[39] G. Guralnik, A. Iorio, R. Jackiw, and S. Y. Pi, “Dimensionally reduced gravitational

Chern-Simons term and its kink,” Ann. Phys. 308 (2003) 222–236, hep-th/0305117.

[40] D. Grumiller and W. Kummer, “The classical solutions of the dimensionally reduced

gravitational Chern-Simons theory,” Ann. Phys. 308 (2003) 211–221,

hep-th/0306036.

[41] T. Hartman and A. Strominger, “Central Charge for AdS2 Quantum Gravity,”

arXiv:0803.3621 [hep-th]. M. Alishahiha and F. Ardalan, “Central Charge for

2D Gravity on AdS(2) and AdS(2)/CFT(1) Correspondence,” arXiv:0805.1861

[hep-th].

[42] E. Witten, “Three-Dimensional Gravity Revisited,” arXiv:0706.3359 [hep-th].

[43] J. Manschot, “AdS3 Partition Functions Reconstructed,” JHEP 10 (2007) 103,

0707.1159.

[44] D. Gaiotto and X. Yin, “Genus Two Partition Functions of Extremal Conformal

Field Theories,” JHEP 08 (2007) 029, 0707.3437.

– 414 –

[45] M. R. Gaberdiel, “Constraints on extremal self-dual CFTs,” JHEP 11 (2007) 087,

0707.4073.

[46] S. D. Avramis, A. Kehagias, and C. Mattheopoulou, “Three-dimensional AdS

gravity and extremal CFTs at c=8m,” JHEP 11 (2007) 022, 0708.3386.

[47] X. Yin, “Partition Functions of Three-Dimensional Pure Gravity,”

arXiv:0710.2129 [hep-th].

[48] X. Yin, “On Non-handlebody Instantons in 3D Gravity,” 0711.2803.

[49] A. Maloney and E. Witten, “Quantum Gravity Partition Functions in Three

Dimensions,” 0712.0155.

[50] S. Giombi, A. Maloney, and X. Yin, “One-loop Partition Functions of 3D Gravity,”

0804.1773.

[51] M. R. Gaberdiel and H. G. Kausch, “A local logarithmic conformal field theory,”

Nucl. Phys. B538 (1999) 631 hep-th/9807091.

[52] H. Eberle and M. Flohr, “Virasoro representations and fusion for general augmented

minimal models,” J. Phys. A39 (2006) 15245 hep-th/0604097.

[53] S. Ertl, D. Grumiller and N. Johansson, “Erratum to ‘Instability in cosmological

topologically massive gravity at the chiral point’,” 0910.1706.

– 415 –

February 8, 2010 9:1 World Scientific Review Volume - 9in x 6in CanCCTMG

Chapter 1

Canonical analysis of cosmological topologically massive

gravity at the chiral point

Daniel Grumiller∗, Roman Jackiw† and Niklas Johansson‡

∗† Massachusetts Institute of Technology,

77 Massachusetts Ave., Cambridge, MA 02139‡ Institutionen for fysik och astronomi, Uppsala universitet,

Box 803, S-751 08 Uppsala, Sweden

Wolfgang Kummer was a pioneer of two-dimensional gravity and a strongadvocate of the first order formulation in terms of Cartan variables. Inthe present work we apply Wolfgang Kummer’s philosophy, the ‘ViennaSchool approach’, to a specific three-dimensional model of gravity, cos-mological topologically massive gravity at the chiral point. Exploitinga new Chern–Simons representation we perform a canonical analysis.The dimension of the physical phase space is two per point, and thusthe theory exhibits a local physical degree of freedom, the topologicallymassive graviton.

1.1. Introduction

Gravity in lower dimensions provides an excellent expedient for testing

ideas about classical and quantum gravity in higher dimensions. The lowest

spacetime dimension where gravity can be described is two, and Wolfgang

Kummer contributed significantly to research on two-dimensional gravity,

see Ref.1 for a review. Those who knew Wolfgang will recall that one

of his main points was to advocate a gauge theoretic approach towards

gravity, see Ref.2 for his last proceedings contributions. Instead of using

the metric, gµν , as fundamental field he insisted on employing the Cartan

variables, Vielbein eaµ and connection ωa

b µ. His approach greatly facilitated

the canonical analysis and the quantization of the theory.

In the present work we shall study gravity in three dimensions along∗E-mail: grumil@hep.itp.tuwien.ac.at†E-mail: jackiw@lns.mit.edu‡E-mail: Niklas.Johansson@fysast.uu.se

417

February 8, 2010 9:1 World Scientific Review Volume - 9in x 6in CanCCTMG

418 D. Grumiller, R. Jackiw and N. Johansson

similar lines. We start by collecting a few well-known features of gravity in

three dimensions. Pure Einstein–Hilbert gravity exhibits no physical bulk

degrees of freedom.3–5 If the theory is deformed by a negative cosmologi-

cal constant it has black hole solutions.6 Another possible deformation is

to add a gravitational Chern–Simons term. The resulting theory is called

topologically massive gravity (TMG) and, remarkably, contains a massive

graviton.7 Including both terms yields cosmological topologically massive

gravity8 (CTMG), a theory that exhibits both gravitons and black holes.

Parameterizing the negative cosmological constant by Λ = −1/ℓ2 the (sec-

ond order) action is given by

ICTMG[g] =

∫d3x

√−g[R +

2

ℓ2+

1

2µελµνΓρ

λσ

(∂µΓσ

νρ +2

3Γσ

µτΓτνρ

)].

(1.1)

In Ref.9 it was advocated to study the theory (1.1) at the chiral point

µℓ = 1 , (1.2)

where the theory exhibits very special properties. We abbreviate this the-

ory by the acronym CCTMG (‘chiral cosmological topologically massive

gravity’). By imposing the Brown–Henneaux boundary conditions Ref.9

argued that CCTMG exhibits no bulk degrees of freedom. On the other

hand Ref.10 found that CCTMG exhibits one bulk degree of freedom. By

slightly relaxing the Brown–Henneaux boundary conditions — still requir-

ing spacetime to be asymptotically AdS — Ref.11 demonstrated that indeed

a physical degree of freedom exists in CCTMG: the topologically massive

graviton. The analyses in Refs.9–11 were focused on the linearized level,

i.e., perturbing around an AdS3 background.

In the present work we go beyond the linearized approximation and

perform a non-perturbative (classical) canonical analysis of CCTMG (see

also Refs.12–14).a Our main goal is to derive the dimension of the physical

phase space, which allows us to deduce the number of physical bulk degrees

of freedom.

This paper is organized as follows. In Section 1.2 we present a new

Chern–Simons formulation of cosmological topologically massive gravity.

In Section 1.3 we focus on the chiral point and establish the Hamiltonian

formulation, identifying all primary, secondary and ternary constraints. In

Section 1.4 we perform a constraint analysis and check the first/second class

properties of all constraints, which allows us to establish the dimension of

the physical phase space. In Section 1.5 we conclude.aFor further recent literature related to CCTMG see Refs.15–21

February 8, 2010 9:1 World Scientific Review Volume - 9in x 6in CanCCTMG

Canonical analysis of cosmological topologically massive gravity at the chiral point 419

Our conventions are as follows. We use Greek spacetime indices and

Latin frame indices. The former are raised and lowered with the spacetime

metric gµν and the latter with the flat metric ηab. Both have signature

−, +, +. For the Dreibein eaµ we choose sign (det e) = 1. When writing

p-forms we usually suppress the spacetime indices, e.g. ea denotes the 1-

form ea = eaµdxµ. We disregard boundary terms in the present work, so

equivalences between actions have to be true only up to total derivatives.

1.2. Chern–Simons formulation

Instead of the action (1.1) which functionally depends on the metric one

can equivalently use the action

ICTMG[e] =

∫ [2ea ∧ Ra(ω) +

1

3ℓ2εabc ea ∧ eb ∧ ec − 1

µCS(ω)

](1.3)

which functionally depends on the Dreibein. The gravitational Chern–

Simons term

CS(ω) := ωa ∧ dωa +1

3εabc ωa ∧ ωb ∧ ωc (1.4)

and the (dualized) curvature 2-form

Ra(ω) := dωa +1

2εabc ωb ∧ ωc (1.5)

depend both exclusively on the (dualized) connection defined by ωa :=12εabcωbc. Note that the connection is not varied independently in the for-

mulation (1.3), but rather it is the Levi-Civita connection, i.e., metric com-

patible ωab = −ωba and torsion-free, T a = 0, where

Ta := dea + εabc ωb ∧ ec (1.6)

is the torsion 2-form. This means that ωa in (1.3) has to be expressed in

terms of ea (and derivatives thereof) before variation.

For our purposes it is very convenient to employ a formulation where we

can vary independently the Dreibein and the connection.22 This is achieved

by supplementing the action (1.3) with a Lagrange multiplier term enforcing

the torsion constraint,

ICTMG[e, ω, λ] =

∫ [2ea ∧Ra +

1

3ℓ2εabc ea ∧ eb ∧ ec − 1

µCS(ω) + λa ∧ Ta

].

(1.7)

The first order action (1.7) is classically equivalent22 to the second order

action (1.1). This can be shown as follows. Varying (1.7) with respect to λa

February 8, 2010 9:1 World Scientific Review Volume - 9in x 6in CanCCTMG

420 D. Grumiller, R. Jackiw and N. Johansson

and ωa establishes the condition of vanishing torsion (1.6) and an algebraic

relation for λa,

1

2εabc λa ∧ eb =

1

µRc − Tc =

1

µRc , (1.8)

in terms of Dreibein, connection and derivatives thereof. Thus, both ωa

and λa can be expressed in terms of the Dreibein, and first and second

derivatives thereof. Varying (1.7) with respect to the Dreibein and plugging

into that equation the relations for λa and ωa in terms of ea yields a set of

third order partial differential equations in ea. Using the defining relation

between Dreibein and metric, gµν = eaµeb

ν ηab, finally establishes

Gµν +1

µCµν = 0 , (1.9)

where

Gµν = Rµν − 1

2gµνR − 1

ℓ2gµν (1.10)

is the Einstein tensor (including cosmological constant) and

Cµν =1

2εµ

αβ ∇αRβν + (µ ↔ ν) (1.11)

is essentially the Cotton tensor. The equations of motion (1.9) also follow

directly from varying the second order action (1.1) with respect to the

metric.

We make now some field redefinitions to further simplify the action

(1.7). We shift the Lagrange multiplier λa → λa − ea/(µℓ2) and obtain

ICTMG[e, ω, λ] =

∫ [2ea∧Ra+

1

3ℓ2εabc ea∧eb∧ec− 1

µCS(ω)+

(λa− ea

µℓ2

)∧Ta

]

(1.12)

In the absence of the λa ∧ Ta-term in (1.12), the well-known field redefini-

tions

Aa := ωa + ea/ℓ , Aa := ωa − ea/ℓ (1.13)

turn the action into a difference of two Chern–Simons terms.23–26 Curiously,

under the same redefinitions (1.13) the Lagrange multiplier term can be

recast into a difference of two Einstein–Hilbert terms, where λ plays the

role of the Dreibein:

2

ℓICTMG[A, A, λ] =

(1− 1

µℓ

)ICS[A]+IEH [λ, A]−

(1+

1

µℓ

)ICS[A]−IEH[λ, A].

(1.14)

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Canonical analysis of cosmological topologically massive gravity at the chiral point 421

We have introduced here the abbreviations

ICS[A] :=

∫CS(A) (1.15)

and

IEH[λ, A] :=

∫λa ∧ Ra(A) (1.16)

and similarly for A.

The reformulation (1.14) of the action (1.7) as difference of Chern–

Simons and Einstein-Hilbert terms seems to be new. It is worthwhile re-

peating that in both Einstein–Hilbert terms the Lagrange multiplier λa

formally plays the role of a ‘Dreibein’. This suggests that λa should be

invertible. We have checked that for pure AdS3 [which obviously solves the

field equations (1.9)] the symmetric tensor λµν = ea(µλν) a is proportional

to the metric. Thus, requiring invertibility of λa is necessary in general to

guarantee invertibility of the metric.

The advantage of the formulation (1.14) is twofold. Because the action

contains only first derivatives (linearly) a canonical analysis is facilitated.

Moreover, at the chiral point µ2ℓ2 = 1 one of the Chern–Simons terms

vanishes.

1.3. Hamiltonian action at the chiral point

We focus now on the theory at the chiral point and assume for sake of

specificity µℓ = 1. The action (1.14) simplifies to

ICCTMG[A, A, λ] =ℓ

2IEH(λ, A)−ℓ ICS(A)− ℓ

2IEH(λ, A) =

∫d3xL (1.17)

To set up the canonical analysis one could now declare the 27 fields λa, Aa,

Aa to be canonical coordinates and calculate their 27 canonical momenta.13

In this way one produces many second class constraints which have to be

eliminated by the Dirac procedure.27 However, this is not the most efficient

way to start the canonical analysis. As realized in Ref.28 if an action is

already in first order form a convenient short-cut exists. In the present case

this short-cut consists basically of picking the appropriate sets of fields as

canonical coordinates and momenta, respectively.

We use the 18 fields λaµ, Aa

0 , Aa1 , A

a0 as canonical coordinates and intro-

duce the notation

qa1 = λa

1 , qa2 = λa

2 , qa3 = Aa

1 , qa1 = λa

0 , qa2 = Aa

0 , qa3 = Aa

0 . (1.18)

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422 D. Grumiller, R. Jackiw and N. Johansson

Like in electrodynamics or non-abelian gauge theory the momenta pai of the

zero components qai are primary constraints. The simplest way to deal with

them is to exclude the pairs qai , pa

i from the phase space and to treat the qai

as Lagrange multipliers for the secondary constraints (“Gauss constraints”).

This reduces the dimension of our phase space to 18. The 9 momenta pai ,

∂L∂∂0λ1 a

= pa1 =

ℓ

2(Aa

2 − Aa2) = ea

2 (1.19)

∂L∂∂0λ2 a

= pa2 = − ℓ

2(Aa

1 − Aa1) = −ea

1 (1.20)

∂L∂∂0A1 a

= pa3 = −2ℓ Aa

2 (1.21)

depend linearly on the fields Aa1 , A

a2 , Aa

2 . These fields are not contained in

our set of canonical coordinates.

The Hamiltonian action is now determined as

ICCTMG[q, p; q] =

∫d3x

(pi aqa

i −H), (1.22)

where the Hamiltonian density

H = qi a Gai (1.23)

is a sum over secondary constraints Gai ≈ 0, as expected on general

grounds.b They are given by

Ga1 = − ℓ

2Ra +

ℓ

2Ra , (1.24)

Ga2 =

ℓ

2Dλa + 2ℓ Ra , (1.25)

Ga3 = − ℓ

2Dλa . (1.26)

We have introduced the following abbreviations

Ra :=(∂1A

a2 − ∂2A

a1

)+

1

2εa

bc

(Ab

1Ac2 − Ab

2Ac1

)(1.27)

and

Dλa :=(∂1λ

a2 − ∂2λ

a1

)+ εa

bc

(Ab

1λc2 − Ab

2λc1

)(1.28)

and similarly for R and Dλ, with A replaced by A in the definitions (1.27)

and (1.28), respectively.

We focus now on the first/second class properties of the constraints and

on their Poisson bracket algebra. We have found 9 secondary constraintsbThe notation ≈ means ‘vanishing weakly’,27 i.e., vanishing on the surface of constraints.

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Canonical analysis of cosmological topologically massive gravity at the chiral point 423

Gai . If all of them were first class then the physical phase space would be

zero-dimensional, because each first class constraint eliminates two dimen-

sions from the phase space, and the dimension of the phase space spanned

by qai , pa

i is 18.

1.4. Constraint analysis

With the canonical Poisson bracket

qai (x), pb

j(x′) = qa

i , p′ bj = δij ηab δ(2)(x − x′) (1.29)

we can now calculate the Poisson brackets of the constraints Gai with each

other and with the Hamiltonian density. The latter,

Gai ,H′ = q′j b Ga

i , G′ bj (1.30)

reduce to a sum over brackets between the secondary constraints. We cal-

culate now these brackets explicitly.

To this end we express the secondary constraints (1.24)-(1.26) in terms

of canonical coordinates and momenta:

Ga1 = −∂1p

a1 − ∂2p

a2 − εa

bc

(2

ℓpb1p

c2 +

1

2ℓpb2p

c3 + qb

3pc1

)(1.31)

Ga2 = Ga

2 + Ga3 = −∂1p

a3 − 2ℓ ∂2q

a3 + εa

bc pbiq

ci (1.32)

Ga3 = − ℓ

2

(∂1q

a2 − ∂2q

a1

)+ εa

bc

(pb1q

c1 + pb

2qc2 −

1

4pb3q

c1 −

ℓ

2qb3q

c2

)(1.33)

Note that instead of Ga2 we use for convenience the linear combination

Ga2 = Ga

2 + Ga3 . Straightforward calculation obtains:

Ga1 , G

′ b1 = Zab

11 δ(2)(x − x′) (1.34)

Ga2 , G

′ b2 = −εab

c Gc2 δ(2)(x − x′) ≈ 0 (1.35)

Ga3 , G

′ b3 = −εab

c Gc3 δ(2)(x − x′) + Zab

33 δ(2)(x − x′) (1.36)

Ga1 , G

′ b2 = εab

c Gc1 δ(2)(x − x′) ≈ 0 (1.37)

Ga2 , G

′ b3 = −εab

c Gc3 δ(2)(x − x′) ≈ 0 (1.38)

Ga1 , G

′ b3 = −εab

c

(Gc

1 −1

4Gc

2

)δ(2)(x − x′) + Zab

13 δ(2)(x − x′) (1.39)

February 8, 2010 9:1 World Scientific Review Volume - 9in x 6in CanCCTMG

424 D. Grumiller, R. Jackiw and N. Johansson

We have used here the abbreviations

Zab11 =

1

2ℓ

(pa2p

b1 − pb

2pa1

)(1.40)

Zab33 =

ℓ

8

(qa2qb

1 − qb2q

a1

)(1.41)

Zab13 = −1

4

(pa1q

b1 + pa

2qb2

)+

1

4ηab

(pc1q1 c + pc

2q2 c

)(1.42)

or, equivalently,

Zab11 = − 1

2ℓ

(ea ∧ eb

)12

(1.43)

Zab33 = − ℓ

8

(λa ∧ λb

)12

(1.44)

Zab13 =

1

4

(ea ∧ λb

)12

− 1

4ηabηcd

(ec ∧ λd

)12

(1.45)

If the quantities Zabij were all vanishing then all secondary constraints would

be first class. Since some of them are non-vanishing we have a certain

number of second class constraints. Namely, not all entries of Zab11 can

vanish because this would lead to a singular Dreibein ea. Similarly, not

all entries Zab33 can vanish because this would lead to a singular Lagrange

multiplier 1-form λa. Since the algebra of constraints does not close we

shall encounter ternary constraints from consistency requirements, namely

the vanishing of the Poisson brackets (1.30).

In the analysis below, the 9 × 9-matrix

Mabij :=

∫

x′

d2x′ Gai , G′ b

j (1.46)

evaluated on the surface of constraints will play a crucial role. First, note

that before imposing the ternary constraints we can establish an upper

bound on the dimension 2n of the physical phase space in terms of the rank

rM of Mabij . We started with a phase space of dimension 18 and accounted

for 9 constraints. The rank rM counts how many of these that are second

class. Thus, before additional constraints are introduced we have

2n ≤ 18 − rM − 2 ∗ (9 − rM ) = rM . (1.47)

Now we turn to the ternary constraints. We note that after imposing these

we are done, since the consistency conditions analog to (1.30) arising from

the T ai do not generate quaternary constraints. Since the algebra (1.34)–

(1.39) closes on δ-functions, requiring vanishing of the brackets (1.30) is

equivalent to requiring

T ai := Mab

ij qj b ≈ 0. (1.48)

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Canonical analysis of cosmological topologically massive gravity at the chiral point 425

Because the ternary constraints T ai contain the canonical partners of the

primary constraints pai complications arise, since some of the latter may lose

their status as first class constraints. Thus we have to include the qai as

canonical variables, giving a phase space of dimension 36 before imposing

the constraints. We determine now the rank of the 27 × 27 matrix

Mabij :=

∫

x′

d2x′ Cai , C′ b

j (1.49)

evaluated on the surface of constraints using the order Cai = (pa

i , Gai , T a

i ).

Because of (1.48) we have

T ai , pb

j = Mabij , (1.50)

and thus M has the block form

M ≈

O O −MT

O M B

M −BT C

, (1.51)

where all the blocks are 9 × 9 matrices. The form of the non-vanishing

matrices B and C is not needed for determining a lower bound for the rank

of M . We can put all copies of M and MT on lower triangular form by

row-operations that do not spoil the block structure of (1.51). This makes

M lower triangular with 3rM non zero anti-diagonal elements. Thus, a

lower bound for the rank rcMof M is 3rM .

We are now in a position to count the number of linearly independent

first- and second-class constraints. We have rcMsecond class constraints.

The total number of constraints is 9(primary) + 9(secondary) + 9(ternary)

= 27, but out of the nine ternary constraints T ai , only rM are linearly

independent. This is so because of (1.48).

Thus, the total number of linearly independent constraints is 9 + 9 +

rM = 18 + rM , and rcMof these are second class. The dimension 2n of the

physical phase space is therefore bounded by

2n = 36 − rcM− 2 ∗ (18 + rM − rcM

) = rcM− 2rM ≥ rM . (1.52)

The two inequalities (1.47) and (1.52) establish 2n = rM .

Thus, all that remains is to determine the rank of M . Using the order

Gai = (Ga

1 , Ga3 , G

a2), M has the block form

M ≈(

A6×6 O6×3

O3×6 O3×3

), A6×6 :=

(Z11 Z13

−ZT13 Z33

). (1.53)

February 8, 2010 9:1 World Scientific Review Volume - 9in x 6in CanCCTMG

426 D. Grumiller, R. Jackiw and N. Johansson

The block entries Ox×y contain x rows and y columns of zeros. From (1.53)

we deduce that the rank of the antisymmetric matrix Mabij must be either

six, four or two. Its nine Eigenvalues n1 . . . n9 are given by

n1 . . . n5 = 0 , n6,7 = ± i

4

(ea ∧ λa

)12

, n8,9 = ± i

4

√P (1.54)

Therefore its rank equals (at most) four, and not six as suggested by naive

counting. The polynomial under the square root in the last expression in

(1.54) is given by

P =2

ℓ2

(ea∧eb

)12

(ea∧eb

)12

+ℓ2

8

(λa∧λb

)12

(λa∧λb

)12

+(ea∧λb

)12

(ea∧λb

)12

(1.55)

The rank of (1.53) is four in general and two if in addition the condition(ea ∧ λa

)12

= −pa1q1 a − pa

2q2 a = 0 (1.56)

holds. Because of (1.8) on-shell we obtain

ea ∧ λa ∝ ea ∧ Rica ∝ Rµν dxµ ∧ dxν = 0 (1.57)

where Rica is the Ricci 1-form with respect to the Levi-Civita connection

(we recall that on-shell torsion vanishes). Thus, the constraint (1.56) must

hold on all classical solutions. Therefore, in the physically relevant sector,

2n = rM = 2.c This completes our constraint analysis.d

To summarize, the dimension of the physical phase space is two and

therefore CCTMG exhibits one physical bulk degree of freedom, which at

the linearized level coincides with the topologically massive graviton.

1.5. Conclusions

In this paper we have reformulated cosmological topologically massive grav-

ity at the chiral point as a Chern–Simons action plus the difference betweencIt is possible, although not necessary, to impose (1.56) as a further constraint. Thisdoes not change anything essential about the counting procedure.dAs a consistency check we investigate now what happens when the torsion constraintis dropped in (1.12). In the current formulation this can be achieved by imposing theconstraints

Ga

4 = qa

1 ≈ 0 , Ga

5 = qa

2 ≈ 0 , Ga

6 = qa

1 ≈ 0 . (1.58)

These constraints render the constraints Ga

3and T a

isuperfluous. Thus, we have now

24 linearly independent constraints, pa

i, Ga

1, bGa

2, Ga

4, Ga

5, Ga

6. The rank of the 24 × 24

matrix analog to (1.51) turns out to be equal to twelve. Therefore, we have now twelvefirst class and twelve second class constraints, which eliminates all dimensions from thephase space. Thus no physical bulk degrees of freedom remain. This is the anticipatedresult.

February 8, 2010 9:1 World Scientific Review Volume - 9in x 6in CanCCTMG

Canonical analysis of cosmological topologically massive gravity at the chiral point 427

two Einstein–Hilbert actions, see (1.17). We have performed a canonical

analysis and recovered the anticipatede result of one physical bulk degree

of freedom, which at the linearized level corresponds to the topologically

massive graviton.

We have also encountered sectors of our first order theory that are not

related to the second order formulation with regular field configurations, but

that may be worthwhile studying in their own right. For instance, if one

imposes by hand the constraints qa1 = 0 = qa

3 then no ternary constraints

arise, but the Dreibein and Lagrange multiplier fail to be invertible.

Finally, we mention that the Poisson bracket algebra of the secondary

constraints (1.34)-(1.39) closes with δ-functions rather than with first

derivatives thereof because of our gauge theoretic reformulation of CCTMG.

The same happens in 1 + 1 dimensions, where this feature was exhibited

and exploited by Wolfgang Kummer and his ‘Vienna School’.1,2

Acknowledgments

We thank Steve Carlip, Stanley Deser, Mu-In Park and Andy Strominger

for correspondence. NJ thanks the CTP at MIT for its kind hospitality dur-

ing parts of this work. This work is supported in part by funds provided

by the U.S. Department of Energy (DoE) under the cooperative research

agreement DEFG02-05ER41360. DG is supported by the project MC-OIF

021421 of the European Commission under the Sixth EU Framework Pro-

gramme for Research and Technological Development (FP6). The research

of NJ was supported in part by the STINT CTP-Uppsala exchange pro-

gram.

References

1. D. Grumiller, W. Kummer, and D. V. Vassilevich, Dilaton gravity in twodimensions, Phys. Rept. 369, 327–429, (2002).

2. W. Kummer, Progress and problems in quantum gravity. (2005).3. S. Weinberg, Gravitation and cosmology: principles and applications of the

general theory of relativity. (Wiley, New York, 1972).

eA recent canonical analysis in the first order formulation13 obtains a 2-dimensionalphysical phase space ‘for each internal index a’, i.e., a 6-dimensional physical phase space.This result disagrees with ours and with previous literature, but it is then interpreted asa single graviton degree of freedom, concurrent with our result. Correspondence with theauthor revealed that he found additional constraints after posting his e-print and thatcurrently he is reconsidering the constraint algebra. Another recent analysis14 agreeswith our results.

February 8, 2010 9:1 World Scientific Review Volume - 9in x 6in CanCCTMG

428 D. Grumiller, R. Jackiw and N. Johansson

4. S. Deser, R. Jackiw, and G. ’t Hooft, Three-dimensional einstein gravity:Dynamics of flat space, Ann. Phys. 152, 220, (1984).

5. S. Deser and R. Jackiw, Three-dimensional cosmological gravity: Dynamicsof constant curvature, Annals Phys. 153, 405–416, (1984).

6. M. Banados, C. Teitelboim, and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69, 1849–1851, (1992).

7. S. Deser, R. Jackiw, and S. Templeton, Three-dimensional massive gaugetheories, Phys. Rev. Lett. 48, 975–978, (1982). Topologically massive gaugetheories, Ann. Phys. 140, 372–411, (1982). Erratum-ibid. 185, 406, (1988).

8. S. Deser, Cosmological Topological Supergravity. Print-82-0692 (Brandeis).9. W. Li, W. Song, and A. Strominger, Chiral Gravity in Three Dimensions,

JHEP 0804 082, (2008).10. S. Carlip, S. Deser, A. Waldron, and D. K. Wise, Cosmological Topologically

Massive Gravitons and Photons. (2008).11. D. Grumiller and N. Johansson, Instability in cosmological topologically mas-

sive gravity at the chiral point. (2008).12. S. Deser and X. Xiang, Canonical formulations of full nonlinear topologically

massive gravity, Phys. Lett. B263, 39–43, (1991).13. M.-I. Park, Constraint Dynamics and Gravitons in Three Dimensions. (2008).14. S. Carlip. in preparation, (2008).15. R. Banerjee, S. Gangopadhyay, and S. Kulkarni, Black Hole Entropy from

Covariant Anomalies. (2008).16. G. Compere and D. Marolf, Setting the boundary free in AdS/CFT. (2008).17. M. Alishahiha and F. Ardalan, Central Charge for 2D Gravity on AdS(2)

and AdS(2)/CFT(1) Correspondence. (2008).18. K. Hotta, Y. Hyakutake, T. Kubota, and H. Tanida, Brown-Henneaux’s

Canonical Approach to Topologically Massive Gravity. (2008).19. W. Li, W. Song, and A. Strominger, Comment on ’Cosmological Topological

Massive Gravitons and Photons’. (2008).20. I. Sachs and S. N. Solodukhin, Quasi-Normal Modes in Topologically Massive

Gravity. (2008).21. D. A. Lowe and S. Roy, Chiral geometries of (2+1)-d AdS gravity. (2008).22. P. Baekler, E. W. Mielke, and F. W. Hehl, Dynamical symmetries in topo-

logical 3d gravity with torsion, Nuovo Cim. 107B, 91–110, (1991).23. A. Achucarro and P. K. Townsend, A Chern-Simons action for three-

dimensional Anti-de Sitter supergravity theories, Phys. Lett. B180, 89,(1986).

24. E. Witten, (2+1)-dimensional gravity as an exactly soluble system, Nucl.

Phys. B311, 46, (1988).25. M. Blagojevic and M. Vasilic, 3D gravity with torsion as a Chern-Simons

gauge theory, Phys. Rev. D68, 104023, (2003).26. S. L. Cacciatori, M. M. Caldarelli, A. Giacomini, D. Klemm, and D. S.

Mansi, Chern-Simons formulation of three-dimensional gravity with torsionand nonmetricity, J. Geom. Phys. 56, 2523–2543, (2006).

27. P. A. M. Dirac, Lectures on Quantum Mechanics. (Belfer Graduate School ofScience, Yeshiva University, New York, 1996).

28. L. D. Faddeev and R. Jackiw, Hamiltonian reduction of unconstrained andconstrained systems, Phys. Rev. Lett. 60, 1692, (1988).

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International Journal of Modern Physics Dc© World Scientific Publishing Company

CONSISTENT BOUNDARY CONDITIONS FOR COSMOLOGICAL

TOPOLOGICALLY MASSIVE GRAVITY AT THE CHIRAL POINT

D. GRUMILLER

Center for Theoretical Physics, Massachusetts Institute of Technology,77 Massachusetts Ave., Cambridge, MA 02139, USA

andInstitute for Theoretical Physics, Vienna University of Technology,

Wiedner Hauptstr. 8–10/136, Vienna, A-1040, Austriagrumil@lns.mit.edu

N. JOHANSSON

Institutionen for Fysik och Astronomi, Uppsala Universitet,Box 803, S-751 08 Uppsala, Sweden

Niklas.Johansson@fysast.uu.se

MIT-CTP 3972, UUITP-18/08, arXiv:0808.2575Received 1. October 2008Accepted 2. October 2008

Communicated by D.V. Ahluwalia

We show that cosmological topologically massive gravity at the chiral point allowsnot only Brown–Henneaux boundary conditions as consistent boundary conditions, butslightly more general ones which encompass the logarithmic primary found in 0805.2610

as well as all its descendants.

Keywords: Cosmological topologically massive gravity, Brown–Henneaux boundary con-ditions, chiral gravity, gravity in three dimensions, logarithmic CFT, AdS/CFT

1. Introduction

Cosmological topologically massive gravity 1 (CTMG) is a 3-dimensional theory of

gravity that exhibits gravitons 2,3 and black holes 4. With the sign conventions of

Ref. 5 its action is given by

ICTMG =1

16πG

∫

d3x√−g

[

R +2

ℓ2+

1

2µελµν Γρ

λσ

(

∂µΓσνρ +

2

3Γσ

µτΓτνρ

)

]

, (1)

where the negative cosmological constant is parameterized by Λ = −1/ℓ2. If the

constants µ and ℓ satisfy the condition

µℓ = 1 (2)

the theory is called “CTMG at the chiral point”. The condition (2) is special because

one of the central charges of the dual boundary CFT vanishes, cL = 0, cR 6= 0.

429

February 1, 2010 18:7 WSPC/INSTRUCTION FILE 0808.2575.new

430 D. GRUMILLER AND N. JOHANSSON

This observation was the motivation for Ref. 6 to consider CTMG at the chi-

ral point in some detail. In that work the theory (1) with (2) was dubbed “chiral

gravity”, assuming that all solutions obey the Brown–Henneaux boundary condi-

tions 7. Moreover, it was conjectured that CTMG at the chiral point is a chiral

theory and that the local physical degree of freedom, the topologically massive

graviton, disappears. These statements were disputed in Ref. 8, which engendered

a lot of recent activity concerning CTMG 9,5,10,11. In particular, the present au-

thors constructed explicitly 5 a physical mode not considered in Ref. 6 using their

formalism. This mode, which we call “logarithmic primary”, violates the Brown–

Henneaux boundary conditions. These results were confirmed very recently 11, where

one of the descendants of the logarithmic primary was considered. It was found that

this descendant (and all successive descendants) can be made consistent with the

Brown–Henneaux boundary conditions by a diffeomorphism. Thus, these modes are

present in classical CTMG (in addition to the standard boundary gravitons), even

if Brown–Henneaux boundary conditions are imposed. The latest development is

a simple classical proof 12 of the chirality of the generators of diffeomorphisms at

µℓ = 1, concurrent with previous results 10.

In the conclusions of Ref. 12 it was speculated that perhaps there are consistent

boundary conditions other than the ones by Brown and Henneaux for CTMG at

the chiral point. It is the purpose of this note to show that this is indeed the case

and that the new set of boundary conditions encompasses the logarithmic primary.

2. Beyond Brown–Henneaux

We follow as closely as possible the notation and the logical flow of Ref. 12. Any met-

ric consistent with the boundary conditions to be imposed below must asymptote

to AdS3, which in Poincare coordinates is given by

gAdSµν dxµ dxν = ℓ2

(

dx+ dx− + dy2

y2

)

, (3)

where the boundary is located at y = 0. The Brown–Henneaux boundary conditions

then require that fluctuations hµν of the metric about (3) must fall off as

h++ = O(1) h+−= O(1) h+y = O(y)

h−−

= O(1) h−y = O(y)

hyy = O(1)

. (4)

By O(x) we mean that the corresponding fluctuation metric component behaves at

most proportional to x in the small y limit.

We define now suitable boundary conditions that encompass the logarithmic

primary and its descendants. Let us first recall the form of the logarithmic pri-

mary and see how the Brown–Henneaux boundary conditions need to be weakened.

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CONSISTENT BOUNDARY CONDITIONS FOR CTMG AT THE CHIRAL POINT 431

Translating the result (3.3) of Ref. 5 into Poincare coordinates yields schematicallya

hnewµν dxµ dxν ∼ O(log y) (dx−)2 + O(y log y) dx− dy + O(y2 log y) dy2 . (5)

Evidently the logarithmic primary behaves as follows:

hnew−−

= O(log y) , hnew−y = O(y log y) , hnew

yy = O(y2 log y). (6)

From (4) we see that the Brown–Henneaux boundary conditions for these three

components are

h−−

= O(1) , h−y = O(y) , hyy = O(1) . (7)

It is clear that (6) is incompatible with (7). However, only the first two conditions

of (3) have to be weakened logarithmically to encompass the logarithmic primary.

Therefore, we propose the following set of boundary conditions:b

hnew++ = O(1) hnew

+−

= O(1) hnew+y = O(y)

hnew−−

= O(log y) hnew−y = O(y log y)

hnewyy = O(1)

(8)

Let us determine the diffeomorphisms

gµν = gAdSµν + hnew

µν → Lζ gµν = gµν = gAdSµν + hnew

µν (9)

that preserve these boundary conditions. I.e., we require that also hnewµν has the fall-

off behavior postulated in (8). Calculating the generator ζµ with this requirement

obtains

ζ+ = ǫ+(x+) − y

2∂2−

ǫ− + O(y4 log y) (10)

ζ− = ǫ−(x−) − y

2∂2+ǫ+ + O(y4) (11)

ζy =y

2

(

∂+ǫ+(x+) + ∂−

ǫ−(x−))

+ O(y3) (12)

Remarkably, the only difference to the Brown–Henneaux-case is the possibility of an

O(y4 log y) behavior for the sub-sub-leading terms in the ζ+ component as opposed

to O(y4), cf. e.g. (5)-(8) in Ref. 12. Thus there are transformations that preserve

the new set of boundary conditions (8) but not the Brown–Henneaux set of bound-

ary conditions (4). These new transformations must still be considered pure gauge

because of their rapid fall-off near the boundary.

Thus we end up with the following situation. The suitable boundary conditions

to encompass the logarithmic primary are given by (8) rather than by (4). These are

preserved by more gauge transformations than the Brown–Henneaux conditions, but

exhibit the same asymptotic symmetries. Since the isometry algebra of AdS3 is part

aThe coordinates in that work are related to the coordinates here as follows: x± = (φ ∓ τ)/2,y ∼ e−ρ.bThe proposal (8) may be compared with footnote 3 in Ref. 12: it is not necessary to weaken theboundary conditions of all components h±± to O(ln y) (see first sentence) and it is not sufficientto take only h−− to be O(ln y) (see second sentence).

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432 D. GRUMILLER AND N. JOHANSSON

of the transformations that preserve (8), and since the descendants are produced

by acting with this algebra, we automatically demonstrated that all descendants of

hnew fulfill (8).

3. Boundary stress tensor and asymptotic symmetry generators

It is also important that all metrics fulfilling (8) have well defined generators of the

asymptotic symmetries. This can be shown as follows. We compute the boundary

stress tensor along the lines of Ref. 5 and find a generalization of the Kraus-Larsen

result 13:c

T++ =1

4πG ℓhnew

++ ∼ O(1) (13)

T−−

= − 1

8πG ℓy∂yhnew

−−

∼ O(1) (14)

T+−= 0. (15)

The off-diagonal contribution T+−vanishes after imposing constraints from the

equations of motion. The generators of the asymptotic symmetry group become

Q[ζ] =1

4πG ℓ

∫

dx+(

hnew++ ǫ+ − 1

2y∂yhnew

−−

ǫ−)

∼ O(1). (16)

Since no divergences arise the generators (16) are well-defined.

Thus, we conclude that there are indeed consistent boundary conditions (8) that

go beyond Brown–Henneaux and that allow for the logarithmic primary and all its

descendants. Because of the analysis in Section 4 of Ref. 5 this result might have

been anticipated: there it was shown that the logarithmic primary is consistent

with the requirement of spacetime being asymptotically AdS and that the ensuing

boundary stress tensor is finite, traceless and conserved.

We close by noting that there are other examples when the Brown–Henneaux

boundary conditions need to be weakened logarithmically to encompass physically

interesting solutions 14. The boundary conditions of Ref. 14 are not identical to the

ones considered in the present note.

Acknowledgment

We thank Stanley Deser, Matthias Gaberdiel, Gaston Giribet, Thomas Hartman,

Roman Jackiw, Matthew Kleban, Alex Maloney, Massimo Porrati, Wei Song and

Andy Strominger for discussion.

This work is supported in part by funds provided by the U.S. Department of

Energy (DoE) under the cooperative research agreement DEFG02-05ER41360. DG

is supported by the project MC-OIF 021421 of the European Commission under

cActually, the result given in the first two versions of Ref. 5 was incorrect due to sign errors. Thefirst appearance of the correct Tij was in Ref. 15. The first correct calculation of the charge Qbelow was in Refs. 16,17. See also our erratum Ref. 18.

February 1, 2010 18:7 WSPC/INSTRUCTION FILE 0808.2575.new

CONSISTENT BOUNDARY CONDITIONS FOR CTMG AT THE CHIRAL POINT 433

the Sixth EU Framework Programme for Research and Technological Development

(FP6).

References

1. S. Deser, “Cosmological Topological Supergravity,” PRINT-82-0692 (Brandeis).2. S. Deser, R. Jackiw and S. Templeton, “Three-Dimensional Massive Gauge Theories,”

Phys. Rev. Lett. 48, 975 (1982).3. S. Deser, R. Jackiw and S. Templeton, “Topologically massive gauge theories,” An-

nals Phys. 140, 372 (1982) [Erratum-ibid. 185, 406.1988 APNYA,281,409 (1988APNYA,281,409-449.2000)].

4. M. Banados, C. Teitelboim and J. Zanelli, “The Black hole in three-dimensional space-time,” Phys. Rev. Lett. 69, 1849 (1992) [arXiv:hep-th/9204099].

5. D. Grumiller and N. Johansson, “Instability in cosmological topologically massivegravity at the chiral point,” JHEP 0807, 134 (2008) [arXiv:0805.2610 [hep-th]].

6. W. Li, W. Song and A. Strominger, “Chiral Gravity in Three Dimensions,” JHEP0804, 082 (2008) [arXiv:0801.4566 [hep-th]].

7. J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization ofAsymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun.Math. Phys. 104, 207 (1986).

8. S. Carlip, S. Deser, A. Waldron and D. K. Wise, “Cosmological Topologically MassiveGravitons and Photons,” arXiv:0803.3998 [hep-th].

9. K. Hotta, Y. Hyakutake, T. Kubota and H. Tanida, “Brown-Henneaux’s Canon-ical Approach to Topologically Massive Gravity,” JHEP 0807, 066 (2008)[arXiv:0805.2005 [hep-th]].W. Li, W. Song and A. Strominger, “Comment on ’Cosmological Topological MassiveGravitons and Photons’,” arXiv:0805.3101 [hep-th].M. I. Park, “Constraint Dynamics and Gravitons in Three Dimensions,”arXiv:0805.4328 [hep-th].I. Sachs and S. N. Solodukhin, “Quasi-Normal Modes in Topologically Massive Grav-ity,” JHEP 0808, 003 (2008) [arXiv:0806.1788 [hep-th]].D. A. Lowe and S. Roy, “Chiral geometries of (2+1)-d AdS gravity,” arXiv:0806.3070[hep-th].D. Grumiller, R. Jackiw and N. Johansson, “Canonical analysis of cosmological topo-logically massive gravity at the chiral point,” arXiv:0806.4185 [hep-th].S. Carlip, S. Deser, A. Waldron and D. K. Wise, “Topologically Massive AdS Gravity,”Phys. Lett. B 666, 272 (2008) [arXiv:0807.0486 [hep-th]].I. Sachs, “Quasi-Normal Modes for Logarithmic Conformal Field Theory,”arXiv:0807.1844 [hep-th].G. W. Gibbons, C. N. Pope and E. Sezgin, “The General Supersymmetric Solution ofTopologically Massive Supergravity,” arXiv:0807.2613 [hep-th].D. Anninos, W. Li, M. Padi, W. Song and A. Strominger, “Warped AdS3 Black Holes,”arXiv:0807.3040 [hep-th].

10. S. Carlip, “The Constraint Algebra of Topologically Massive AdS Gravity,”arXiv:0807.4152 [hep-th].

11. G. Giribet, M. Kleban and M. Porrati, “Topologically Massive Gravity at the ChiralPoint is Not Unitary,” arXiv:0807.4703 [hep-th].

12. A. Strominger, “A Simple Proof of the Chiral Gravity Conjecture,” arXiv:0808.0506[hep-th].

13. P. Kraus and F. Larsen, “Holographic gravitational anomalies,” JHEP 0601, 022

February 1, 2010 18:7 WSPC/INSTRUCTION FILE 0808.2575.new

434 D. GRUMILLER AND N. JOHANSSON

(2006) [arXiv:hep-th/0508218].14. M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, “Asymptotically anti-de Sitter

spacetimes and scalar fields with a logarithmic branch,” Phys. Rev. D 70, 044034(2004) [arXiv:hep-th/0404236].

15. K. Skenderis, M. Taylor and B. C. van Rees, “Topologically Massive Gravity and theAdS/CFT Correspondence,” JHEP 0909 (2009) 045 [arXiv:0906.4926 [hep-th]].

16. M. Henneaux, C. Martinez and R. Troncoso, “Asymptotically anti-de Sitter spacetimesin topologically massive gravity,” Phys. Rev. D 79 (2009) 081502R [arXiv:0901.2874[hep-th]].

17. A. Maloney, W. Song and A. Strominger, “Chiral Gravity, Log Gravity and ExtremalCFT,” arXiv:0903.4573 [hep-th].

18. S. Ertl, D. Grumiller and N. Johansson, “Erratum to ‘Instability in cosmologicaltopologically massive gravity at the chiral point’, 0805.2610,” arXiv:0910.1706 [hep-th].

MIT–CTP 4011, TUW 09–03, YITP–SB–09–04

Holographic counterterms from local

supersymmetry without boundary conditions

Daniel Grumiller a and Peter van Nieuwenhuizen b

aCenter for Theoretical Physics, Massachusetts Institute of Technology,77 Massachusetts Ave, Cambridge, MA 02139, USA

andInstitute for Theoretical Physics, Vienna University of Technology,

Wiedner Hauptstr. 8–10/136, Vienna, A-1040, AustriabC.N. Yang Institute for Theoretical Physics, Math 6-118, Stony Brook University,

Stony Brook, NY 11794-3840, USA

Abstract

We show in some lower-dimensional supergravity models that the holographic coun-terterms which are needed in the AdS/CFT correspondence to make the theoryfinite, coincide with the counterterms that are needed to make the action super-symmetric without imposing any boundary conditions on the fields.

Key words: holographic renormalization, holographic counterterms, supergravity,supersymmetry without boundary conditions, AdS/CFT correspondence

1 Introduction

Holographic renormalization is the by now well-known procedure of subtracting boundarycounterterms from the action in order to render the variational principle well-defined. Asan additional benefit the boundary-stress tensor usually becomes finite by this procedure,which provided the original motivation for holographic renormalization [1–9]. The full action

I = Ibulk + IGHY − Icounter (1)

consists of three parts: a bulk action Ibulk that generates the equations of motion (EOMs),a Gibbons–Hawking–York (GHY) boundary term IGHY that leads to a Dirichlet boundary

Email addresses: grumil@hep.itp.tuwien.ac.at (Daniel Grumiller),vannieu@insti.physics.sunysb.edu (Peter van Nieuwenhuizen).

Preprint submitted to Physics Letters B 1 February 2010

value problem, and the holographic counterterm Icounter. The latter guarantees that the firstvariation of the full action vanishes for all variations that preserve the boundary conditionsfor the fields:

δI = 0 (2)

While there are different techniques — for instance, the Hamilton–Jacobi method [10–13]— to implement the procedure of holographic renormalization, all of them have one featurein common: they require the specification of precise boundary conditions for the fields.

In applications this procedure works well, but conceptually it is not entirely satisfying, forthe following logical cycle arises: in order to determine the appropriate boundary conditionsfor the fields one should know already the behavior of classical solutions near the boundary.This requires knowledge of the EOMs, which are derived from the action. But in order toconsistently derive the EOMs by virtue of a variational principle from the action, one hasto know the action including all boundary terms. If the derivation of the latter appeals toappropriate boundary conditions we are back to square one.

In this note we show that supersymmetry (SUSY) can break the logical cycle. Namely, weapply the main credo of [14–16]: an action should be SUSY-invariant, even in the presence ofboundaries, without imposing any boundary conditions on the fields. In other words, off-shellthere are no boundary conditions needed for maintaining SUSY. Since many of the theorieswhere the issue of holographic renormalization arises are supergravity (SUGRA) theories,the requirement of SUSY is pertinent. Moreover, the main credo jibes well with our desireto avoid imposing boundary conditions on the fields in order to evade the logical cycle.However, at this point it is not at all clear if SUSY has anything to say about holographicrenormalization. The main purpose of our note is to exhibit that this is indeed the case.

For sake of specificity we restrict ourselves initially to a study of 3-dimensional SUGRAtheories. The simplest example is pure SUGRA with negative cosmological constant, whosebosonic version of the action I is given by the bulk action

Ibulk =1

2

∫

M

d3x√−g

(R− 2

ℓ2

)(3)

and the boundary action Ib = IGHY − Icounter, with [1–4, 7]

IGHY =∫

∂Md2x

√−hK (4a)

Icounter =∫

∂Md2x

√−h 1

ℓ(4b)

Here h is the determinant of the induced metric at the boundary ∂M of the spacetime Mand K is the trace of extrinsic curvature. Our goal is to derive the boundary action (4)from the knowledge of the bulk action (3) by imposing local SUSY. It is non-trivial andinteresting that this is possible.

To set up the stage we summarize in section 2 the results of [15], which lead to the GHYboundary term (4a). In section 3 we show that SUSY without boundary conditions au-

436

tomatically leads to the correct holographic counterterm (4b). To investigate whether ourconclusions apply also to other cases we consider generic 2-dimensional dilaton SUGRA insection 4. We find again that imposing SUSY without boundary conditions establishes thecorrect holographic counterterm.

2 Review of SUGRA without boundary conditions

We review now briefly the results of [15], whose conventions we adopt: Our Ricci-scalar ispositive for AdS. We set 8πGN = 1 and use signature (−,+,+). Upper case indices refer tothe bulk theory and lower case indices to the boundary theory. Indices from the beginningof the alphabet (A,B, . . . and a, b, . . . ) refer to an anholonomic frame (“flat indices”) andindices from the middle of the alphabet (M,N, . . . and m,n, . . . ) refer to a holonomic frame(“curved indices”). The boundary ∂M is a surface of constant x3, located at x3 = 0 (in

the bulk x3 > 0). The Lorenz gauge ea3 = 0 is imposed (and thus em

3 = 0). Defining

ǫ± = 12(1± γ 3)ǫ where γ 3 is constant, the unbroken half of SUSY is generated by ǫ+. When

considering SUSY transformations in this paper we always mean the “modified ǫ+ SUSY” of[15], a specific linear combination of SUSY and compensating Lorentz-transformations that

preserves the Lorenz gauge condition em3 = 0. The quantity ω and related hatted curvature

quantities such as extrinsic curvature K or gravitino field strength ψMN = DMψN − DNψM

always refer to supercovariant objects.

We are interested in constructing SUGRA actions of the form

I =∫

M

d3xLF −∫

∂Md2xLA (5)

and require that half of SUSY and diffeomorphisms along the boundary are preserved:

δξI = 0 δ′ǫ+I = 0 (6)

Here ξ refers to 2-dimensional diffeomorphisms within the boundary and δ′ǫ+ refers to theunbroken modified SUSY transformations. It is crucial for the credo stated in the introduc-tion that (6) holds without imposing boundary conditions on the fields. If this is the casethen we call an action (5) with the property (6) SUSY-invariant. 1 Suppose that in additionto the SUGRA multiplet (

eMA, ψM , S

)(7)

we have a (composite or fundamental) scalar multiplet

(A, χ, F

)(8)

1 By “SUSY-invariant” in this note we always mean “locally SUSY-invariant with respect tomodified ǫ+ SUSY and without imposing any boundary conditions on the (off-shell) fields”.

437

The main result of [15] is that the SUSY-invariant action is given by

I =∫

M

d3x e3

(F +

1

2ψMγ

Mχ +1

4AψMγ

MNψN + AS

)

︸ ︷︷ ︸LF

−∫

∂Md2x e2 A︸ ︷︷ ︸

LA

(9)

The bulk action in 3 + 1 dimensions was obtained in [17] while the 2 + 1 and 1 + 1 caseswere obtained in [18]. In the case of pure SUGRA the relevant multiplet realizing (8) is thethe scalar curvature multiplet

(S,

1

2γMN ψMN − 1

2γMψMS,

1

2R(ω) − 1

2ψMγN ψMN +

1

4SψMψM − 3

4S2

)(10)

The SUSY-invariant action (9) entails an ambiguity. Namely, consider a co-dimension 1spinor multiplet (χ′, A′) whose highest component contributes to the boundary action.This implies a shift of the boundary Lagrange-density LA → LA + LA′. Such a shift ispossible because LA′ is SUSY-invariant by itself. As demonstrated in [15] the ambiguity ofadding a SUSY-invariant boundary term can be fixed uniquely for pure SUGRA: withoutan appropriate co-dimension 1 multiplet the boundary action would contain the auxiliaryfield S linearly, which in turn would imply the (unphysical) boundary EOM e2 = 0. Theboundary term linear in S can be cancelled with a boundary term constructed from theextrinsic curvature multiplet (γaψa−, K + S). The result for the SUSY invariant action of3-dimensional pure SUGRA is

ISUGRA =1

2

∫

M

d3xe3

(R(ω) + ψMγ

MNKDNψK +1

2S2

)+

∫

∂Md2xe2

(K +

1

2ψa+γ

aγbψb−

)(11)

Setting the gravitino ψM to zero and eliminating the auxiliary field S by means of its EOMleads to the Einstein–Hilbert action (3) with the GHY boundary term (4a). The result(11) was derived without imposing any boundary conditions on the fields [15]. There is nocounterterm of the form (4b) in this example because we are not yet in AdS space.

3 Holographic counterterms from SUGRA without boundary conditions

The issue of holographic renormalization typically arises for theories where the EOMs leadto solutions for the metric that asymptote to AdS [7]. We are therefore led to consider3-dimensional SUGRA theories that allow for asymptotically AdS solutions. The simplestone is obtained from pure SUGRA by adding a cosmological constant multiplet

(1

ℓ, 0, 0

)(12)

438

where ℓ is the AdS radius. Inserting A = 1ℓ, χ = 0, F = 0 into the result for the SUSY-

invariant action (9) yields the SUSY-invariant cosmological constant action

IΛ =∫

M

d3xe3

(1

4ℓψMγ

MNψN +1

ℓS

)−

∫

∂Md2xe2

1

ℓ(13)

The total SUSY-invariant action IΛSUGRA is obtained by adding the cosmological constantaction (13) to the SUGRA action (11). The result is 2

IΛSUGRA =1

2

∫

M

d3xe3

(R(ω) + ψMγ

MNKDNψK +1

2ℓψMγ

MNψN +1

2S2 +

2

ℓS

)

+∫

∂Md2xe2

(K − 1

ℓ+

1

2ψa+γ

aγbψb−

)(14)

Setting the gravitino ψM to zero and eliminating the auxiliary field S by means of its EOMleads to the bosonic action

IΛEH =1

2

∫

M

d3x√−g

(R − 2

ℓ2

)+

∫

∂Md2x

√−h

(K − 1

ℓ

)(15)

This is the cosmological Einstein–Hilbert action with the GHY boundary term (4a) and theholographic counterterm (4b). 3 Thus, we have reached our goal to derive the result for theholographic counterterm from requiring SUSY-invariance of the action. We achieved thiswithout imposing any boundary conditions on the fields.

4 Two dimensional dilaton SUGRA

So far we have treated SUGRA in three spacetime dimensions. To investigate whether ourconclusions apply also to other cases we consider here 2-dimensional dilaton SUGRA. DilatonSUGRA in two dimensions was introduced by Park and Strominger [21], based upon thework by Howe [22]. It was studied in detail e.g. in [23–27].

We need the following multiplets. The 2-dimensional curvature multiplet [18]

(S,

1

2γMN ψMN − 1

2γMψMS

︸ ︷︷ ︸:=ζ

,1

2R(ω) − 1

2ψMγN ψMN +

1

4SψMψM − S2

)(16)

2 Using a similar philosophy as in the present work, Luckock and Moss derived the action (14) toorder fermion-squared [19] (see also [20]), which happens to be the complete result as shown in thepresent work.3 Of course, one ambiguity always remains: we can add an arbitrary (finite or infinite) constantto the action, like the logarithmic subtraction linked with the Weyl anomaly [6]. This ambiguitycannot be fixed at the level of the action, but only upon appealing to specific solutions of theEOM, e.g. by demanding that the free energy of the ground state solution vanishes.

439

the dilaton multiplet (X, χ, F

)(17)

and the pre-potential multiplet

(u(X), u′(X)χ, u′(X)F − 1

2u′′(X)χχ

)(18)

The bosonic bulk action without auxiliary fields is of the form

Ibulk

DG=

1

2

∫

M

d2x√−g

(XR− 2u(X)u′(X)

)(19)

In [28] the full boundary action Ib = IDGHY − IDcounter was derived using the Hamilton–Jacobimethod of holographic renormalization:

IDGHY =∫

∂Md2x

√−hXK (20a)

IDcounter =∫

∂Md2x

√−h u(X) (20b)

Our goal to derive the boundary terms (20) from SUSY-invariance.

The product of the curvature multiplet (16) and the dilaton multiplet (17) leads to

(SX, Sχ+Xζ,

1

2XR(ω) −XS2 + SF − χζ − 1

2XψMγN ψMN +

1

4XSψMψM

)(21)

For a scalar multiplet (A, χ, F ) the 2-dimensional version of the SUSY-invariant action (9)is given by

I =∫

M

d2xe2

(F +

1

2ψMγ

M χ +1

4AψMγ

MNψN + AS

)−

∫

∂Mdxe1 A (22)

We plug now the multiplet (21) into the 2-dimensional SUSY-invariant action (22), thenwe do the same with the pre-potential multiplet (18) and add both contributions. Thisprocedure yields an action IDSG that contains a boundary term linear in the auxiliary field.Such a term is problematic, because elimination of the auxiliary field S implies an un-physical boundary EOM, e1X = 0. 4 In order to cancel the offending term we employ theco-dimension 1 multiplet

(Xγaψa−, X(K + S) − χ−γ

aψa−

)(23)

and add the corresponding SUSY-invariant boundary action to the action IDSG. This obtains

4 Setting e1 = 0 leads to a degenerate boundary metric. Setting X = 0 eliminates the space ofsolutions and implies infinite gravitational coupling at the boundary.

440

uniquely the SUSY invariant dilaton SUGRA action

IDSG =1

2

∫

M

d2xe2

(XR(ω) + 2SF + 2u(X)S + 2u′(X)F − χγMN ψMN

+1

2u(X)ψMγ

MNψN + u′(X)ψMγMχ− u′′(X)χχ)

+∫

∂Md2xe1

(XK − u(X) +

1

2Xψa+γ

aγbψb− − χ−γaψa−

)(24)

The bulk part of the action (24) up to notational changes and integrating out auxiliaryfields coincides with the actions used in [21, 23–27]. The boundary part of the action (24)required to maintain SUSY-invariance is a new result.

Integrating out the auxiliary field density e2S leads to a functional delta-function whose ar-gument implies the constraint F = −u(X) for the dilaton auxiliary field F upon integratingout the latter. No Jacobians arise from these integrations. Setting all spinors to zero andintegrating out S and F , the action (24) reduces to

IDG =1

2

∫

M

d2x√−g

(XR− 2u(X)u(X)′

)+

∫

∂Md2x

√−h

(XK − u(X)

)(25)

Comparison of the bulk action in (25) with the bulk action (19) shows that they coincide.Comparison of the boundary action in (25) with the boundary action (20) establishes againthe remarkable result that SUSY-invariance automatically leads to the GHY boundary term(20a) and to the holographic counterterm (20b). The result (25) was derived without im-posing any boundary conditions on the fields.

For simplicity we have neglected a kinetic term for the dilaton. It can be introduced througha dilaton-dependent Weyl rescaling, see for instance section 5 in [29] for details. It can bechecked easily that our procedure leading to (25) generalizes to models containing a kineticterm for the dilaton field. In this way we recover (up to notational changes) Eq. (7.1) of [28]. 5

5 Conclusions

We have demonstrated in this note that imposing SUSY without boundary conditions onthe fields automatically entails the correct holographic counterterms, at least in the lower-dimensional examples considered here. This result is reminiscent of the findings by Larsenand McNees [12], who showed for inflationary spacetimes that the requirement of diffeomor-phism invariance leads to the correct holographic counterterms at the late time boundary.We intend to apply our procedure to other supersymmetric theories that require holographicrenormalization, like pure gravity in AdS4, AdS5 or cosmological topological supergravity

5 The relation between pre-potential u(X) and various functions of the dilaton is as follows:e−Q(X)w(X) = u2(X). For the case of interest Q(X) = U(X) = 0 we have the simple relationsV (X) = −u(X)u′(X) and w(X) = u2(X).

441

in three dimensions. One might also apply our program to theories with rigid SUSY onan AdS background, or SUGRA theories with local superconformal invariance. Yet anotherinteresting case are branes in the presence of a Born-Infeld action. Bosonic systems can betreated in the same way by viewing them as truncations of supersymmetric systems.

Some important open questions are: Why does our program work? Does it work in higherdimensions? Does SUSY require finiteness of response functions like the Brown–York stresstensor? Concerning the last question we recall that the infrared divergences near the AdSboundary are related by duality to the ultraviolet divergences in the boundary theory. SUSYhas been quite successful in curing ultraviolet divergences in various theories, and perhapsthis is why local SUSY without boundary conditions is capable to produce holographiccounterterms.

Acknowledgments

We thank Dmitry Belyaev, Simone Giombi, Robert McNees, Leonardo Rastelli and KostasSkenderis for discussion.

DG was supported by the project MC-OIF 021421 of the European Commission underthe Sixth EU Framework Programme for Research and Technological Development (FP6).Research at the Massachusetts Institute of Technology is supported in part by funds pro-vided by the U.S. Department of Energy (DoE) under the cooperative research agreementDEFG02-05ER41360. During the final stage DG was supported by the START projectY435-N16 of the Austrian Science Foundation (FWF).

The research of PvN is supported by the NSF grant PHY-0653342.

References

[1] V. Balasubramanian and P. Kraus, Commun. Math. Phys. 208 (1999) 413, hep-th/9902121.

[2] R. Emparan, C.V. Johnson and R.C. Myers, Phys. Rev. D60 (1999) 104001, hep-th/9903238.

[3] R.C. Myers, Phys. Rev. D60 (1999) 046002, hep-th/9903203.

[4] S. de Haro, S.N. Solodukhin and K. Skenderis, Commun. Math. Phys. 217 (2001) 595, hep-th/0002230.

[5] M. Bianchi, D.Z. Freedman and K. Skenderis, Nucl. Phys. B631 (2002) 159, hep-th/0112119.

[6] M. Henningson and K. Skenderis, JHEP 07 (1998) 023, hep-th/9806087. Fortsch. Phys. 48(2000) 125, hep-th/9812032.

[7] K. Skenderis, Class. Quant. Grav. 19 (2002) 5849, hep-th/0209067.

442

[8] I. Papadimitriou and K. Skenderis, (2004), hep-th/0404176.

[9] R.B. Mann, Phys. Rev. D60 (1999) 104047, hep-th/9903229.

[10] J. de Boer, E.P. Verlinde and H.L. Verlinde, JHEP 08 (2000) 003, hep-th/9912012.

[11] D. Martelli and W. Muck, Nucl. Phys. B654 (2003) 248, hep-th/0205061.

[12] F. Larsen and R. McNees, JHEP 07 (2004) 062, hep-th/0402050.

[13] A. Batrachenko et al., JHEP 05 (2005) 034, hep-th/0408205.

[14] D.V. Belyaev, JHEP 01 (2006) 047, hep-th/0509172.

[15] D.V. Belyaev and P. van Nieuwenhuizen, JHEP 02 (2008) 047, 0711.2272.

[16] D.V. Belyaev and P. van Nieuwenhuizen, JHEP 04 (2008) 008, 0801.2377.

[17] S. Ferrara and P. van Nieuwenhuizen, Phys. Lett. B78 (1978) 573. Phys. Lett. B76 (1978)404.

[18] T. Uematsu, Z. Phys. C29 (1985) 143.

[19] H. Luckock and I. Moss, Class. Quant. Grav. 6 (1989) 1993.

[20] I.G. Moss, Nucl. Phys. B729 (2005) 179, hep-th/0403106.

[21] Y.C. Park and A. Strominger, Phys. Rev. D47 (1993) 1569, arXiv:hep-th/9210017.

[22] P.S. Howe, J. Phys. A12 (1979) 393.

[23] J.M. Izquierdo, Phys. Rev. D59 (1999) 084017, arXiv:hep-th/9807007.

[24] M. Ertl, W. Kummer and T. Strobl, JHEP 01 (2001) 042, arXiv:hep-th/0012219.

[25] D. Grumiller, W. Kummer and D.V. Vassilevich, Phys. Rept. 369 (2002) 327, hep-th/0204253.

[26] L. Bergamin and W. Kummer, JHEP 05 (2003) 074, hep-th/0209209.

[27] L. Bergamin, D. Grumiller and W. Kummer, J. Phys. A37 (2004) 3881, hep-th/0310006.

[28] D. Grumiller and R. McNees, JHEP 04 (2007) 074, hep-th/0703230.

[29] L. Bergamin and W. Kummer, Phys. Rev. D68 (2003) 104005, hep-th/0306217.

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Preprint typeset in JHEP style - HYPER VERSION MIT–CTP 4079, LMU–ASC 45/09

TUW–09–13, ESI 2188

AdS3/LCFT2 – Correlators in Cosmological

Topologically Massive Gravity

Daniel Grumiller

Institute for Theoretical Physics, Vienna University of Technology,

Wiedner Hauptstr. 8–10/136, A-1040 Vienna, Austria

and

Center for Theoretical Physics, Massachusetts Institute of Technology,

77 Massachusetts Ave, Cambridge, MA 02139, USA

Email: grumil@hep.itp.tuwien.ac.at

Ivo Sachs

Arnold Sommerfeld Center for Theoretical Physics,

Theresienstrasse 37, D-80333 Munich, Germany

Email: Ivo.Sachs@physik.uni-muenchen.de

Abstract: For cosmological topologically massive gravity at the chiral point we

calculate momentum space 2- and 3-point correlators of operators in the postulated

dual CFT on the cylinder. These operators are sourced by the bulk and boundary

gravitons. Our correlators are fully consistent with the proposal that cosmological

topologically massive gravity at the chiral point is dual to a logarithmic CFT. In the

process we give a complete classification of normalizable and non-normalizeable left,

right and logarithmic solutions to the linearized equations of motion in global AdS3.

Keywords: logarithmic CFT, AdS/CFT, gravity in three dimensions,

cosmological topologically massive gravity, new massive gravity.

Contents

1. Introduction 447

2. Perturbative expansion of the action 449

2.1 Post-variational identities 450

2.2 Second variation of the action 451

2.3 Third variation of the action 452

3. Linearized solutions 453

3.1 Classification of linearized solutions 453

3.2 Normalizable modes, primaries and descendants 454

3.3 Generic massive solutions 456

3.4 Non-normalizable left and right solutions 458

3.5 Non-normalizable logarithmic solutions 462

4. Correlators 466

4.1 Two-point correlators 466

4.1.1 Einstein gravity 466

4.1.2 Cosmological topologically massive gravity at the chiral point 468

4.2 Three-point correlators 472

4.2.1 Boundary terms 472

4.2.2 Correlators without log insertions 474

4.2.3 Single log insertions 474

4.2.4 Multiple log insertions and limit of large weights 476

4.3 Comparison with Euclidean logarithmic CFT correlators 480

5. Discussion 482

A. Boundary conditions 485

B. Some hypergeometric identities 489

C. Non-normalizable modes 493

C.1 Left and right branch 493

C.2 Logarithmic branch 496

D. Correlation functions in Euclidean logarithmic CFT 501

– 446 –

1. Introduction

Cosmological topologically massive gravity [1] (CTMG) is a 3-dimensional theory of

gravity that exhibits gravitons [2, 3] and black holes [4]. Its action is given by

SCTMG = − 1

κ2SEH − 1

κ2SCS (1.1)

The Einstein–Hilbert action with negative cosmological constant Λ = −1/ℓ2

SEH =

∫

d3y√−g

[

R +2

ℓ2

]

(1.2)

is supplemented by the Chern–Simons action for the Christoffel-connection

SCS =1

2µ

∫

d3y ǫλµνΓρσλ

[

∂µΓσρν +

2

3Γσ

κµΓκ

σν

]

(1.3)

If the coupling constant µ and the AdS radius ℓ satisfy the condition

µℓ = 1 (1.4)

the theory is called “CTMG at the chiral point” (CCTMG). The condition (1.4) is

special because one of the central charges of the asymptotic isometry group vanishes,

cL = 0, cR 6= 0 [5].

This observation together with the fact that CTMG supports asymptotically

AdS solutions was the motivation for Li, Song and Strominger to consider CTMG at

the chiral point, dubbed “chiral gravity” [6]. They conjectured that CTMG at the

chiral point is dual to a chiral CFT. However, there are examples of CFTs that have

vanishing left-moving central charge without being chiral, namely logarithmic CFTs

(LCFTs), see [7–9] and references therein. The defining property of a LCFT is that

the Virasoro generator L0 is not diagonalizable. For instance,

L0

(

ψlog

ψL

)

=

(

2 12

0 2

) (

ψlog

ψL

)

(1.5)

In the parlance of LCFT literature the mode ψlog is the logarithmic partner of the

mode ψL. Interestingly, precisely the form (1.5) was found for CTMG at the chiral

point [10]. The mode ψL is the left-moving boundary graviton and its logarithmic

partner ψlog is essentially the bulk graviton, a propagating spin-2 excitation that is

present for all (finite) values of µ and ℓ [11–13]. Moreover, it was shown that ψlog is

compatible with asymptotic AdS behavior [10]. This was confirmed independently

in [14–17]. For additional recent literature on CTMG cf. e.g. [18–60].

Given that (1.5) is realized in CCTMG, it appears that the dual CFT is not

chiral but logarithmic [10], although a chiral CFT might be obtained as a consistent

truncation [16]. So far no good gravity duals for LCFTs are known, see [61–65]. If

– 447 –

an AdS3/LCFT2 dictionary could be established, we can use CCTMG as a gravity

dual for certain (strongly coupled) LCFTs, with potential applications in condensed

matter physics where LCFTs are applied. It is thus of importance to provide further

evidence for the proposal1 [10] that the CFT dual to CCTMG, if it exists, is a LCFT.

The LCFT conjecture can be tested as follows: calculate correlators on the grav-

ity side, relate them by the purported AdS3/LCFT2 correspondence to correlators

on the CFT side and check if these correlators really have the properties as required

by a LCFT. Conformal symmetry poses particularly stringent constraints on 2- and

3-point correlators [66], so if the conjecture is true then these correlators must have

an essentially unique form in CCTMG. On the other hand, if the conjecture is wrong

then it is suggestive that some of the stringent LCFT constraints may be violated

already at the level of 2- or 3-point correlators. The calculation of 2- and 3-point

correlators on the gravity side is thus a major step towards establishing the LCFT

conjecture. Such a check was carried out recently by Skenderis, Taylor and van

Rees [17] for 2-point correlators. They found perfect agreement with the LCFT 2-

point correlators, which supports the conjecture that CTMG at the chiral point is

dual to a LCFT.

In this paper we provide the basis for the calculation of arbitrary correlators

on the gravity side. In particular, we construct all regular non-normalizable left,

right and logarithmic modes in global coordinates in terms of elementary functions

and show how these modes are organized in SL(2,R) representations. We then plug

these modes into the second and third variation of the action to calculate 2- and 3-

point correlators. We find perfect agreement with the behavior expected from LCFT

correlators. Thus, we corroborate the conjecture that CCTMG is dual to a LCFT.

This paper is organized as follows: In section 2 we calculate the first three

variations of the action (1.1). In section 3 we discuss generic solutions of the linearized

equations of motion, find all regular non-normalizable left, right and logarithmic

modes in global coordinates, and discuss their properties. In section 4 we calculate on

the gravity side 2- and 3-point correlators in CCTMG. We conclude with a discussion

in section 5, where we address the status of CCTMG as a possible gravity dual to

LCFTs and as a tentative toy model for quantum gravity. We mention also spin-offs

and generalizations of our calculations.

Before starting we mention some of our conventions, which coincide with the

conventions used in [10]. Our signature is (−,+,+). The overall sign in front of the

CTMG action (1.1) is irrelevant in the present work, but for sake of completeness we

mention that we have chosen it such that black hole solutions have positive energy

and graviton excitations negative energy [6]. In three dimensions the Riemann tensor

is determined uniquely from the Ricci tensor Rµν = Rσµσν as follows: Rσρµν =

(

Rσµgρν +Rρνgσµ −Rσνgρµ −Rρµgσν

)

− 12R

(

gσµgρν − gσνgρµ

)

. The sign of the Ricci

1This possibility was pointed out first by John McGreevy during a talk by Andy Strominger at

MIT in May 2008, four days before the posting of Ref. [10].

– 448 –

tensor is defined by Rµν = ∂σΓσµν + . . . The Levi-Civita symbol is denoted by ǫαβγ

(its sign is fixed by ǫτφρ = +1), and the Levi-Civita tensor by εαβγ = ǫαβγ/√−g.

Our gravitational coupling constant κ is related to Newton’s constant GN by κ2 =

16π GN .

2. Perturbative expansion of the action

In this section we calculate the first three variations of the CTMG action (1.1). To

fix our remaining notations we briefly review the first variation of the CTMG action

(1.1), modulo boundary terms, which will be taken into account in due course:

δSCCTMG = − 1

κ2

∫

d3y√−g δgµν

[

Gµν +1

µCµν

]

(2.1)

Setting δSCCTMG = 0 leads to the equations of motion (EOM)

Gµν +1

µCµν = 0 (2.2)

Here Gµν is the modified Einstein tensor

Gµν = Rµν −1

2gµνR− 1

ℓ2gµν (2.3)

and Cµν is the Cotton tensor

Cµν = εµκσ∇κ

(

Rσν −1

4gσνR

)

(2.4)

We are interested in the second and third variation of the action with respect to

metric variations of the form

δgµν = hµν δgµν = −hµν (2.5)

Indices of h are raised and lowered with the background metric g. In addition to the

second variation of the metric

δ(2)gµν = 0 δ(2)gµν = −δhµν = 2hµαh

αν (2.6)

we are going to need the variation of various geometric quantities, which we col-

lect in the following. The variation of the volume element is given by 2 δ√−g =√−g gµνhµν =

√−g gµνhµν . The variations of the Riemann tensor

δRαµβν = ∇βδΓ

αµν −∇νδΓ

αβµ (2.7)

and the Ricci tensor δRµν = δRαµαν are determined from the variation of the

Christoffel connection. It is useful to have a formula valid for arbitrary variations of

the Christoffel connection:

δ(n)Γρσλ =

n

2

(

∇σhλκ + ∇λhσκ −∇κhσλ

)

δ(n−1)gρκ (2.8)

– 449 –

The variation of the Cotton tensor density is given by

δ(√−g Cµν

)

=√−g

(

δgµλCλ

ν

)

−gµλ ǫλσκ

(

δ(∇κRσν)−1

4(δgσν∂κR+gσν∂κ δR)

)

(2.9)

For our purposes we are going to need only the first and second variation of various

quantities, including the second variation of the Ricci tensor

δ(2)Rµν = ∇αδ(2)Γα

µν −∇µδ(2)Γα

να + 2δΓκµν δΓ

λκλ − 2δΓκ

λµ δΓλ

κν (2.10)

and of the Cotton tensor density

δ(2)(√−g Cµν

)

= 2 δgµλ δ(√−g Cλ

ν

)

− gµλ ǫλσκ δ(2)

(

∇κRσν

)

+1

4Ξµν (2.11)

where Ξµν = 2gµλ ǫλσκ δgσν ∂κ δR+ gµλgνσ ǫ

λσκ ∂κ δ(2)R. In the formula for the second

variation of the Cotton tensor density (2.11) we have separated two contributions in

the tensor Ξµν , because they vanish either after contraction with a symmetric tensor

like δgµν or due to background and gauge fixing identities. We describe now these

identities in detail.

2.1 Post-variational identities

The identities above are valid generically in three dimensions. Now we consider

identities that can be used only after performing all variations.

The first set of such identities comes from assuming that the background metric

is pure AdS

ds2 = gµν dxµ dxν = ℓ2(

dρ2 − 1

4cosh2ρ (du+ dv)2 +

1

4sinh2ρ (du− dv)2

)

(2.12)

and therefore maximally symmetric.

R = − 6

ℓ2Rµν = − 2

ℓ2gµν Cµν = 0 (2.13)

The second set of identities comes from assuming that the linear fluctuations obey

the transverse-traceless gauge conditions [6, 67, 68]:

hµνgµν = 0 ∇µhµν = 0 (2.14)

Both sets together imply further useful identities, which we collect here:

hµνRµν = 0 gµνδRµν = 0 δR = 0 δΓανα = 0 δ

√−g = 0 (2.15)

The second derivative of the metric variation simplifies to

∇κ∇σhκλ = [∇κ,∇σ] h

κλ = − 3

ℓ2hσλ (2.16)

– 450 –

The variations of the Ricci tensor δRµν = ∇αδΓα

µν = −12

(

∇2hµν + 6ℓ2hµν

)

and of the

modified Einstein tensor

δGµν = −1

2

(

∇2hµν +2

ℓ2hµν

)

(2.17)

also simplify considerably. The second variation of the modified Einstein tensor

contracted with the fluctuation hµν simplifies to

hµν δ(2)Gµν = hµν δ(2)Rµν = hµν∇α δ(2)Γα

µν − hµν∇µ δ(2)Γα

να − 2hµν δΓκλµ δΓ

λκν

(2.18)

Finally, the variation of the Riemann tensor can be expressed in terms of variations

of the Einstein tensor and the metric.

δRσρµν = δGσµ gρν + δGρν gσµ − δGσν gρµ − δGρµ gσν

− 1

ℓ2(

δgσµ gρν + δgρν gσµ − δgσν gρµ − δgρµ gσν

)

(2.19)

2.2 Second variation of the action

It is convenient for later applications to denote the two variations of the metric by

δgµν and by hµν , respectively. For our purposes we can neglect terms that vanish on

the background. The second variation of the action

δ(2)SCCTMG = − 1

κ2

∫

d3y√−g δgµν

[

δGµν(h) +1

µδCµν(h)

]

=1

κ2

∫

d3y δL(2) (2.20)

leads to the linearized EOM for the fluctuations hµν in the gauge (2.14)

δGµν(h) +1

µδCµν(h) = −1

2

(

∇2 +2

ℓ2)(

hµν +1

µεµ

αβ∇αhβν

)

= 0 (2.21)

Defining the mutually commuting first order operators

(

DM)β

µ= δβ

µ +1

µεµ

αβ∇α

(

DL/R)β

µ= δβ

µ ± ℓ εµαβ∇α (2.22)

the linearized EOM can be reformulated as [6]

(DM δG(h))µν = (DMDLDRh)µν = 0 (2.23)

Here we have taken advantage of the identities δGµν(h)+ 1µδCµν(h) = (DM δG(h))µν

and

2ℓ2 δGµν(h) = (DLDRh)µν (2.24)

A mode annihilated by DM (DL) [DR] is called massive (left-moving) [right-moving]

and is denoted by ψM (ψL) [ψR].

– 451 –

2.3 Third variation of the action

In analogy with section 2.2 we parameterize the third variation in terms of three

independent fields δg, h and k. The third variation of the action is then given by

δ(3)SCCTMG = − 1

κ2

∫

d3y√−g δgµν

[

δ(2)Rµν(h, k) +1

µδ(2)Cµν(h, k)

]

(2.25)

where we have used (2.18). In (2.25) and all formulas below we always assume

that the metric is given by the AdS background (2.12), all fluctuations around it

are transverse-traceless (2.14), and solve the linearized EOM (2.21). Therefore, we

exploit all the identities derived above to simplify expressions like the second variation

of the Cotton tensor density (2.11). The second variation of the Cotton tensor

δ(2)Cµν(h, k) = hµλ gλσ δCσν(k) + kµλ g

λσ δCσν(h) − εµσκδ(2)

(

∇κRσν

)

(h, k) (2.26)

can be re-expressed using the linearized EOM (2.21) and the second variation of the

Ricci-tensor (2.10):

δ(2)Cµν(h, k) = µ∆µν(h, k) − εµσκ∇κ δ

(2)Rσν(h, k) (2.27)

with

∆µν(h, k) = − 1

2ℓ2kσ

µ

(

DLDRh)

σν+

1

2µℓ2εµ

σκ δΓακν(k)

(

DLDRh)

σα+ h↔ k (2.28)

We have exploited the identity (2.24) to bring ∆µν(h, k) into the form above. The

second variation of the Ricci tensor yields

δ(2)Rµν(h, k) = hαβ δR

βµνα(k)+

1

4

(

2∇λhκµ∇λkκν −2∇λh

κµ∇κk

λν +∇µh

κλ∇νk

λκ

)

+h↔ k

(2.29)

The definition (2.22) of the linear operator DM allows us to provide a convenient

re-formulation of the third variation of the action:

δ(3)SCCTMG(δg, h, k) = − 1

κ2

∫

d3y√−g δgµν

[

(

DM δ(2)R(h, k))

µν+ ∆µν(h, k)

]

=1

κ2

∫

d3y δL(3) (2.30)

With the formula for the variation of the Riemann tensor (2.19) and the definitions

(2.22), (2.28) and (2.29) we obtain explicitly:

δL(3) = −δgµν[

(

DM)

µβ(1

4(2∇λh

κβ∇λkκν − 2∇λh

κβ∇κk

λν + ∇βh

κλ∇νk

λκ) − 1

ℓ2hα

βkαν

)

+1

2µℓ2εµ

σκ(

∇σ(kακ(DLDRh)αν) − δΓνκα(k)(DLDRh)α

σ

)

+ h↔ k]√−g (2.31)

Our final result for the third variation of the action, (2.30) with (2.31), requires the

definitions (2.8), (2.22) but otherwise is explicit in the three variations δg, h and k.

If h and k are both linear combinations of only left- and right-moving modes the

ε-term in the second line of (2.31) vanishes.

– 452 –

3. Linearized solutions

According to the standard AdS/CFT dictionary CFT correlators are obtained upon

insertion of non-normalizable solutions of the linearized EOM into variations of the

action [69]. We have considered the first three variations of the action in the previous

section. In this section we find all solutions to the linearized EOM (2.23). We start

by classifying them in section 3.1. We review normalizable modes, primaries and

their descendants in section 3.2. We then discuss the massive branch in section 3.3,

since the corresponding solutions encompass all other solutions as special (sometimes

singular) limits. We construct explicitly all regular non-normalizable left and right

modes in sections 3.4, and show how to obtain regular non-normalizable logarithmic

modes in section 3.5. We also unravel their algebraic properties. For simplicity we

set ℓ = 1 from now on.

3.1 Classification of linearized solutions

The linearized EOM (2.23) contain three mutually commuting first order operators

(2.22). If they are non-degenerate we can build the general solution from three

branches: massive, left and right branches, whose modes are annihilated, respectively,

by DM , DL, and DR. At the chiral point DM and DL degenerate with each other and

we obtain instead the following three branches: logarithmic, left and right, where the

logarithmic modes are annihilated by (DL)2 but not by DL.

For each branch the linearized solutions can be regular or singular at the origin

ρ = 0. A mode ψ is called singular if at least one of its components diverges at

ρ = 0; clearly, perturbation theory breaks down for such a solution near ρ = 0 since

the AdS background metric remains bounded at the origin, and thus the linearized

solution no longer is a small perturbation there. In the absence of point particles or

black holes the singular modes should be discarded for consistency. Of main interest

to us are therefore regular modes.

The asymptotic (large ρ) behavior allows us to classify modes ψ into normalizable

and non-normalizable ones. This classification is very simple in Gaussian normal

coordinates2 for the perturbed metric g = g + ψ, where g is the background metric

(2.12)(

gµν + ψGNC

µν

)

dxµ dxν = dρ2 +(

gij(xk, ρ) + ψij(x

k, ρ))

dxi dxj (3.1)

For modes that allow a Fefferman–Graham expansion (all left and right modes)

ψij(xk, ρ) = ψ

(0)ij (xk) e2ρ + ψ

(1)ij (xk)ρ+ ψ

(2)ij (xk) + . . . (3.2)

2Below we are not going to use Gaussian normal coordinates, but rather modes ψ in transverse-

traceless gauge. The associated coordinate transformations are very simple asymptotically. We are

not going to provide them explicitly. See appendix A for a summary of boundary conditions.

– 453 –

normalizability means ψ(0)ij (xk) = 0. The non-normalizable left and right modes have

a non-zero leading term in the Fefferman–Graham expansion (3.2), ψ(0)ij 6= 0. Ac-

cording to the AdS/CFT dictionary these non-normalizable modes act as sources for

the operators associated with the left- and right-moving boundary gravitons. Mas-

sive modes in general are not compatible with the expansion (3.2). The logarithmic

modes discovered in [10] are compatible with the Fefferman–Graham expansion (3.2)

with ψ(0)ij = 0 and thus they are normalizable. One of the goals of this section is

to find their non-normalizable counterparts, because they are needed for correlators

involving insertions of logarithmic modes [17].

3.2 Normalizable modes, primaries and descendants

Li, Song and Strominger exploited the SL(2,R)L × SL(2,R)R isometry algebra of

the AdS3 background (2.12) in their construction of normalizable regular primaries

for the massive, left and right branches. The normalizable modes are descendants

of primaries with respect to the isometry algebra. The properties of the isometry

algebra will also be useful for the non-normalizable modes constructed in section 3.4.

We summarize now briefly the relevant formulas.

The SL(2,R)L generators read [we recall that u = t+ φ, v = t− φ, see (2.12)]

L0 = i∂u (3.3)

L+ = ie−iu(cosh 2ρ

sinh 2ρ∂u − 1

sinh 2ρ∂v +

i

2∂ρ

)

(3.4)

L− = ieiu(cosh 2ρ

sinh 2ρ∂u − 1

sinh 2ρ∂v −

i

2∂ρ

)

(3.5)

with algebra[

L0, L±

]

= ±L± ,[

L−, L+

]

= 2L0 (3.6)

and quadratic Casimir

L2 =1

2

(

L−L+ + L+L−

)

− L20 . (3.7)

The SL(2,R)R generators L0, L+, L− satisfy the same algebra and are given by

(3.3)-(3.5) with u↔ v and L↔ L. The equivalence

(

DLDRψ)

µν= 0 ↔ (L2 + L2 + 2)ψµν = 0 (3.8)

is useful to construct solutions of the linearized Einstein equations. Namely, for

primaries L−ψ0 = 0, L−ψ0 = 0 we can generate new solutions of the linearized

EOM by acting on ψ0 with the ladder operators L+, L+. It should be noted that

general solutions to the linearized EOM are not descendants of primaries. The latter

– 454 –

correspond to modes that are regular and normalizable. Similar remarks apply to

the logarithmic modes. Starting from the equivalence3

(

(DL)2(DR)2ψ)

µν= 0 ↔ (L2 + L2 + 2)2ψµν = 0 (3.9)

all regular normalizable logarithmic modes can be written as descendants of the

logarithmic primary algebraically. We recall now in a bit more detail how this works.

Starting point is the separation Ansatz

ψµν(h, h) = e−ihu−ihv

Fuu(ρ) Fuv(ρ) Fuρ(ρ)

Fvv(ρ) Fvρ(ρ)

Fρρ(ρ)

(3.10)

so that the modes ψ are Eigenvectors of L0 and L0:

L0ψ = hψ L0ψ = hψ (3.11)

The Eigenvalues h, h are the weights of the state ψ if it is a primary, and otherwise

they are sums of weights and levels. For sake of brevity we shall always refer to them

as “weights”, even when ψ is not a primary. Periodicity in the angular coordinate

requires that the difference of the weights, the angular momentum, is an integer. We

shall always assume that this is the case. If additionally the sum of the weights (and

therefore the weights) are integer then additionally periodicity in time is guaranteed.

We do not necessarily assume that this is the case.

We focus now on the regular normalizable left branch, DLψL = 0. For primaries,

L−ψ = 0 = L−ψ, it is then required that h = 2, h = 0. The angular momentum

equals to 2 and the excitation is a (boundary) graviton. The corresponding primary

ψL is then given by

ψLµν(2, 0) =

e−2iu

cosh4ρ

14

sinh2(2ρ) 0 i2sinh (2ρ)

0 0 0i2sinh (2ρ) 0 −1

µν

(3.12)

Descendants of the primary are obtained by acting on it repeatedly with L+ and/or

L+. For instance we have

ψLµν(2 + n, 0) =

(

(L+)nψL(2, 0))

µν∝ e−inu tanhnρψL

µν(2, 0) (3.13)

The action of L+ on the left primary is more complicated. For later purposes we note

that the vµ-components vanish for the primary (3.12) and consequently the first L+

descendant has a vanishing vv-component, but all further descendants have ψLvv 6= 0:

ψLvv(h, 0) = ψL

vv(h, 1) = 0 ψLvv(h, h) 6= 0 if h ≥ 2 (3.14)

3The linearized equation in (3.9) contains not only linearized solutions of CCTMG, but also

linearized solutions of New Massive Gravity at a chiral point [70]. However, for the same reason

that left and right modes do not mix, the logarithmic modes do not mix: CCTMG solutions always

generate other CCTMG solutions when acting on them with ladder operators L±, L±.

– 455 –

The right modes are obtained from the left modes by exchanging u ↔ v and

h ↔ h. We do not address them separately. Both the left and the right modes are

pure gauge in the bulk and therefore do not constitute a local physical degree of

freedom. The only physical bulk degree of freedom comes therefore from the massive

or the logarithmic branch. The massive modes will be discussed extensively in section

3.3 below.

The regular normalizable logarithmic modes, (DL)2ψlog = 0, are related to the

regular normalizable left modes by [10]

ψlogµν = −1

2(i(u+ v) + ln cosh2ρ)ψL

µν (3.15)

A particular example for a logarithmic mode is the normalizable regular logarithmic

primary: ψlogµν (2, 0) = −1

2(i(u + v) + ln cosh2ρ)ψL

µν(2, 0). By construction, the loga-

rithmic primary ψlogµν (2, 0) is annihilated by L− and L−. Acting on it with L+ and

L+ produces a tower of descendants. The logarithmic modes obtained in this way

are not Eigenstates of L0 and L0, but only of their difference [cf. (1.5)]:

L0 ψlogµν = hψlog

µν +1

2ψL

µν L0 ψlogµν = hψlog

µν +1

2ψL

µν (3.16)

All the modes above are regular at the origin ρ = 0. In appendix A we discuss

point particle modes as particular examples of singular modes. We do not further

dwell on this case and consider from now on exclusively regular modes. Having

classified the normalizable modes, we now describe the most general set of regular

modes. To this end it is sufficient to consider the massive branch and extract the

other branches as certain limits thereof.

3.3 Generic massive solutions

In this subsection we discuss generic solutions to the EOM that are not necessarily

descendants of the primaries. This is tantamount to giving up the normalizabil-

ity condition. Non-normalizable modes play the role of sources in the AdS/CFT

dictionary.

We make again the separation Ansatz (3.10) and solve the equation DMψ = 0,

viz.,

ψµν +1

µεµ

αβ∇αψβν = 0 (3.17)

If µ = 1 (µ = −1) we obtain left (right) solutions. Note that solutions of (3.17) are

necessarily traceless and transversal. The six independent EOM (3.17) are sufficient

to determine all components Fµν for any given set of weights h, h. Four of these

– 456 –

equations are algebraic:

hFuu − hFuv =µ− 1

4isinh (2ρ)Fuρ (3.18a)

hFuv − hFvv =µ+ 1

4isinh (2ρ)Fvρ (3.18b)

hFuρ − hFvρ =i

sinh (2ρ)

(

Fvv(µ+ 1) + Fuu(µ− 1) − 2µ cosh(2ρ)Fuv

)

(3.18c)

Fρρ =4

sinh2(2ρ)

(

2 cosh(2ρ)Fuv − Fuu − Fvv

)

(3.18d)

The remaining two provide a coupled set of linear first order differential equations

for Fuv and Fvv:

dFuv

dρ=

µ+ 1

sinh(2ρ)

(

Fuv

( 4hh

(µ+ 1)2− cosh(2ρ)

)

+ Fvv

(

1 − 4h2

(µ+ 1)2

)

)

(3.19a)

dFvv

dρ= − µ+ 1

sinh(2ρ)

(

Fvv

( 4hh

(µ+ 1)2− cosh(2ρ)

)

+ Fuv

(

1 − 4h2

(µ+ 1)2

)

)

(3.19b)

For later purposes we parameterize the coupling constant µ by

µ+ 1

2= 1 − ε (3.20)

At the moment ε need not be small or positive. We also define

x := cosh(2ρ) (3.21)

Decoupling the two differential equations leads to a second order equation for Fvv

(prime denotes derivative with respect to x):

F ′′vv +

2x

x2 − 1F ′

vv −αx2 − 2hh x+ h2 + h2 − α

(x2 − 1)2Fvv = 0 (3.22)

with α = (1−ε)+(1−ε)2. The differential equation (3.22) can be transformed easily

into a hypergeometric differential equation. Its most general solution is given by

Fvv = a1(x− 1)(h−h)/2(x+ 1)(h+h)/22F1

(

2 + h− ε, −1 + h + ε, 1 + h− h;1 − x

2

)

+ a2(x− 1)(h−h)/2(x+ 1)(h+h)/22F1

(

2 + h− ε, −1 + h+ ε, 1 + h− h;1 − x

2

)

(3.23)

provided the difference between the weights is not integer, h−h 6∈ Z. Useful identities

for the Gauss hypergeometric function 2F1 are collected in appendix B.

We consider now regularity at the origin x = 1. Near the origin we can expand

2F1 = 1 + O(x − 1). The singular modes are those where a2 = 0, a1 6= 0 and the

– 457 –

regular modes are those where a1 = 0, a2 6= 0. We are exclusively interested in

regular modes and therefore set a1 = 0. The regular solution for Fvv is given by

Fvv = a2(x−1)(h−h)/2(x+1)(h+h)/22F1(2+h−ε,−1+h+ε, 1+h− h; 1 − x

2) (3.24)

Similar considerations yield the regular solution for Fuv.

Fuv = a(x− 1)(h−h)/2(x+ 1)−(h+h)/22F1(−h+ ε, 1 − h− ε, 1 + h− h;

1 − x

2) (3.25)

The constant a is determined uniquely by the choice of the overall normalization

a2, see appendix C for explicit results. The solutions for Fuu and Fuv, (3.24) and

(3.25), also solve the first order system (3.19), apart from certain degenerate cases

that we shall address separately. All other components of Fµν are obtained from the

algebraic relations (3.18).

The case h > h is recovered upon exchanging a1 ↔ a2 and h ↔ h in the

discussion and the formulas above. The case h = h leads to one regular mode given

in (3.24) and another mode that turns out to be singular at x = 1. The same issue

arises if the difference between the weights is integer, h − h ∈ Z. Thus, the most

general regular case is covered by (3.24).

3.4 Non-normalizable left and right solutions

The solutions of DLψ = 0 are recovered from the massive solutions above in the limit

ε → 0. If the weights are integers then we obtain elementary functions instead of

hypergeometric ones. We assume that this is the case. For concreteness we demand

h > h (3.26)

and address the other cases in the end. Solutions of DRψ = 0 are obtained from the

left modes by replacing everywhere u↔ v, h↔ h and L with R.

We classified solutions into normalizable (all components Fµν are bounded for

large x) and non-normalizable ones (not all components Fµν are bounded for large

x). Since we require regularity there is no freely adjustable parameter anymore in

our solution. For any given set of weights the component Fvv must take the form

(3.24). Thus, for any given set of weights only three possibilities exist: there is

only a normalizable mode, there is only a non-normalizable mode, or there is both

a normalizable and a non-normalizable mode and the former has Fvv = 0. We have

mentioned in equation (3.14) that normalizable modes with non-vanishing component

Fvv exist for any h ≥ 2. Therefore, a necessary condition for non-normalizable regular

modes is the inequality h ≤ 1. In the following paragraph we establish conditions

that are necessary and sufficient.

For h = 1 or h = 0 we find that there are no regular non-normalizable solutions,

see the end of appendix B. Modes with weights h ≤ −2 and negative h are normal-

izable. Thus, regular non-normalizable modes exist if and only if the weights obey

– 458 –

the inequalities

h ≥ −1 h ≤ −1 (3.27)

The only-if-part of the statement is clear from the previous discussion. The if-part

will be shown explicitly by constructing all the modes. Note that the inequalities

(3.27), if saturated, include the mode h = h = −1, which turns out to be non-

normalizable and regular (C.7). We keep this mode in the discussion below, even

though it does not obey the strict inequality (3.26).

The results for Fvv and Fuv can be extracted from (B.8) and (B.9) plugged into

(3.24) and (3.25) (with ε = 0), respectively. All other components follow algebraically

from (3.18). The results are presented in detail in appendix C.1. As an example we

present here the result for the non-normalizable left moving boundary graviton:

ψLµν(1,−1) = e−iu+iv

0 0 0

0 x− 1 −2i√

x−1x+1

0 −2i√

x−1x+1

− 4x+1

µν

(3.28)

For generic values h ≥ −1 ≥ h the results in appendix C lead to the following

asymptotic expansion:

Fvv = x+ hh+ O(1

x

)

(3.29a)

Fuv = 1 − h2 + O( ln x

x

)

(3.29b)

Fuu = −hh

(h2 − 1) + O( ln x

x

)

(3.29c)

Fvρ = −2ih + O( ln x

x

)

(3.29d)

Fuρ =2i(1 − h2)

h x

(

h+ h− 2hh (lnx

2− ψ(h) − ψ(1 − h) − 2γ)

)

+ O( ln x

x2

)

(3.29e)

Fρρ =4(1 − 2h2)

x+ O

( lnx

x2

)

(3.29f)

The quantity γ is the Euler–Mascheroni constant and ψ = Γ′/Γ is the digamma-

function. We note in passing that the asymptotic expansion (3.29) is valid for non-

integer values of h, h as well. The uu-component is non-polynomial in the weight h.

It will become clear in section 4 how this non-locality is related to the dynamics of

CFT.

Let us now discuss algebraic properties of the non-normalizable left modes. These

properties are very useful to relate solutions of different weights, analogous to the

discussion in section 3.2 for normalizable modes. We could again exploit the property

(3.8), but we use a slightly stronger statement here. Namely, when acting on left

– 459 –

modes the operator DL commutes with the generators L± (3.4)-(3.5) and L±:4

[

DL, L±

]

ψL = 0[

DL, L±

]

ψL = 0 (3.30)

Starting from a regular non-normalizable left mode we can therefore create alge-

braically further left modes (not necessarily regular or non-normalizable). We find

the algebraic relations

L0 ψL(h, h) = hψL(h, h) (3.31a)

L− ψL(h, h) = (h + 1)ψL(h− 1, h) + δh,−1N(−2, h) (3.31b)

L+ ψL(h, h) = (h− 1)ψL(h+ 1, h) (3.31c)

L0 ψL(h, h) = h ψL(h, h) (3.31d)

L− ψL(h, h) = (h− 1)ψL(h, h− 1) (3.31e)

L+ ψL(h, h) = (h + 1)ψL(h, h+ 1) + δh,−1N(h, 0) (3.31f)

where N(h, h) with |h| > 1 are left modes that are regular and normalizable (if

|h| ≤ 1 the quantity N(h, h) vanishes) and ψL(h, h) are regular non-normalizable

modes. We have fixed the normalization of all left modes in such a way that it

is compatible with the asymptotic expansion (3.29). Note that the normalizable

modes never mix with the non-normalizable modes: if the right hand side of one of

the relations (3.31) contains a normalizable contribution, the non-normalizable one

automatically vanishes.

To prove the relations (3.31) it is essentially sufficient to consider the action of

L± on the vv-component and the action of L± on the uu-component. This produces

immediately the relations above, but without the terms proportional to normaliz-

able modes N . As long as it is non-vanishing, it is sufficient to consider the vv-

component (uu-component) and compare it with the corresponding component of

regular non-normalizable modes: since we know that the modes ψL must be solu-

tions of DLψL = 0 knowledge of one non-vanishing component determines the other

components algebraically. The only caveat is that a normalizable mode with the

same weights h, h could mix with the regular non-normalizable mode, provided the

former has a zero vv-component (uu-component). However, for generic weights no

such modes exist; the vv-component (uu-component) of normalizable modes is non-

vanishing, with only a few exceptions. These exceptions lead precisely to the terms

proportional to the modes N in the algebraic relations (3.31).

4The quickest way to show this is as follows: Take the tensor identity D(g)LψL = 0 and make a

coordinate change D(g+Lξg)L(ψL +Lξψ

L) = 0, where Lξ is the Lie-derivative along a vector field

ξ. If ξ is one of the Killing vectors L±, L± we obtain D(g)L(LξψL) = 0, which is equivalent to the

statement in (3.30). We thank Niklas Johansson for providing this argument.

– 460 –

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

u u u u u

u u u u u

u u u u u

u u u u u

u u u u u

u u u u u

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

X

X

X

6

-

h

h

- - - - - - - -

- - - - - - - -

- - - - - - - -

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

- - - -

- - - -

- - - -

-

-

-

-

-

-

-

-

-

6 6 6 6 6 6 6 6 6

6 6 6 6 6 6 6 6 6

6 6 6 6 6

6 6 6 6 6

6 6 6 6 6

6 6 6 6 6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

? ? ? ? ?

? ? ? ? ?

? ? ? ? ?

? ? ? ? ?

? ? ? ? ?

? ? ? ? ?

Figure 1: Display of the ladder operators L+ (up), L− (down), L+ (right) and L− (left).

Notation: • = normalizable regular, = non-normalizable regular, X = singular

A simple example is provided by the boundary graviton (3.28). Inserting it into

the algebraic relations (3.28) we obtain

L0 ψL(1,−1) = ψL(1,−1) (3.32a)

L− ψL(1,−1) = 2ψL(0,−1) (3.32b)

L+ ψL(1,−1) = 0 (3.32c)

L0 ψL(1,−1) = −ψL(1,−1) (3.32d)

L− ψL(1,−1) = −2ψL(1,−2) (3.32e)

L+ ψL(1,−1) = 0 (3.32f)

The validity of the relations (3.32) can be checked explicitly with the formulas in

appendix C.1. Note that the boundary graviton is annihilated by L+ and L+. In

that sense it is the non-normalizable analog of the primary (3.12). However, as

opposed to the situation in the normalizable case we cannot generate from it all non-

normalizable modes: the modes with h > 1 are not accessible by repeatedly acting

with the generators L±, L± on the boundary graviton. We obtain in this way only

modes with weights (1, h), (0, h) and (−1, h). The modes with h ≥ 2 can be obtained

algebraically starting from the mode with weights (2,−1) given explicitly in (C.10).

– 461 –

We have started the discussion in this subsection by assuming h > h (3.26). If

we assume instead h = h then nearly all modes are regular and normalizable. There

are three exceptions. If h = h = 0 we obtain singular modes. Additionally, there are

two non-normalizable regular modes, namely h = h = −1 (C.7) and h = h = 1. We

have included the former already in the discussion above, while the latter is included

in the following case. If we assume h < h we just have to replace h → −h and

h → −h in the discussion in this subsection. We recall that the right modes are

obtained from the left modes by replacing everywhere u ↔ v, h↔ h and L with R.

Additionally, we assumed that both weights are integer. If only their difference is

an integer, but not their sum, then we cannot express the hypergeometric functions

appearing in (3.24) and (3.25) in terms of more elementary functions. Moreover, the

algebra analog to (3.31) will no longer be “osmotic” between non-normalizable and

normalizable modes, i.e., the figure 1 does not apply to this more general situation.

In conclusion, the only regular non-normalizable left modes must have weights

(h ≥ −1, h ≤ −1) or (h ≤ 1, h ≥ 1). The general result for regular non-normalizable

left modes is given in appendix C.1. The asymptotic expansion for general regular

non-normalizable left modes is provided in (3.29). The algebraic properties (3.31)

depicted in figure 1 relate the modes and allow to generate all regular left modes

starting e.g. from (C.10).

3.5 Non-normalizable logarithmic solutions

The logarithmic modes emerge from the linear combination [10]

ψlogµν = lim

ε→0

ψMµν(ε) − ψL

µν

ε=

dψMµν

dε

∣

∣

∣

ε=0(3.33)

For finite values of ε the middle expression in (3.33) is annihilated by DMDL, but

not by either of DM/L. After taking the limit ε → 0 all three expressions in (3.33)

are annihilated by (DL)2, but not by DL. This is the defining property of logarithmic

modes.

We focus again first on the case h ≥ h. The discussion is analogous to the one of

left modes. Generic non-normalizable logarithmic modes exist for the same weights as

non-normalizable left modes, h ≥ −1 ≥ h. One can use relations between contiguous

functions similar to (B.2) to establish recursion relations resembling (C.16). This

procedure allows to express all hypergeometric functions appearing in the logarithmic

modes in terms of rational functions and logarithms. However, there is a simpler way

to obtain generic logarithmic modes: make the shifts h→ h+ε, h→ h+ε and define

Fµν = FLµν + εF log

µν + O(ε2), where FLµν is a non-normalizable left mode. Expanding

– 462 –

the algebraic equations (3.18) to O(ε) yields

hF loguu − hF log

uv = FLuv

(

1 − h+x− h

h

)

+ FLvv

h2 − 1

h(3.34a)

hF loguv − hF log

vv =

√x2 − 1

2iF log

vρ + FLvv(h + 1) − FL

uv(h+ 1) (3.34b)

hF loguρ − hF log

vρ =2i√x2 − 1

(

F logvv − xF log

uv − FLuu − FL

vv + 2xFLuv

)

+ FLuρ − FL

vρ (3.34c)

F logρρ =

4

x2 − 1

(

2xF loguv − F log

uu − F logvv

)

(3.34d)

The differential equations (3.19) yield to O(ε)

dF loguv

dx=

1

x2 − 1

(

F loguv (hh− x) + F log

vv (1 − h2) + FLuv(x+ h+ h+ hh) − FL

vv(1 + h)2)

(3.35a)

dF logvv

dx=

1

1 − x2

(

F logvv (hh− x) + F log

uv (1 − h2) + FLvv(x+ h+ h+ hh) − FL

uv(1 + h)2)

(3.35b)

We know the homogeneous solutions to the system (3.35): they are precisely the

non-normalizable left modes provided in appendix C.1. Solving the differential equa-

tions (3.35) is straightforward (if lengthy) and yields the components F loguv and F log

vv ,

up to two integration constants. One of the integration constants parameterizes the

expected ambiguity corresponding to an addition of left modes to the logarithmic

modes. We may fix this ambiguity for instance by demanding that F logvv asymptot-

ically has no contribution linear in x. The other integration constant is fixed by

demanding regularity at x = 1. The remaining components F logµν follow algebraically

from (3.34). The logarithmic mode for weights h ≥ −1 ≥ h is then given by

ψlogµν = i(u+ v)ψL

µν − F logµν e

−ihu−ihv (3.36)

We have fixed the overall normalization constant in a convenient way. We provide

explicit results for logarithmic modes in appendix C.2. As an example we present

here the logarithmic partner of the non-normalizable left moving boundary graviton

– 463 –

(3.28):

ψlogvv (1,−1) = e−iu+iv (i(u+ v) + ln

x+ 1

2) (x− 1) (3.37a)

ψloguv (1,−1) = 4e−iu+iv

(

1 − 2ln x+1

2

x− 1

)

(3.37b)

ψloguu (1,−1) = −4e−iu+iv

(

1 − 2ln x+1

2

x− 1

)

(3.37c)

ψlogvρ (1,−1) = −2ie−iu+iv

(

(i(u+ v) + lnx+ 1

2)

√

x− 1

x+ 1− 2

x2 − 4x+ 3 + 4 ln x+12

(x− 1)√x2 − 1

)

(3.37d)

ψloguρ (1,−1) = 16ie−iu+iv x− 1 − (x+ 1) ln x+1

2

(x− 1)√x2 − 1

(3.37e)

ψlogρρ (1,−1) = −4

e−iu+iv

x+ 1

(

i(u+ v) + lnx+ 1

2− 4

(2x+ 1)(x− 1 − 2 ln x+12

)

(x− 1)2

)

(3.37f)

It is a straightforward exercise to check that ψlogµν (1,−1) is indeed annihilated by

(DL)2. Acting on it with DL only once we obtain

(

DLψlog(1,−1))

µν= (L0 + L0)ψ

logµν (1,−1) = −2ψL

µν(1,−1) (3.38)

where the regular non-normalizable left moving boundary graviton ψLµν(1,−1) is given

explicitly in (3.28). The logarithmic mode (3.37) has angular momentum 2:

(L0 − L0)ψlogµν (1,−1) = 2ψlog

µν (1,−1) (3.39)

Therefore, we call the mode (3.37) “non-normalizable bulk graviton”.

It is straightforward to obtain an asymptotic expansion for F logµν by solving

asymptotically the differential equations (3.35). This obtains

F logvv = −(x+ hh) ln x+ h + h+ hh+ O

( ln x

x

)

(3.40a)

F loguv = (h2 − 1) lnx+ (1 − 3h)(h+ 1) + O

( lnx

x

)

(3.40b)

Inserting this result (and the asymptotic expansions (B.14), (B.15) for the left modes)

into the algebraic relations (3.34) yields asymptotic results for the non-normalizable

– 464 –

logarithmic modes with arbitrary weights h ≥ −1 ≥ h:5

ψlogvv ∼ e−i(hu+hv)

(

(x+ hh)(ln x+ i(u+ v)) − h− h− hh)

+ O( ln x

x

)

(3.41a)

ψloguv ∼ −e−i(hu+hv)

(

(h2 − 1)(lnx+ i(u+ v)) + (1 − 3h)(h+ 1))

+ O( ln x

x

)

(3.41b)

ψloguu ∼ −e−i(hu+hv)

(h

h(1 − h2)(lnx− i(u+ v))

+ 2h

h(h2 − 1)

(

ψ(h− 1) + ψ(−h) − 3

2+ 2γ + ln 2

)

)

+ O( ln x

x

)

(3.41c)

ψlogvρ ∼ −2i e−i(hu+hv)

(

h (ln x+ i(u+ v)) − (h+ 1))

+ O( ln x

x

)

(3.41d)

ψloguρ ∼ O

( lnx

x

)

(3.41e)

ψlogρρ ∼ −4

xe−i(hu+hv)

(

(2h2 − 1) (lnx+ i(u+ v)) + 2(1 − 3h)(h+ 1))

+ O( lnx

x2

)

(3.41f)

The notation ∼ indicates that all equalities above are true up to the addition of non-

normalizable left modes with the same weights. The vv-component grows asymp-

totically like x ln x ∼ e2ρρ, which is logarithmically stronger growth as compared to

the Fefferman-Graham expansion (3.2). As anticipated, the non-normalizable loga-

rithmic modes are not asymptotically AdS (see appendix A). Particularly the O(1)

term of the uu-component will be important for the 2-point correlators in section 4,

since it is non-polynomial in the weights.

We consider now algebraic properties of logarithmic modes, starting with the

identities[

(DL)2, L±

]

ψlog = 0[

(DL)2, L±

]

ψlog = 0 (3.42)

We find the algebraic relations

L0 ψlog(h, h) = hψlog(h, h) − ψL(h, h) (3.43a)

L− ψlog(h, h) ∼ (h+ 1)ψlog(h− 1, h) + δh,−1N(−2, h) (3.43b)

L+ ψlog(h, h) ∼ (h− 1)ψlog(h+ 1, h) (3.43c)

L0 ψlog(h, h) = h ψlog(h, h) − ψL(h, h) (3.43d)

L− ψlog(h, h) ∼ (h− 1)ψlog(h, h− 1) (3.43e)

L+ ψlog(h, h) ∼ (h+ 1)ψlog(h, h+ 1) + δh,−1N(h, 0) (3.43f)

where N(h, h) with |h| > 1 are normalizable logarithmic modes (if |h| ≤ 1 the

quantity N(h, h) vanishes). The sign ∼ denotes equivalence up to the addition of

left modes, the standard ambiguity for logarithmic modes. The relations above can be

proven in a similar way as for the left modes. The algebraic properties (3.43) relate

the modes and allow to generate all regular logarithmic modes starting e.g. from

(C.22).5For h = 0,±1 the following limits are needed: limh→0 hψ(h−1) = limh→±1(h∓1)ψ(h−1) = −1.

– 465 –

4. Correlators

We have now collected nearly all ingredients to calculate all momentum space 2-

and 3-point correlators of operators in the postulated dual CFT on the cylinder: in

section 2 we presented the first three variations of the bulk action and reviewed the

linearized EOM. In section 3 we constructed the regular non-normalizable solutions

of the linearized EOM that act as sources for the operators corresponding to bulk and

boundary gravitons. The only missing ingredient for the calculation of correlators

are boundary terms in the action and their variation. However, we shall prove that

for 3-point correlators these boundary terms are not needed, while for the 2-point

correlators we shall employ a convenient short-cut. In section 4.1 we derive all 2-

point correlators and in section 4.2 we derive all 3-point correlators (in two cases

qualitatively, in the other eight cases exactly). We work in Lorentzian signature and

postpone a comparison to the more familiar Euclidean LCFT correlators in the short

distance limit to section 4.3.

4.1 Two-point correlators

We recall first 2-point correlators in a CFT with Lorentzian signature on the cylinder

S1 ×R (see e.g. [71] for a review). Next we explain how to obtain 2-point correlators

in the momentum representation on the gravity side for Einstein gravity. Finally

we derive all 2-point correlators in cosmological topologically massive gravity at the

chiral point.

4.1.1 Einstein gravity

As a warm-up as well as to fix the notation let us consider the stress tensor correlation

function in a CFT dual to Einstein gravity in the momentum representation. The

details of this CFT are irrelevant for this correlator since it is uniquely determined

by the conformal Ward-identities. We shall refer to this CFT as “Einstein-CFT”

below. Consider the time ordered 2-point function

GF (u, v) = 〈TTuu(u)Tuu(0)〉 (4.1)

We can obtain its momentum representation by noting that, quite generally, we have

for k ∈ Z, ∆ ∈ N and ω > 0

0∫

−2π

dφ

∞∫

−∞

dteiωt+ikφ

sin2∆( t+φ2

− iǫ sgn(t))(4.2)

= 22∆ 2πi(iω)2∆−1

(2∆ − 1)!

∑

n≥0

eiω 2πn

0∫

−2π

dφ ei(k−ω)φ

=(−4)∆2πi

Γ(2∆)

ω2∆−1

2h(4.3)

– 466 –

where we have regularized the φ-integral through ω → ω + i 0+. We recall that the

coordinates φ and t are related to the light-cone coordinates by u = t+φ, v = t−φ.

The momentum and frequency are related to h and h through h = ω+k2

and h = ω−k2

.

Now, since the 2-point function of the stress tensor in Einstein-CFT dual to Einstein

gravity on global AdS is given by [71]

〈TTuu(u)Tuu(0)〉 =( C

4 sin4( t+φ2

− iǫ sgn(t))+

C6 sin2( t+φ

2− iǫ sgn(t))

)

(4.4)

we can apply the general formula above to get

GF (h, h) = 2πiC3

ω3 − ω

h

= 2πiC3

h3 − h

h+ contact terms (4.5)

The constant C is related to the central charge cBH through C = cBH

8(2π)2, with the

Brown–Henneaux central charge given by [72]

cBH =3

2GN(4.6)

Let us now compare the result (4.5) with the momentum space 2-point correlator

obtained from the Einstein action with cosmological constant (1.2). The source terms

for Tuu are the left-moving non-normalizable boundary gravitons in section (3.4).

These modes are not only solutions of linearized CCTMG, but also of linearized

Einstein gravity around AdS3. We thus substitute these into the second variation

of the Einstein–Hilbert action, together with some as yet unspecified normalization

constant α. The second variation of the Einstein–Hilbert action δ2SEH is given by

the first term in (2.20)

δ(2)SEH(ψL, ψL) = − α2

16πGN

∫

d3y√−g ψµν ∗

L δGµν(ψL) + boundary terms (4.7)

with the boundary terms determined by demanding that the second order action leads

to a well-defined variational principle. Including these terms the on-shell action is

then given by

δ2SEH(ψL, ψL) =α2

32π GNlimρ→∞

t1∫

t0

dt

2π∫

0

dφ√−g ψL

ij∗(h, h) gikgjl∇ρψ

Lkl(h

′, h′) (4.8)

Here gij is the induced metric at the boundary and g its determinant. We explain

now how to fix the normalization constant α. We demand standard coupling of the

metric to the stress tensor:

S(ψu Lv , T v

u ) =1

2

∫

dt dφ√

−g(0) ψuuL Tuu =

∫

dt dφ e−ihu−ihv Tuu (4.9)

– 467 –

Here S is either some CFT action with background metric g(0) or a dual gravitational

action with boundary metric g(0). The non-normalizable mode ψL is the source

for the energy-momentum flux component Tuu. The requirement (4.9) leads to the

normalization α = 14.

If h 6= h′ the integrand in the second variation of the on-shell action (4.8) has an

oscillating factor in φ that integrates to zero. Similarly, if h 6= h′ the integrand has

an oscillating factor in t, which vanishes if integrated over a periodic time interval

(in the non-compact case one can argue that it vanishes in a distributional sense).

Therefore, the weights must match.

h = h′ h = h′ (4.10)

Then the oscillating terms cancel precisely. Now we use the asymptotic expansion

(3.29) together with the definition (3.21) and keep only the leading terms:

δ2SEH(ψL, ψL) =α2

16GN

t1∫

t0

dt1

4e2ρ ψL u

v∗(h, h) ∂ρψ

L vu (h, h) (1 + O(e−2ρ)) (4.11)

Collecting all factors, inserting α = 14, taking the limit and replacing GN by means

of the Brown–Henneaux result (4.6) yields

δ2SEH(ψL, ψL) =cBH

24

h

h(h2 − 1)

t1∫

t0

dt (4.12)

In comparing (4.12) with the Einstein-CFT result (4.5) we should note that an extra

factor of 2π∫

dt arises due to the fact that the gravitational computation corre-

sponds to a double Fourier transform with respect to both coordinates of the 2-point

function. The extra factor i in (4.5) comes about because we work in Lorentzian

signature, where the action is multiplied by i. The result on the gravity side (4.12)

agrees therefore exactly with the result on the Einstein-CFT side (4.5), provided the

central charge takes the Brown–Henneaux value (4.6). This is of course well-known.

For the right modes the same calculation goes through, upon exchanging u↔ v

and h ↔ h. For mixed correlators, i.e., correlators that contain one left and one

right mode, the result for the correlator turn out to vanish up to a polynomial in the

weights which corresponds to contact terms. Thus, all left-right 2-point correlators

in Einstein gravity vanish up to contact terms in agreement with the predictions of

the postulated dual Einstein-CFT.

4.1.2 Cosmological topologically massive gravity at the chiral point

At the chiral point the left-moving central charge cL vanishes [5], whereas the right-

moving central charge cR = 2cBH is twice that of the Einstein-CFT. From the CFT

– 468 –

point of view the 2-point function (4.4) should thus vanish while its right moving

companion should take twice the value compared to Einstein gravity. This structure

is precisely encoded in CCTMG as we shall now see.

Generically the 2-point correlators on the gravity side between two modes ψ1(h, h)

and ψ2(h′, h′) in momentum space are determined by

〈ψ1(h, h)ψ2(h′, h′)〉 =1

2

(

δ(2)SCCTMG(ψ1, ψ2) + δ(2)SCCTMG(ψ2, ψ1))

(4.13)

where 〈ψ1 ψ2〉 stands for the correlation function of the CFT operators dual to

the graviton modes ψ1 and ψ2. On the right hand side one has to plug the non-

normalizable modes ψ1 and ψ2 into the second variation of the on-shell action (2.20)

and symmetrize with respect to the two modes. However, one should be careful and

collect all boundary terms, because the whole contribution to the correlator (4.13)

turns out to be a boundary term evaluated at the asymptotic boundary. This was

done in great detail in [17]. We use instead a short-cut to calculate the 2-point corre-

lators in momentum space that does not require the construction of these boundary

terms. To see how this works we take the second variation of the on-shell action

(2.20) which can be written as

δ(2)SCCTMG = − 1

16πGN

∫

d3y√−g

(

DLδg)µν

δGµν(h) + boundary terms (4.14)

We then see that the bulk term on the right hand side has the same form as in

Einstein theory with δg replaced by DLδg. Now, consider the variation of this action.

Possible obstructions to a well-defined boundary value problem can come only from

the variation δGµν(h). Thus any boundary terms appearing in (4.14) containing

normal derivatives must be identical with those in Einstein gravity. In addition

there can be boundary terms which do not contain normal derivatives of the metric.

However, from the asymptotic expansion (3.29) we infer that such terms can at most

lead to contact terms in the holographic computation of 2-point functions.6 Since we

ignore such contributions these additional boundary terms are irrelevant (although

they will be needed below in order to obtain a finite result). The upshot of this

discussion is that we can reduce the calculation of all possible 2-point functions in

CCTMG to the equivalent calculation in Einstein gravity with suitable source terms.

To continue we go on-shell.

DLψL = 0 DLψR = 2ψR (4.15)

By virtue of the linearized EOM (4.15) we obtain immediately

〈ψR(h, h)ψR(h′, h′)〉CCTMG ∼ 2〈ψR(h, h)ψR(h′, h′)〉EH (4.16a)

〈ψL(h, h)ψL(h′, h′)〉CCTMG ∼ 0 (4.16b)

〈ψL(h, h)ψR(h′, h′)〉CCTMG ∼ 〈ψL(h, h)ψR(h′, h′)〉EH ∼ 0 (4.16c)

6As mentioned in the last paragraph of section 4.1.1 in the momentum representation contact

terms correspond to contributions which are polynomial in h and h.

– 469 –

The results on the gravity side (4.16) exactly match the LCFT prediction.

Let us now consider the logarithmic modes. According to the AdS/CFT corre-

spondence they should source an operator (see e.g. [73, 74]) t = bcL

Tuu + b2T where

T is a second operator with conformal weight ∆ = 2 such that 〈Tuu(u) T (0)〉 = 0.

The parameter b is fixed by the precise nature of the logarithmic CFT. Now, while

〈Tuu(u)Tuu(0)〉 vanishes for cL → 0, the correlator

〈Tuu(u) t(0)〉 = b〈Tuu(u)Tuu(0)〉

cL(4.17)

is finite and determines the parameter b. This structure is again beautifully realized

in CCTMG. Indeed using again (4.15) together with the on-shell relation

DLψlog = −2ψL (4.18)

we obtain upon substitution in (4.14) immediately7

〈ψL(h, h)ψlog(h′, h′)〉CCTMG ∼ −2 〈ψL(h, h)ψL(h′, h′)〉EH (4.19a)

and after a bit of calculation also

〈ψR(h, h)ψlog(h′, h′)〉CCTMG ∼ 0 (4.19b)

Again the results on the gravity side (4.19) exactly match the LCFT prediction.

Finally we address the 2-point correlator with two logarithmic insertions. In

principle one can again obtain the 〈t t〉-correlator using conformal Ward-identities

in logarithmic CFT by expanding the 〈T T 〉-correlator about cL = 0 [73, 75]. As a

consequence of the conformal Ward-identities 〈T T 〉 is again given by a generalization

of (4.2) for conformal weights (∆, ∆) = (2+δ, δ). The 〈t t〉-correlator is then obtained

upon differentiating with respect to δ at δ = 0. In practice we face the problem that

(4.2) is valid for ∆ and ∆ ∈ N. Its analytic continuation to non-integer values of ∆

is ambiguous and therefore not conclusive. However, below we will argue on general

grounds that the result predicted by CCTMG is the correct one. In order to obtain

it we use again the on-shell relations (4.15), (4.18). This does not lead to a correlator

known from Einstein gravity, because there remains still one logarithmic mode, and

these modes do not exist in Einstein gravity. However, we can still use the relation

(4.18) to convert the second variation of the CCTMG action (4.14) into a variation

that takes the form of the second variation of the Einstein–Hilbert action (4.7). The

only missing ingredient are boundary counterterms that make the correlator finite:

δ(2)SCCTMG(ψlog, ψlog) =β2

8π GN

∫

d3y√−g ψL µνδGµν(ψ

log)+boundary terms (4.20)

7We have to take care of the normalizations of modes. We found above that we should use αψL

for left (or right) modes by demanding that these modes be correctly normalized sources for the

energy momentum tensor, with α = 14 . We make the Ansatz βψlog for the normalization of the

logarithmic modes. At the end of our calculations we require a standard form of the LCFT 2-point

correlators and find β = α = 14 .

– 470 –

Again, it is clear from the asymptotic expansion (3.41) that boundary counterterms

can at most be contact terms. Analogous to the Einstein–Hilbert case (4.8) we obtain

δ(2)SCCTMG(ψlog, ψlog) = − 4β2

16πGN

limρ→∞

t1∫

t0

dt

2π∫

0

dφ√−g ψL

ij∗(h, h) gikgjl∇ρψ

logkl (h′, h′)

+ contact terms (4.21)

To avoid oscillating integrals the weights must match (4.10). Keeping only terms

that do not vanish in the limit ρ→ ∞ we get, modulo contact terms,

δ(2)SCCTMG(ψlog, ψlog) ∼ − limρ→∞

4β2

GN

t1∫

t0

dt(

ψLvv

∗ ∂ρ(ψloguu e

−2ρ) + ψLuu

∗ ∂ρ(ψlogvv e

−2ρ))

(4.22)

Inserting β = 14

as well as the asymptotic expansions (3.29) and (3.41) yields

δ(2)SCCTMG(ψlog, ψlog) ∼ − limρ→∞

1

4GN

t1∫

t0

dt(h

h(h2 − 1)

(

ψ(h− 1) + ψ(−h))

+(

α(ρ+ it) + β) h

h(h2 − 1)

)

(4.23)

valid for h ≥ −1, h ≤ −1. Again ψ is the digamma function and α, β are weight-

independent constants. The expression in the second line of (4.23) diverges as the cut-

off for ρ tends to infinity. To cancel it we could introduce an appropriate boundary

counterterm, as it was done in [17]. However, there is an alternative possibility.

The factor h(h2 − 1)/h is precisely the factor that arises in the correlator between

logarithmic and left modes, see (4.19) with (4.12). For each finite value of the

cutoff ρ we can exploit the shift ambiguity ψlog → ψlog + γ ψL to cancel such terms.

Therefore, they play no physical role and can be dropped, even without introducing

new boundary counterterms. The term in the first line of (4.23) is finite, analytic in

h, h and non-trivial. This is our final result for the 2-point correlator on the gravity

side between two logarithmic modes on the cylinder.

In order to show that our result (4.23) provides the correct answer for a LCFT

we first note that in the short-distance limit we should recover the continuum result.

To this end we evaluate the first line of (4.23) in the limit of large weights h → ∞,

h→ −∞ and obtain by virtue of (B.16) the asymptotic result

limh,−h→∞

δ(2)SCCTMG(ψlog, ψlog) ∼ − 1

2GN

h3

hln

√

−hht1

∫

t0

dt (4.24)

– 471 –

Here the sign ∼ means equality up to contact terms and up to additional terms that

can be absorbed by shifting ψlog → ψlog + γ ψL with some constant γ.

Comparison with 2-point correlators of logarithmic modes in a LCFT reveals

that the momentum space correlator (4.23) has the correct short-distance behavior

(see also section 4.3 below). For generic values of h, h with hh 6= 0, CFT-momentum

space correlators on a finite cylinder should have no cuts or poles. Our result (4.23)

is the only such function with the correct asymptotic behavior. Of course, there is a

freedom of adding a polynomial of degree 2 or less in h to (4.23) without spoiling the

asymptotic behavior (4.24). Equation (4.2) then shows that such terms correspond

to adding non-logarithmic operators of conformal weight ∆ < 2 to t. Similarly

a contribution of the form f(h)h

with analytic f(h) is compatible with conformal

invariance. Such terms arise in the right moving sector involving Tvv.

We summarize now the results of this section so far. All 2-point correlators

in CCTMG match precisely with corresponding 2-point correlators in a LCFT, in

agreement with the analysis in [17]. Many of the correlators could be reduced to

correlators known from Einstein gravity by exploiting specific features of CCTMG

and the second variation of its action. The most interesting correlator, the 2-point

correlator between two logarithmic modes (4.23), is analytic in h and h and only

develops a branch cut in the continuum limit (4.24).

4.2 Three-point correlators

The 3-point correlators on the gravity side between three modes ψ1(h, h), ψ2(h′, h′)

and ψ3(h′′, h′′) in momentum space are determined by

〈ψ1(h, h)ψ2(h′, h′)ψ3(h′′, h′′)〉 =1

6

(

δ(3)SCCTMG(ψ1, ψ2, ψ3)+5 permutations)

(4.25)

On the right hand side one has to plug the non-normalizable modes ψ1, ψ2 and ψ3

into the third variation of the on-shell action (2.30) and symmetrize with respect

to all three modes. The Witten diagram corresponding to the correlator (4.25) is

depicted in Fig. 2.

4.2.1 Boundary terms

Before proceeding with calculations we prove that all boundary terms can be ne-

glected. This considerably simplifies our calculations. Since all our modes are regular

at the origin there are only asymptotic boundary terms. We can therefore exploit

the asymptotic results (3.29) for the left modes (and right modes upon exchanging

u ↔ v, h ↔ h) and the logarithmic modes (3.41). Actually, it is sufficient to keep

track of the exponential behavior in ρ, so there is no essential difference between loga-

rithmic and left modes for our proof. Thus, in this paragraph we do not discriminate

between logarithmic and left modes. Whenever a statement is valid for left modes it

is also valid for logarithmic modes (up to irrelevant polynomial terms in ρ). We only

– 472 –

Ψ1

Ψ3

Ψ2

Figure 2: Witten diagram for three graviton correlator (4.25)

have to keep contributions that are not contact terms. As explained below (4.14) this

implies that each expression must contain at least one component ψLuu or one com-

ponent ψRvv. With one index raised these terms behave asymptotically like O(e−2ρ).

On the other hand, each tri-linear expression of the form δgνµh

λνk

µλ must contain at

least two terms of order of unity and at most one term that decays asymptotically

like e−2ρ. This is so, because such tri-linear terms are multiplied by√−g ∼ e2ρ. If

a tri-linear term decays faster than e−2ρ the corresponding boundary term vanishes

in the limit ρ → ∞. Consider for example the leading order contribution of two

left and one right mode. The only expression that does not decay faster than e−2ρ

is given by ψLuv ψR v

u ψLuu . However, this expression asymptotically is a contact term,

since it does neither contain ψL vu nor ψR u

v :

limρ→∞

ψL νµ ψL λ

ν ψR µλ = O(e−4ρ) + contact terms (4.26)

The same result applies for L ↔ R. Considering only left moving modes even leads

to faster decay:

limρ→∞

ψL νµ ψL λ

ν ψL µλ = O(e−4ρ) (4.27)

The same result applies to right modes. Insertion of covariant derivatives, ε-tensors

or background metrics into tri-linear expressions does not change anything essential

about the conclusions above. For instance, the only non-contact term that decays

like e−2ρ constructed solely out of left modes schematically must be of the form

T ρvu(ψ

L uv ψL u

v ψLvu ), with some O(1) tensor T constructed out of covariant derivatives,

ε-tensors or the background metric. Any such T ρvu is not O(1), but must decay at

least like e−2ρ. This is not completely obvious, but it can be shown straightforwardly

by considering the asymptotic behavior of the Christoffel symbols and of the back-

ground metric. The same considerations apply to terms with any other combination

of left and right modes. Generic boundary terms related to the bulk expression (2.30)

– 473 –

are all of the type just discussed: they contain tri-linear expressions in the modes,

possibly with some insertions of covariant derivatives, ε-tensors or background met-

rics.

In conclusion, all asymptotic boundary terms that emerge from partial integra-

tions in 3-point correlators either vanish or yield contact terms. We are therefore

free to partially integrate at will and to drop all boundary terms in the calculation

of 3-point correlators.

4.2.2 Correlators without log insertions

Partially integrating the first term in the third variation of the action (2.30) leads to

a useful result:

δ(3)SCCTMG ∼ − 1

κ2

∫

d3y√−g

[

(

DLδg)µν

δ(2)Rµν(h, k) + δgµν ∆µν(h, k)]

(4.28)

This expression allows us to use the same trick as for 2-point correlators: 3-point

correlators that involve no logarithmic modes can be reduced to 3-point correlators

calculated in Einstein gravity [68]:

δ(3)SEH = − 1

κ2

∫

d3y√−g δgµν δ(2)Rµν(h, k) (4.29)

To show this we recall the properties (4.15), (4.18) and the fact that ∆µν(h, k) = 0

if both h and k are left or right modes. Therefore, we have the following results:

〈ψR(h, h)ψR(h′, h′)ψR(h′′, h′′)〉CCTMG ∼ 2 〈ψR(h, h)ψR(h′, h′)ψR(h′′, h′′)〉EH (4.30a)

〈ψL(h, h)ψR(h′, h′)ψR(h′′, h′′)〉CCTMG ∼ − 4

3κ2

∫

d3y√−g ψR µν δ(2)Rµν(ψ

R, ψL) ∼ 0

(4.30b)

〈ψL(h, h)ψL(h′, h′)ψR(h′′, h′′)〉CCTMG ∼ − 2

3κ2

∫

d3y√−g ψR µν δ(2)Rµν(ψ

L, ψL) ∼ 0

(4.30c)

〈ψL(h, h)ψL(h′, h′)ψL(h′′, h′′)〉CCTMG ∼ 0 (4.30d)

The first and the last result are compatible with the CCTMG values of the central

charges. The other expressions reduce to terms that arise already in Einstein gravity,

where they are found to be zero up to contact terms. The results (4.30) coincide

with the results in a LCFT.

4.2.3 Single log insertions

Of course, the really interesting 3-point correlators contain one or more insertions

of logarithmic modes. We discuss now correlators that contain exactly one such

insertion.

– 474 –

We start with the correlator between two left and one logarithmic mode. If δg

is the logarithmic mode we obtain from (4.28) the contribution

δ(3)SCCTMG(ψlog, ψL1 , ψ

L2 ) ∼ 2

κ2

∫

d3y√−g ψLµν δ(2)Rµν(ψ

L1 , ψ

L2 ) (4.31)

where we have used the identity (4.18) and the vanishing of ∆µν(ψL1 , ψ

L2 ) = 0 (2.28).

Up to the overall factor −2 this coincides precisely with the corresponding expression

in Einstein gravity appearing in the 3-point correlator of three left modes. If δg is

one of the left modes we obtain instead

δ(3)SCCTMG(ψL1 , ψ

L2 , ψ

log) ∼ − 1

κ2

∫

d3y√−g ψL µν

1 ∆µν(ψL2 , ψ

log) (4.32)

To calculate ∆µν a helpful formula is(

DL(ψL ψL2 )

)

µν= εµ

στ ψα Lτ ∇σψ

L2 αν (4.33)

We obtain the intermediate result

δ(3)SCCTMG(ψL1 , ψ

L2 , ψ

log) ∼ 2

κ2

∫

d3y√−g ψL µν

1

(1

2εµ

σκ ψL ασ (∇νψ

L2 ακ −∇αψ

L2 νκ)

− ψL σ2 µ ψ

Lσν

)

(4.34)

The definition (2.28) of ∆µν makes it transparent why only left modes remain in the

result (4.34). Using on-shell manipulations like ψLαβ = −εα

µν∇µψLνβ or

ψL σ2 µ ψ

Lσν = (∇αψ

L2 µβ)(∇βψLα

ν ) − (∇αψL2 µβ)(∇αψL β

ν ) (4.35)

it is straightforward to show the identity [see (2.29)]

δ(3)SCCTMG(ψL1 , ψ

L2 , ψ

log) + δ(3)SCCTMG(ψL2 , ψ

L1 , ψ

log) ∼2

κ2

∫

d3y√−g ψL µν

1 δ(2)Rµν(ψL2 , ψ

L) +2

κ2

∫

d3y√−g ψL µν

2 δ(2)Rµν(ψL1 , ψ

L) (4.36)

The results (4.31) and (4.36) together imply that the correlator between two left

and one logarithmic mode in CCTMG is reduced to the correlator between three left

modes in Einstein gravity, multiplied by a factor −2. Similar considerations apply

to correlators with one or two right modes instead of the left modes, where all ex-

pressions turn out to be contact terms. Though this result is simple and transparent

from the CFT point of view, it is quite involved to derive it on the gravity side. In

section 4.2.4 we shall calculate this correlator explicitly in the limit of large weights,

where considerable simplifications arise. Let us summarize our results for 3-point

correlators with one logarithmic insertion:

〈ψR(h, h)ψR(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ 0 (4.37a)

〈ψL(h, h)ψR(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ 0 (4.37b)

〈ψL(h, h)ψL(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ −2 〈ψL(h, h)ψL(h′, h′)ψL(h′′, h′′)〉EH

(4.37c)

The results (4.37) coincide with the results in a LCFT.

– 475 –

4.2.4 Multiple log insertions and limit of large weights

The remaining correlators contain at least two logarithmic insertions and therefore

are more complicated. In particular, the vanishing of the correlator between a right

mode and two logarithmic ones is important for the consistency of the interpretation

of CCTMG as gravity dual to some LCFT. We have not found a closed expression

for these correlators, although we can evaluate them for any given set of weights.

Nevertheless we can check the consistency with a LCFT on the infinite plane by

considering the limit of small distance on the cylinder or, equivalently, large weights,

h,−h → ∞ while keeping h+ h finite. By the UV/IR connection [76] the coincidence

limit corresponds to the IR regime on the gravity side. It is therefore sufficient to

substitute the asymptotic large-x expansion of the non-normalizable modes (source

terms) in the 3-point correlator (4.25). See also the discussion at the end of appendix

B affirming the UV/IR connection we are exploiting.

We then proceed as follows. We start with the asymptotic expansions (3.29) and

(3.41) keeping only the leading and subleading terms in the large x expansion. Next

we evaluate by brute-force the missing three correlators 〈ψlog ψlog ψL/R/log〉 with com-

puter algebra [77]. We shall only keep terms that are of leading order in the weights,

since by scale-invariance these should reproduce the momentum space correlator on

the infinite plane. In addition, we are free to drop all polynomial terms in the weights

as they correspond to contact terms. The remaining non-polynomial terms are then

either rational functions in the weights or rational functions times logarithms in the

weights, similar to our results for 2-point correlators.

In order to avoid oscillating behavior of the integrand only two pairs of weights

can be chosen freely, while the third one is determined by the others. For complex

modes ψ1(h, h), ψ2(h′, h′), ψ3(h

′′, h′′) this condition is

h + h′ + h′′ = 0 h+ h′ + h′′ = 0 (4.38)

For simplicity we consider at the moment the complex modes of appendix C, rather

than real ones.8 This means that the 3-point correlators calculated below will have

a real and an imaginary part, corresponding to two specific linear combinations of

3-point correlators with real mode insertions. The main features of the 3-point cor-

relators are captured by these calculations. For completeness we address correlators

with real mode insertions in the end.

We need one more ingredient before we can start with the calculations. Namely,

as we shall see below there are two types of rational functions that we can get.

Either we obtain a sum of poles in all three weights (here and below quantities like

P (h, h′, h, h′) denote polynomials in the weights):

〈ψ1(h, h)ψ2(h′, h′)ψ3(h

′′, h′′)〉 ∼ P (h, h′, h, h′)

hh′(h+ h′)6= 0 (4.39)

8An overall factor 164 arises in all 3-point correlators due to the normalization factor 1

4 explained

in section 4.1. Additional numerical factors arise if we use real instead of complex modes.

– 476 –

Or we obtain a sum of poles in only two of the three weights:

〈ψ1(h, h) ψ2(h′, h′) ψ3(h

′′, h′′)〉 ∼ P (h, h′, h, h′)

hh′∼ 0 (4.40)

The latter case has the following simple interpretation in the dual CFT: All operators

appearing in the operator product expansion (OPE) of the operators O1 and O2

sourced by ψ1 and ψ2 are such that their 2-point function with the third operator

O3, sourced by ψ3, is a contact term. Thus, we shall equate correlators of the form

(4.40) to zero modulo contact terms in the calculations below.

To test this procedure we start with the calculation of correlators for which we

have obtained exact expressions already. We use the relations (4.38) between the

weights to eliminate h′′ and h′′ in terms of the other weights and find the following

results

lim|weights|→∞

〈ψL(h, h)ψL(h′, h′)ψL(h′′, h′′)〉CCTMG ∼ 0 (4.41a)

lim|weights|→∞

〈ψL(h, h)ψL(h′, h′)ψR(h′′, h′′)〉CCTMG ∼ 0 (4.41b)

lim|weights|→∞

〈ψL(h, h)ψR(h′, h′)ψR(h′′, h′′)〉CCTMG ∼ 0 (4.41c)

lim|weights|→∞

〈ψR(h, h)ψR(h′, h′)ψR(h′′, h′′)〉CCTMG ∼ PR(h, h, h′, h′)

hh′(h+ h′)(4.41d)

lim|weights|→∞

〈ψL(h, h)ψL(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ PL(h, h, h′, h′)

hh′(h+ h′)(4.41e)

lim|weights|→∞

〈ψL(h, h)ψR(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ 0 (4.41f)

lim|weights|→∞

〈ψR(h, h)ψR(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ 0 (4.41g)

with the polynomials

PR(h, h, h′, h′) ∝(

hh′(h+ h′)2(h3 + h′ 3) − h(h+ h′)h′ 2(h3 − (h + h′)3)

− h′(h + h′)h2(h′ 3 − (h+ h′)3))

(4.41h)

PL(h, h, h′, h′) = −PR(h′, h′, h, h) (4.41i)

The polynomials are symmetric under the exchanges h, h ↔ h′, h′ and h, h ↔ h′′ =

−(h + h′), h′′ = −(h + h′). Of course, this must be the case by the trivial exchange

symmetry 〈ψ ψ′ ψ′′〉 = 〈ψ′ ψ ψ′′〉, so this merely provides a check on the correctness

of the calculations. As evident from the last equation (4.41i) the polynomials are

related to each other by an exchange of weights and by a proportionality constant −1.

In coordinate space these properties correspond to an exchange u↔ v and a certain

proportionality between the coefficients in the 3-point correlators. It is easy to check

that these are precisely the properties required by the first correlator in (4.30) and

by the last correlator in (4.37), and that the proportionality constant −1 matches.

– 477 –

The poles appearing in the non-vanishing correlators are located precisely where they

should be, so the procedure above indeed works. On the other hand we should note

that the polynomials PR/L are of degree seven, whereas dimensional analysis shows

that the correct degree is five. The origin of this mismatch is that we integrated

over all of AdS3 while the correct region of integration should be supported only

at large x in order to be consistent with the asymptotic x-dependence assumed for

non-normalizable modes (source terms) inserted into the 3-point correlators (4.25).

At a qualitative level this can be taken into account by imposing an infrared cut-off

x > λ2, where λ is of the order of h, h, h′, . . . . The resulting polynomial is then

∼ λ−2PL/R which now has the correct dimension. Since we are interested only in the

location of the poles we may ignore this issue at present.

Of course, we did not need to invoke this procedure for the correlators just dis-

cussed, since we have reduced their calculation to a calculation of similar correlators

in Einstein gravity in sections 4.2.2 and 4.2.3. So, the example just considered pro-

vides a consistency test for our approach to evaluate 3-point correlators involving

logarithmic modes. The calculation of the last two correlators in (4.41) is still some-

what involved, because one has to take into account the leading order term, whose

integrand (2.31) decays only like 1/x, and the next-to-leading order term, whose in-

tegrand (2.31) decays like lnx/x2, t/x2 or 1/x2. All these terms independently turn

out to be contact terms for large weights.

Comparing our results in the large weight limit (4.41) with the exact results

(4.30) and (4.37) shows that they agree in this limit in the following sense: we

reproduce the vanishing of all correlators that should vanish and the correct location

of the poles (but not their residues) of the correlators that do not vanish. This

provides a consistency check that the large weight limit indeed produces the correct

results for the correlators in the near-coincidence limit, in the sense just explained.

We consider now the correlator between one right and two logarithmic modes in

the limit of large weights.9 We find the following result

lim|weights|→∞

〈ψlog(h, h)ψlog(h′, h′)ψR(h′′, h′′) 〉CCTMG ∼∫

dx

∫

dt[f1

x+f2 ln2 x

x2

+f3 t ln x

x2+f4 t

2

x2+f5 ln x

x2+f6 t

x2+f7

x2+ O

( ln2 x

x3

)

]

(4.42a)

where f1, f2, f3, f4 ∼ 0. The expressions f5, f6, f7 (up to contact terms) are given by

f5 ∝ f6 ∝h3h′ 2

h+h2h′ 3

h′f7 ∝

h3h′ 2

hln (−hh) +

h2h′ 3

h′ln (−h′h′) + αf5 (4.42b)

where α is a constant. The expressions (4.42) contain poles in the weights h and h′,

but not in h′′ = −h−h′. By the reasoning below equation (4.40) these contributions

9Details are available at http://quark.itp.tuwien.ac.at/∼grumil/Rloglog.html.

– 478 –

are equal to zero up to contact terms. Therefore the correlator between one right and

two logarithmic modes vanishes up to contact terms.

We consider next qualitatively the correlator between one left and two logarith-

mic modes in the limit of large weights. We find the following result

lim|weights|→∞

〈ψlog(h, h)ψlog(h′, h′)ψL(h′′, h′′) 〉CCTMG ∼ P log(h, h′, h, h′)

hh′(h+ h′)(4.43a)

where the quantity P log (up to contact terms) is given by

P log(h, h′, h, h′) = P1 (h, h′, h, h′) ln (−hh)+P1 (h′, h, h′, h) ln (−h′h′)+P2 (h, h′, h, h′)

(4.43b)

The polynomials P1 and P2 are again of degree seven in the weights.10 The expres-

sions (4.43) contain poles in the weights h, h′ and h′′ = −h − h′. The correlator

between one left and two logarithmic modes does not vanish, even after dropping all

contact terms.

The missing correlator between three logarithmic modes is lengthy, even in the

limit of large weights. We do not present any formulas for this case, and just mention

that we have checked that this correlator is non-vanishing, even after dropping all

contact terms, and has the expected poles.11 If we use real modes instead of complex

modes we reproduce all the results above, but with slightly different numerical co-

efficients. The qualitative features of the correlators do not change. All correlators

that vanish above still vanish, including the crucial correlator between a right and

two logarithmic modes.

We summarize our results for 3-point correlators with at least two logarithmic

insertions:

lim|weights|→∞

〈ψR(h, h)ψlog(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ 0 (4.44a)

lim|weights|→∞

〈ψL(h, h)ψlog(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ P log(h, h′, h, h′)

hh′(h+ h′)(4.44b)

lim|weights|→∞

〈ψlog(h, h)ψlog(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ lengthy

hh′(h+ h′)(4.44c)

In particular, we have demonstrated the vanishing of the correlator between a right

mode and two logarithmic ones, which is a non-trivial result on the gravity side. We

do not provide explicit expressions for the last two correlators in (4.44), because we

do only trust the locations of the poles, but not their residues. The proportionality

constants depend on the weights, but do not contain further poles. We see already

from the qualitative behavior of the penultimate correlator (4.43) (and similar ex-

pressions for the last correlator) that these correlators are non-vanishing and exhibit

10Details are available at http://quark.itp.tuwien.ac.at/∼grumil/Lloglog.html.11Details are available at http://quark.itp.tuwien.ac.at/∼grumil/logloglog.html.

– 479 –

the required features for LCFT correlators: poles in all three weights and expressions

that contain logarithms in appropriate pairs of weights in the large weight expansion.

Let us emphasize the main results of section 4.2. We have reduced the correlators

between three right modes (4.30) and between two left and one logarithmic mode

(4.37) to correlators known from Einstein gravity, thence establishing their exact

form. Moreover, we have found that all the correlators which should vanish in a LCFT

indeed do vanish for CCTMG (4.30), (4.37), including the crucial correlator between

one right and two logarithmic modes (4.44). In the latter case we had to invoke a

procedure that involved a large weight expansion. While we trust that procedure

to generate the correct location of poles (or absence thereof), we do not expect it

to yield the correct residues, since we exploited a certain scaling between weights

and the coordinate x to evaluate the hypergeometric functions and the integrals. We

have not found a simple way to extract the residues of the poles or to avoid the large

weight expansion altogether. This will be necessary for obtaining the correlators

〈ψL ψlog ψlog〉 and 〈ψlog ψlog ψlog〉 quantitatively for arbitrary weights. Qualitatively

we found that these correlators have poles at the correct places (4.44) and contain

the anticipated terms logarithmic in the weights (4.43b).

4.3 Comparison with Euclidean logarithmic CFT correlators

For completeness we relate now our results for 2- and 3-point correlators in sections

4.1 and 4.2 to the more familiar form of various LCFT correlators on the complex

plane collected in appendix D. The vanishing correlators on the gravity side also

vanish on the LCFT side (and vice versa), so we focus solely on the non-vanishing

ones. To this end we transform the results (4.16), (4.19) and (4.24) to coordinate

space, fixing the integral over time to 2πi so that we can compare with the Euclidean

correlators in appendix D.12

〈ψR(z, z)ψR(0)〉 =iπ

2GN

∂3

∂δ(2)(z, z) (4.45a)

〈ψL(z, z)ψlog(0)〉 = − iπ

2GN

∂3

∂δ(2)(z, z) (4.45b)

〈ψlog(z, z)ψlog(0)〉 = −2iπ

GN

ln(

m2√

−∂∂) ∂3

∂δ(2)(z, z) (4.45c)

Evaluating the distributions in (4.45) with standard methods and keeping only the

12We use Fourier transformations with respect to t, φ, not with respect to light cone coordinates

u, v or z, z. The factor 2 in the relation δ(2)(t, φ) = 2 δ(2)(z, z) enters in all formulas (4.45).

– 480 –

most singular terms yields

〈ψR(z, z)ψR(0)〉 =cR2z4

(4.46a)

〈ψL(z, z)ψlog(0)〉 =b

2z4(4.46b)

〈ψlog(z, z)ψlog(0)〉 = −b ln (m2|z|2)z4

(4.46c)

provided we use the values

cL = 0 cR =3

GNb = − 3

GN(4.47)

These are exactly the values for central charges cL, cR [5] and new anomaly b [17]

found before. The results (4.46) are equivalent to the LCFT results (D.1).

We compare now the 3-point correlators. The vanishing correlators on the gravity

side also vanish on the LCFT side (and vice versa), so we focus again on the non-

vanishing ones. All non-vanishing 3-point correlators with at most one logarithmic

insertion can be reduced to 3-point correlators known from Einstein gravity.

〈ψR(z, z)ψR(z′, z′)ψR(0, 0)〉EH =cBH

z2z′ 2(z − z′)2(4.48)

〈ψL(z, z)ψL(z′, z′)ψL(0, 0)〉EH =cBH

z2z′ 2(z − z′)2(4.49)

The results (4.30) and (4.37) then imply

〈ψR(z, z)ψR(z′, z′)ψR(0, 0)〉 =cR

z2z′ 2(z − z′)2(4.50a)

〈ψL(z, z)ψL(z′, z′)ψlog(0, 0)〉 =b

z2z′ 2(z − z′)2(4.50b)

with the same values of cR and b as before (4.47). The results (4.50) are equivalent to

the LCFT results (D.2), (D.3). The non-vanishing 3-point correlators with at least

two logarithmic insertions were calculated only qualitatively on the gravity side, see

for instance the schematic result (4.43). The appearance of ln (−hh) terms on the

gravity side in momentum space is in qualitative agreement with the appearance of

ln |z|2 terms in (D.4).

In conclusion all six 2-point correlators and the eight of ten 3-point correlators

that we calculated on the gravity side coincide precisely with corresponding LCFT

correlators. With the tools provided in this work also the remaining two correlators

〈ψL/log ψlog ψlog〉 can be checked in principle, though we have not found an efficient

way to do so. Therefore, we have considered them in the limit of large weights and

found qualitative agreement, notably the correct location of poles, so we expect that

they coincide as well with corresponding LCFT correlators. It would be of interest

to calculate the correlator 〈ψlog ψlog ψlog〉 in full detail since this determines another

defining parameter of the LCFT, denoted by a in [74].

– 481 –

5. Discussion

Summary In this paper we confirmed the conjecture [10] that CTMG (1.1) at

the chiral point (1.4) is dual to a logarithmic CFT: we constructed all regular non-

normalizable left, right and logarithmic modes in global coordinates, see section 3

and appendix C. We plugged these modes into the second (2.20) and third variation

of the action (2.30) to evaluate 2- and 3-point correlators on the gravity side in

section 4. We found that they agree with correlators in a logarithmic CFT. This

is concurrent with recent calculations by Skenderis, Taylor and van Rees [17], who

constructed 2-point correlators on the gravity side on the Poincare patch and also

found agreement with logarithmic CFT correlators.

Generalizations It would be interesting to calculate the correlator (4.44c) between

three logarithmic modes not just qualitatively, but in full detail. This will allow to

extract another parameter in addition to central charges and new anomaly (4.47),

sometimes denoted by a, that determines properties of the LCFT [74]. Exploiting our

results for the non-normalizable modes also higher order correlators can be calculated.

Of course these calculations are rather involved. Some of the tricks we have used

simplify these calculations. For example, one can probably show that all boundary

terms vanish for all higher order correlators, by analogy to our proof for 3-point

correlators. If this is true one just has to vary the bulk action and can partially

integrate freely to simplify expressions, e.g. by applying the on-shell relations (4.15),

(4.18). It might be of interest to check the 4-point correlators, since they contain

the first non-trivial information about the CFT beyond that implied by conformal

invariance and the values of central charges cL/R, new anomaly b and the parameter

a.

Our analysis essentially applies also to New Massive Gravity [70] at a chiral

point: its linearized EOM around an AdS3 background for a particular tuning of

parameters take the form (DLDR)2ψ = 0, with the same operators DL/R as in (2.22).

The construction of all non-normalizable modes is therefore contained in the present

work already: they are given by regular non-normalizable left, right, logarithmic, and

flipped logarithmic modes. The latter are constructed from the logarithmic modes in

the same way as the right modes are constructed from the left modes: by exchanging

the light-like coordinates u ↔ v and switching the weights h ↔ h. Consistent AdS

boundary conditions analog to the ones for CCTMG [14] were constructed for New

Massive Gravity in [78]. These boundary conditions include all normalizable left,

right, logarithmic and flipped logarithmic modes. The main missing ingredient for the

calculation of 2- and 3-point correlators in New Massive Gravity is the construction

of the second and third variation of the action, which is a straightforward exercise.

Another interesting generalization is the inclusion of supersymmetry [52, 79].

Perhaps an AdS/LSCFT correspondence can be established (see [80–82] for some

– 482 –

LSCFT literature). Of course, once the AdS/L(S)CFT conjecture is taken for granted

one can utilize the full power of conformal symmetry to simplify the calculations of

correlators. For instance, one can then exploit the conformal Ward identities, which

was not possible in the present work whose goal was to substantiate the conjecture

by calculating correlators on the gravity side.

Comments on CCTMG as quantum gravity Since CCTMG apparently is dual

to a logarithmic CFT it is neither chiral nor unitary. We address now consequences

for its status as a toy model for quantum gravity. One possible option is to truncate

the logarithmic modes, either by imposing periodicity conditions [10] or by imposing

boundary conditions that are stricter than the requirement of asymptotic AdS, like

Brown–Henneaux boundary conditions [30]. The dual CFT, if it exists, would be

extremal [83], chiral [6,30], and avoids the difficulties with holomorphic factorization

encountered in [84]. That theory contains black holes but no (bulk) gravitons. An

indication for the viability of this option is the existence of a consistent modular

invariant partition function for cR = 24k with k ∈ Z+ [16, 83]. A counter-indication

for the viability of this option is the potential non-existence of extremal CFTs for

large central charges [85]. A related option is that chiral gravity does not have

its own dual CFT, but that a chiral truncation of the LCFT dual to CCTMG can

be achieved by restricting the latter to a superselection sector with vanishing left

charges [16]. Since we know many examples of non-unitary theories with zero-norm

states (gauge degrees of freedom) and negative norm states (ghosts) that can be

truncated consistently to unitary theories by superselecting to ghost number zero, it

is conceivable that the same construction is possible for CCTMG. The assessment of

CCTMG and its chiral truncation as a toy model for quantum gravity still remains

inconclusive.

An alternative option not addressed extensively in the literature (see however

[11]) is the possibility to reverse the sign of the action (1.1), truncate the black

holes by some mechanism and keep the gravitons. We can achieve this by imposing

boundary conditions that are stricter than AdS boundary conditions, see the chiral

boundary conditions in appendix A. This procedure breaks VirasoroR × VirasoroL

to U(1)R × VirasoroL. TMG (or CTMG with the black holes truncated) may serve

as a suitable quantum gravity toy model with an S-matrix (or some AdS-analog

thereof) for gravity wave scattering. This alternative was not considered in [83] and

subsequent work, mainly because pure Einstein gravity in three dimensions does not

provide any degrees of freedom for scattering. By contrast, (C)TMG does provide

them in the form of topologically massive gravitons.

It is possible that neither of the truncations mentioned above works consistently

at the quantum level, see [56] for a recent work that addresses some of these issues.

In that case another option could be pursued [10], namely to unitarily complete the

theory. For practical purposes this could mean a lift to a sector of string theory,

– 483 –

though it is not obvious if and how this works in practice. While reasonable, this

option clearly goes against the original intention [83] to use pure three dimensional

gravity as a suitable quantum gravity toy model.

Prospects for AdS3/LCFT2 CCTMG could serve as a relatively simple gravity

dual to certain logarithmic CFTs. To the best of our knowledge this would be the

first explicit gravity dual of this type. Besides the AdS/LCFT literature [61–65] the

only other context where gravity and LCFTs appear together so far seems to be two-

dimensional gravity coupled to matter [86]. It would be nice to find some condensed

matter applications, like some strongly coupled systems described by a logarithmic

CFT with cL = 0, cR > 0. Many logarithmic CFT examples require negative central

charge, including the physically interesting examples of turbulence [87], the fractional

Quantum Hall effect at filling factor ν = 5/2 [88–90] or dense polymers [91]. It is

clear that these systems cannot be dual to CCTMG, though they may have other

gravity duals, including possibly CTMG with µℓ < 1 or New Massive Gravity. Some

logarithmic CFT systems have indeed vanishing central charge, like quenched random

magnets [92] or other critical systems with quenched disorder, dilute self-avoiding

polymers, percolation etc. [8, 9]. It would be of interest to check whether theses

systems, at least at strong coupling, allow for a dual description in terms of CCTMG.

We have gained now sufficient confidence in the AdS3/LCFT2 correspondence so that

we may start looking for applications.

Acknowledgments

We are grateful to Melanie Becker, Steve Carlip, Sean Downes, Matthias Gaberdiel,

Gaston Giribet, Roman Jackiw, Niklas Johansson, Per Kraus, Max Kreuzer, David

Lowe, Alex Maloney, John McGreevy, Massimo Porrati, Radoslav Rashkov, Stephen

Shenker, Kostas Skenderis, Wei Song, Andy Strominger, Marika Taylor and Stefan

Theisen for discussions. We thank Rene Sedmik for drawing figure 2 on p. 473.

DG was supported by the project MC-OIF 021421 of the European Commis-

sion under the Sixth EU Framework Programme for Research and Technological

Development (FP6). Research at the Massachusetts Institute of Technology is sup-

ported in part by funds provided by the U.S. Department of Energy (DoE) under

the cooperative research agreement DEFG02-05ER41360. During the final stage DG

was supported by the START project Y435-N16 of the Austrian Science Founda-

tion (FWF). DG thanks the Arnold-Sommerfeld Center for Theoretical Physics for

repeated hospitality while part of this work was conceived.

IS was supported by the Transregional Collaborative Research Centre TRR 33,

the DFG cluster of excellence “Origin and Structure of the Universe” as well as the

DFG project Ma 2322/3-1. IS would like to thank the Erwin-Schrodinger Institute in

– 484 –

Vienna for hospitality and financial support during the workshop “Gravity in three

dimensions” in April 2009.

A. Boundary conditions

For generic solutions of the linearized EOM we summarize here various asymptotic

boundary conditions and regularity conditions at the origin. The full metric g =

gAdS + ψ is the sum of the global AdS background (2.12) and solutions ψ of the

linearized EOM (2.23). We assume in this appendix that the modes ψ are brought

into Gaussian normal coordinates (3.1). Modes that are asymptotically AdS3 must

have a Fefferman-Graham expansion (3.2). For the time being we assume that all

modes are regular at the origin.

Normalizable modes By definition these modes are not allowed to modify the

boundary metric γ(0)ij in the expansion (3.2). We neglect terms that fall off asymp-

totically (ρ → ∞) and find the following expansions for left, right and logarithmic

modes:

ψLij = γ

L (2)ij + . . . (A.1)

ψRij = γ

R (2)ij + . . . (A.2)

ψlogij = γ

(1)ij ρ+ γ

log (2)ij + . . . (A.3)

All normalizable modes are asymptotically AdS, including the bulk graviton encoded

in the logarithmic modes.

Non-normalizable modes Solutions to the linearized EOM (2.23) that are not

normalizable according to the criterion above are called “non-normalizable”. We

neglect terms that fall off asymptotically (ρ→ ∞) and find the following expansions

for left, right and logarithmic modes:

ψLij = γ

L (0)ij e2ρ + γ

L (2)ij + . . . (A.4)

ψRij = γ

R (0)ij e2ρ + γ

R (2)ij + . . . (A.5)

ψlogij = γ

(−1)ij ρ e2ρ + γ

log (0)ij e2ρ + γ

(1)ij ρ+ γ

log (2)ij + . . . (A.6)

The left and right non-normalizable modes are asymptotically AdS, while the loga-

rithmic non-normalizable modes are not asymptotically AdS.

Brown–Henneaux and beyond For completeness we mention that in 3-dimensional

Einstein gravity (but not in CCTMG) asymptotically AdS boundary conditions are

equivalent to Brown–Henneaux boundary conditions [72]

ψ ≃

ψuu = O(1) ψuv = O(1) ψuρ = O(e−2ρ)

ψvv = O(1) ψvρ = O(e−2ρ)

ψρρ = O(e−2ρ)

(A.7)

– 485 –

In CCTMG these are replaced by slightly weaker conditions [14, 15]

ψ ≃

ψuu = O(ρ) ψuv = O(1) ψuρ = O(ρ e−2ρ)

ψvv = O(1) ψvρ = O(e−2ρ)

ψρρ = O(e−2ρ)

(A.8)

We stress again that the boundary conditions (A.8) are compatible with asymptotic

AdS behavior, cf. e.g. [58, 93].

Point particle modes In the body of the paper we consider exclusively modes

that are regular at the origin. For sake of completeness, but also because they can

be of relevance in other contexts, we address now particular singular modes. We call

them ‘point particle modes’ for reasons that will become apparent. The linearized

EOM (2.23) admit solutions ξ that are locally pure gauge and obey the Brown–

Henneaux boundary conditions (A.7), but which blow up at the origin ρ = 0. We

focus again first on the left modes. These modes have weights h ≤ 1, h ≥ 0.

ξLµν(h, 0) = e−ihu tanhhρ

1 0 2isinh (2ρ)

0 0 02i

sinh (2ρ)0 − 4

sinh2(2ρ)

µν

(A.9a)

ξLµν(h, h) =

(

(L+)hξL(h, 0))

µν(A.9b)

All modes ξL(h, h) are annihilated by DL and L−, transverse ∇µξµνL = 0 and traceless

ξµL µ = 0. The simplest of these modes χL = ξL(0, 0) depends only on the radial

coordinate ρ:

χLµν =

1 0 2isinh (2ρ)

0 0 02i

sinh (2ρ)0 − 4

sinh2(2ρ)

µν

(A.10)

We see that some of the components of the mode (A.10) diverge for ρ → 0. Con-

sequently the perturbative gravitational energy given by the 00 component of the

pseudo tensor tµν obtained from (2.31) through

E =

∫

d2x√−g t00 =

∫

d2x√−g gµ0 δL(3)

δgµ0(A.11)

diverges.13 This divergence is an artifact of perturbation theory. Indeed these modes

correspond to infinitesimal point sources which in three dimensions cause a conical

singularity. This is not a small deformation of the metric.

In order to exhibit the relation of the modes χL to point particles in AdS we

consider the metric for a point particle with mass M and angular momentum J in

global AdS-coordinates

ds2 = dρ2 − cosh2ρ (r+ dτ − r− dφ)2 + sinh2ρ (r+ dφ− r− dτ)2 (A.12)

13We thank Wei Song for pointing this out to us.

– 486 –

with

1 −M = r2+ + r2

− − J = 2r+r− (A.13)

Global AdS (2.12) is obtained for M = J = 0 while BTZ black holes would corre-

spond to M ≥ 1. Let us now consider a perturbation g → g + 12h of (A.12) around

M = J = 0 with small values of mass M ≪ 1 and angular momentum J ≪ 1. In

(τ, φ, ρ)-coordinates such a perturbation is given by

hµν =

2M cosh2ρ −J 0

−J −2M sinh2ρ 0

0 0 0

µν

(A.14)

The perturbation h in (A.14) is not in the transverse-traceless gauge (2.14). The trace

is non-zero, gµνhµν = −4M. Two components of the divergence vanish, ∇µhµτ =

∇µhµφ = 0, but we also have a non-vanishing component: ∇µh

µρ = 4M coth (2ρ). We

can bring hµν into the transverse-traceless form (2.14) by means of an infinitesimal

diffeomorphism ξµ with (∇2 − 2) ξν = −∇µhµν . The solution to these conditions is

given by ξτ = ξφ = 0 and

ξρ =M2

(coth ρ+ tanh ρ) (A.15)

Applying ξµ to hµν we get h→ h with

hµν =

M −J 0

−J M 0

0 0 − 4Msinh2(2ρ)

µν

(A.16)

In the (u, v, ρ)-coordinates (2.12) the perturbation h becomes diagonal,

hµν =

12(M−J ) 0 0

12(M + J ) 0

0 0 − 4Msinh2(2ρ)

µν

(A.17)

=1

2(M−J ) Re(χL) +

1

2(M + J ) Re(χR) (A.18)

The quantity χL is given in (A.10), and χR is its right handed pendant, which is ob-

tained from χL by exchanging u↔ v. The result (A.18) then establishes the relation

between the singular modes (A.10) and localized point sources. Therefore, we call

these modes “point particle modes”. The ADM mass of these perturbations is finite

(in fact zero at linear order). This result is in agreement with the energy computed in

the Hamiltonian formalism. The divergence of the perturbative gravitational energy

(A.11) comes about because back-reaction has not been taken into account.

– 487 –

Chiral boundary conditions With the overall sign of the action as in (1.1) black

holes in CCTMG have positive energy [6] and gravitons negative energy [10]. This is

problematic for the theory. It is suggestive, therefore, to truncate the theory, either

by eliminating negative energy gravitons or by reversing the sign of the action and

eliminating the now negative energy black holes. We discuss these two possibilities

in some detail and show at the linearized level that one can achieve either of them by

imposing suitable boundary conditions, which we call “chiral boundary conditions”.

We propose now boundary conditions on the fluctuation of the metric ψ that

eliminate ψL, ψlog and all their descendants.

ψ ≃

ψuu = m ψuv = O(1) ψuρ = O(e−2ρ)

ψvv = O(1) ψvρ = O(e−2ρ)

ψρρ = O(e−2ρ)

(A.19)

Here m is a fixed constant. The boundary conditions (A.19) are more restrictive

than the Brown–Henneaux boundary conditions (A.7) because ψuu = m cannot

vary. Consequently, the asymptotic symmetry group does not consist of two Virasoro

copies VirasoroL×VirasoroR, but is broken to U(1)L×VirasoroR. This can be shown

as usual, by considering the generators of diffeomorphisms ζµ that preserve the fall-off

behavior postulated in (A.19):

ζu = ζu0 + 2∂2

vζv0 (v) e−2ρ + O(e−4ρ) (A.20a)

ζv = ζv0 (v) + O(e−4ρ) (A.20b)

ζρ = −1

2∂vζ

v0 (v) + O(e−2ρ) (A.20c)

The U(1)L is generated by the constant ζu0 and the VirasoroR is generated by the

function ζv0 (v). All subleading terms are also independent of the light-cone coordinate

u because ∂uζµ = 0 is required to all orders in e−2ρ. It is also easy to see why (A.19)

eliminates ψL and ψlog but not the right-moving primary ψR. This is a consequence

of requiring m to be fixed. The descendants are obtained by acting (repeatedly)

with the remaining VirasoroR generators, L−n [72], on the primaries. The vanishing

uu-component of ψR changes under Lie-derivative along a vector field ζ as follows:

LζψRuu = 2ψR

uµ ∂uζµ (A.21)

The vector fields associated with the generators L−n are all independent of the light-

like coordinate u. Therefore all descendants of ψR have a vanishing uu-component,

and we can generalize our conclusions to all descendants: the left and the logarithmic

sector are eliminated, while the right sector remains intact. Thus, at the linearized

level we do have a consistent chiral theory where all energies are positive.

– 488 –

Essentially the same story is true for boundary conditions that eliminate only

the right moving sector.

ψ ≃

ψuu = O(ρ) ψuv = O(1) ψuρ = O(ρ e−2ρ)

ψvv = m ψvρ = O(e−2ρ)

ψρρ = O(e−2ρ)

(A.22)

The ensuing theory contains the left-moving boundary graviton ψL as well as the

logarithmic mode ψlog and their appropriate descendants. Comparison with the

Brown–Henneaux case (A.7) exhibits two differences: The component ψvv in (A.22)

is required to be fixed to some constant of order of unity, while in the Brown–

Henneaux case it is allowed to vary at order of unity. This restriction eliminates the

right-moving boundary graviton and its descendants, analog to the previous case. In

addition, the components ψuu, ψuρ are linearly divergent in ρ as compared to their

Brown–Henneaux counterpart. This generalization allows for the logarithmic mode

and its descendants. The asymptotic symmetry group is broken to U(1)R×VirasoroL.

The generators of diffeomorphisms ζµ that preserve the fall-off behavior postulated

in (A.22) are given by

ζu = ζu0 (u) + O(e−4ρ) (A.23a)

ζv = ζv0 + 2∂2

uζu0 (u)e−2ρ + O(ρ e−4ρ) (A.23b)

ζρ = −1

2∂uζ

u0 (u) + O(e−2ρ) (A.23c)

where the subleading terms are such that the conditions ∂vζµ = 0 hold. With

reversed overall sign of the action as compared to (1.1) all modes have now non-

negative energy.

B. Some hypergeometric identities

Frequently we use the Euler transformation

2F1(a, b, c; z) = (1 − z)c−a−b2F1(c− a, c− b, c; z) (B.1)

Often we use relations between contiguous functions, for instance

(a− 1 − (n− b)z) 2F1(a, b, n+ 1; z) + (n + 1 − a) 2F1(a− 1, b, n+ 1; z)

= n(1 − z) 2F1(a, b, n; z) (B.2)

Similar relations can be found in standard literature, like [94].

For integer values h ≤ −1, h ≥ −1 we obtain from (3.25):

Fuv = a(x− 1)(h−h)/2(x+ 1)−(h+h)/22F1(−h, −h + 1, −h + 1 + h;

1 − x

2

)

(B.3)

– 489 –

The case h = −1 is treated in detail in appendix C.1. For h < −1 identities exist

that allow a simple expansion of the hypergeometric function appearing in (B.3) in

terms of elementary functions. For instance, if h = −2 we obtain

2F1

(

2, 3, 3 + h;1 − x

2

)

= − 2

(x+ 1)3

(

2h(h+ 1)(x+ h) 2F1

(

1, 1, 3 + h;1 − x

2

)

− x(h2 + 3h+ 2) − 2h3 − 3h2 + h− 2)

(B.4)

and

2F1

(

1, 1, 3 + h; z)

=(h+ 2)z

(z − 1)2

(

h+2∑

k=2

(1 − 1/z)k

h+ 3 − k− (1 − 1/z)h+3 ln (1 − z)

)

(B.5)

For other integer values of h < −2 similar identities exist. For integer values h ≤ −1,

h ≥ −1 we obtain from (3.24):

Fvv = a22h+h (x−1)(h−h)/2(x+1)−(h+h)/2

2F1

(

−h−1, −h+2, h−h+1;1 − x

2

)

(B.6)

The case h = −1 is treated in detail in appendix C.1. For h < −1 identities exist

that allow a simple expansion of the hypergeometric function appearing in (B.3) in

terms of elementary functions. For instance, if h = −2 we obtain

2F1

(

1, 4, 3 + h;1 − x

2

)

=2

3(x+ 1)3

(

2h(1 − h2) 2F1

(

1, 1, 3 + h,1 − x

2

)

+ x2(2 + h) + x(−h2 + h+ 6) + 2h3 + h2 − 4h+ 4)

(B.7)

and can again use (B.5) to evaluate (B.7) in terms of elementary functions. For other

integer values of h < −2 similar identities exist. These identities are consequences

of relations between contiguous functions.

If h = 0,±1 the hypergeometric function appearing in (3.24) for ε = 0 becomes

a (Jacobi) polynomial of degree −h − 1. If h ≤ −2 and h ≥ 2 we can exploit the

following representation of the hypergeometric function:

2F1

(

h− 1, h+ 2, h+ 1 − h;1 − x

2

)

=(−1)−h+1(−2)h−1(h− h)!

(−2 − h)!(1 − h)!(h− 2)!(h+ 1)!

· dh+1

dxh+1

(

(x+ 1)1−h d−2−h

dx−2−h

( ln x+12

x− 1

))

(B.8)

This formula allows to express all regular non-normalizable left modes in terms of

elementary functions, except for the special cases discussed already. Trivially, we can

also express the hypergeometric function in (3.25) for ε = 0 as follows:

2F1

(

− h, 1 − h, −h + h + 1;1 − x

2

)

=(−1)h(−2)−h(h− h)!

(−1 − h)!(−h)!(h− 1)!h!

· d−h

dx−h

(

(x+ 1)h dh−1

dxh−1

( ln x+12

x− 1

))

(B.9)

– 490 –

if h ≤ −1 and h ≥ 1.

Checking regularity of modes requires an expansion around x = 1. This is

straightforward:

2F1

(

a, b, c;1 − x

2

)

= 1 − ab

2c(x− 1) + O(x− 1)2 (B.10)

Therefore, the regularity or singularity at the origin is entirely due to the behavior

of the polynomial pre-factors in (3.24) and (3.25).

Checking (non-)normalizability requires an asymptotic expansion. For the mas-

sive branch with h ≥ h and non-integer values of the Chern–Simons coupling constant

µ we obtain

limx→∞

2F1

(

a, b, h− h+ 1;1 − x

2

)

∝ x−c c = min a, b (B.11)

For the left branch and integer weights h ≥ h the hypergeometric function of interest

takes the form

limx→∞

2F1

(

− h− 1, −h + 2, h− h + 1;1 − x

2

)

∝ xh+1 if h ≤ 1 and h ≥ −1

(B.12)

limx→∞

2F1

(

− h− 1, −h + 2, h− h + 1;1 − x

2

)

∝ xh−2 otherwise (B.13)

Thus, for h ≥ 2 the left modes are normalizable, while for h ≤ 1 they are non-

normalizable as long as h ≥ −1. However, the three cases h = 0,±1 have to be

treated separately because the solutions of the second order differential equation

(3.22) are not necessarily compatible with the first order system (3.19a), (3.19b) for

these values of h or with the algebraic system (3.18). The case h = −1 is treated in

appendix C.1 and turns out to be consistent. We treat here the other two cases and

assume again h ≥ h. If h = 0 then the first three algebraic conditions (3.18) (for

µ = 1) establish Fuv = 0 and h = 1. Thus, there is only one non-normalizable left

mode if h = 0. This mode, however, has a singular ρρ component, so there are no

regular non-normalizable left modes if h = 0. If h = 1 then the first order equation

(3.19b) decouples and yields Fvv ∝ (x + 1)(h+1)/2(x − 1)(1−h)/2. The ensuing modes

are non-normalizable, but not regular at the origin in the vv component unless h = 1.

Thus, there is no regular non-normalizable left mode if h = 1, unless h = 1. Suppose

that h > 1 and h < −1 (otherwise we recover one of the special cases discussed

separately above). Then we can exploit the asymptotic expansions

2F1

(

− h, −h + 1, h− h + 1;1 − x

2

)

=(h− h)!

h!(−h)!

(x− 1

2

)h(

1

+2hh

x

(

lnx

2− ψ(h) − ψ(1 − h) + 1 − 2γ

)

+ O(ln x/x2))

(B.14)

– 491 –

and

2F1

(

h− 1, h+ 2, h− h + 1;1 − x

2

)

=2(h− h)!

(h+ 1)!(1 − h)!

(x− 1

2

)1−h(

1

− (h− 1)(1 − h)

x− 1+h(h− 1)(1 − h)(−h)

(x− 1)2− 2(1 − h)(−h)(−h− 1)(h+ 1)h(h− 1)

3x3

·(

lnx

2− ψ(h + 2) − ψ(−h− 1) +

11

6− 2γ

)

+ O(ln x/x4))

(B.15)

Here γ is the Euler–Mascheroni constant and ψ(z) = Γ′(z)/Γ(z) is the digamma-

function, with the asymptotic expansion for large weights

ψ(h) = lnh− 1

2h+ O(1/h2) (B.16)

In some considerations the limit for large weights is of interest. To obtain a

formula for hypergeometric functions in this limit we proceed as follows. We shift

h→ h+ λ, h→ h− λ and let λ→ ∞. The hypergeometric function in (3.24) leads

to the following limit

limλ→∞

2F1

(

λ+h+2, λ+h−1, 2λ+h− h+1;1 − x

2

)

= Ξ(λ, x)(

1+O(1/λ))

(B.17)

The function Ξ(λ, x) was derived by Dr. Watson in 1918 [95]:

Ξ(λ, x) = f(x)(x− 1

2

)−λ

λ−1/2 Γ(2λ+ h− h+ 1)

Γ(λ+ h− 1) Γ(λ+ 2 − h)

(

1 +4 − 2

√

2(x+ 1)

x− 1

)λ

(B.18)

where f(x) is a known λ-independent function of x. The factor (x − 1)−λ cancels

with the polynomial pre-factors multiplying the hypergeometric function in (3.24).

The Γ-functions and powers of 2 cancel mostly with factors from the normalization

constant a2, see footnote 14 on p. 493. In the end a factor λ5/2 remains, multiplied by

finite terms and by the last bracket in (B.18). A key observation is that in the range

of definition x ∈ [1,∞) the last bracket in (B.18) is always smaller than 1, and thus

goes to zero rapidly as λ tends to ∞ so that the whole expression vanishes. The only

exception arises in the limit of large x, which has to be treated separately. This is

an explicit realization of the UV/IR connection [76]: the limit of large weights/small

distances (UV) on the CFT side implies the limit of large x (IR) on the gravity side.

If x scales like λ2+ε the function f(x) in (B.18) behaves as

limλ→∞

f(λ2+εx) ∝ λ−5(2+ε)/4 (B.19)

whereas the last bracket in (B.18) behaves as

limλ→∞

(

1 +4 − 2

√

2(λ2+εx+ 1)

λ2+εx− 1

)λ

=

1 if ε > 0,

0 if − 2 < ε < 0,

e−2√

2/x if ε = 0

(B.20)

– 492 –

C. Non-normalizable modes

C.1 Left and right branch

The right modes are obtained from the left modes by exchanging u ↔ v and h ↔ h

in all formulas below. The left modes are obtained as follows. We start from the

separation Ansatz

ψLµν = e−ihu−ihv Fµν(ρ) (C.1)

and solve the EOM (3.18), (3.19) with µ = 1, assuming regularity at x = cosh (2ρ) =

1. Below we provide explicit results for the tensor Fµν .

Generic case We summarize here our results for generic regular non-normalizable

left modes with weights h ≥ 2 and h ≤ −2.14

Fvv =(−1)h−h(x− 1)(h−h)/2(x+ 1)(h+h)/2

2 (−2 − h)! (h− 2)!

dh+1

dxh+1

(

(x+ 1)1−h d−2−h

dx−2−h

( ln x+12

x− 1

))

(C.2a)

Fuv =(−1)h−h(1 − h2)(x− 1)(h−h)/2(x+ 1)−(h+h)/2

(−1 − h)! (h− 1)!

d−h

dx−h

(

(x+ 1)h dh−1

dxh−1

( ln x+12

x− 1

))

(C.2b)

Fuu =h

hFuv (C.2c)

Fvρ =2i√x2 − 1

(

hFuv − hFvv

)

(C.2d)

Fuρ =2i

h

√x2 − 1

dFuv

dx(C.2e)

Fρρ =4

x2 − 1

(

(2x− h

h)Fuv − Fvv

)

(C.2f)

The results above follow from (B.8), (B.9) and (3.18).

Special case h = 1 Modes with weights h = 1 and h ≤ −2 are given by:

Fvv = (x− 1)(1−h)/2(x+ 1)(1+h)/2 (C.3a)

Fuv = 0 (C.3b)

Fuu = 0 (C.3c)

Fvρ = −2i(x− 1)−h/2(x+ 1)h/2 (C.3d)

Fuρ = 0 (C.3e)

Fρρ = −4(x− 1)(−1−h)/2(x+ 1)−(1−h)/2 (C.3f)

14 We have chosen the overall normalization such that Fvv = x+O(1) asymptotically. This choice

leads to a2 = 2−h(h+ 1)!(1 − h)!/(h− h)! and a = 2h(1 − h2)h!(−h)!/(h− h)! in (3.24), (3.25).

– 493 –

h: -1 0 1 2 3 4 5 6 7 8 9

a: -4 2 0 0 24 100 260 539 4872/5 8028/5 17316/7

Table 1: Values of the integration constant a in (C.6g) for small weights h

Special case h = 0 Modes with weights h = 0 and h ≤ −2 are given by:

Fvv = (x− h)(x− 1)−h/2(x+ 1)h/2 (C.4a)

Fuv = (x− 1)−h/2(x+ 1)h/2 (C.4b)

Fuu = 0 (C.4c)

Fvρ = 2ih (x− 1)(−1−h)/2(x+ 1)−(1−h)/2 (C.4d)

Fuρ = −2i(x− 1)(−1−h)/2(x+ 1)−(1−h)/2 (C.4e)

Fρρ = 4(x+ h)(x− 1)−1−h/2(x+ 1)−1+h/2 (C.4f)

Special case h = −1 Modes with weights h = −1 and h ≤ −2 are given by:

Fvv = (x2 − 2hx+ 2h2 − 1)(x− 1)(−1−h)/2(x+ 1)−(1−h)/2 (C.5a)

Fuv = −4h(x− 1)(−1−h)/2(x+ 1)−(1−h)/2 (C.5b)

Fuu = 4(x− 1)(−1−h)/2(x+ 1)−(1−h)/2 (C.5c)

Fvρ = 2i (x2 − 2hx− 2h2 − 1)(x− 1)−1−h/2(x+ 1)−1+h/2 (C.5d)

Fuρ = 8i (x+ h)(x− 1)−1−h/2(x+ 1)−1+h/2 (C.5e)

Fρρ = −4(x2 + 6hx+ 2h2 + 3)(x− 1)(−3−h)/2(x+ 1)−(3−h)/2 (C.5f)

Special case h = −1 If h = −1 the first order equations (3.19) decouple for ε = 0.

Solving the homogeneous first order equation for Fvv yields Fvv = (x− 1)(1+h)/2(x+

1)(1−h)/2. The first order equation (3.19a) determines Fuv in terms of elementary

functions. The integration constant is determined uniquely by the requirement of

regularity at x = 1. We obtain

Fvv = (x− 1)(1+h)/2(x+ 1)(1−h)/2 (C.6a)

Fuv = (x− 1)(1+h)/2(x+ 1)(1−h)/2H(x) (C.6b)

Fuu = −h (x− 1)(1+h)/2(x+ 1)(1−h)/2 H(x) (C.6c)

Fvρ = −2i (x− 1)h/2(x+ 1)−h/2(

H(x) + h)

(C.6d)

Fuρ = 2i (x− 1)h/2(x+ 1)−h/2(

(x+ h)H(x) + h2 − 1)

(C.6e)

Fρρ = 4 (x− 1)(−1+h)/2(x+ 1)−(1+h)/2(

(2x+ h)H(x) − 1)

(C.6f)

The function H(x) is given by

H(x) = (1 − h2)(x+ 1)h−1(x− 1)−h−1(

x+ 1 − 2h lnx+ 1

2

−h

∑

k=2

(

h

k

)

(−2)k

k − 1(x+ 1)1−k

)

− a (x+ 1)h−1(x− 1)−h−1 (C.6g)

– 494 –

with the integration constant a determined uniquely from the requirement of regu-

larity at the origin.15

a = 2(1 − h2)(

1 −h

∑

k=2

(−1)kh!

k!(h− k)!(k − 1)

)

− 4 δh,−1 (C.6h)

See table 1 for small values of the weight h. Note that in the sums above we have the

usual convention that they evaluate to zero if h < 2. Near the origin the function

H(x) behaves like (1 − h)/2 + O(x− 1). Asymptotically we obtain limx→∞H(x) =

(1 + 2h)(1 − h2)/x+ O(lnx/x2).

Examples with h = −1 For convenience some explicit examples are presented

below. All modes below have weight h = −1.

h = −1:

Fvv = x+ 1 (C.7a)

Fuv =4

x+ 1(C.7b)

Fuu =4

x+ 1(C.7c)

Fvρ = 2i(x+ 3)

√

x− 1

(x+ 1)3(C.7d)

Fuρ = 8i

√

x− 1

(x+ 1)3(C.7e)

Fρρ = −4(x− 5)

(x+ 1)2(C.7f)

h = 0:

Fvv =√x2 − 1 (C.8a)

Fuv =

√

x− 1

x+ 1(C.8b)

Fuu = 0 (C.8c)

Fvρ = − 2i

x+ 1(C.8d)

Fuρ = − 2i

x+ 1(C.8e)

Fρρ = 4

√

x− 1

(x+ 1)3(C.8f)

15The form of (C.6g) does not make it completely evident that all poles at x = 1 can be cancelled

by a single choice of parameter. However, it is clear that this must be possible since the alternative

form of H(x) in terms of a hypergeometric function, (3.25), manifestly is regular at x = 1 for h ≥ h.

– 495 –

h = 1:

Fvv = x− 1 (C.9a)

Fuv = 0 (C.9b)

Fuu = 0 (C.9c)

Fvρ = −2i

√

x− 1

x+ 1(C.9d)

Fuρ = 0 (C.9e)

Fρρ = − 4

x+ 1(C.9f)

The mode h = 1, h = −1 is of particular interest, because it has angular momentum

2 (“non-normalizable boundary graviton”), a property it shares with all primaries.

h = 2:

Fvv = (x− 1)

√

x− 1

x+ 1(C.10a)

Fuv = −3

√

x+ 1

(x− 1)3(x+ 1 − 4 ln

x+ 1

2− 4

x+ 1) (C.10b)

Fuu = 6

√

x+ 1

(x− 1)3(x+ 1 − 4 ln

x+ 1

2− 4

x+ 1) (C.10c)

Fvρ = −2i(7 + x2(−9 + 2x) + 12(1 + x) ln x+12

)

(x− 1)2(1 + x)(C.10d)

Fuρ =6i(5 + (2 − 7x)x+ 4(1 + x)(2 + x) ln x+1

2)

(x− 1)2(1 + x)(C.10e)

Fρρ = 419 + 3x− 15x2 − 7x3 + 24(1 + x)2 ln x+1

2

(x− 1)5/2(x+ 1)3/2(C.10f)

For h = 2 logarithmic (asymptotically subleading) terms appear in most of the

components. This is a generic feature of all modes with h ≥ 2. It is worthwhile

emphasizing that all these modes are regular at x = 1, despite of the appearance of

1/(x − 1) factors in various components (e.g. the factor 1/(x − 1)5/2 in Fρρ above).

Using the algebraic relations (3.31) we can generate all regular non-normalizable left

modes from the (2,−1) mode above.

C.2 Logarithmic branch

The logarithmic modes

ψlogµν = i(u+ v)ψL

µν − F logµν e

−ihu−ihv (C.11)

– 496 –

are specified uniquely — up to addition of left modes and overall rescalings — by pro-

viding the tensor F logµν . Once the components F log

vv and F loguv are known the remaining

ones follow algebraically from the relations (3.34). The components F logvv and F log

uv are

determined from solutions of linear ordinary first order differential equations (3.35).

Generic modes Using the algebraic relations (3.43) we can generate all regular

non-normalizable logarithmic modes from the (2,−1) mode, see (C.22) below. We

do not list generic modes explicitly.

Special case h = −1 We assume here h = −1 after taking the limit in (3.33). In

order to evaluate (3.33) we start with the massive branch solution (3.10)-(3.25) and

assume that ε is small and positive. In addition we vary the weights with ε, so that

the following two conditions hold

h = −1 + ε (C.12)

h− h = n n ∈ N (C.13)

The first condition ensures that we have the same periodicity properties as for cor-

responding normalizable massive modes (see (47) in [6]).16 It implies that the differ-

ential equations decouple for any ε, so that we obtain a first order equation for Fvv.

The second condition is necessary since we want to keep periodicity in the angle φ

at each step, which requires the difference h − h to be an integer. The assumption

h ≥ h implies that n is a natural number. The attribute “non-normalizable” refers

to growth faster than asymptotically AdS, i.e., the modes ψ are not compatible with

the Fefferman–Graham expansion (3.2) but violate it logarithmically, see (A.6). As

explained in [17] this is the expected behavior for the source terms of the operators

associated with the logarithmic modes.

We obtain the following expression for the massive mode:

ψεµν = −e−i(n−1+ε)u+i(1−ε)v (x− 1)n/2 (x+ 1)1−n/2−ε Hε

µν(x) (C.14a)

where

Hεvv = 1 (C.14b)

Hεuv = Hε(x) (C.14c)

Hεuu = −βHε(x) + εn(n− 2) + O(ε2) (C.14d)

Hεvρ = − 2i√

x2 − 1

(

Hε(x) + n− 1 + εn+ O(ε2))

(C.14e)

Hεuρ =

2i√x2 − 1

(

(x+ β)Hε(x) + n(n− 2) + 2εn(n− 1) + O(ε2))

(C.14f)

Hερρ =

4

x2 − 1

(

(2x+ β)Hε(x) − 1 − εn(n− 2) + O(ε2))

(C.14g)

16Unlike normalizable ones there exist non-normalizable solutions with h, h independent from ε.

– 497 –

with β = n− 1 − ε(x− 1) and

Hε(x) =1 − n/2 − ε

1 − ε2F1

(

1, 2(1 − ε), n+ 1;1 − x

2

)

(C.14h)

In the limit ε → 0 the function Hε(x) approaches the function H(x) given in (C.6g),

while the parameter β approaches h = n− 1. Consequently, the solution (C.14a) up

to normalization coincides with the non-normalizable regular left modes (C.6) with

weights h = −1 and h = n− 1.

The regular non-normalizable logarithmic modes are determined from (C.14a)-

(C.14h) using the definition (3.33). Our final result is

ψlogµν (h = n− 1, h = −1) =

(

i(u+ v) + lnx+ 1

2

)

ψLµν

− e−i(n−1)u+iv (x− 1)n/2 (x+ 1)1−n/2dHε

µν

dε

∣

∣

∣

ε=0(C.15)

where ψLµν are the regular non-normalizable left modes with weights h = n − 1,

h = −1 given in (C.6). One can add to ψlog a regular non-normalizable left moving

mode with the same weights without changing anything essential. We fix this shift

ambiguity in a convenient way in (C.15), but in the body of the paper we freely add

or subtract such left moving modes to simplify some expressions. This is related to

a well-known ambiguity in LCFTs. The first line of our solution for the logarithmic

modes (C.15) coincides with the relation between normalizable logarithmic and left

modes (see Eqs. (3.1), (3.2) in [10]). The second line contains the function Hε(x)

(C.14h) to next to leading order in ε. In order to obtain Hε(x) for arbitrary n we

use a relation between contiguous functions (B.2) and exploit 2F1(0, b, n; z) = 1 to

establish a recursion relation

2F1

(

1, 2(1 − ε), n + 1;1 − x

2

)

=2n

(1 − x)(n− 2(1 − ε))

(

1 − x+ 1

22F1

(

1, 2(1 − ε), n;1 − x

2

)

)

(C.16)

Thus, we need to evaluate by hand only the starting point n = 1, which is very

simple:

2F1

(

1, 2(1 − ε), 1;1 − x

2

)

=(x+ 1

2

)−2(1−ε)(C.17)

Higher values of n are then obtained recursively from (C.16). From our result for

regular non-normalizable logarithmic modes (C.15) we obtain

L0ψlog = (n− 1)ψlog − ψL L0ψ

log = −ψlog − ψL (C.18)

This result essentially coincides with the result (3.16) for normalizable modes for

h = n− 1, h = −1. See also the algebraic relations (3.43).

– 498 –

Examples with h = −1 For convenience some explicit examples are presented

below. All modes below have weight h = −1.

h = −1:

F logvv = −(x+ 1) ln

x+ 1

2(C.19a)

F loguv =

4 ln x+12

x+ 1(C.19b)

F loguu =

4(

x+ ln x+12

− 1)

x+ 1(C.19c)

F logvρ = −2i

(

x2 + 2x+ 5)

ln x+12

(x+ 1)√x2 − 1

(C.19d)

F loguρ =

8i(x− 1)(

ln x+12

− 1)

(x+ 1)√x2 − 1

(C.19e)

F logρρ =

4(

(x2 + 10x− 3) ln x+12

− 4(x− 1))

(x+ 1)2(x− 1)(C.19f)

h = 0:

F logvv = −

√x2 − 1 ln

x+ 1

2(C.20a)

F loguv =

−(x+ 3) ln x+12

+ x− 1√x2 − 1

(C.20b)

F loguu = −2

√x2 − 1

x+ 1(C.20c)

F logvρ = 2i

(x+ 3) ln x+12

− x2 − x+ 2

x2 − 1(C.20d)

F loguρ = −2i

(3x+ 1) ln x+12

− x+ 1

x2 − 1(C.20e)

F logρρ = 4

−(x2 + 6x+ 1) ln x+12

+ 2(x2 − 1)

(x2 − 1)3/2(C.20f)

– 499 –

h = 1:

F logvv = −(x− 1) ln

x+ 1

2(C.21a)

F loguv = −4

x− 2 ln x+12

− 1

x− 1(C.21b)

F loguu = 4

x− 2 ln x+12

− 1

x− 1(C.21c)

F logvρ = 2i

(x2 − 2x− 7) ln x+12

− 2(x− 1)(x− 3)

(x− 1)√x2 − 1

(C.21d)

F loguρ = 16i

(x+ 1) ln x+12

− x+ 1

(x− 1)√x2 − 1

(C.21e)

F logρρ = 4

(x2 + 14x+ 9) ln x+12

− 8x2 + 4x+ 4

(x− 1)2(x+ 1)(C.21f)

h = 2:

F logvv = −(x− 1)3/2 ln x+1

2√x+ 1

(C.22a)

F loguv =

2(x+ 1) ln2 x+12

− 3(x2 + 14x+ 9) ln x+12

(x− 1)√x2 − 1

+2(7x+ 11)√

x2 − 1(C.22b)

F loguu =

−4(x+ 1) ln2 x+12

+ 2(7x2 + 42x+ 23) ln x+12

(x− 1)√x2 − 1

+x2 − 38x− 35√

x2 − 1(C.22c)

F logvρ = −2i

2(x+ 1) ln2 x+12

− (2x+ 1)(x2 − 2x+ 25) ln x+12

(x− 1)2(x+ 1)− 2i(3x2 + 8x+ 25)

x2 − 1

(C.22d)

F loguρ = 2i

2(x+ 1)(x+ 2) ln2 x+12

− (6x3 + 47x2 + 120x+ 43) ln x+12

(x− 1)2(x+ 1)

+4i(14x2 + 15x+ 25)

x2 − 1(C.22e)

F logρρ = 4

4(x+ 1)2 ln2 x+12

− (5x3 + 101x2 + 135x+ 47) ln x+12

(x− 1)5/2(x+ 1)3/2+

4(27x2 + 82x+ 35)

(x2 − 1)3/2

(C.22f)

For h = 2 squared logarithmic (asymptotically subleading) terms appear in most of

the components. This is a generic feature of all modes with h ≥ 2. It is worthwhile

emphasizing that all these modes are regular at x = 1, despite of the appearance of

1/(x− 1) factors in various components (e.g. the factor 1/(x− 1)5/2 in F logρρ above).

Using the algebraic relations (3.43) we can generate all regular non-normalizable

logarithmic modes from the (2,−1) mode above.

– 500 –

D. Correlation functions in Euclidean logarithmic CFT

LCFTs arise if there are fields with degenerate scaling dimensions having a Jordan

block structure like in (1.5). In any LCFT one of these degenerate fields becomes a

zero norm state coupled to a logarithmic partner [80]. For our purposes the following

consideration is sufficient. Suppose the operators OL and Olog form a logarithmic

pair with conformal weights (2, 0), where OL corresponds to the zero norm state.

Suppose that we have an additional operator OR with conformal weights (0, 2) that

commutes with OL and Olog. With a standard normalization for Olog and keeping

only the most singular pieces in the coincidence limit the correlators between these

operators take the form [7–9,80]

〈OR(z, z)OR(0, 0)〉 =cR2z4

(D.1a)

〈OL(z, z)OL(0, 0)〉 = 0 (D.1b)

〈OL(z, z)OR(0, 0)〉 = 0 (D.1c)

〈OR(z, z)Olog(0, 0)〉 = 0 (D.1d)

〈OL(z, z)Olog(0, 0)〉 =b

2z4(D.1e)

〈Olog(z, z)Olog(0, 0)〉 = −b ln (m2|z|2)z4

(D.1f)

The scale m is arbitrary and has no significance. It can be changed by a shift

Olog → Olog+γOL. Without loss of generality we setm = 1. The right central charge

cR and the “new anomaly” b have a significant meaning and define key properties of

the LCFT. The left central charge cL is assumed to vanish.

The 3-point correlators without logarithmic insertions are given by [66]

〈OR(z, z)OR(z′, z′)OR(0, 0)〉 =cR

z2z′ 2(z − z′)2(D.2a)

〈OL(z, z)OR(z′, z′)OR(0, 0)〉 = 0 (D.2b)

〈OL(z, z)OL(z′, z′)OR(0, 0)〉 = 0 (D.2c)

〈OL(z, z)OL(z′, z′)OL(0, 0)〉 = 0 (D.2d)

The 3-point correlators with one logarithmic insertion are given by [75]

〈OR(z, z)OR(z′, z′)Olog(0, 0)〉 = 0 (D.3a)

〈OL(z, z)OL(z′, z′)Olog(0, 0)〉 =b

z2z′2(z − z′)2(D.3b)

〈OL(z, z)OR(z′, z′)Olog(0, 0)〉 = 0 (D.3c)

– 501 –

The 3-point correlators with at least two logarithmic insertion are given by [75]

〈OR(z, z)Olog(z′, z′)Olog(0, 0)〉 = 0 (D.4a)

〈OL(z, z)Olog(z′, z′)Olog(0, 0)〉 = − 2b ln |z′|2 + b2

z2z′2(z − z′)2(D.4b)

〈Olog(z, z)Olog(z′, z′)Olog(0, 0)〉 =lengthy

z2z′2(z − z′)2(D.4c)

Again we have kept only the contributions that are most singular in the coincidence

limit.

References

[1] S. Deser, Cosmological Topological Supergravity, in Quantum Theory Of Gravity

(S. M. Christensen, ed.), pp. 374–381. Adam Hilger, Bristol, 1984. Print-82-0692

(Brandeis).

[2] S. Deser, R. Jackiw and S. Templeton, Three-dimensional massive gauge theories,

Phys. Rev. Lett. 48 (1982) 975–978.

[3] S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories, Ann.

Phys. 140 (1982) 372–411. Erratum-ibid. 185 (1988) 406.

[4] M. Banados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional

space-time, Phys. Rev. Lett. 69 (1992) 1849–1851 [hep-th/9204099].

[5] P. Kraus and F. Larsen, Holographic gravitational anomalies, JHEP 01 (2006) 022

[hep-th/0508218].

[6] W. Li, W. Song and A. Strominger, Chiral Gravity in Three Dimensions, JHEP 04

(2008) 082 [0801.4566].

[7] V. Gurarie, Logarithmic operators in conformal field theory, Nucl. Phys. B410

(1993) 535–549 [hep-th/9303160].

[8] M. Flohr, Bits and pieces in logarithmic conformal field theory, Int. J. Mod. Phys.

A18 (2003) 4497–4592 [hep-th/0111228].

[9] M. R. Gaberdiel, An algebraic approach to logarithmic conformal field theory, Int. J.

Mod. Phys. A18 (2003) 4593–4638 [hep-th/0111260].

[10] D. Grumiller and N. Johansson, Instability in cosmological topologically massive

gravity at the chiral point, JHEP 07 (2008) 134 [0805.2610].

[11] S. Carlip, S. Deser, A. Waldron and D. K. Wise, Cosmological Topologically Massive

Gravitons and Photons, Class. Quant. Grav. 26 (2009) 075008 [0803.3998].

– 502 –

[12] D. Grumiller, R. Jackiw and N. Johansson, Canonical analysis of cosmological

topologically massive gravity at the chiral point, in Fundamental Interactions - A

Memorial Volume for Wolfgang Kummer. World Scientific, 2009. 0806.4185.

[13] S. Carlip, The Constraint Algebra of Topologically Massive AdS Gravity, JHEP 10

(2008) 078 [0807.4152].

[14] D. Grumiller and N. Johansson, Consistent boundary conditions for cosmological

topologically massive gravity at the chiral point, Int. J. Mod. Phys. D17 (2009)

2367–2372 [0808.2575].

[15] M. Henneaux, C. Martinez and R. Troncoso, Asymptotically anti-de Sitter spacetimes

in topologically massive gravity, Phys. Rev. D79 (2009) 081502R [0901.2874].

[16] A. Maloney, W. Song and A. Strominger, Chiral Gravity, Log Gravity and Extremal

CFT, 0903.4573.

[17] K. Skenderis, M. Taylor and B. C. van Rees, Topologically Massive Gravity and the

AdS/CFT Correspondence, 0906.4926.

[18] K. Hotta, Y. Hyakutake, T. Kubota and H. Tanida, Brown-Henneaux’s Canonical

Approach to Topologically Massive Gravity, JHEP 07 (2008) 066 [0805.2005].

[19] W. Li, W. Song and A. Strominger, Comment on ’Cosmological Topological Massive

Gravitons and Photons’, 0805.3101.

[20] M.-i. Park, Constraint Dynamics and Gravitons in Three Dimensions, JHEP 09

(2008) 084 [0805.4328].

[21] I. Sachs and S. N. Solodukhin, Quasi-Normal Modes in Topologically Massive

Gravity, JHEP 08 (2008) 003 [0806.1788].

[22] D. A. Lowe and S. Roy, Chiral geometries of (2+1)-d AdS gravity, Phys. Lett. B668

(2008) 159–162 [0806.3070].

[23] Y. S. Myung, H. W. Lee and Y.-W. Kim, Entropy of black holes in topologically

massive gravity, 0806.3794.

[24] S. Carlip, S. Deser, A. Waldron and D. K. Wise, Topologically Massive AdS Gravity,

Phys. Lett. B666 (2008) 272–276 [0807.0486].

[25] H. W. Lee, Y.-W. Kim and Y. S. Myung, Quasinormal modes for topologically

massive black hole, 0807.1371.

[26] I. Sachs, Quasi-Normal Modes for Logarithmic Conformal Field Theory, JHEP 09

(2008) 073 [0807.1844].

[27] G. W. Gibbons, C. N. Pope and E. Sezgin, The General Supersymmetric Solution of

Topologically Massive Supergravity, Class. Quant. Grav. 25 (2008) 205005

[0807.2613].

– 503 –

[28] D. Anninos, W. Li, M. Padi, W. Song and A. Strominger, Warped AdS3 Black Holes,

JHEP 03 (2009) 130 [0807.3040].

[29] G. Giribet, M. Kleban and M. Porrati, Topologically Massive Gravity at the Chiral

Point is Not Unitary, JHEP 10 (2008) 045 [0807.4703].

[30] A. Strominger, A Simple Proof of the Chiral Gravity Conjecture, 0808.0506.

[31] G. Compere and S. Detournay, Semi-classical central charge in topologically massive

gravity, Class. Quant. Grav. 26 (2009) 012001 [0808.1911].

[32] Y. S. Myung, Logarithmic conformal field theory approach to topologically massive

gravity, Phys. Lett. B670 (2008) 220–223 [0808.1942].

[33] S. de Haro, Dual Gravitons in AdS4/CFT3 and the Holographic Cotton Tensor,

JHEP 01 (2009) 042 [0808.2054].

[34] K. A. Stevens, K. Schleich and D. M. Witt, Non-existence of Asymptotically Flat

Geons in 2+1 Gravity, Class. Quant. Grav. 26 (2009) 075012 [0809.3022].

[35] S. Deser, Distended Topologically Massive Electrodynamics, in Fundamental

Interactions - A Memorial Volume for Wolfgang Kummer. World Scientific, 2009.

0810.5384.

[36] K. Hotta, Y. Hyakutake, T. Kubota, T. Nishinaka and H. Tanida, The

CFT-interpolating Black Hole in Three Dimensions, JHEP 01 (2009) 010

[0811.0910].

[37] H. Quevedo and A. Sanchez, Geometric description of BTZ black holes

thermodynamics, Phys. Rev. D79 (2009) 024012 [0811.2524].

[38] J. J. Oh and W. Kim, Absorption Cross Section in Warped AdS3 Black Hole, JHEP

01 (2009) 067 [0811.2632].

[39] A. Garbarz, G. Giribet and Y. Vasquez, Asymptotically AdS3 Solutions to

Topologically Massive Gravity at Special Values of the Coupling Constants, Phys.

Rev. D79 (2009) 044036 [0811.4464].

[40] W. Kim and E. J. Son, Thermodynamics of warped AdS3 black hole in the brick wall

method, Phys. Lett. B673 (2009) 90–94 [0812.0876].

[41] R. B. Mann, J. J. Oh and M.-I. Park, The Role of Angular Momentum and Cosmic

Censorship in the (2+1)-Dimensional Rotating Shell Collapse, Phys. Rev. D79

(2009) 064005 [0812.2297].

[42] M. Blagojevic and B. Cvetkovic, Canonical structure of topologically massive gravity

with a cosmological constant, JHEP 05 (2009) 073 [0812.4742].

[43] S. Nam and J.-D. Park, Hawking radiation from covariant anomalies in 2+1

dimensional black holes, Class. Quant. Grav. 26 (2009) 145015 [0902.0982].

– 504 –

[44] S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, 0902.2790.

[45] E. Sezgin and Y. Tanii, Witten-Nester Energy in Topologically Massive Gravity,

0903.3779.

[46] D. Anninos, M. Esole and M. Guica, Stability of warped AdS3 vacua of topologically

massive gravity, 0905.2612.

[47] G. Compere and S. Detournay, Boundary conditions for spacelike and timelike

warped AdS3 spaces in topologically massive gravity, JHEP 08 (2009) 092

[0906.1243].

[48] K. Hotta, Y. Hyakutake, T. Kubota, T. Nishinaka and H. Tanida, Left-Right

Asymmetric Holographic RG Flow with Gravitational Chern-Simons Term, Phys.

Lett. B680 (2009) 279–285 [0906.1255].

[49] D. Anninos, Sailing from Warped AdS3 to Warped dS3 in Topologically Massive

Gravity, 0906.1819.

[50] S. Carlip, Chiral Topologically Massive Gravity and Extremal B-F Scalars, JHEP 09

(2009) 083 [0906.2384].

[51] D. D. K. Chow, C. N. Pope and E. Sezgin, Exact solutions of topologically massive

gravity, 0906.3559.

[52] M. Becker, P. Bruillard and S. Downes, Chiral Supergravity, 0906.4822.

[53] M. Blagojevic and B. Cvetkovic, Asymptotic structure of topologically massive

gravity in spacelike stretched AdS sector, JHEP 09 (2009) 006 [0907.0950].

[54] Y. Vasquez, Charged Black Holes in Three Dimensional Einstein Theory with

Torsion and Chern Simons Terms, 0907.4165.

[55] J. F. Duncan and I. B. Frenkel, Rademacher sums, Moonshine and Gravity,

0907.4529.

[56] T. Andrade and D. Marolf, No chiral truncation of quantum log gravity?, 0909.0727.

[57] O. Miskovic and R. Olea, Background-independent charges in Topologically Massive

Gravity, 0909.2275.

[58] K. Skenderis, M. Taylor and B. C. van Rees, AdS boundary conditions and the

Topologically Massive Gravity/CFT correspondence, 0909.5617.

[59] S. Ertl, D. Grumiller and N. Johansson, Erratum to ‘Instability in cosmological

topologically massive gravity at the chiral point’, arXiv:0805.2610, 0910.1706.

[60] H. R. Afshar, M. Alishahiha and A. Naseh, On three dimensional bigravity,

0910.4350.

– 505 –

[61] A. M. Ghezelbash, M. Khorrami and A. Aghamohammadi, Logarithmic conformal

field theories and AdS correspondence, Int. J. Mod. Phys. A14 (1999) 2581–2592

[hep-th/9807034].

[62] Y. S. Myung and H. W. Lee, Gauge bosons and the AdS(3)/LCFT(2)

correspondence, JHEP 10 (1999) 009 [hep-th/9904056].

[63] I. I. Kogan, Singletons and logarithmic CFT in AdS/CFT correspondence, Phys.

Lett. B458 (1999) 66–72 [hep-th/9903162].

[64] A. Lewis, Logarithmic operators in AdS(3)/CFT(2), Phys. Lett. B480 (2000)

348–354 [hep-th/9911163].

[65] S. Moghimi-Araghi, S. Rouhani and M. Saadat, Correlation functions and

AdS/LCFT correspondence, Nucl. Phys. B696 (2004) 492–502 [hep-th/0403150].

[66] P. H. Ginsparg, Applied conformal field theory, hep-th/9108028.

[67] H. Liu and A. A. Tseytlin, D = 4 super Yang-Mills, D = 5 gauged supergravity, and

D = 4 conformal supergravity, Nucl. Phys. B533 (1998) 88–108 [hep-th/9804083].

[68] G. Arutyunov and S. Frolov, Three-point Green function of the stress-energy tensor

in the AdS/CFT correspondence, Phys. Rev. D60 (1999) 026004 [hep-th/9901121].

[69] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, Large N field

theories, string theory and gravity, Phys. Rept. 323 (2000) 183–386

[hep-th/9905111].

[70] E. A. Bergshoeff, O. Hohm and P. K. Townsend, Massive Gravity in Three

Dimensions, Phys. Rev. Lett. 102 (2009) 201301 [0901.1766].

[71] D. Boyanovsky and C. M. Naon, An introduction to conformal invariance in

quantum field theory and statistical mechanics, Riv. Nuovo Cim. 13N2 (1990) 1–76.

[72] J. D. Brown and M. Henneaux, Central Charges in the Canonical Realization of

Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun.

Math. Phys. 104 (1986) 207–226.

[73] V. Gurarie and A. W. W. Ludwig, Conformal algebras of 2D disordered systems, J.

Phys. A35 (2002) L377–L384 [cond-mat/9911392].

[74] I. I. Kogan and A. Nichols, Stress energy tensor in LCFT and the logarithmic

Sugawara construction, JHEP 01 (2002) 029 [hep-th/0112008].

[75] I. I. Kogan and A. Nichols, Stress energy tensor in c = 0 logarithmic conformal field

theory, hep-th/0203207.

[76] L. Susskind and E. Witten, The holographic bound in anti-de Sitter space,

hep-th/9805114.

– 506 –

[77] “GRTensorII.” This is a package which runs within Maple but distinct from

packages distributed with Maple. It is distributed freely on the World-Wide-Web

from the address: http://grtensor.org.

[78] Y. Liu and Y.-W. Sun, Consistent Boundary Conditions for New Massive Gravity in

AdS3, JHEP 05 (2009) 039 [0903.2933].

[79] R. Andringa et. al., Massive 3D Supergravity, 0907.4658.

[80] J. S. Caux, I. Kogan, A. Lewis and A. M. Tsvelik, Logarithmic operators and

dynamical extension of the symmetry group in the bosonic SU(2)0 and SUSY SU(2)2

WZNW models, Nucl. Phys. B489 (1997) 469–484 [hep-th/9606138].

[81] M. Khorrami, A. Aghamohammadi and A. M. Ghezelbash, Logarithmic N = 1

superconformal field theories, Phys. Lett. B439 (1998) 283–288 [hep-th/9803071].

[82] N. E. Mavromatos and R. J. Szabo, D-brane dynamics and logarithmic

superconformal algebras, JHEP 10 (2001) 027 [hep-th/0106259].

[83] E. Witten, Three-Dimensional Gravity Revisited, 0706.3359.

[84] A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three

Dimensions, 0712.0155.

[85] M. R. Gaberdiel, S. Gukov, C. A. Keller, G. W. Moore and H. Ooguri, Extremal

N=(2,2) 2D Conformal Field Theories and Constraints of Modularity, 0805.4216.

[86] A. Bilal and I. I. Kogan, On gravitational dressing of 2-D field theories in chiral

gauge, Nucl. Phys. B449 (1995) 569–588 [hep-th/9503209].

[87] M. A. I. Flohr, Two-dimensional turbulence: A novel approach via logarithmic

conformal field theory, Nucl. Phys. B482 (1996) 567–578 [hep-th/9606130].

[88] V. Gurarie, M. Flohr and C. Nayak, The Haldane-Rezayi quantum Hall state and

conformal field theory, Nucl. Phys. B498 (1997) 513–538 [cond-mat/9701212].

[89] A. Cappelli, L. S. Georgiev and I. T. Todorov, A unified conformal field theory

description of paired quantum Hall states, Commun. Math. Phys. 205 (1999)

657–689 [hep-th/9810105].

[90] N. Read and D. Green, Paired states of fermions in two-dimensions with breaking of

parity and time reversal symmetries, and the fractional quantum Hall effect, Phys.

Rev. B61 (2000) 10267 [cond-mat/9906453].

[91] E. V. Ivashkevich, Correlation functions of dense polymers and c = -2 conformal

field theory, J. Phys. A32 (1999) 1691–1699 [cond-mat/9801183].

[92] J. L. Cardy, Logarithmic Correlations in Quenched Random Magnets and Polymers,

cond-mat/9911024.

– 507 –

[93] K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19

(2002) 5849–5876 [hep-th/0209067].

[94] A. Erdelyi, ed., Higher Transcendental Functions, vol. I. McGraw-Hill, 1953.

[95] G. N. Watson, Asymptotic expansions of hypergeometric functions, Trans. Cambridge

Philos. Soc. 22 (1918) 277–308.

– 508 –

MIT-CTP-4089

TUW–09–18

AdS3/LCFT2 – Correlators in New Massive Gravity

Daniel Grumiller 1 and Olaf Hohm 2

1 Institute for Theoretical Physics,

Vienna University of Technology,

Wiedner Hauptstr. 8–10/136, A-1040 Vienna, Austria

email: grumil@hep.itp.tuwien.ac.at

2 Center for Theoretical Physics,

Massachusetts Institute of Technology,

77 Massachusetts Ave., Cambridge, MA 02139, USA

email: ohohm@mit.edu

ABSTRACT

We calculate 2-point correlators for New Massive Gravity at the chiral point and

find that they behave precisely as those of a logarithmic conformal field theory, which

is characterized in addition to the central charges cL = cR = 0 by ‘new anomalies’

bL = bR = −σ 12ℓGN

, where σ is the sign of the Einstein–Hilbert term, ℓ the AdS radius

and GN Newton’s constant.

November 2009

1 Introduction

In the recent two years there has been increasing interest in a possible AdS/CFT rela-

tion for gravity in three dimensions. After the proposal of [1] for pure gravity, another

model has been investigated in [2], namely cosmological topologically massive gravity

(CTMG) [3]. The latter extends the pure Einstein-Hilbert theory with negative cos-

mological constant by a parity-violating, third-order, gravitational Chern-Simons term.

Due to this Chern-Simons term, the left- and right-moving central charges cL and cR of

the dual CFT are different, and accordingly the parameters of the theory can be tuned

such that precisely one of the central charges vanishes. It has been conjectured that the

theory at this ‘chiral point’ is dual to a chiral CFT [2]. Soon afterwards [4] it has been

realized, however, that there are logarithmic modes in the bulk [5] that violate chiral-

ity [6] but are compatible with asymptotic AdS behavior [7, 8], and so it remains as an

open question whether there is a consistent truncation to a quantum theory of ‘chiral

gravity’ [9]. Irrespective of whether this will turn out to be true or not, the full gravita-

tional theory, i.e. untruncated CTMG at the chiral point, may itself have a description

in terms of a dual CFT. Provided this is the case, the dual CFT has to be a so-called

logarithmic CFT (LCFT) [5]. Despite being non-unitary, these theories are of interest in

their own right [10], with potential applications in condensed matter physics, see [11,12]

and Refs. therein.

Recently, it has been shown that the 2-point [13] and 3-point correlators [14] of CTMG

at the chiral point (CCTMG) are indeed of the form of a LCFT. It is the aim of this

note to verify the same at the level of the 2-point correlators for yet another theory

of gravity in three dimensions, the so-called ‘new massive gravity’ (NMG) [15], which

has several features in common with CTMG. They differ, however, in the respect that

NMG extends the Einstein-Hilbert term by a parity-preserving fourth-order term instead.

Consequently, at the chiral point both central charges are zero, leading to a LCFT both

for the left- and right-moving sector, thereby potentially providing a novel gravitational

dual to LCFTs of this type.

2 New massive gravity in AdS backgrounds

The action for NMG with a cosmological parameter is given by [15]

S =1

κ2

∫d3x

√−g[σR +

1

m2

(RµνRµν −

3

8R2

)− 2λm2

], (1)

where m is a mass parameter, λ a dimensionless cosmological parameter and σ = ±1 the

sign of the Einstein-Hilbert term. This action leads to equations of motion that have as

particular solutions maximally symmetric vacua for λ ≥ −1. One special feature of this

model is that it propagates unitarily massive graviton modes about some of its (A)dS

510

vacua, provided the sign of the Einstein-Hilbert term is chosen to be the opposite of the

sign in higher dimensions, σ = −1.1 (See also [17].)

In this note we focus on the special case where the vacuum is global AdS3, for which

the AdS radius ℓ is determined by real solutions of

1/ℓ2 = 2m2(σ ±

√1 + λ

). (2)

In the following we will focus on λ > 0, for which there is always a unique AdS vacuum.

The AdS3 metric reads

ds2AdS3

= ℓ2(dρ2 − cosh2ρ dt2 + sinh2ρ dφ2

), (3)

and the boundary cylinder on which the dual CFT will be defined corresponds to ρ→ ∞.

For the computation of the 2-point correlators according to the AdS/CFT recipe, the

quadratic fluctuations about AdS3 are required. These fluctuations (bulk and boundary

gravitons), which we collectively denote by ψ, need to solve the linearized field equations,

which are fourth order linear partial differential equations. In transverse-traceless gauge

for the fluctuations they are given by [18]

(DLDRDMDMψ)µν = 0 , (4)

with the mutually commuting first order operators

(DM/M

)µ

β = δµβ ± α εµ

αβ∇α ,(DL/R

)µ

β = δµβ ± ℓ εµ

αβ∇α , (5)

where α is determined from the parameters in the action. We tune now the parameters

according to

λ = 3 ⇒ m2 = − σ

2ℓ2. (6)

We observe that this special point, which defines the ‘chiral point’, exists for σ = −1 and

m2 > 0 or for σ = 1 and m2 < 0. Although this latter choice leads to ghost modes at

the chiral point, we will analyze this case as well since the computation below does not

depend on the actual sign of m2. For the choice (6) the parameter in (5) is determined

to be α = ℓ. Consequently, the operators DM and DL degenerate, and analogously for

DM and DR. In CTMG at the chiral point a similar degeneration led to the structure of

a logarithmic CFT [5], with central charges and ‘new anomaly’ given by [13, 14]

CCTMG : cL = 0 , cR =3ℓ

GN

, bL = − 3ℓ

GN

, bR = 0 . (7)

More precisely, the parameters bL and bR denoting the new anomalies together with

the central charges completely determine a LCFT at the level of 2-point correlators, see

1To be more precise, if one assumes that a Breitenlohner-Freedman-type bound is consistent with

unitarity, then on certain AdS backgrounds away from the chiral point there are also unitary graviton

modes for σ = +1 and m2

< 0 [16].

511

discussion below. In NMG the central charges of the dual CFT (if it exists) are given

by [16, 18]

New Massive Gravity : cL = cR =3ℓ

2GN

(σ +

1

2ℓ2m2

). (8)

They vanish at the chiral point (6), which provides another hint that the dual CFT might

be logarithmic. (The consistency of log boundary conditions has been demonstrated

in [19].) It is therefore fair to inquire if NMG (1) at the chiral point (6) is dual to a

LCFT as well. The purpose of this note is to show that this is the case, at least at the

level of 2-point correlators, to which we turn now.

3 Two-point correlation functions

In order to calculate the 2-point correlators on the gravity side we proceed exactly as

in [14] by following the AdS/CFT recipe. The starting point are solutions ψ of the

linearized equations of motion (4), which we expand in Fourier modes

ψµν(h, h) = e−ih(t+φ)−ih(t−φ) Fµν(ρ) . (9)

Next, we have to analyze their asymptotic (large ρ) behavior for any given set of weights

h, h. If the tensor F has components that grow exponentially with 2ρ, the corresponding

mode is called non-normalizable and acts as a source for the corresponding operator in

the dual CFT. Using the standard AdS/CFT dictionary we insert these sources into the

second variation of the on-shell action and obtain in this way 2-point correlators between

the corresponding operators. In fact, we can reduce the calculation up to pre-factors to

calculations that were performed in detail in [14], and thus we shall exhibit below only

the points where NMG differs from CCTMG.

As we focus on the 2-point correlators, it is sufficient to consider the action quadratic

in the fluctuations. This action has been determined in [16] for a formulation involving

an auxiliary field fµν ,

S =1

κ2

∫d3x

√−g[σR + fµνGµν −

1

4m2

(fµνfµν − f 2

)− 2λm2

], (10)

which can be seen to be equivalent to (1) upon integrating out fµν . We define the

fluctuations to linear order to be

gµν = gµν + hµν , fµν = − 1

m2ℓ2(gµν + hµν + ℓ2kµν

). (11)

We note that, in contrast to [16], here we have not rescaled the fluctuations by κ. The

quadratic piece of the Lagrangian, at the chiral point, is then given by [16]

L2 = − 1

m2κ2kµνGµν(h) −

1

4m2κ2

(kµνkµν − k2

), (12)

512

where G is the linearization of the Einstein tensor modified by the cosmological constant.

This bilinear Lagrangian contains the same information as the second variation of the

action, which formally differs from the former only in that it corresponds to a quadratic

form with two different arguments, c.f. eq. (15) below. In the following we perform a

gauge-fixing to transverse-traceless gauge, which implies in particular that on-shell k is

traceless as well. The result for the quadratic action at the chiral point then reads

S(2) = − 1

16π GN m2

∫d3x

√−g[kµνGµν(h) +

1

4kµνkµν

]+ boundary terms , (13)

where

Gµν(h) =1

2ℓ2(DLDRh)µν , (14)

and we have introduced Newton’s constant via κ2 = 16πGN .

By analogy to the Einstein–Hilbert case or to CCTMG, the second variation of the

on-shell action is given by2

δ(2)S(ψ1, ψ2) ∼ 1

32πGN m2limρ→∞

t1∫

t0

dt

2π∫

0

dφ√−g k1 ∗

ij gikgjl∇ρψ

2kl , (15)

which we have evaluated in the coordinates (3). Here, k1 is related to the mode ψ1 by

virtue of the linearized equations of motion

k1µν = −2Gµν(ψ

1) = − 1

ℓ2(DLDRψ1)µν . (16)

The remaining linearized equations of motion, (DLDR)2ψ = 0, lead to four branches of

solutions: ψL (ψR) [ψlog] ψflog, which are annihilated by the linear differential operators

DL (DR) [(DL)2] (DR)2. The modes ψL (ψR) [ψlog] ψflog are called left (right) [loga-

rithmic] flipped logarithmic modes. The left and logarithmic modes were constructed

in [14]. The right modes are obtained from the left modes by exchange of the light-cone

coordinates and of the weights, which amounts to the substitutions φ→ −φ and h↔ h,

see again [14] for details. Analogously, the flipped logarithmic modes are obtained from

the logarithmic modes by exchange of the light-cone coordinates and of the weights. The

modes obey the following identities:

DLψL = 0 , DLψR = 2ψR , DLψlog = −2ψL , (17)

DRψR = 0 , DRψL = 2ψL , DRψflog = −2ψR . (18)

The identities (17)-(18) allow relevant simplifications in the calculations of correlators.

Generically the 2-point correlators on the gravity side between two modes ψ1(h, h)

and ψ2(h′, h′) in momentum space are determined by

〈ψ1(h, h)ψ2(h′, h′)〉 =1

2

(δ(2)S(ψ1, ψ2) + δ(2)S(ψ2, ψ1)

), (19)

2The formula (15) is the analog of (4.8) in [14]. The asterisk denotes complex conjugation. The sign

∼ denotes equivalence up to boundary counterterms, which turn out to be contact terms.

513

where 〈ψ1 ψ2〉 stands for the correlation function of the CFT operators dual to the (bulk

and/or boundary) graviton modes ψ1 and ψ2. Below we present the Fourier-transformed

version of the momentum space correlators (19), i.e., the correlators in ordinary space.

The results (15)-(18) allow to determine immediately the vanishing of the correlators

between left and right modes:

〈ψL(z)ψL/R(0)〉 = 〈ψR(z)ψL/R(0)〉 = 0 (20)

The result (20) is consistent with the corresponding correlators in a LCFT with cL =

cR = 0.

The remaining correlators involve also the (flipped) logarithmic modes. All non-

vanishing ones essentially can be reduced to correlators that have been calculated already

in CCTMG, see section 4.1 in [14]. We provide here results for all non-vanishing 2-point

correlators in the near coincidence limit:

〈ψlog(z)ψL(0)〉 =bL2z4

(21)

〈ψflog(z)ψR(0)〉 =bR2z4

(22)

〈ψlog(z)ψlog(0)〉 = −bL ln (m2L|z|2)

z4(23)

〈ψflog(z)ψflog(0)〉 = −bR ln (m2

R|z|2)z4

(24)

The new anomalies bL and bR will be calculated below. The mass scales mL and mR play

no physical role and can be rescaled to any finite value by redefining ψlog → ψlog + γψL

and ψflog → ψ

flog + γψR, which corresponds to a well-known ambiguity in LCFTs. The

results (21)–(24) coincide precisely with the non-vanishing 2-point correlators in a LCFT

with cL = cR = 0, cf. e.g. [22]. Thus, at the level of 2-point correlators NMG (1) at the

chiral point (6) is indeed dual to a LCFT with vanishing central charges.

We close with a derivation of the result for the new anomalies bL and bR. After taking

into account the linearized equations of motion (16)-(18) the overall factor in front of the

second variation of the on-shell action (15) differs by a factor 4σ from the corresponding

expression in CCTMG, equations (4.8) and (4.19a) in [14], provided we use the same

normalizations of the modes as in that work. Therefore, all normalizations being equal,

the new anomalies bL = bR must be given by 4σ times the value of bL in CCTMG.

Inserting the result (7) finally establishes

bL = bR = −σ 12ℓ

GN. (25)

We note that the new anomalies are positive only upon choosing the negative sign in

front of the Einstein-Hilbert term, σ = −1.

514

4 Discussion and comments

In this note we calculated the 2-point correlators (21) of the CFT dual to new massive

gravity (1) at the chiral point (6). We found that the dual CFT, if it exists, takes the

form of a logarithmic CFT, with new anomalies given by (25).

We address now a particular consequence of our results. For generic values of the

parameters it was found in [16] that the propagating degrees of freedom about AdS3

are massive spin-2 modes that are unitary whenever the central charges of the dual

CFT are negative. The only exception is the chiral point (6) at which the bulk modes

become massive spin-1 modes while the central charges and the mass of BTZ black

holes are zero [16, 23]. The central charges determine the entropy of black holes and

the number of microstates via Cardy’s formula and should therefore be positive. The

requirements of positivity of central charges and of positive-energy graviton modes are

mutually exclusive, which is problematic for the consistency/stability of the AdS vacua,

analogous to the problems unravelled in [2] for cosmological topologically massive gravity.

In a logarithmic CFT with vanishing central charges the new anomalies bL and bR take

over the role of the parameters that measure the number of degrees of freedom [20], and

here we see from (25) that they are positive only for σ = −1. It should be noted, however,

that there are physically interesting CFTs with negative central charges and LCFTs with

negative new anomalies, like polymers with b = −5/8 [21]. If we nevertheless take at face

value the interpretation of the new anomalies as a measure for the number of microstates

we can draw an interesting conclusion. Positivity of the number of microstates in the dual

CFT and positive-energy bulk modes now both require the ‘wrong-sign’ Einstein-Hilbert

term. Moreover, at the chiral point black hole solutions are known whose mass is also

positive only provided one chooses σ = −1 [24]. Thus it appears to be promising that

NMG at the chiral point is fully consistent precisely for the ‘wrong-sign’ Einstein-Hilbert

term.

This research can be extended into various directions. An obvious but technically

challenging extension is to calculate 3-point functions as it has been done in [14] for

CCTMG, which requires the calculation of the third variation of the action (1). Apart

from ‘general massive gravity’, which involves both the gravitational Chern-Simons term

and the curvature-square combination of NMG [15, 16, 25], another interesting case is

the N = 1 supergravity constructed in [26]. Due to the presence of non-trivial curvature

couplings of an auxiliary field, the values of the central charges turn out to be unmodified

as compared to the Brown-Henneaux values [26]. Consequently, after including the N = 1

super-invariant of the gravitational Chern-Simons term, a chiral point appears exactly

as for CTMG, and it might be interesting to see in which respects this model deviates

from CCTMG. Finally, it is important to investigate whether there are applications of the

LCFTs dual to the gravitational theories considered here, say, in the context of condensed

matter physics described by strongly coupled LCFTs.

515

Acknowledgments

We acknowledge helpful discussions with Eric Bergshoeff, Gaston Giribet, Roman Jackiw,

Niklas Johansson, Ivo Sachs, Erik Tonni and Paul Townsend.

DG was supported by the START project Y435-N16 of the Austrian Science Foun-

dation (FWF). DG thanks the CTP at MIT for hospitality while part of this work was

completed. The work of OH is supported by the DFG – The German Science Foundation

and in part by funds provided by the U.S. Department of Energy (DoE) under the co-

operative research agreement DE-FG02-05ER41360. OH thanks the Erwin-Schrodinger

Institute in Vienna for hospitality and financial support during the workshop “Gravity

in three dimensions” in April 2009.

References

[1] E. Witten, “Three-Dimensional Gravity Revisited,” arXiv:0706.3359 [hep-th].

[2] W. Li, W. Song and A. Strominger, “Chiral Gravity in Three Dimensions,” JHEP

0804 (2008) 082 [arXiv:0801.4566 [hep-th]].

[3] S. Deser, R. Jackiw and S. Templeton, “Topologically massive gauge theories,” An-

nals Phys. 140 (1982) 372 [Erratum-ibid. 185:406, 1988, Annals Phys. 281:409-449,

2000].

[4] S. Carlip, S. Deser, A. Waldron, and D. K. Wise, “Cosmological Topologically Mas-

sive Gravitons and Photons,” Class. Quant. Grav. 26 (2009) 075008, 0803.3998.

[5] D. Grumiller and N. Johansson, ”Instability in cosmological topologically massive

gravity at the chiral point”, JHEP 07 (2008) 134 [0805.2610].

[6] G. Giribet, M. Kleban and M. Porrati, “Topologically Massive Gravity at the Chiral

Point is Not Chiral,” JHEP 0810 (2008) 045 [arXiv:0807.4703 [hep-th]].

[7] D. Grumiller and N. Johansson, “Consistent boundary conditions for cosmological

topologically massive gravity at the chiral point,” Int. J. Mod. Phys. D17 (2009)

2367–2372, 0808.2575.

[8] M. Henneaux, C. Martinez, and R. Troncoso, “Asymptotically anti-de Sitter space-

times in topologically massive gravity,” Phys. Rev. D79 (2009) 081502R, 0901.2874.

[9] A. Maloney, W. Song and A. Strominger, “Chiral Gravity, Log Gravity and Extremal

CFT,” arXiv:0903.4573 [hep-th].

[10] V. Gurarie, “Logarithmic operators in conformal field theory,” Nucl. Phys. B410

(1993) 535–549, hep-th/9303160.

516

[11] M. Flohr, “Bits and pieces in logarithmic conformal field theory,” Int. J. Mod. Phys.

A18 (2003) 4497–4592, hep-th/0111228.

[12] M. R. Gaberdiel, “An algebraic approach to logarithmic conformal field theory,”

Int. J. Mod. Phys. A18 (2003) 4593–4638, hep-th/0111260.

[13] K. Skenderis, M. Taylor and B. C. van Rees, ”Topologically Massive Gravity and

the AdS/CFT Correspondence”, 0906.4926.

[14] D. Grumiller and I. Sachs, ”AdS3/LCFT2 – Correlators in Cosmological Topologi-

cally Massive Gravity”, 0910.5241.

[15] E. A. Bergshoeff, O. Hohm and P. K. Townsend, ”Massive Gravity in Three Dimen-

sions”, Phys. Rev. Lett. 102 (2009) 201301 [0901.1766].

[16] E. A. Bergshoeff, O. Hohm and P. K. Townsend, ”More on Massive 3D Gravity”,

Phys. Rev. D79 (2009) 124042 [0905.1259].

[17] E. A. Bergshoeff, O. Hohm and P. K. Townsend, “On Higher Derivatives in 3D

Gravity and Higher Spin Gauge Theories,” [arXiv:0911.3061 [hep-th]].

[18] Y. Liu and Y. w. Sun, “Note on New Massive Gravity in AdS3,” JHEP 0904 (2009)

106 [arXiv:0903.0536 [hep-th]].

[19] Y. Liu and Y. W. Sun, “Consistent Boundary Conditions for New Massive Gravity

in AdS3,” JHEP 0905 (2009) 039 [arXiv:0903.2933 [hep-th]].

[20] V. Gurarie, “c-Theorem for Disordered Systems,” Nucl. Phys. B546 (1999) 765

[arXiv:cond-mat/9808063].

[21] V. Gurarie and A. W. W. Ludwig, “Conformal algebras of 2D disordered systems,”

J. Phys. A35 (2002) L377 [arXiv:cond-mat/9911392].

[22] I. I. Kogan and A. Nichols, “Stress energy tensor in LCFT and the logarithmic

Sugawara construction,” JHEP 0201 (2002) 029 [Int. J. Mod. Phys. A18 (2003)

4771] [arXiv:hep-th/0112008]. IMPAE,A18,4771;

[23] G. Clement, “Warped AdS3 black holes in new massive gravity,” Class. Quant. Grav.

26 (2009) 105015 [arXiv:0902.4634 [hep-th]].

[24] G. Clement, “Black holes with a null Killing vector in new massive gravity in three

dimensions,” arXiv:0905.0553 [hep-th].

[25] Y. Liu and Y. W. Sun, “On the Generalized Massive Gravity in AdS3,” Phys. Rev.

D 79 (2009) 126001 [arXiv:0904.0403 [hep-th]].

[26] R. Andringa, E. A. Bergshoeff, M. de Roo, O. Hohm, E. Sezgin and P. K. Townsend,

“Massive 3D Supergravity,” arXiv:0907.4658 [hep-th].

517

Gravity duals for logarithmic conformal field theories

Daniel Grumiller and Niklas Johansson

Institute for Theoretical Physics, Vienna University of TechnologyWiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria

E-mail: grumil@hep.itp.tuwien.ac.at, niklasj@hep.itp.tuwien.ac.at

Abstract. Logarithmic conformal field theories with vanishing central charge describe systemswith quenched disorder, percolation or dilute self-avoiding polymers. In these theories the energymomentum tensor acquires a logarithmic partner. In this talk we address the construction ofpossible gravity duals for these logarithmic conformal field theories and present two viablecandidates for such duals, namely theories of massive gravity in three dimensions at a chiralpoint.

Outline

This talk is organized as follows. In section 1 we recall salient features of 2-dimensional conformalfield theories. In section 2 we review a specific class of logarithmic conformal field theories wherethe energy momentum tensor acquires a logarithmic partner. In section 3 we present a wish-listfor gravity duals to logarithmic conformal field theories. In section 4 we discuss two examplesof massive gravity theories that comply with all the items on that list. In section 5 we addresspossible applications of an Anti-deSitter/logarithmic conformal field theory correspondence incondensed matter physics.

1. Conformal field theory distillate

Conformal field theories (CFTs) are quantum field theories that exhibit invariance under anglepreserving transformations: translations, rotations, boosts, dilatations and special conformaltransformations. In two dimensions the conformal algebra is infinite dimensional, and thustwo-dimensional CFTs exhibit a particularly rich structure. They arise in various contexts inphysics, including string theory, statistical mechanics and condensed matter physics, see e.g. [1].

The main observables in any field theory are correlation functions between gauge invariantoperators. There exist powerful tools to calculate these correlators in a CFT. The operatorcontent of various CFTs may differ, but all CFTs contain at least an energy momentum tensorTµν . Conformal invariance requires the energy momentum tensor to be traceless, T µ

µ = 0,in addition to its conservation, ∂µT

µν = 0. In lightcone gauge for the Minkowski metric,ds2 = 2dz dz, these equations take a particularly simple form: Tzz = 0, Tzz = Tzz(z) := OL(z)and Tzz = Tzz(z) := OR(z). Conformal Ward identities determine essentially uniquely the formof 2- and 3-point correlators between the flux components OL/R of the energy momentum tensor:

Gravity duals for logarithmic conformal field theories 520

〈OR(z)OR(0)〉 =cR2z4

(1a)

〈OL(z)OL(0)〉 =cL2z4

(1b)

〈OL(z)OR(0)〉 = 0 (1c)

〈OR(z)OR(z′)OR(0)〉 =cR

z2z′ 2(z − z′)2(1d)

〈OL(z)OL(z′)OL(0)〉 =cL

z2z′ 2(z − z′)2(1e)

〈OL(z)OR(z′)OR(0)〉 = 0 (1f)

〈OL(z)OL(z′)OR(0)〉 = 0 (1g)

The real numbers cL, cR are the left and right central charges, which determine key properties ofthe CFT. We have omitted terms that are less divergent in the near coincidence limit z, z → 0 aswell as contact terms, i.e., contributions that are localized (δ-functions and derivatives thereof).

If someone provides us with a traceless energy momentum tensor and gives us a prescriptionhow to calculate correlators,1 but does not reveal whether the underlying field theory is a CFT,then we can perform the following check. We calculate all 2- and 3-point correlators of the energymomentum tensor with itself, and if at least one of the correlators does not match precisely withthe corresponding correlator in (1) then we know that the field theory in question cannot be aCFT. On the other hand, if all the correlators match with corresponding ones in (1) we havenon-trivial evidence that the field theory in question might be a CFT. Let us keep this stringentcheck in mind for later purposes, but switch gears now and consider a specific class of CFTs,namely logarithmic CFTs (LCFTs).

2. Logarithmic CFTs with an energetic partner

LCFTs were introduced in physics by Gurarie [2]. We focus now on some properties of LCFTsand postpone a physics discussion until the end of the talk, see [3,4] for reviews. There are twoconceptually different, but mathematically equivalent, ways to define LCFTs. In both versionsthere exists at least one operator that acquires a logarithmic partner, which we denote by Olog.We focus in this talk exclusively on theories where one (or both) of the energy momentumtensor flux components is the operator acquiring such a partner, for instance OL. We discussnow briefly both ways of defining LCFTs.

According to the first definition “acquiring a logarithmic partner” means that theHamiltonian H cannot be diagonalized. For example

H

(

Olog

OL

)

=

(

2 10 2

)(

Olog

OL

)

(2)

The angular momentum operator J may or may not be diagonalizable. We consider only theorieswhere J is diagonalizable:

J

(

Olog

OL

)

=

(

2 00 2

)(

Olog

OL

)

(3)

The eigenvalues 2 arise because the energy momentum tensor and its logarithmic partner bothcorrespond to spin-2 excitations.

1 This is exactly what the AdS/CFT correspondence does: given a gravity dual we can calculate the energymomentum tensor and correlators.

Gravity duals for logarithmic conformal field theories 521

The second definition makes it more transparent why these CFTs are called “logarithmic”in the first place. Suppose that in addition to OL/R we have an operator OM with conformalweights h = 2 + ε, h = ε, meaning that its 2-point correlator with itself is given by

〈OM (z, z)OM (0, 0)〉 =B

z4+2εz2ε(4)

The correlator of OM with OL vanishes since the latter has conformal weights h = 2, h = 0, andoperators whose conformal weights do not match lead to vanishing correlators. Suppose nowthat we send the central charge cL and the parameter ε to zero, and simultaneously send B toinfinity, such that the following limits exist:

bL := limcL→0

2cLε

6= 0 B := limcL→0

(

B +2

cL

)

(5)

Then we can define a new operator Olog that linearly combines OL/M .

Olog = bLOL

cL+bL2

OM (6)

Taking the limit cL → 0 leads to the following 2-point correlators:

〈OL(z)OL(0, 0)〉 = 0 (7a)

〈OL(z)Olog(0, 0)〉 =bL2z4

(7b)

〈Olog(z, z)Olog(0, 0)〉 = −bL ln (m2L|z|2)

z4(7c)

These 2-point correlators exhibit several remarkable features. The flux component OL of theenergy momentum tensor becomes a zero norm state (7a). Nevertheless, the theory does notbecome chiral, because the left-moving sector is not trivial: OL has a non-vanishing correlator(7b) with its logarithmic partner Olog. The 2-point correlator (7c) between two logarithmicoperators Olog makes it clear why such CFTs have the attribute “logarithmic”. The constantbL, sometimes called “new anomaly”, defines crucial properties of the LCFT, much like thecentral charges do in ordinary CFTs. The mass scale mL appearing in the last correlator abovehas no significance, and is determined by the value of B in (5). It can be changed to any finitevalue by the redefinition Olog → Olog +γOL with some finite γ. We set mL = 1 for convenience.

Conformal Ward identities determine again essentially uniquely the form of 2- and 3-pointcorrelators in a LCFT. For the specific case where the energy momentum tensor acquires alogarithmic partner the 3-point correlators were calculated in [5]. The non-vanishing ones aregiven by

〈OL(z, z)OL(z′, z′)Olog(0, 0)〉 =bL

z2z′2(z − z′)2(8a)

〈OL(z, z)Olog(z′, z′)Olog(0, 0)〉 = −2bL ln |z′|2 + bL

2

z2z′2(z − z′)2(8b)

〈Olog(z, z)Olog(z′, z′)Olog(0, 0)〉 =lengthy

z2z′2(z − z′)2(8c)

If also OR acquires a logarithmic partner Oflog then the construction above can be repeated,changing everywhere L → R, z → z etc. In that case we have a LCFT with cL = cR = 0 and

Gravity duals for logarithmic conformal field theories 522

bL, bR 6= 0. Alternatively, it may happen that only OL has a logarithmic partner Olog. In thatcase we have a LCFT with cL = bR = 0 and bL, cR 6= 0. This concludes our brief excursion intothe realm of LCFTs.

Given that LCFTs are interesting in physics (see section 5) and that a powerful way todescribe strongly coupled CFTs is to exploit the AdS/CFT correspondence [6] it is natural toinquire whether there are any gravity duals to LCFTs.

3. Wish-list for gravity duals to LCFTs

In this section we establish necessary properties required for gravity duals to LCFTs. Weformulate them as a wish-list and explain afterwards each item on this list.

(i) We wish for a 3-dimensional action S that depends on the metric gµν and possibly on furtherfields that we summarily denote by φ.

(ii) We wish for the existence of AdS3 vacua with finite AdS radius ℓ.

(iii) We wish for a finite, conserved and traceless Brown–York stress tensor, given by the firstvariation of the full on-shell action (including boundary terms) with respect to the metric.

(iv) We wish that the 2- and 3-point correlators of the Brown–York stress tensor with itself aregiven by (1).

(v) We wish for central charges (a la Brown–Henneaux [7]) that can be tuned to zero, withoutrequiring a singular limit of the AdS radius or of Newton’s constant. For concreteness weassume cL = 0 (in addition cR may also vanish, but it need not).

(vi) We wish for a logarithmic partner to the Brown–York stress tensor, so that we obtain aJordan-block structure like in (2) and (3).

(vii) We wish that the 2- and non-vanishing 3-point correlators of the Brown–York stress tensorwith its logarithmic partner are given by (7) and (8) (and the right-handed analog thereof).

We explain now why each of these items is necessary. (i) is required since the AdS/CFTcorrespondence relates a gravity theory in d+ 1 dimensions to a CFT in d dimensions, and wechose d = 2 on the CFT side. (ii) is required since we are not merely looking for a gauge/gravityduality, but really for an AdS/CFT correspondence, which requires the existence of AdS solutionson the gravity side. (iii) is required since we desire consistency with the AdS dictionary, whichrelates the vacuum expectation value of the renormalized energy momentum tensor in the CFT〈Tij〉 to the Brown–York stress tensor TBY

ij :

〈Tij〉 = TBYij =

2√−gδS

δgij

∣

∣

∣

EOM

(9)

The right hand side of this equation contains the first variation of the full on-shell action withrespect to the metric, which by definition yields the Brown–York stress tensor. (iv) is requiredsince the 2- and 3-point correlators of a CFT are fixed by conformal Ward identities to takethe form (1). (v) is required because of the construction presented in section 2, where a LCFTemerges from taking an appropriate limit of vanishing central charge, so we need to be ableto tune the central charge without generating parametric singularities. Actually, there aretwo cases: either left and right central charge vanish and both energy momentum tensor fluxcomponents acquire a logarithmic partner, or only one of them acquires a logarithmic partner,which for sake of specificity we always choose to be left. (vi) is required, since we considerexclusively LCFTs where the energy momentum tensor acquires a logarithmic partner. (vii) isrequired since the 2- and 3-point correlators of a LCFT are fixed by conformal Ward identities totake the form (7), (8). If any of the items on the wish-list above is not fulfilled it is impossible thatthe gravitational theory under consideration is a gravity dual to a LCFT of the type discussed

Gravity duals for logarithmic conformal field theories 523

in section 2.2 On the other hand, if all the wishes are granted by a given gravitational theorythere are excellent chances that this theory is dual to a LCFT. Until recently no good gravityduals for LCFTs were known [8–12].

Before addressing candidate theories that may comply with all wishes we review briefly howto calculate correlators on the gravity side [6], since we shall need such calculations for checkingseveral items on the wish-list. The basic identity of the AdS/CFT dictionary is

〈O1(z1)O2(z2) . . .On(zn)〉 =δ(n)S

δj1(z1)δj2(z2) . . . δjn(zn)

∣

∣

∣

ji=0(10)

The left hand side is the CFT correlator between n operators Oi, where Oi in our case comprisethe left- and right-moving flux components of the energy momentum tensor and their logarithmicpartners. The right hand side contains the gravitational action S differentiated with respect toappropriate sources ji for the corresponding operators. According to the AdS/CFT dictionary“appropriate sources” refers to non-normalizable solutions of the linearized equations of motion.We shall be more concrete about the operators, actions, sources and non-normalizable solutionsto the linearized equations of motion in the next section. For now we address possible candidatetheories of gravity duals to LCFTs.

The simplest candidate, pure 3-dimensional Einstein gravity with a cosmological constantdescribed by the action

SEH = − 1

8πGN

∫

Md3x

√−g[

R+2

ℓ2

]

− 1

4π GN

∫

∂Md2x

√−γ[

K − 1

ℓ

]

(11)

does not comply with the whole wish list. Only the first four wishes are granted: The 3-dimensional action (12) depends on the metric. The equations of motion are solved by AdS3.

ds2AdS3= gAdS3

µν dxµ dxν = ℓ2(

dρ2 − 1

4cosh2ρ (du+ dv)2 +

1

4sinh2ρ (du− dv)2

)

(12)

The Brown–York stress tensor (9) is finite, conserved and traceless. The 2- and 3-pointcorrelators on the gravity side match precisely with (1). However, the central charges are givenby [7]

cL = cR =3ℓ

2GN(13)

and therefore allow no tuning to cL = 0 without taking a singular limit. Moreover, there is nocandidate for a logarithmic partner to the Brown–York stress tensor. Thus, pure 3-dimensionalEinstein gravity cannot be dual to a LCFT.

Adding matter fields to Einstein gravity does not help neither. While this may lead to otherkinds of LCFTs, it cannot produce a logarithmic partner for the energy momentum tensor. Thisis so, because the energy momentum tensor corresponds to graviton (spin-2) excitations in thebulk, and the only field producing such excitations is the metric.

Therefore, what we need is a way to provide additional degrees of freedom in the gravitysector. The most natural way to do this is by considering higher derivative interactions of themetric. The first gravity model of this type was constructed by Deser, Jackiw and Templeton [13]who introduced a Chern–Simons term for the Christoffel connection.

SCS = − 1

16πGN µ

∫

d3x ǫλµνΓρσλ

[

∂µΓσρν +

2

3Γσ

κµΓκσν

]

(14)

2 Other types of LCFTs exist, e.g. with non-vanishing central charge or with logarithmic partners to operatorsother than the energy momentum tensor. The gravity duals for such LCFTs need not comply with all the itemson our wish list.

Gravity duals for logarithmic conformal field theories 524

Here µ is a real coupling constant. Adding this action to the Einstein–Hilbert action (11)generates massive graviton excitations in the bulk, which is encouraging for our wish list sincewe need these extra degrees of freedom. The model that arises when summing the actions (11)and (14),

SCTMG = SEH + SCS (15)

is known as “cosmological topologically massive gravity” (CTMG) [14]. It was demonstrated byKraus and Larsen [15] that the central charges in CTMG are shifted from their Brown–Henneauxvalues:

cL =3ℓ

2GN

(

1 − 1

µℓ

)

cR =3ℓ

2GN

(

1 +1

µℓ

)

(16)

This is again good news concerning our wish list, since cL can be made vanishing by a (non-singular) tuning of parameters in the action.

µℓ = 1 (17)

CTMG (15) with the tuning above (17) is known as “cosmological topologically massive gravityat the chiral point” (CCTMG). It complies with the first five items on our wish list, but we stillhave to prove that also the last two wishes are granted. To this end we need to find a suitablepartner for the graviton.

4. Keeping logs in massive gravity

4.1. Login

In this section we discuss the evidence for the existence of specific gravity duals to LCFTs thathas accumulated over the past two years. We start with the theory introduced above, CCTMG,and we end with a relatively new theory, new massive gravity [16].

4.2. Seeds of logs

Given that we want a partner for the graviton we consider now graviton excitations ψ aroundthe AdS background (12) in CCTMG.

gµν = gAdS3µν + ψµν (18)

Li, Song and Strominger [17] found a nice way to construct them, and we follow their constructionhere. Imposing transverse gauge ∇µψ

µν = 0 and defining the mutually commuting first orderoperators

(

DM)β

µ= δβ

µ +1

µεµ

αβ∇α

(

DL/R)β

µ= δβ

µ ± ℓ εµαβ∇α (19)

allows to write the linearized equations of motion around the AdS background (12) as follows.

(DMDLDRψ)µν = 0 (20)

A mode annihilated by DM (DL) [DR] (DL)2 but not by DL is called massive (left-moving)[right-moving] logarithmic and is denoted by ψM (ψL) [ψR] ψlog. Away from the chiralpoint, µℓ 6= 1, the general solution to the linearized equations of motion (20) is obtained fromlinearly combining left, right and massive modes [17]. At the chiral point DM degenerates withDL and the general solution to the linearized equations of motion (20) is obtained from linearlycombining left, right and logarithmic modes [18]. Interestingly, we discovered in [18] that themodes ψlog and ψL behave as follows:

(L0 + L0)

(

ψlog

ψL

)

=

(

2 10 2

)(

ψlog

ψL

)

(21)

Gravity duals for logarithmic conformal field theories 525

where L0 = i∂u, L0 = i∂v and

(L0 − L0)

(

ψlog

ψL

)

=

(

2 00 2

)(

ψlog

ψL

)

(22)

If we define naturally the Hamiltonian by H = L0 + L0 and the angular momentum byJ = L0 − L0 we recover exactly (2) and (3), which suggests that the CFT dual to CCTMG(if it exists) is logarithmic, as conjectured in [18]. It was further shown with Jackiw that theexistence of the logarithmic excitations ψlog is not an artifact of the linearized approach, butpersists in the full theory [19].

Thus, also the sixth wish is granted in CCTMG. The rest of this section discusses the lastwish.

4.3. Growing logs

We assume now that there is a standard AdS/CFT dictionary [6] available for LCFTs and checkif CCTMG indeed leads to the correct 2- and 3-point correlators. To this end we have to identifythe sources ji that appear on the right hand side of the correlator equation (10). Following thestandard AdS/CFT prescription the sources for the operators OL (OR) [Olog] are given by left(right) [logarithmic] non-normalizable solutions to the linearized equations of motion (20). Thus,our first task is to find all solutions of the linearized equations of motion and to classify theminto normalizable and non-normalizable ones, where “normalizable” refers to asymptotic (largeρ) behavior that is exponentially suppressed as compared to the AdS background (12).

A construction of all normalizable left and right solutions was provided in [17], and thenormalizable logarithmic solutions were constructed in [18].3 The non-normalizable solutionswere constructed in [25]. It turned out to be convenient to work in momentum space

ψL/R/logµν (h, h) = e−ih(t+φ)−ih(t−φ) FL/R/log

µν (ρ) (23)

The momenta h, h are called “weights”. All components of the tensor Fµν are determinedalgebraically, except for one that is determined from a second order (hypergeometric) differentialequation. In general one of the linear combinations of the solutions is singular at the origin ρ = 0,while the other is regular there. We keep only regular solutions. For each given set of weights h, hthe regular solution is either normalizable or non-normalizable. It turns out that normalizablesolutions exist for integer weights h ≥ 2, h ≥ 0 (or h ≤ −2, h ≤ 0). All other solutions arenon-normalizable.

An example for a normalizable left mode is given by the primary with weights h = 2, h = 0

ψLµν(2, 0) =

e−2iu

cosh4ρ

14 sinh2(2ρ) 0 i

2 sinh (2ρ)0 0 0

i2 sinh (2ρ) 0 −1

µν

(24)

Note that all components of this mode behave asymptotically (ρ→ ∞) at most like a constant.The corresponding logarithmic mode is given by

ψlogµν (2, 0) = −1

2(i(u + v) + ln cosh2ρ)ψL

µν(2, 0) (25)

Evidently, it behaves asymptotically like its left partner (24), except for overall linear growth inρ. It is also worthwhile emphasizing that the logarithmic mode (25) depends linearly on time

3 All these modes are compatible with asymptotic AdS behavior [20,21], and they appear in vacuum expectationvalues of 1-point functions. Indeed, the 1-point function 〈T ij〉 involves both ψlog and ψR [21–24].

Gravity duals for logarithmic conformal field theories 526

t = (u + v)/2. Both features are inherent to all logarithmic modes. All other normalizablemodes can be constructed from the primaries (24), (25) algebraically.

An example for a non-normalizable left mode is given by the mode with weights h = 1,h = −1

ψLµν(1,−1) =

1

4e−iu+iv

0 0 0

0 cosh (2ρ) − 1 −2i√

cosh (2ρ)−1cosh (2ρ)+1

0 −2i√

cosh (2ρ)−1cosh (2ρ)+1 − 4

cosh (2ρ)+1

µν

(26)

Note that all components of this mode behave asymptotically (ρ→ ∞) at most like a constant,except for the vv-component, which grows like e2ρ. The corresponding logarithmic mode growsagain faster than its left partner (26) by a factor of ρ and depends again linearly on time.

Given a non-normalizable solution ψL obviously also αψL is a non-normalizable solution,with some constant α. To fix this normalization ambiguity we demand standard coupling of themetric to the stress tensor:

S(ψu Lv , T v

u ) =1

2

∫

dt dφ

√

−g(0) ψuuL Tuu =

∫

dt dφ e−ihu−ihv Tuu (27)

Here S is either some CFT action with background metric g(0) or a dual gravitational action withboundary metric g(0). The non-normalizable mode ψL is the source for the energy-momentumflux component Tuu. The requirement (27) fixes the normalization. The discussion abovefocussed on left modes. For the right modes essentially the same discussion applies, but withthe substitutions L↔ R, h↔ h and u↔ v.

4.4. Logging correlators

Generically the 2-point correlators on the gravity side between two modes ψ1(h, h) and ψ2(h′, h′)in momentum space are determined by

〈ψ1(h, h)ψ2(h′, h′)〉 =1

2

(

δ(2)SCCTMG(ψ1, ψ2) + δ(2)SCCTMG(ψ2, ψ1))

(28)

where 〈ψ1 ψ2〉 stands for the correlation function of the CFT operators dual to the gravitonmodes ψ1 and ψ2. On the right hand side one has to plug the non-normalizable modes ψ1

and ψ2 into the second variation of the on-shell action and symmetrize with respect to the twomodes. The second variation of the on-shell action of CCTMG

δ(2)SCCTMG = − 1

16π GN

∫

d3x√−g

(

DLψ1 ∗)µν

δGµν(ψ2) + boundary terms (29)

turns out to be very similar to the second variation of the on-shell Einstein–Hilbert action

δ(2)SEH = − 1

16π GN

∫

d3x√−g ψ1 µν ∗δGµν(ψ2) + boundary terms (30)

This similarity allows us to exploit results from Einstein gravity for CCTMG, as we now explain.4

The bulk term in CCTMG (29) has the same form as in Einstein theory (30) with ψ1 replacedby DLψ1. Now, consider boundary terms. Possible obstructions to a well-defined Dirichletboundary value problem can come only from the variation δGµν(ψ2), since DL is a first orderoperator. Thus any boundary terms appearing in (29) containing normal derivatives must be

4 Alternatively, one can follow the program of holographic renormalization, as it was done by Skenderis, Taylorand van Rees [23]. Their results for 2-point correlators agree with the results presented here.

Gravity duals for logarithmic conformal field theories 527

identical with those in Einstein gravity upon substituting ψ1 → DLψ1. In addition there can beboundary terms which do not contain normal derivatives of the metric. However, it turns outthat such terms can at most lead to contact terms in the holographic computation of 2-pointfunctions. The upshot of this discussion is that we can reduce the calculation of all possible 2-point functions in CCTMG to the equivalent calculation in Einstein gravity with suitable sourceterms. To continue we go on-shell.5

DLψL = 0 DLψR = 2ψR DLψlog = −2ψL (31)

These relations together with the comparison between CCTMG (29) and Einstein gravity (30)then establish

〈ψR(h, h)ψR(h′, h′)〉CCTMG ∼ 2〈ψR(h, h)ψR(h′, h′)〉EH (32a)

〈ψL(h, h)ψL(h′, h′)〉CCTMG ∼ 0 (32b)

〈ψL(h, h)ψR(h′, h′)〉CCTMG ∼ 0 (32c)

〈ψR(h, h)ψlog(h′, h′)〉CCTMG ∼ 0 (32d)

〈ψL(h, h)ψlog(h′, h′)〉CCTMG ∼ −2 〈ψL(h, h)ψL(h′, h′)〉EH (32e)

Here the sign ∼ means equality up to contact terms. Evaluating the right hand sides in Einsteingravity yields

〈ψL(h, h)ψL(h′, h′)〉EH = δh,h′ δh,h′

cBH

24

h

h(h2 − 1)

t1∫

t0

dt (33)

and similarly for the right modes, with h ↔ h. The quantity cBH is the Brown–Henneauxcentral charge (13). The calculation of the 2-point correlator between two logarithmic modescannot be reduced to a correlator known from Einstein gravity. The result is given by [25]

〈ψlog(h, h)ψlog(h′, h′)〉CCTMG ∼ −δh,h′ δh,h′

ℓ

4GN

h

h(h2 − 1)

(

ψ(h− 1) + ψ(−h))

t1∫

t0

dt (34)

where ψ is the digamma function. An ambiguity in defining ψlog, viz., ψlog → ψlog + γ ψL, wasfixed conveniently in the result (34). This ambiguity corresponds precisely to the ambiguity ofthe LCFT mass scale mL in (7c) (see also the discussion below that equation).

To compare the results (32)-(34) with the Euclidean 2-point correlators in the short-distance limit (1), (7) we take the limit of large weights h,−h → ∞ (e.g. limh→∞ ψ(h) =lnh+O(1/h)) and Fourier-transform back to coordinate space (e.g. h3/h is Fourier-transformedinto ∂4

z/(∂z∂z) δ(2)(z, z) ∝ ∂4

z ln |z| ∝ 1/z4). Straightforward calculation establishes perfectagreement with the LCFT correlators (1), (7), provided we use the values

cL = 0 cR =3ℓ

GNbL = − 3ℓ

GN(35)

These are exactly the values for central charges cL, cR [15] and new anomaly bL [23, 25] foundbefore. Thus, at the level of 2-point correlators CCTMG is indeed a gravity dual for a LCFT.

5 Above by “on-shell” we meant that the background metric is AdS3 (12) and therefore a solution of the classicalequations of motion. Here by “on-shell” we mean additionally that the linearized equations of motion (20) hold.

Gravity duals for logarithmic conformal field theories 528

Ψ1

Ψ3

Ψ2

Figure 1. Witten diagram for three graviton correlator

We evaluate now the Witten diagram in Fig. 1, which yields the 3-point correlator on thegravity side between three modes ψ1(h, h), ψ2(h′, h′) and ψ3(h′′, h′′) in momentum space.

〈ψ1(h, h)ψ2(h′, h′)ψ3(h′′, h′′)〉 =1

6

(

δ(3)SCCTMG(ψ1, ψ2, ψ3) + 5 permutations)

(36)

On the right hand side one has to plug the non-normalizable modes ψ1, ψ2 and ψ3 into the thirdvariation of the on-shell action and symmetrize with respect to all three modes.

δ(3)SCCTMG ∼ − 1

16π GN

∫

d3x√−g

[

(

DLψ1)µν

δ(2)Rµν(ψ2, ψ3) + ψ1 µν ∆µν(ψ

2, ψ3)]

(37)

The quantity δ(2)Rµν(ψ2, ψ3) denotes the second variation of the Ricci-tensor and the tensor

∆µν(ψ2, ψ3) vanishes if evaluated on left- and/or right-moving solutions. All boundary terms

turn out to be contact terms, which is why only bulk terms are present in the result (37) for thethird variation of the on-shell action. We compare again with Einstein gravity.

δ(3)SEH ∼ − 1

16π GN

∫

d3x√−g ψ1 µν δ(2)Rµν(ψ2, ψ3) (38)

Once more we can exploit some results from Einstein gravity for CCTMG, and we find thefollowing results [25] for 3-point correlators without log-insertions:

〈ψR(h, h)ψR(h′, h′)ψR(h′′, h′′)〉CCTMG ∼ 2 〈ψR(h, h)ψR(h′, h′)ψR(h′′, h′′)〉EH (39a)

〈ψL(h, h)ψR(h′, h′)ψR(h′′, h′′)〉CCTMG ∼ 0 (39b)

〈ψL(h, h)ψL(h′, h′)ψR(h′′, h′′)〉CCTMG ∼ 0 (39c)

〈ψL(h, h)ψL(h′, h′)ψL(h′′, h′′)〉CCTMG ∼ 0 (39d)

with one log-insertion:

〈ψR(h, h)ψR(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ 0 (40a)

〈ψL(h, h)ψR(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ 0 (40b)

〈ψL(h, h)ψL(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ −2 〈ψL(h, h)ψL(h′, h′)ψL(h′′, h′′)〉EH (40c)

Gravity duals for logarithmic conformal field theories 529

and with two or more log-insertions:

lim|weights|→∞

〈ψR(h, h)ψlog(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ 0 (41a)

lim|weights|→∞

〈ψL(h, h)ψlog(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ δh′′,−h−h′δh′′,−h−h′

P log(h, h′, h, h′)

hh′(h+ h′)(41b)

lim|weights|→∞

〈ψlog(h, h)ψlog(h′, h′)ψlog(h′′, h′′)〉CCTMG ∼ δh′′,−h−h′δh′′,−h−h′

lengthy

hh′(h+ h′)(41c)

The last two correlators so far could be calculated qualitatively only (P log is a known polynomialin the weights and also contains logarithms in the weights, as expected on general grounds),and it would be interesting to calculate them exactly. They are in qualitative agreement withcorresponding LCFT correlators. All other correlators have been calculated exactly [25], andthey are in precise agreement with the LCFT correlators (1), (8), provided we use again thevalues (35) for central charges and new anomaly.

In conclusion, also the seventh wish is granted for CCTMG.6 Thus, there are excellent chancesthat CCTMG is dual to a LCFT with values for central charges and new anomaly given by (35).

4.5. Logs don’t grow on trees

From the discussion above it is clear that possible gravity duals for LCFTs are sparse in theoryspace: Einstein gravity (11) does not provide a gravity dual for any tuning of parameters andCTMG (15) does potentially provide a gravity dual only for a specific tuning of parameters (17).Any candidate for a novel gravity dual to a LCFT is therefore welcomed as a rare entity.

Very recently another plausible candidate for such a gravitational theory was found [26].That theory is known as “new massive gravity” [16].

SNMG =1

16π GN

∫

d3x√−g

[

σR+1

m2

(

RµνRµν − 3

8R2

)

− 2λm2]

(42)

Here m is a mass parameter, λ a dimensionless cosmological parameter and σ = ±1 the sign ofthe Einstein-Hilbert term. If they are tuned as follows

λ = 3 ⇒ m2 = − σ

2ℓ2(43)

then essentially the same story unfolds as for CTMG at the chiral point. The main differenceto CCTMG is that both central charges vanish in new massive gravity at the chiral point(CNMG) [27,28].

cL = cR =3ℓ

2GN

(

σ +1

2ℓ2m2

)

= 0 (44)

Therefore, both left and right flux component of the energy momentum tensor acquire alogarithmic partner. It is easy to check that CNMG grants us the first six wishes from section3. The seventh wish requires again the calculation of correlators. The 3-point correlators havenot been calculated so far, but at the level of 2-point correlators again perfect agreement witha LCFT was found, provided we use the values [26]

cL = cR = 0 bL = bR = −σ 12ℓ

GN(45)

6 The sole caveat is that two of the ten 3-point correlators were calculated only qualitatively. It would beparticularly interesting to calculate the correlator between three logarithmic modes (41c), since it contains anadditional parameter independent from the central charges and new anomaly that determines LCFT properties.

Gravity duals for logarithmic conformal field theories 530

It is likely that a similar story can be repeated for general massive gravity [16], which combinesnew massive gravity (42) with a gravitational Chern–Simons term (14). Thus, even though theyare sparse in theory space we have found a few good candidates for gravity duals to LCFTs:cosmological topologically massive gravity, new massive gravity and general massive gravity. Inall cases we have to tune parameters in such a way that a “chiral point” emerges where at leastone of the central charges vanishes.

4.6. Chopping logs?

So far we were exclusively concerned with finding gravitational theories where logarithmic modescan arise. In this subsection we try to get rid of them. The rationale behind the desire toeliminate the logarithmic modes is unitarity of quantum gravity. Gravity in 2+1 dimensions issimple and yet relevant, as it contains black holes [29], possibly gravity waves [13] and solutionsthat are asymptotically AdS. Thus, it could provide an excellent arena to study quantum gravityin depth provided one is able to come up with a consistent (unitary) theory of quantum gravity,for instance by constructing its dual (unitary) CFT. Indeed, two years ago Witten suggested aspecific CFT dual to 3-dimensional quantum gravity in AdS [30]. This proposal engendered alot of further research (see [31–37] for some early references), including the suggestion by Li,Song and Strominger [17] to construct a quantum theory of gravity that is purely right-moving,dubbed “chiral gravity”. To make a long story [18,19,24,38–81] short, “chiral gravity” is nothingbut CCTMG with the logarithmic modes truncated in some consistent way.

We discuss now two conceptually different possibilities of implementing such a truncation.The first option was proposed in [18]. If one imposes periodicity in time for all modes, t→ t+β,then only the left- and right-moving modes are allowed, while the logarithmic modes areeliminated since they grow linearly in time, see e.g. (25). The other possibility was pursuedin [22]. It is based upon the observation that logarithmic modes grow logarithmically faster ine2ρ than their left partners, see e.g. (25). Thus, imposing boundary conditions that prohibit thislogarithmic growth eliminates all logarithmic modes.

Currently it is not known whether chiral gravity has its own dual CFT or if it exists merelyas a zero-charge superselection sector of the logarithmic CFT. In the latter case it is unclearwhether or not the zero-charge superselection sector is a fully-fledged CFT. Another alternativeis that neither the LCFT nor its chiral truncation dual to chiral gravity exists. In that caseCTMG is unlikely to exist as a consistent quantum theory on its own. Rather, it would requirea UV completion, such as string theory.

4.7. Logout

We summarize now the key results reviewed in this section as well as some open issues.Cosmological topologically massive gravity (15) at the chiral point (17) is likely to be dualto a LCFT with a logarithmic partner for one flux component of the energy momentum tensorsince 2- [23] and 3-point correlators [25] match. The values of central charges and new anomalyare given by (35). The detailed calculation of the correlator with three log-insertions (41c)still needs to be performed and will determine another parameter of the LCFT. New massivegravity (42) at the chiral point (43) is likely to be dual to a LCFT with a logarithmic partnerfor both flux components of the energy momentum tensor since 2-point correlators match [26].The central charges vanish and the new anomalies are given by (45). The calculation of 3-point correlators still needs to be performed and will provide a more stringent test of theconjectured duality to a LCFT. A similar story is likely to repeat for general massive gravity(the combination of topologically and new massive gravity) at a chiral point, and it could berewarding to investigate this issue. Finally we addressed possibilities to eliminate the logarithmicmodes and their partners, since such an elimination might lead to a chiral theory of quantumgravity [17], called “chiral gravity”. The issue of whether chiral gravity exists still remains open.

Gravity duals for logarithmic conformal field theories 531

5. Towards condensed matter applications

In this final section we review briefly some condensed matter systems where LCFTs do arise,see [3, 4] for more comprehensive reviews. We focus on LCFTs where the energy-momentumtensor acquires a logarithmic partner, i.e., the class of LCFTs for which we have found possiblegravity duals.7 Condensed matter systems described by such LCFTs are for instance systemsat (or near) a critical point with quenched disorder, like spin glasses [83]/quenched randommagnets [84, 85], dilute self-avoiding polymers or percolation [86]. “Quenched disorder” arisesin a condensed matter system with random variables that do not evolve with time. If theamount of disorder is sufficiently large one cannot study the effects of disorder by perturbingaround a critical point without disorder — standard mean field methods break down. Thesystem is then driven towards a random critical point, and it is a challenge to understand itsprecise nature. Mathematically, the essence of the problem lies in the infamous denominatorarising in correlation functions of some operator O averaged over disordered configurations (seee.g. chapter VI.7 in [87])

〈O(z)O(0)〉 =

∫

DV P [V ]

∫

Dφ exp(

− S[φ] −∫

d2z′V (z′)O(z′))

O(z)O(0)∫

Dφ exp(

− S[φ] −∫

d2z′V (z′)O(z′)) (46)

Here S[φ] is some 2-dimensional8 quantum field theory action for some field(s) φ and V (z) is arandom potential with some probability distribution. For white noise one takes the Gaussianprobability distribution P [V ] ∝ exp

(

−∫

d2zV 2(z)/(2g2))

, where g is a coupling constant thatmeasures the strength of the impurities. If it were not for the denominator appearing on the righthand side of the averaged correlator (46) we could simply perform the Gaussian integral overthe impurities encoded in the random potential V (z). This denominator is therefore the sourceof all complications and to deal with it requires suitable methods, see e.g. [88]. One possibility isto eliminate the denominator by introducing ghosts. This so-called “supersymmetric method”works well if the original quantum field theory described by the action S[φ] is very simple, like afree field theory. Another option is the so-called replica trick, where one introduces n copies ofthe original quantum field theory, calculates correlators in this setup and takes the limit n→ 0in the end, which formally reproduces the denominator in (46). Recently, Fujita, Hikida, Ryuand Takayanagi combined the replica method with the AdS/CFT correspondence to describedisordered systems [89] (see [90,91] for related work), essentially by taking n copies of the CFT,exploiting AdS/CFT to calculate correlators and taking formally the limit n → 0 in the end.Like other replica tricks their approach relies on the existence of the limit n→ 0.

One of the results obtained by the supersymmetric method or replica trick is that correlatorslike the one in (46) develop a logarithmic behavior, exactly as in a LCFT [84]. In fact, inthe n → 0 limit prescribed by the replica trick, the conformal dimensions of certain operatorsdegenerate. This produces a Jordan block structure for the Hamiltonian in precise parallel tothe µℓ → 1 limit of CTMG. More concretely, LCFTs can be used to compute correlators ofquenched random systems!

This suggests yet-another route to describe systems with quenched disorder, and our presentresults add to this toolbox. Namely, instead of taking n copies of an ordinary CFT we maystart directly with a LCFT. If this LCFT is weakly coupled we can work on the LCFT sideperturbatively, using the results mentioned above [3,4,84–86]. On the other hand, if the LCFTbecomes strongly coupled, perturbative methods fail. To get a handle on these situations wecan exploit the AdS/LCFT correspondence and work on the gravity side. Of course, to this end

7 A well-studied alternative case is a LCFT with c = −2 [2, 82]. There is no obvious way to construct a gravitydual for such LCFTs, even when considering CTMG or new massive gravity away from the chiral point. We thankIvo Sachs for discussions on this issue.8 Analog constructions work in higher dimensions, but we focus here on two dimensions.

Gravity duals for logarithmic conformal field theories 532

one needs to construct gravity duals for LCFTs. The models reviewed in this talk are simpleand natural examples of such constructions.

Acknowledgments

We thank Matthias Gaberdiel, Gaston Giribet, Olaf Hohm, Roman Jackiw, David Lowe, HongLiu, Alex Maloney, John McGreevy, Ivo Sachs, Kostas Skenderis, Wei Song, Andy Stromingerand Marika Taylor for discussions. DG thanks the organizers of the “First MediterraneanConference on Classical and Quantum Gravity” for the kind invitation and for all their efforts tomake the meeting very enjoyable. DG and NJ are supported by the START project Y435-N16of the Austrian Science Foundation (FWF). During the final stage NJ has been supported byproject P21927-N16 of FWF. NJ acknowledges financial support from the Erwin-Schrodinger-Institute (ESI) during the workshop “Gravity in three dimensions”.

References[1] Di Francesco P, Mathieu P and Senechal D 1997 Conformal Field Theory (Springer)[2] Gurarie V 1993 Nucl. Phys. B410 535–549 (Preprint hep-th/9303160)[3] Flohr M 2003 Int. J. Mod. Phys. A18 4497–4592 (Preprint hep-th/0111228)[4] Gaberdiel M R 2003 Int. J. Mod. Phys. A18 4593–4638 (Preprint hep-th/0111260)[5] Kogan I I and Nichols A 2002 JHEP 01 029 (Preprint hep-th/0112008)[6] Aharony O, Gubser S S, Maldacena J M, Ooguri H and Oz Y 2000 Phys. Rept. 323 183–386 (Preprint

hep-th/9905111); for a review focussed on the condensed matter perspective seeMcGreevy J, (Preprint 0909.0518)

[7] Brown J D and Henneaux M 1986 Commun. Math. Phys. 104 207–226[8] Ghezelbash A M, Khorrami M and Aghamohammadi A 1999 Int. J. Mod. Phys. A14 2581–2592 (Preprint

hep-th/9807034)[9] Myung Y S and Lee H W 1999 JHEP 10 009 (Preprint hep-th/9904056)

[10] Kogan I I 1999 Phys. Lett. B458 66–72 (Preprint hep-th/9903162)[11] Lewis A 2000 Phys. Lett. B480 348–354 (Preprint hep-th/9911163)[12] Moghimi-Araghi S, Rouhani S and Saadat M 2004 Nucl. Phys. B696 492–502 (Preprint hep-th/0403150)[13] Deser S, Jackiw R and Templeton S 1982 Phys. Rev. Lett. 48 975–978

Deser S, Jackiw R and Templeton S 1982 Ann. Phys. 140 372–411; 1988 Erratum-ibid. 185 406[14] Deser S 1984 Quantum Theory Of Gravity ed Christensen S M (Bristol: Adam Hilger) pp 374–381 (Preprint

Print-82-0692 (Brandeis))[15] Kraus P and Larsen F 2006 JHEP 01 022 (Preprint hep-th/0508218)[16] Bergshoeff E A, Hohm O and Townsend P K 2009 Phys. Rev. Lett. 102 201301 (Preprint 0901.1766)[17] Li W, Song W and Strominger A 2008 JHEP 04 082 (Preprint 0801.4566)[18] Grumiller D and Johansson N 2008 JHEP 07 134 (Preprint 0805.2610)[19] Grumiller D, Jackiw R and Johansson N 2009 Fundamental Interactions - A Memorial Volume for Wolfgang

Kummer (World Scientific) (Preprint 0806.4185)[20] Grumiller D and Johansson N 2009 Int. J. Mod. Phys. D17 2367–2372 (Preprint 0808.2575)[21] Henneaux M, Martinez C and Troncoso R 2009 Phys. Rev. D79 081502R (Preprint 0901.2874)[22] Maloney A, Song W and Strominger A 2009 (Preprint 0903.4573)[23] Skenderis K, Taylor M and van Rees B C 2009 JHEP 09 045 (Preprint 0906.4926)[24] Ertl S, Grumiller D and Johansson N 2009 (Preprint 0910.1706)[25] Grumiller D and Sachs I 2009 (Preprint 0910.5241)[26] Grumiller D and Hohm O 2009 (Preprint 0911.4274)[27] Liu Y and Sun Y W 2009 JHEP 04 106 (Preprint 0903.0536)[28] Bergshoeff E A, Hohm O and Townsend P K 2009 Phys. Rev. D79 124042 (Preprint 0905.1259)[29] Banados M, Teitelboim C and Zanelli J, 1992 Phys. Rev. Lett. 69 1849 (Preprint hep-th/9204099)[30] Witten E 2007 (Preprint 0706.3359)[31] Manschot J 2007 JHEP 10 103 (Preprint 0707.1159)[32] Gaiotto D and Yin X 2007 JHEP 08 029 (Preprint 0707.3437)[33] Gaberdiel M R 2007 JHEP 11 087 (Preprint 0707.4073)[34] Yin X 2007 (Preprint arXiv:0710.2129 [hep-th])[35] Yin X 2008 JHEP 09 120 (Preprint 0711.2803)[36] Maloney A and Witten E 2007 (Preprint 0712.0155)

Gravity duals for logarithmic conformal field theories 533

[37] Manschot J and Moore G W 2007 (Preprint 0712.0573)[38] Carlip S, Deser S, Waldron A and Wise D K 2009 Class. Quant. Grav. 26 075008 (Preprint 0803.3998)[39] Carlip S 2008 JHEP 10 078 (Preprint 0807.4152)[40] Hotta K, Hyakutake Y, Kubota T and Tanida H 2008 JHEP 07 066 (Preprint 0805.2005)[41] Li W, Song W and Strominger A 2008 (Preprint 0805.3101)[42] Park M I 2008 JHEP 09 084 (Preprint 0805.4328)[43] Sachs I and Solodukhin S N 2008 JHEP 08 003 (Preprint 0806.1788)[44] Lowe D A and Roy S 2008 Phys. Lett. B668 159–162 (Preprint 0806.3070)[45] Myung Y S, Lee H W and Kim Y W 2008 (Preprint 0806.3794)[46] Carlip S, Deser S, Waldron A and Wise D K 2008 Phys. Lett. B666 272–276 (Preprint 0807.0486)[47] Lee H W, Kim Y W and Myung Y S 2008 (Preprint 0807.1371)[48] Sachs I 2008 JHEP 09 073 (Preprint 0807.1844)[49] Gibbons G W, Pope C N and Sezgin E 2008 Class. Quant. Grav. 25 205005 (Preprint 0807.2613)[50] Anninos D, Li W, Padi M, Song W and Strominger A 2009 JHEP 03 130 (Preprint 0807.3040)[51] Giribet G, Kleban M and Porrati M 2008 JHEP 10 045 (Preprint 0807.4703)[52] Strominger A 2008 (Preprint 0808.0506)[53] Compere G and Detournay S 2009 Class. Quant. Grav. 26 012001 (Preprint 0808.1911)[54] Myung Y S 2008 Phys. Lett. B670 220–223 (Preprint 0808.1942)[55] de Haro S 2009 JHEP 01 042 (Preprint 0808.2054)[56] Stevens K A, Schleich K and Witt D M 2009 Class. Quant. Grav. 26 075012 (Preprint 0809.3022)[57] Deser S 2009 Fundamental Interactions - A Memorial Volume for Wolfgang Kummer (World Scientific)

(Preprint 0810.5384)[58] Hotta K, Hyakutake Y, Kubota T, Nishinaka T and Tanida H 2009 JHEP 01 010 (Preprint 0811.0910)[59] Quevedo H and Sanchez A 2009 Phys. Rev. D79 024012 (Preprint 0811.2524)[60] Oh J J and Kim W 2009 JHEP 01 067 (Preprint 0811.2632)[61] Garbarz A, Giribet G and Vasquez Y 2009 Phys. Rev. D79 044036 (Preprint 0811.4464)[62] Kim W and Son E J 2009 Phys. Lett. B673 90–94 (Preprint 0812.0876)[63] Mann R B, Oh J J and Park M I 2009 Phys. Rev. D79 064005 (Preprint 0812.2297)[64] Blagojevic M and Cvetkovic B 2009 JHEP 05 073 (Preprint 0812.4742)[65] Nam S and Park J D 2009 Class. Quant. Grav. 26 145015 (Preprint 0902.0982)[66] Hellerman S 2009 (Preprint 0902.2790)[67] Sezgin E and Tanii Y 2009 Class. Quant. Grav. 26 235005 (Preprint 0903.3779)[68] Anninos D, Esole M and Guica M 2009 JHEP 10 083 (Preprint 0905.2612)[69] Compere G and Detournay S 2009 JHEP 08 092 (Preprint 0906.1243)[70] Hotta K, Hyakutake Y, Kubota T, Nishinaka T and Tanida H 2009 Phys. Lett. B680 279–285 (Preprint

0906.1255)[71] Anninos D 2009 (Preprint 0906.1819)[72] Carlip S 2009 JHEP 09 083 (Preprint 0906.2384)[73] Chow D D K, Pope C N and Sezgin E 2009 (Preprint 0906.3559)[74] Becker M, Bruillard P and Downes S 2009 JHEP 10 004 (Preprint 0906.4822)[75] Blagojevic M and Cvetkovic B 2009 JHEP 09 006 (Preprint 0907.0950)[76] Vasquez Y 2009 (Preprint 0907.4165)[77] Duncan J F and Frenkel I B 2009 (Preprint 0907.4529)[78] Andrade T and Marolf D 2009 (Preprint 0909.0727)[79] Miskovic O and Olea R 2009 JHEP 12 046 (Preprint 0909.2275)[80] Skenderis K, Taylor M and van Rees B C 2009 (Preprint 0909.5617)[81] Afshar H R, Alishahiha M and Naseh A 2009 (Preprint 0910.4350)[82] Gaberdiel M R and Kausch H G 1996 Phys. Lett. B386 131–137 (Preprint hep-th/9606050)[83] Binder K and Young A P 1986 Rev. Mod. Phys. 58 801–976[84] Cardy J L 1999 (Preprint cond-mat/9911024)[85] Reza Rahimi Tabar M 2000 Nucl. Phys. B588 630–637 (Preprint cond-mat/0002309)[86] Gurarie V and Ludwig A W W 2002 J. Phys. A35 L377–L384 (Preprint cond-mat/9911392)[87] Zee A 2003 Quantum field theory in a nutshell (Princeton University Press)[88] Bernard D 1995 (Preprint hep-th/9509137)[89] Fujita M, Hikida Y, Ryu S and Takayanagi T 2008 JHEP 12 065 (Preprint 0810.5394)[90] Kiritsis E and Niarchos V 2009 Nucl. Phys. B812 488–524 (Preprint 0808.3410)[91] Myers R C and Wapler M C 2008 JHEP 12 115 (Preprint 0811.0480)

CURRICULUM VITAE - DANIEL GRUMILLER

Dr. Daniel Grumiller

Professional Address Institute for Theoretical Physics (ITP), Vienna University of Technology (VUT), Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austriae-mail: grumil@hep.itp.tuwien.ac.at, web: http://quark.itp.tuwien.ac.at/~grumil

Personal Data Date of BirthLanguagesMarital Status

4th of May 1973 in Vienna, Austria (Nationality: Austrian)German (native), English, Spanishmarried since 1997, 2 children (born 1997 and 1999)

Scientific Career

ITP, VUT, Austriasince 08/2009: Scientific staff member through Austrian START prize

European Space Agency (ESA), Advanced Concepts Team, NetherlandsScientific Coordinator of Ariadna project (ID 07/1301) 12/2007 - 09/2009

ITP, VUT, AustriaScientific staff member and Marie-Curie fellow 08/2006 - 07/2009

Center for Theoretical Physics (CTP), Massachusetts Institute of Technology (MIT), USAPost-doc and Marie-Curie fellow 08/2006 - 07/2008

Institute for Theoretical Physics, University of Leipzig, GermanyPost-doc and Erwin-Schrödinger fellow 01/2004 - 08/2006

ITP, VUT, AustriaScientific staff member and PhD fellow 10/1998 - 12/2003

Salztorzentrum, Salvation Army, AustriaCivil service 10/1997 - 09/1998

Experiment NA48, CERN, Switzerland and Institute of High Energy Physics, AustriaSoftware development and monitoring of electronics 06/1997 - 08/1997

ITP, VUT, AustriaTeaching tutor 10/1993 - 02/1997

Scientific Education

01/06/01 Graduation with distinction, thesis: “Quantum dilaton gravity in two dimen-sions with matter”, advisor: Wolfgang Kummer, degree: Dr. Techn. (PhD)

1998 - 2001 Postgraduate studies “Theoretical physics” at ITP, VUT

1997 CERN Summer Lectures

Selected Ad-Personam Grants and Awards

START-prize by Austrian Ministry of Science&Research and Austrian Science Foundation “Black Holes in Anti-deSitter, the Universe and Analog Systems”Amount: 1.090.000 €, Date of award: 11/2008

Marie-Curie Outgoing International Fellowship of the European Commission, post-doc grant“Gravity, Chern-Simons extensions and solid-state physics applications”Amount: 260.000 €, Date of award: 08/2006, project ID: MC-OIF-021421

Erich-Schmid award by Austrian Academy of Sciences, for excellent contributions to gravityAmount: 3.700 €, Date of award: 10/2005

Erwin-Schrödinger fellowship of the Austrian Science Foundation (FWF), post-doc grant“Nonperturbative aspects of 2d dilaton gravity”Amount: 60.000 €, Date of award: 01/2004 , project ID: project J-2330-N08

Victor-Hess prize by the Austrian Physical Society, for excellent PhD thesisAmount: 500 €, Date of award: 10/2003

Summary of Research

40 publications in peer reviewed international journals, 16 proceedings contributions

16 invited talks at international conferences

Research interests: gravitational physics, quantum gravity, cosmology, AdS/CFT

Teaching

since 1999 Supervision of 5 master theses and 1 PhD thesis at VUT and ITP Leipzig

since 2009 Lectures “Black Holes” at VUT

since 2008 Organizer of post-graduate student seminars at VUT

2005 - 2008 Lectures on lower-dimensional gravity in seminars at CTP, MIT and at ITP Leipzig

1996 - 2004 Supervision of numerous student projects on quantum field theory at VUT

1993 - 1997 Teaching tutor at the Institute for Experimental Physics and ITP, VUT

Funding ID [total amount since 1998: more than 1.800.000 €]

START project Y435-N16, six year research project on black holes and AdS/CFT at ITP, VUTAmount: 1.090.000 €, Date of activity: since 08/2009, Principal Investigator

European Commission FP6 project MC-OIF-021421, post-doc at CTP, MIT and ITP, VUTAmount: 260.000 €, Date of activity: 08/2006 – 07/2009, Principal Investigator

Austrian Science Foundation project P21927-N16, on 3-dimensional gravity at ITP, VUTAmount: 220.000 €, Date of activity: as of 02/2010, Principal Investigator

German Research Foundation project GR-3157/1-1, on black holesAmount: 100.000 €, Date of activity: 01/2006 – 08/2006, Principal Investigator

Erwin-Schrödinger Institute (ESI) grants, for workshops on 2- and 3-dimensional gravityAmount: 85.000 €, Dates of activity: 04/2009 (3d) & 09/2003 – 12/2003 (2d), Scientific Organizer

Austrian Science Foundation project J-2330-N08, on dilaton gravityAmount: 60.000 €, Date of activity: 01/2004 – 12/2005, Principal Investigator

Austrian Science Foundation projects P-14650 and P-12815-TPH, for PhD on 2d gravityAmount: 43.000 €, Date of Activity: 10/1998 – 12/2003

ESA project Ariadna ID 07/1301, on non-perturbative effects in gravityAmount: 29.000 €, Date of activity: 10/2007 – 09/2009, Scientific Coordinator

Editorial Duties and Refereeing

2007 - 2009 Editor of “Fundamental Interactions - A Memorial Volume for Wolfgang Kummer”

2004 - 2007 Co-editor for December Special Issues of Int.J.Mod.Phys.D

since 2001 Referee: CQG, EPJC, GRG, IJMPA, IJMPD, JHEP, MPLA, PLA, PLB, PRD, RMP

Functions

since 2010 Chairman of Nuclear and Particle Physics division of the Austrian Physical Society

since 2009 MIT exchange student-coordinator for Austria

since 2009 Chairman of outreach webpage teilchen.at

2008 - 2009 Chairman of workshop “Gravity in three dimensions” at ESI, Vienna

2002 - 2003 Chairman of workshop “Gravity in two dimensions” at ESI, Vienna

Selected memberships: American Physical Society, Austrian Physical Society (ÖPG)