Special Geometry, Black Holes, and Instantonsmedia/blackholes_supergravity/... · Special Geometry,...
Transcript of Special Geometry, Black Holes, and Instantonsmedia/blackholes_supergravity/... · Special Geometry,...
Special Geometry, Black Holes, and Instantons
Thomas Mohaupt
Department of Mathematical SciencesUniversity of Liverpool
Inaugural Workshop on Black Holes in Supergravity andM/Superstring Theory, September 2010
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 1 / 59
Introduction
Approach to special geometry (here: N = 2 vector multiplets) basedon a combination of old and new ideas
Special coordinates, homogeneity (conformal calculus), as in B.de Wit and A. Van Proeyen (1984).
More recent work in differential geometry (D. Freed, N. Hitchin, V.Cortés), see ‘Special complex manifolds’, D.V. Alekseevsky, V.Cortés and C. Devchand (1999).
In this talk:
Modifications: Euclidean supersymmetry and para-complexgeometry.
Generalizations: non-supersymmetric theories with target spacesencoded by potentials (not necessarily symmetric orhomogeneous spaces).
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 2 / 59
References
‘Euclidean special geometry’:
Rigid vector multiplets: V. Cortés, C. Mayer, T.M. and F.Saueressig, JHEP 03 (2004) 028.
Rigid hypermultiplets (bosonic sector, only): V. Cortés, C. Mayer,T.M. and F. Saueressig, JHEP 06 (2005) 025.
Local vector multiplets (bosonic sector, only): V. Cortés and T.M.,JHEP 07 (2009) 066.
Applications:
Multi-centered extremal black hole solutions: T.M. and K. Waite,JHEP 10 (2009) 058.
Black hole solutions and instantons. T.M. and K. Waite, inpreparation.
Single-centered non-extremal black hole solutions. T.M. and O.Vaughan, arXiv:1006.3439 and work in progress.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 3 / 59
Part I
Special geometry of vector multiplets: modificationsand generalizations
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 4 / 59
Special real geometry
5d vector multiplets (M. Gunaydin, G. Sierra, P. Townsend (1984)).
Prelude: rigid case
gIJ(σ) =∂2V
∂σI∂σJ ,
Susy: Hesse potential V(σ) is a polynomial of degree three.
Hessian metric: flat, torsion-free connection ∇, such that ∇g iscompletely symmetric. Locally (∇-affine coordinates):
∂IgJK completely symmetric ⇔ ΓIJ|K completely symmetric .
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 5 / 59
Special real geometry
Local case: scalar manifold is a hypersurface
M = V(σ) = 1 ⊂ M
in Hessian manifold M with metric
gIJ(σ) =∂2V
∂σI∂σJ , V ≃ − log V(σ)
and where prepotential V(σ) is a homogeneous polynomial of degree3.
Generalization (loose Susy): allow V to be a homogeneous function ofarbitrary degree p. M is ‘conical Hessian’.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 6 / 59
r-map (dimensional reduction)
Rigid case. Reduce
L ∼ −12
gIJ(σ)∂µσI∂µσJ − 14
gIJ(σ)F IµνF J|µν
over space/time:
L ∼ −12
gIJ(σ)(∂mσI∂mσJ ± ∂mbI∂mbJ) + · · ·
= −12
gIJ(X , X )∂mX I∂mX J + · · ·
where bI = AI∗, and
Space-like: X I = σI + ibI , X I = σI − ibI , i2 = −1Time-like: X I = σI + ebI , X I = σI − ebI , e2 = 1
e = para-complex unit (called ‘hyperbolic complex’ by G. Gibbons, M.Green and M. Perry (1995)).
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 7 / 59
Para-complex geometry
Almost para-complex structure J = tensor field of type (1, 1) such that
J2 = 1 , with ‘balanced’ eigenvalues .
Integrability, para-Hermitian, para-Kähler etc. can be definedanalogous to complex geometry.
Remark: para-Hermitian metrics are indefinite (split signature).
Additional references: several articles in V. Cortés (ed.), Handbook ofPseudo-Riemannian Geometry and Supersymmetry, EMS (2010).
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 8 / 59
Euclidean supersymmetry and para-complexstructures
Zumino (1977): Euclidean supersymmetry implies ‘non-compact chiraltransformation’ due to change of R-symmetry group with space-timesignature:
SU(2)R × U(1)R → SU(2)R × SO(1, 1)R
Generator of abelian factor = complex/para-complex structure onvector multiplet scalar manifold.
Special para-Kähler geometry (para-holomorphic prepotential).Rigid case: V. Cortés, C. Mayer, T.M., F. Saueressig (2004)Local case (bosonic sector, only) : V. Cortés, T.M. (2009).
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 9 / 59
Minkowksi or Euclidean?
δX I = i ǫ+ λI+ ,
δX I = i ǫ− λI− ,
δλiI+ = −1
4γmnF I
−mnǫi+ − i
2∂/X Iǫi
− − Y ij Iǫ+ j ,
δλiI− = −1
4γmnF I
+ mnǫi− − i
2∂/X Iǫi
+ − Y ij Iǫ− j ,
δAIm =
12
(
ǫ+γmλI− + ǫ−γmλI
+
)
,
δY ij I = −12
(
ǫ(i+∂/λ
j) I− + ǫ
(i−∂/λ
j) I+
)
.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 10 / 59
Comments
i → e only for those i corresponding to the complex structure ofthe scalar manifold, e.g. σI + ibI → σI + ebI.
To obtain uniform expressions, need to use reality conditions forspinors which appy to both signatures. Here: symplecticMajorana.
‘Chiral projections’ for spinors, and ‘(anti-)selfduality projections’ ofgauge fields involve factors of e. Example:
F I±|mn =
12(F I
mn ± eF Imn)
Geometry: use para-complexified tangent space of scalarmanifold.
Can do with light cone coordinates X I = σI ± bI, but loose analogy.
Complex form.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 11 / 59
r-maps with Susy
spatial reduction : special real → special Kähler.M. Gunaydin, G. Sierra, P. Townsend (1984), B. de Wit and A. VanProeyen (1992).
temporal : special real → special para-Kähler.V. Cortés and T.M., JHEP 07 (2009) 058, and work in progress.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 12 / 59
r-maps without Susy
When dropping Susy, we loose the ‘special’ properties in 4d (flatconnection, prepotential), but retain a (para-)Kähler potential.
Hessian → (para-)Kähler (with isometries)
X I = σI + iǫbI, iǫ = i , e. Hesse potential → (para-)Kähler potential:
gIJ =∂2V(σ)
∂σI∂σJ = 4∂2V(X + X )
∂X I∂X J
D. Alekseevsky and V. Cortés, arXiv:0811.1658, T.M. and K. Waite,JHEP 10 (2009) 058, V. Cortés and T.M. work in progress.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 13 / 59
Part II
Multi-centered extremal black holes
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 14 / 59
Some references
Generating stationary solutions via dimensional reduction over time.
G. Neugebauer und D. Kramer (1969)P. Breitenlohner, D. Maison and G. Gibbons (1988)
Branes and higher-dimensional theories:G. Clement and D. Gal’tsov PRD 54 (1996) 6136, D. Gal’tsov and O.A.Rychkov PRD 58 (1998) 122001,E. Cremmer, I. Lavrinenko, H. Lü, C. Pope, K. Stelle and T. Tran, NPB534 (1998) 40, . . .
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 15 / 59
Some more recent references
Many recent applications, including
M. Gunaydin, A. Neitzke, B. Pioline and A. Waldron, Phys. Rev. D 73(2006) 084019, JHEP 09 (2007) 056, A. Ceresole and G. Dall’Agata,JHEP 03 (2007) 110, G.L. Cardoso, A. Ceresole, G. Dall’Agata, J.M.Oberreuter and J. Perz, JHEP 10 (2007) 063, D. Gaiotto, W. Li and M.Padi, JHEP 12 (2007) 093, J. Perz, P. Smyth, T. Van Riet and B.Vercnocke, JHEP 03 (2009) 150, E. Bergshoeff, W. Chemissany, A.Ploegh, M. Trigiante and T. Van Riet, NPB 812 (2009) 343,Bossard’s talk, . . .
Most work assumes symmetric or homogeneous target spaces.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 16 / 59
The 5d theory
Lagrangian
L ≃ 12
R(5) −34
(
aIJ(σ)∂µσI∂µσJ)
V=1− 1
4aIJ(σ)F I
µνF J|µν + · · ·
n gauge fields, n − 1 scalars valued in V(σ) = 1 ⊂ M. Scalar metricis ‘conical Hessian’:
aIJ(σ) = ∂2I,JV(σ) , V(σ) = −1
plog V(σ) ,
V(σ) homogeneous of degree p. (Susy: p = 3.) Goal: find static,purely electric solutions (black holes).
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 17 / 59
Dimensional reduction over time:
ds2(5) = −e2σ(dt + AKK )2 + e−σds2
(4)
Absorb KK scalar: σI → eσσI . Reduced Lagrangian
L ≃ 12
R(4) − aIJ(σ)(∂mσI∂mσJ − ∂mbI∂mbJ) + · · ·
2n independent real scalar fields σI , bI = AI0.
X I = σI + ebI: Hesse potential → Para-Kähler potential:
aIJ(σ) =∂2V(σ)
∂σI∂σJ = 4∂V(X + X )
∂X I∂X J
‘Special’ for p = 3. Need homogeneity (scaling properties. Not:homogeneous space).
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 18 / 59
4d equations of motion
1√
|h|∂m(
√
|h|aIJ(σ)∂mσJ) − 12
∂IaJK (∂mσI∂mσJ − ∂mbI∂mbJ) = 0
∂m(√
|h|aIJ(σ)∂mbJ) = 014
aIJ(σ)(∂mσI∂nσJ − ∂mbI∂nbJ) =
16
Rmn
Drastic simplification when imposing ‘extremality’ ∂mσI = ±∂mbI.Rmn = 0 ⇒ can take hmn = δmn.Remaining field equations:
∂m(aIJ(σ)∂mσJ) = 0 .
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 19 / 59
Extremality
What does ∂mσI = ±∂mbI mean?For ds2
(4) = δmndxmdxn, the lifted line element corresponds to anextremal black hole:
ds2(5) = −e2σdt2 + e−σds2
(4) , where epσ = V(σ).
(We will discuss regularity of eσ later. )Susy case (p = 3). Euclidean BPS condition for purely scalar fieldconfigurations. (Lifts to 5d BPS condition).When dualizing axions bI into tensors Bmn|I,
Hmnp|I = 3!∂[mBnp]I = 3!aIJǫmnpq∂qbJ
then the dual, positive definite Euclidean action
L ≃ 12
aIJ(σ)∂mσI∂mσJ +1
2 · 3!aIJ(σ)Hmnp|IH
mnpJ
has a genuine Bogomol’nyi bound saturated by
∂mσI = ±ǫmnpqaIJ∂nBpqJ
(Observation frequently used for axionic theories.)Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 20 / 59
Axions vs Tensors
Wormholes: S.B. Giddings and A. Strominger, NPB 306 (1988)890, C.P. Burgess and A. Kshirsagar, NPB 324 (1989) 157, S.Coleman and K. Lee, NPB 329 (1990) 387, . . .
type-IIB D-instanton: G.W. Gibbons, M.B. Green and M.J. Perry,PLB 370 (1996) 37, M.B. Green and M. Gutperle, NPB 498 (1997)195.
Hypermultiplet instantons: M. Gutperle and M. Spalinski, JHEP 06(2000) 037, NPB 598 (2001) 509, U. Theis and S. Vandoren,JHEP 09 (2002) 059, M. Davidse, U. Theis and S. Vandoren, NPB697 (2004) 48, . . .
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 21 / 59
Extremality (cont’d)
What does ∂mσI = ±∂mbI mean?
Geometrically, solution is parallel to the eigendirections of thepara-complex structure J.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 22 / 59
Harmonic maps
Equations of motion of non-linear sigma model on E with target spaceN ⇔ Harmonic map
Φ : E → N .
Construct solutions using totally geodesic submanifolds S
Φ : E → S ⊂ N
Extremal solutions ⇔ E flat, S isotropic submanifold.
P. Breitenlohner, D. Maison and G. Gibbons (1988), . . .
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 23 / 59
Using para-complex geometry
N is para-Kähler, with para-complex structure J, S is defined byeigendirections of para-complex structure.
Eigendirections define a submanifold (J is integrable).
S is totally geodesic (J is parallel).
S is totally isotropic (J is an anti-isometry).
S is flat wrt pulled-back Levi-Civita connection of N (algebraicstructure of Riemann tensor). Therefore
Φ : E → S ⊂ N
is a harmonic map between flat manifolds and solutions can beexpressed in terms of harmonic functions.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 24 / 59
Harmonic functions
Remaining field equations:
∂m(aIJ(σ)∂mσJ) = 0
Define ‘dual scalars’:
σI := ∂IV(σ) ⇒ ∂mσI = aIJ(σ)∂mσj
so that field equations become
∆σI = 0 .
Solution given by n harmonic functions HI(x)! Can be multi-centered.Solving σI = ∂IV = HI for the σI is equivalent to solving the black holeattractor equations.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 25 / 59
Lifting to 5d
By lifting to 5d we recover (for p = 3) and generalize (for p 6= 3) theresults of A. Chamseddine and W. Sabra PLB 426 (1998) 36, PLB 460(1990) 63.
Attractor equations (‘global’ form), setting σI = eσhI:
σI = HI ⇒ e−σ ∂V(h)
∂hI = HI
determine 5d metric and scalars in terms of n harmonic functions.Asymptotics at a center located at r = 0:
HI ≈qI
r2 , e−σ ≈ Zr2
Attractor equations (fixed point):
Z∂V(h)
∂hI
∣
∣
∣
∣
∣
∗
= qI
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 26 / 59
Z generalizes the central charge: Z = 1p qIhI
∗.
Mass:
MADM =32
∮
d3Σme−σ∂mσ = σI(∞)qI ,
Entropy:
SBH =π2
2Z 3/2∗
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 27 / 59
Part III
Black holes and instantons
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 28 / 59
Instantons?
Expect correspondence between 0-branes (black holes/solitons) and(−1)-branes (instantons).I.p. Electric charge ≃ axionic charge, Mass (Tension) ≃ Action.
But can the Euclidean solutions we used to generate black holesolutions be regarded as instantons?Consistent saddle point approximation requires (i) finite positiveinstanton action, (ii) damped Gaussian around the saddle point.Problems: (i) Instanton action is zero, (ii) Action functional is indefinite(damping?)
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 29 / 59
Euclidean actions for axions
Should axions be ‘Wick rotated’, aI → bI = iaI?
Euclidean supersymmetry: B. Zumino PLB 69 (1977) 369, P. VanNieuwenhuizen and A. Waldron, PLB 389 (1996) 29, . . .
Wormholes: C.P. Burgess and A. Kshirsagar, NPB 324 (1989)157, S. Coleman and K. Lee, NPB 329 (1990) 387, . . .
D-instanton: G.W. Gibbons, M.B. Green and M.J. Perry, PLB 370(1996) 37, M.B. Green and M. Gutperle, NPB 498 (1997) 195, . . .
Complex actions and reality conditions: E. Bergshoeff, J. Hartong,A. Ploegh, J. Rosseel and D. Van den Bleecken, JHEP 07 (2007)067, T.M. and K. Waite, JHEP 10 (2009) 058 and to appear.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 30 / 59
Indefinite Euclidean action
Indefinite Euclidean action (obtained by dimensional reduction overtime or modified Wick rotation)
L ≃ aIJ(σ)(∂mσI∂mσJ − ∂mbI∂mbJ)
Instanton candidate ∂mσI = ±∂mbI.
‘Instanton’ is a solution of the field equations.
‘Instanton’ lifts to ‘soliton’.
Problem: zero instanton action.
Problem: action functional is indefinite.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 31 / 59
Definite Euclidean action
Definite Euclidean action (obtained by standard Wick rotation)
L ≃ aIJ(σ)(∂mσI∂mσJ + ∂maI∂maJ)
Instanton candidate ∂mσI = ±i∂maI.
Problem: ‘instanton’ is not a solution (Derrick’s theorem),
Problem: zero instanton action.
At least, this action functional is positive definite.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 32 / 59
Amplitude calculation
Semi-classical evaluation of transition amplitude, adapting calculationsof S. Coleman and K. Lee, NPB 329 (1990) 387. Hypermultiplets: M.Chiodaroli and M. Gutperle, PRD 79 (2009) 085023.
Amplitude (schematically):
A ≃∫
Da e−Sdef[a]−Sbd[a]
Note boundary term.Consistent saddle point approximation:
a = ib∗ + a
Imaginary saddle point, real fluctuation.
A ≃ e−Sbd[ib∗]
∫
Da e−Sdef[a]
Analogy: one-dimensional real integral dominated by complex saddlepoint.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 33 / 59
Observation: can re-write, using ‘rotated’ variable (a = ib):
A ≃∫
Db e−Sindef[b]−Sbd[b]
Consistent saddle point approximation:
a = ib∗ + a ⇔ b = b∗ + b = b∗ − i a
Real saddle point, imaginary fluctuation.
Same physical amplitude expressed in different variables.Consistency of saddle point approximation.
Can’t avoid to consider complex field configurations.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 34 / 59
Instanton action and ADM mass
Investigate relation between mass of 0-brane (black hole) and action of(-1)-brane (instanton).Instanton action (boundary term)
Sinst = iaI∗(∞)qI = bI
∗(∞)qI .
Axionic shift symmetry broken to discrete subgroup aI → aI + 2πZ.ADM mass
MADM = σI∗(∞)qI
Extremality condition ∂mσI = ±∂mbI does not fix constant part of bI.Instanton action = ADM mass if σI(∞) = bI(∞) (up to discrete shifts).Global shift symmetry of axions vs local gauge symmetries of vectorfields.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 35 / 59
Axions vs tensors, quantum
Similar observation when comparing instanton solutions of axionictheory and dual tensor field theory:
Saxioninst = iaI
∗(∞)qI = bI∗(∞)qI .
Stensorinst = σI
∗(∞)qI .
Quantum equivalence ‘up to zero modes’, C.P. Burgess and A.Kshirsagar, NPB 324 (1989) 157, . . .
Can relate physical quanitites, but need to be careful regarding zeromodes, boundary conditions.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 36 / 59
Part IV
Non-extremal Black Holes
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 37 / 59
Some previous and related work
Higher-dimensional black holes: F. Tangherlini, Nuovo Cimento 77(1963) 636, R. Myers and R. Perry, Ann. Phys. 172 (1986) 304.
Non-extremal black holes in supergravity/string theory (entropy,U-duality, Hawking radiation, etc.): C. Callan and J. Maldacena,NPB 472 (1996) 591, G. Horowitz and A. Strominger, PRL 77(1996) 2368, M. Cvetic and D. Youm, PRD 54 (1996) 2612, K.Behrndt, M. Cvectic and W. Sabra, PRD 58 (1997) 084018, . . .
Non-extremal branes, instantons: G. Gibbons and K. Maeda NPB298 (1988), G. Horowitz and A. Strominger PRD 43 NPB 360(1991) 197, M.J. Duff, H. Lü and C. Pope, PLB 382 (1996) 73, M.Cvetic and A. Tseytlin, NPB 478 (1996) 181, E. Cremmer, I.Lavrinenko, H. Lü, C. Pope, K. Stelle and T. Tran, NPB 534 (1998)40, . . .
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 38 / 59
Recently: 1st order equations, integrability for symmetric targetsH. Lu, C. Pope and J. Vazquez-Poritz, NPB 709 (2003) 47, C.Miller, K. Schalm and E. Weinberg, PRD 76 044001, L.Andrianapoli, R. D’Auria and E. Orazi JHEP 11 (2007) 032, B.Janssen, P. Smyth, T. Van Riet and B. Vercnocke, JHEP 04 (2008)007, G. Cardoso and V. Grass, NPB 803 (2008) 209, J. Perz, P.Smyth, T. Van Riet and B. Vercnocke, JHEP 03 (2009) 150, W.Chemissany, J. Rosseel, M. Trigiante and T. Van Riet NPB 830(2010) 391, W. Chemissany, P. Fré, J. Rosseel, A. Sorin, M.Trigiante and T. Van Riet, arXiv:1007.3209.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 39 / 59
Clues
5d version of Reissner-Nordström solution (isotropic coordinates):
ds2(5) = −W (ρ)
H2(ρ)dt2 +
H(ρ)
W 1/2(ρ)
[
dρ2
W 1/2(ρ)+ W 1/2(ρ)ρ2dΩ2
(3)
]
, where
H = 1 +qρ2 , W = 1 − 2c
ρ2
Solution is build up from harmonic function (on E = R4).
For extremal solutions, W = 1, dimensional reduction over time
ds2(5) = −e2σdt2 + e−σds2
(4)
results in a flat Euclidean 4d metric.
The non-extremal solution is obtained by ‘dressing’ the extremalsolution with the harmonic function W (ρ). (Similar results knownfor black branes.)
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 40 / 59
Equations of motion
Remember
1√
|h|∂m(
√
|h|aIJ(σ)∂mσJ) − 12
∂IaJK (∂mσI∂mσJ − ∂mbI∂mbJ) = 0
∂m(√
|h|aIJ(σ)∂mbJ) = 014
aIJ(σ)(∂mσI∂nσJ − ∂mbI∂nbJ) =
16
Rmn
Impose spherical symmetry:
ds2(4) = e6A(τ)dτ2 + e2A(τ)dΩ2
(3) ,
τ = radial coordinate. Solve Einstein equation (regularity).
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 41 / 59
Universality
The four-dimensional geometry is the time-reducedReissner-Nordstöm metric, irrespective of the matter sector:
ds2(4) =
c3
sinh3(2cτ)dτ2 +
csinh(2cτ)
dΩ2(3)
=dρ2
W (ρ)1/2+ W 1/2(ρ)ρ2dΩ2
(3) , W (ρ) = 1 − 2cρ2 = e−4cτ .
where
ρ2 =ce2cτ
sinh(2cτ)→c→0
12τ
.
τ → 0 ⇔ ρ → ∞: asymptotically flat.τ → ∞ ⇔ ρ →
√2c: outer horizon.
ρ → 0: inner horizon (extension!).Note: Lifted (5d) metric is real for all ρ.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 42 / 59
Equations of motion
Remaining field equations:
ddτ
(aIJ(σ)σJ ) − 12∂IaJK (σJ σK − bJ bK ) = 0
ddτ
(aIJ(σ)bJ) = 0
aIJ(σ)(σJ σK − bJ bK ) = 4c2
Radial coordinate τ = affine parameter on geodesic curve C ⊂ N.Non-extremality parameter c related to (constant) length of tangentvectors.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 43 / 59
4d solution
b-equations solved in terms of conserved charges:
aIJ(σ)bJ = qI = const.
Using dual coordinates, remaining equation is
σI +12
∂IaJK (σJ σK − qJqK ) = 0
Still complicated, but contraction with σI gives
aIJσI σJ = 4c2 .
Crucial identity: aIJσIσJ = 1. Parametrize our ignorance:
4c2σI = σI + XI , σIXI = 0
Are there solutions where XI = 0?
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 44 / 59
Solutions with XI = 0
The equation obtained by setting XI = 0,
σI = 4c2σI
is solved by
σI = AI cosh(2cτ) +1
2cBI sinh(2cτ) →c→0 AI + BIτ .
Re-substitution: constraints on the integration constants AI, BI .
Can always recover 5d non-extremal Reissner-Nordström withconstant 5d scalars.
For models with diagonal metric and connection, obtain black holesolution with general scalar profile.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 45 / 59
Example
Para-version of dilaton/axion system, (σI = e−2φI).
Hesse potential:V ≃ − log(σ1σ2 · · · σp)
Manifold:
N =
(
SL(2,R)
SO(1, 1)
)p
Then
σI = AI cosh(2cτ) +12c
BI sinh(2cτ)
is the general solution.
p = 3: supersymmetric STU model.General p: ‘Seed solution’ for symmetric target spaces. P.Breitenlohner, D. Maison and G. Gibbons, CMP 120 (1988) 295, . . .
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 46 / 59
Extend solution to inner horizon
Replace affine parameter τ by standard radial coordinate ρ.
σI = AI cosh(2cτ) +12c
BI sinh(2cτ) =HI(ρ)
W 1/2(ρ),
where
HI(ρ) = AI +QI
ρ2 , W (ρ) = 1 − 2cρ2
Integration constants: AI = σI(ρ → ∞),
QI =BI − 2cAI
2=
electric charges qI , for c = 0‘dressed’ charges, for c 6= 0
Solution works indeed by (twofold) ‘dressing’!
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 47 / 59
Lifting to 5d
Metric ds2(5) = −e2σdt2 + e−σds2
(4)
ds2(5) = − W
(H1H2 · · ·Hp)2/pdt2 +
(H1H2 · · ·Hp)1/p
W 1/2ds2
(4)
= − W(H1H2 · · ·Hp)2/p
dt2 + (H1H2 · · ·Hp)1/p[W−1dρ2 + ρ2dΩ2
(3)]
p = 3: recover non-extremal type-IIB D5-D1-pp solution of C. Callanand J. Maldacena, NPB 472 (1996) 591.
For H1 ∝ H2 ∝ · · · ∝ Hp ∝ H, recover non-extremalReissner-Nordström solution:
ds2(5) = −W
H2 dt2 + H[W−1dρ2 + ρ2dΩ2(3)]
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 48 / 59
5d scalars/dressed attractors
Same expression for 5d scalars as in non-extremal case
hI =(H1 · · ·Hp)1/p
HI
Inner horizon limit:
hI →ρ→0(Q1 · · ·Qp)1/p
QI.
‘Dressed attractor formula’.
QI =BI − 2cAI
2→c→0
BI
2= qI .
Not a true attractor for c 6= 0, since QI depend on AI ∼ hI(∞).Similar formula at outer horizon, with differently dressed charges.What does this tell us?
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 49 / 59
Black holes vs geodesics
A generic geodesic has 4n parameters: σI(∞), σI(∞), bI(∞), bI(∞).
Black hole solutions only have 2n:
σI(∞) , σI(∞) or σIHorizon ∼
qI ∼ bI(∞) , for c = 0QI(σ
J (∞), qJ) , for c 6= 0
Interpretation: Regularity of lifted solution ⇔ fine tuning of asymptoticsof geodesic (relating σI and bI) ⇔ (dressed) attractors, geodesicequations reduces to gradient flow equation.
Extremal limit:
σI = ±bI ⇒ σI = ∂IW , W = qIσI
Non-extremal case:
σI = ∂IW ⇒ W(σ(τ)) = −V(σ(τ)) + 4c2τ + W0
Can find W(σ) explicitly for STU model.Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 50 / 59
Integrability
pI := aIJ σJ = σI are the canonical momenta associated to σI .
qI := aIJ bJ = electric charges = canonical momanta associated to bI.
For extremal black holes, σI = ±bI, we have manifest integrability forpara-Kähler target spaces without assuming them to be homogeneousor symmetric spaces, because pI and qI are conserved.
Symmetric spaces: geodesic equations fully integrable (Liouville andHamilton/Jacobi), regular black hole solutions ≃ vanishingHamiltonians. (W.Chemissany, P. Fré, J. Rosseel, A.S. Sorin, M.Trigiante and T. Van Riet, arXiv:1007.3209).
What to expect for (non-homogeneous) para-Kähler targets?
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 51 / 59
Brief Summary
Modifications and generalizations of special geometry can beobtained systematically and have interesting applications.
Para-complex geometry is a useful tool, i.p. in combination withtemporal reduction and harmonic maps.
Can go beyond homogeneous/symmetric target spaces usingpotentials (integrability properties), special coordinates (definedintrinsically in terms of special data, like flat connections) andhomogeneity properties (homothetic Killing vector fields).
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 52 / 59
Outlook
Non-extremal black holes, gradient flow equations, integrability.
(generalized) hypermultiplet geometries, c-map
4d/3d reduction, inclusion of magnetic charges. (For purelyelectric charges, 5d/4d reduction is straightforward to extend tod + 1/d .)
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 53 / 59
Appendix
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 54 / 59
Complexification
Basic dilaton/axion system: X = e−2φ + iǫa, iǫ = i , e.
L ∼ ∂mX∂mXRe(X )2 ∼ −(∂mφ∂mφ ± 1
4e4φ∂ma∂ma)
Two real forms of one complex-Riemannian space:
SL(2,R)
SO(2)⊂ SL(2,C)
GL(1,C)⊃ SL(2,R)
SO(1, 1)
Change of real form via intermediate complexification is useful forhandling dualities, constructing solutions. Maximal 10d supergravity:E. Bergshoeff, J. Hartong, A. Ploegh, J. Rosseel and D. Van denBleecken, JHEP 07 (2007) 067Not restricted to symmetric spaces (assuming analyticity/integrabilityproperties): T.M. and K. Waite, JHEP 10 (2009) 058 and to appear.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 55 / 59
Complexifying the complex
Complexify a complex field:
σI + ibI → (σI + j σI) + j(bI + jβI)
Note the second complex structure/unit j .Since ij = ji and i2 = −1 = j2, we get a para-complex structure for free:e := ij , with e2 = 1, and a real form carrying a para-complex structure:
σI + ijβI = σI + eβI .
Analytical continuation from complex to para-complex.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 56 / 59
Comment: non-BPS extremal solutions
If aIJ has discrete isometries
aKLRKI RL
J = aIJ
then ∂mσI = ±∂mbI can be replaced by
∂mσI = RIJ∂mbJ
A. Ceresole and G. Dall’Agata, JHEP 03 (2007) 110, G. LopesCardoso, A. Ceresole, G. Dall’Agata, J. Oberreuter and J. Perz, JHEP10 (2007) 063.Such ‘non-BPS extremal’ solutions mix eigendirections of J, isometryguarantees existence of totally geodesic submanifold.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 57 / 59
Comment on the role of discrete isometries
aKLRKI RL
J = aIJ
allowing relaxed ansatz
∂mσI = RIJ∂mbJ
For general (non-symmetric) target spaces such isometriesappear somewhat non-generic. But there are examples: P. Kauraand A. Misra, Fortsch. Phys. 54 (2006) 1109, S. Belluci, S.Ferrara, S. Marrani and A. Yeranyan, Riv. Nuovo Cimento 29N5(2006) 1.
For symmetric spaces there are many isometries beyond thegeneric ones we assume. Huge literature on non-BPS extremalblack holes, see review S. Belluci, S. Ferrara, M. Gunaydin and A.Marrani, arXiv:0905.3451.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 58 / 59
BPS instantons of 4d Euclidean STU models
Stringy interpretation:
Instanton action ∼ g−2: wrapped five-brane (space-timeinstanton)!
ds2StrFrame = −dt2 + (dy1)2 + · · · + (dy5)2
+H(x)((dx1)2 + · · · + (dx4)2)
e2(Φ−Φ∞) = H(x) , dB = ⋆4dH(x)
is the 0 + 4 → 1 + 9 lift of the basic SL(2,R)/SO(1, 1) solutionwith X = S = e−2φ + eb = 1
g2 + eb.
Same solution, X = T , U geometrical modulus.Instanton action ∼ g0: must be world-sheet instanton.
Thomas Mohaupt (University of Liverpool) Special Geometry, Black Holes, . . . Workshop, September 2010 59 / 59