PHYSICAL REVIEW D VOLUME 46, NUMBER 10 15 NOVEMBER 1992

BRST cohomology and invariants of four-dimensional gravity in Ashtekar variables

Lay Nam Chang and Chopin Soo Institute for High Energy Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0435

(Received 13 April 1992)

We discuss the Becchi-Rouet-Stora-Tyutin (BRST) cohomologies of the invariants associated with the description of classical and quantum gravity in four dimensions, using the Ashtekar variables. These in- variants are constructed from several BRST cohomology sequences. They provide a systematic and clear characterization of nonlocal observables in general relativity with unbroken diffeomorphism invari- ance, and could yield further differential invariants for four-manifolds. The description includes fluctua- tions of the vierbein fields, but there exists a nontrivial phase which can be expressed in terms of Witten's topological quantum field theory. In this phase, the descent sequences are degenerate, and the corre- sponding classical solutions can be identified with the conformally self-dual sector of Einstein manifolds. The full theory includes fluctuations which bring the system out of this sector while preserving diffeomorphism invariance.

PACS number(s): 04.20.Cv, 04.60. +n

A great deal has been learned recently about two- manifolds, and it is now possible to give a complete description in terms of the gravitational fields in two di- mensions (2D) [I]. Similar results have also been ob- tained in three dimensions [2]. However, it is not entirely clear how such results can be extended to four-manifolds. There are several reasons for suspecting that this general- ization will not be straightforward. First, unlike in two and three dimensions, in four dimensions pure gravity can exist in a phase wherein there are two propagating degrees of freedom. Second, the work of Donaldson and Kronheimer [3] shows that four-manifolds can have far richer differential structures than in other dimensions, so their complete characterization can prove to be a daunt- ing task indeed.

Witten has suggested [4,5] that 4D gravity possesses a phase describable by a topological quantum field theory (TQFT) [6] in that the observables consist entirely of glo- bal invariants. Some of these invariants have been identified. In particular, it can be shown that by consid- ering the moduli space of (anti-)instantons, the Donald- son maps [7] can be identified as Becchi-Rouet-Stora- Tyutin (BRST) invariants of the corresponding TQFT [4].

What happens when we have four-manifolds which can support 4D gravity with propagating degrees of freedom? To extend Witten's analysis to such instances, it will be necessary to describe these degrees of freedom using vari- ables that are related naturally to those employed in TQFT. In particular, we must require that they are suit- able for implementing both diffeomorphism and gauge in- variance.

In this paper, we show that in many respects the vari- ables introduced by Ashtekar satisfy these requirements [8]. In place of the metric of general relativity, the classi- cal Ashtekar variables corresponding to specific Einstein manifolds consist of densitized triad fields, and their con- jugate momenta, which turn out to be simple SO(3) gauge potentials satisfying (anti-)self-duality conditions. We

shall analyze the classical and BRST symmetry of 4D gravity in terms of these variables, determine their coho- mology descent equations, and identify the invariants which can be constructed out of them. These quantities can then be used to describe the diffeomorphism and gauge-invariant observables of the theory, and to furnish characterizations of the differential structures of such four-manifolds. While such issues have been discussed earlier using conventional variables 191, we shall see that the Ashtekar variables provide for finer characterizations of the properties of four-manifolds than have been possi- ble in the past.

We shall work with Riemannian manifolds [lo] and start with the action proposed by Samuel [ l l ] :

where the anti-self dual two-form

and e,, A =0, . . . , 3 denote the vierbein one-forms in four dimensions. The Latin indices run from 1 to 3 and label internal SO(3) indices. Fa is the curvature two-form of the SO(3) Ashtekar connection A , , while h is the cosmological constant. In the Ashtekar formalism, the metric is considered to be a derived quantity, expressed in terms of B through

In applying the canonical formalism to Eq. (I) , it is convenient, albeit not essential [8], to work in the spatial gauge in which the vierbein can be written as

where the Greek index p runs from 0 to 3. The form as-

4257 @ 1992 The American Physical Society

4258 LAY NAM CHANG AND CHOPIN SO0 46

sumed in (4) is compatible with the Arnowitt-Deser- with the spatial metric g i j = e ~ e a j . Thus we see that Misner (ADMI [12] decomposition of the metric choice (4) in no way compromises the values of the lapse

ds2=eApeA,,dxpdxv and shift functions N a n d N' which have geometrical in- terpretations in hypersurface deformations. With this

= ~ ~ ( d x ~ ~ ~ + ~ ~ ~ ( d x ' + ~ ' d x ~ ) ( d x ~ + ~ ~ d x ~ ) , (5) decomposition, it is straightforward to rewrite Eq. (1) as

1 .A =--d4x ( 2 d i a k l a + 2 ~ , , ~ ~ , d ' ~ + 2 ~ j d ' " ~ , , ~ j 16rG

-- I I d4x { N(C,~,C'"~'~F; + + ~ ~ E ~ ~ ~ ~ ~ ~ ~ B ~ ~ c J ~ ~ ~ ~ ) / +boundary terms , 1 6 ~ G

with d and &' defined as

The tildes above and below the variables indicate that they are tensor densities of weights 1 and - 1, respective- ly. Thus 2CLa is readily identified as the conjugate vari- able to A,. (We shall suppress the factor 16 rG for con- venience.) The variables A,,, N', and & are clearly Lagrange multipliers for the Ashtekar constraints. These constraints can be identified as Gauss' law generating SO(3) gauge invariance

and the "supermomentum" and "super-Hamiltonian" constraints

Ashtekar showed that these constraints, despite their re- markable simplicity, are equivalent in content to the con- straints and constraint algebra of 4D general relativity. The equations of motion that are obtained can be written succinctly as [13,14]

with

Sab=Sba and T r S = - A . ( 1 0 ~ )

The last two conditions solve the "supermomentum" and "super-Hamiltonian" constraints, and can be taken to be the general solution for nondegenerate metrics. For metrics of Euclidean signature, S is a real symmetric ma- trix with three eigenvectors. The classical solutions can then be classified according to the number of distinct ei- genvalues of S , and are called types I, D, or 0, depending on there being three, two, or one distinct eigenvalues, re- spectively. The set of equations (10) can be shown to be equivalent to the equations for Einstein manifolds in four dimensions, i.e., Rpv = kgpv [15]. One can, in principle, eliminate S from the theory by using the relation implied in ( lob) and obtain

with

and

where * is the Hodge dual operator. S plays an impor- tant role in characterizing the possible phases of the theory. However, it is prudent not to eliminate Z in favor of F by inverting S, because there are various in- teresting cases for which S is noninvertible, and yet the vierbein and Z remain regular. Explicit examples, espe- cially those involving the F = O sector, as well as cases with Albelian anti-instantons, are described in [13]. In what follows, we shall construct invariants that are dependent on F and on Z, thereby demonstrating that they are both essential in the description of 4D gravity.

We begin by analyzing the symmetries of the action and the associated BRST invariance. It is easy to see that the action is invariant under SO(3) gauge transformations as well as four-dimensional diffeomorphisms. Working with .A, which is explicitly gauge invariant, we can con- sider a diffeomorphism d, of M into itself, generated by the vector field 8. On the one-form variables A, and e A , the induced variations are Lie derivatives

and

with i p denoting interior multiplication or contraction with the vector field 0 , which in local coordinates can be written as BFap. If we denote the Lagrangian four-form as L, then

I M ( L -~*L)=o== JM&,L = I aMiPL = O . (16)

The action is thus invariant if M is closed; otherwise in- variance can be maintained provided B vanishes at the boundary. Since SO(3) gauge invariance is a symmetry of the theory, one can also consider diffeomorphisms for which the exterior derivative operator in the Lie deriva- tive is replaced by the covariant derivative to make it compatible with the canonical analysis.

46 - BRST COHOMOLOGY AND INVARIANTS OF FOUR- . . . 4259

In the BRST formalism, the classical gauge and diffeomorphism symmetries are mimicked by transforma- tions with the parameters replaced by ghosts 77; and 6. Thus the BRST transformations of the variables are (henceforth 6 shall mean hBRST)

For a general differential form X,

There is a sign difference in the second term because, un- like a normal vector field, 6 carries a ghost number of 1. The above BRST transformation of the vierbein implies that the anti-self-dual two form 2, transforms according to

The standard procedure of splitting the ghost 77' into q l = q - i 5 A allows us to write the BRST transformations in a gauge-covariant manner,

The BRST transformations for the ghosts are

and

It can be verified that the BRST transformations above are nilpotent, i.e., h2=0. The variables (A, 7 , 2, g), which are (1,0,2) forms and a vector field, respectively, are assigned ghost numbers (0,1,0,1) and carry a grading equal to the form degree plus the ghost number. By this we mean that if x,,, are p,,,-forms with ghost numbers g1,2, then

The effective quantum action will consist of the classi- cal action and a piece from a gauge-fixing Lagrangian of the form L g f = 6 x which will involve antighosts and auxi- liary fields. Since the BRST transformation is nilpotent, the effective action will be BRST invariant provided the boundary term vanishes.

The transformation rules for A and 7 may also be ob- tained by considering the multiplet (A,7) as a connection x of the universal bundle over M X e / S . Here e is the space of connections A, while 9 is the group of gauge transformations. 2 carries a grading of 1, and can be decomposed into its (1,O) and (0,l) components as

Its curvature is given by

The transformation rules for A and q are equivalent to the statement

Fa =eexp(i5)~, = [ I + i c + ( 1 /2! ) ic iC]~, , (26)

which reduces to the "soul-flatness" condition F=F in ordinary gauge theories when diffeomorphism invariance generated by 6 is absent. Here the concept of "horizon- tality" in curved space is the statement that F can be ex- panded in terms of F and its contractions with the ghost 5. The BRST transformation of F as a consequence of (20a) is

I t can be verified that the curvature F satisfies the Bian- chi identity

DF, = ( d +6)Fa ++E,b'Jb A F ~ = O . (28)

As a consequence T~(F") obeys

Since the gauge group is S0(3), it suffices to consider n =2. One can expand Fa AFa in terms of the ghost number, i.e., writing

4

Fa A A, = 2 w ; = ~ - ~ . (30) g =O

The resulting BRST cohomology descent equations from (29), i.e.,

are

with

Consequently, there is an off-shell descent sequence in- volving the curvature of the Ashtekar connection. In view of the fact that, on shell, the covariant curl of Z is zero and Z transforms in the same way as F, one expects that a BRST cohomology sequence analogous to the W descent exists for 2 as well. This is indeed true if one also makes use of the other equations of motion (lob) and ( 1 0 ~ ) . This suggests that, even off shell, a descent involv- ing 2 could be realized. However, for off-shell computa- tions we should keep all terms involving DH. We thus find that the BRST transformations of Z and g imply

4260 LAY NAM CHANG AND CHOPIN SO0 46

So instead of the Bianchi identity for F, we have

Thus we see that the consistency condition is the require- ment that 6% can be expanded in terms of D Z and its contractions with the ghost <. For dF, this expansion is trivial because of the Bianchi identity for F. By consider- ing D (%, A %, ) we have

and expanding 2, A z , in terms of ghost number,

4

E , A % , = v g g , g = O

and using (20b), (32) , and (331, Eq. 134) yields the identi- ties

d ( Z , A Z , ) = Z D Z , A Z , , (35aj

S ( Z , A t , )=-d [ i c ( Z , A Z , ) ] + i 5 d ( Z a A Z , ) , i35b)

(350

It is remarkable that although Z , unlike F, has a nonzero covariant curl off shell, the nonexact terms on the right- hand side (RHSj of (35b)-(350 actually vanish because they are all contractions with < of d (2 , A 8, j , which is

zero in four dimensions. Hence, we do have another BRST cohomology sequence with descent equations

d v ? = O , (36a)

6 v f - , = - d ~ j ' ? ; , for g=1,2,3 , (36b)

~ v ; = o , ( 3 6 ~ )

where

V ? = Z , A Z , , V : = i < ( Z , A Z , ) ,

The existence of the two descents above naturally leads us to ask whether a further descent can be constructed from %, A Fa by considering 8 ( z a AF,). It is straight- forward to verify that from

( d + 6 j ( % , A F ~ ~ = D E , A F ,

we arrive at

6 ( Z a A F , ) = - d [ i j ( Z a A F a ) ] + i c d ( 2 , A F , ) , (37a)

The nonexact terms on the R H S are again contractions with the ghost < of d ( 2 , A Fa j , which also vanish in four dimensions. There is thus a third BRST cohomology des- cent sequence. While the two previous descents involve either F o r Z but not both together, this third set of des- cent equations involves both I: and F, i.e., the conjugate variables d and A . The corresponding descent equations can be written as

! I . . . . 6 ~ l E 1 6 1 E ~ S ( Z a AF , )

I . . . . . = - j l l c l p ~ S l c l c d ( Z a AF , j .

BRST COHOMOLOGY AND INVARIANTS OF FOUR- . . . 426 1

The elements of the BRST descents can be used to con- struct invariants by integrating over the appropriate cy- cles. Observables of the theory should be expressed in terms of these invariants. To obtain the invariants, con- sider y p E H p ( M ) , p =0, . . . ,4. Picking an element y;-~ of the W, V, or U descent, we see that

6 J y y;-p=- J y dy;:f P P

y5-P =-.I-,, P - 1

Moreover, the BRST cohomology class of Y;-~( y p ) S Y;-P depends only on the homology class

Y P of yp because the descent equations guarantee that for yP, g p = ~ p + a ~ p + l ~ ~ p ( ~ ) , p =o, . . . ,3 ,

y4-P y;-p(cP ) = s y p + ~ ~ p + , P

= Y;-P( y p ) + j- dY;-P O P + l

= ~ p 4 - ~ ( ~ , ) - 8 ~ ~ p 3 + P . Op+1

Finally, consider the transformation

Provided the boundary term vanishes (which is automatic if M has no boundary), a further global invariant J M s a b ~ , A Fb will be present. It is possible to construct a descent for which sabFa A Fb is the zero ghost number four-form. This can be achieved by using the identity

b ( S a b F a A F ~ ) = ( B s a b ) ~ F a A F ~

= [ ( D S ) ~ ~ + ~ ~ ( D S ) " ~ ] A F ~ (40)

with Sab = -$*(Fa A Z b + Za A Fb) . The descent equa- tions take the form

Unlike the previous descents, this set of equations explic- itly involves the duality operator, ., and hence the in- verse of the metric gp,, defined through Eq. (3) 1161. Since degenerate metrics could not be ruled out in quan- tum fluctuations, and the descent involves complicated products of noncommuting operators, it remains to be seen whether the descent (41) survives regularization. However, even in the classical context, it would be in- teresting to investigate the interplay between degenerate metrics and the invariants defined by (41).

In TQFT, the form with a zero ghost number in the descent equations, called the top-form, can be regarded as the action before gauge fixing. For our purposes, the ac- tion ( 1 ) is also a combination of top-forms from two of the three sequences, and correspond to u:, and the cosmological (volume) four-form v:. The last top-form W : can be added on to the action density, and will give a term analogous to the 6 term in QCD.

A Dirac quantization of the theory requiring that physical quantum states be annihilated by all the con- straints has to face the problem of having a consistent or- dering of the quantum constraints. We choose one specific scheme of ordering to make the discussion con- crete. While such procedure is not the final word on such a complicated theory, it is nevertheless instructive, and one expects the broad features of the theory to be present for all schemes. An ordering with a formal closure of the quantum constraint algebra exists and it can be shown [17,18] that the resulting physical states selected by

has a sector described by

where

1 3 M~ [ 3 C ] b e x p [ i ~ ] &=-/ d 3 x 6 A i a exp - I -

h 2i8Aia

and C denotes the Chern-Simons functional

Indeed in the reduced phase-space analysis, the restricted sector of the theory corresponding to (43) is described by the Ashtekar-Renteln ansatz 1191

~ i a - A-ia 3 O . (46)

This ansatz is the set of initial data for conformally self- dual Einstein manifolds and they are described by

h F =--Z a 3 " '

(47)

Sob = -( h / 3 M a , is precisely type 0 for this sector of the full theory [20]. A naive counting of the number of con-

4262 LAY NAM CHANG AND CHOPIN S O 0 46

straints tells us that , if we a re restricted t o this T Q F T phase, there a r e n o local degrees of freedom. In this phase, t h e reduced action is

T h e resulting constraints a r e Eq. (46) and Gauss' law. This set of constraints, however, is reducible since Eq . (46) generates deformations of the gauge potential, and, if i t holds, so will Gauss' law. T h e BRST analysis of this T Q F T action has been performed (see [6] for details) a n d the BRST invariants f rom the descent equations were successfully identified with the Donaldson maps by Wit- ten [4]. If we let 6TqFT be the B R S T transformation, then the action is invariant under

Comparing with Eqs. (20) a n d (211, this restricted (type 0) phase of t h e full theory can be identified with the T Q F T of Eq. (48) with Donaldson-Witten invariants, if we make the substitution

In this sector, t h e role of C has been eliminated in terms of F. F r o m the vantage point of the BRST invariants a n d t h e observables we have constructed so far, we see tha t in

this phase the elements of the descents a re degenerate be- cause of (47).

Equation (42) can have solutions which a re not annihi- lated by the topological charge Q of Eqs. (44) and (45). When this happens, we a re n o longer in the T Q F T phase, and classically a t least, we will have solutions which a re not of type 0 . Outside of this phase, t h e variables 2 and F are n o longer proportional t o each other, and so there can be full-fledged canonical degrees of freedom in the theory. A naive counting of the number of constraints minus t h e number of conjugate pairs shows tha t we now have up t o two unconstrained degrees of freedom. These represent local fluctuations in the vierbeins a n d the Ashtekar connections [13]. However, it is important t o remember that we still maintain complete diffeomorphism invariance, so it is not clear these degrees of freedom can be directly identified with t h e graviton [4,5]. In any case, the BRST descent sequences are now independent of one another , and the observables a re characterized by many more invariants. Despite t h e lo- cal nature of the fluctuations in C and F , the observables described by the invariants a re nonlocal, since exact diffeomorphism symmetry remains unbroken. T h e physi- cal implications of these observables when this invariance is broken, as well as the important question of what hap- pens in the presence of matter couplings, will be dis- cussed elsewhere.

W e thank Marek Grabowski and Waichi Ogura for helpful discussions. T h e research for this work was sup- ported by the DOE under G r a n t N o DE-FGO5- 92ER40709.

[ I] A. M. Polyakov, Mod. Phys. Lett. A 11, 801 (1987). [2] E. Witten, Nucl. Phys. B311, 46 (1988). [3] For a review, see S. K. Donaldson and P. B. Kronheimer,

The Geometry of Four-Manifolds (Clarendon, Oxford, 1990).

[4] E. Witten, Commun. Math. Phys. 117, 353 (1988). [5] E. Witten, Commun. Math. Phys. 118,411 (1988). [6] This subject is reviewed in D. Birmingham, M. Blau, M.

Rakowski, and G. Thompson, Phys. Rep. 209, 129 (1991). [7] S. K . Donaldson, Topology 29, 257 (1990). [8] A. Ashtekar, Phys. Rev. Lett. 57, 2244 (1986); Phys. Rev.

D 36, 1587 (1986); New Perspectives in Canonical Gravity (Bibliopolis, Naples, 1988); Lectures on Non-Perturbatiue Canonical Gravity (World Scientific, Singapore, 1991).

[9] L. Baulieu and M. Bellon, Phys. Lett. 161B, 96 (19851; Nucl. Phys. B266, 75 (1986).

[lo] We will be concentrating on Riemannian manifolds in this paper. In the case of pseudo Riemannian manifolds, we need to make sure that the Ashtekar potentials satisfy a set of reality conditions [8].

[11] J. Samuel, PramZna J. Phys. 28, L429 (1987); Class. Quan- tum Grav. 5, L123 (1988).

[12] R. Arnowitt, S. Deser, and C. W. Misner, Phys. Rev. 120, 313 (1960); Ann. Phys. (N.Y.) 33, 33 (1965).

[13] L. N. Chang and C. P. Soo, Report No. VPI-IHEP 92-5 (unpublished).

[14] Such equations have been considered elsewhere; see for ex-

ample, R. Capovilla, J . Dell, and T. Jacobson, Class. Quantum Grav. 7, L1 (1990); R. Capovilla, J. Dell, T. Jacobson, and L. Mason, ibid. 8, 41 (1991). In these pa- pers, the fundamental variables are the self-dual two- forms 2, which are to be contrasted with the vierbein fields we use in this paper. The papers also do not address the issue of the BRST invariance we shall be describing later on.

[15] We show elsewhere [13] that all classical Einstein mani- folds can be described as anti-instantons in the Ashtekar variables. Indeed, we can establish a relationship between the reduced phase space of these variables and the moduli space of reducible and irreducible anti-instantons, and thence construct new topologically nontrivial solutions.

[16] Further invariants involving torsion elements can be simi- larly constructed.

[17] H. Kodama, Phys. Rev. D 42,2548 (19901. [18] L. N. Chang and C. P. Soo, in Ashtekar's Variables and the

Topological Phase of Quantum Gravity, Proceedings of the Twentieth Conference on Differential Geometric Methods in Physics, edited by S. Catto and A. Rocha (World Scientific, Singapore, 199 11, Report No. VPI-IHEP 9 1-2.

[19] A. Ashtekar and P. Renteln, Lecture Notes on New Vari- ables (Astrophysics Group, Mathematics Department, University of Poona, 1987).

[20] For earlier work on such type 0 solutions using Ashtekar variables, see C. G. Torre, Phys. Lett. B 252, 242 (1990).