Boundary dynamics of higher dimensional AdS spacetime

6 April 2000

Physics Letters B 478 Ž2000. 327–332

Boundary dynamics of higher dimensional AdS spacetime J. Gegenberg

a,1

, G. Kunstatter

b,2

a

b

Dept. of Mathematics and Statistics, UniÕersity of New Brunswick, Fredericton, NB E3B 5A3, Canada Dept. of Physics and Winnipeg Institute of Theoretical Physics, UniÕersity of Winnipeg, Winnipeg, Man. R3B 2E9, Canada Received 1 February 2000; accepted 24 February 2000 Editor: H. Georgi

Abstract We examine the dynamics induced on the four dimensional boundary of a five dimensional anti-deSitter spacetime by the five dimensional Chern-Simons theory with gauge group the direct product of SO Ž4,2. with UŽ1.. We show that, given boundary conditions compatible with the geometry of 5d AdS spacetime in the asymptotic region, the induced surface theory is the WZW4 model. q 2000 Elsevier Science B.V. All rights reserved.

It may be possible to understand the physics of fields which propagate through space in terms of physics on the boundary of space w1x. For example, even though 3d Einstein gravity has no local dynamics, black holes possess thermodynamic properties. The resolution of this seeming paradox is that the dynamical processes responsible reside on the boundary of 3d spacetime. This idea was broached by Carlip w2x with the boundary dynamics occurring on the event horizon. Later Strominger w3x, motivated by earlier work of Brown and Henneaux w4x and by string theory, showed that states on the boundary at infinity could also account for 3d black hole thermodynamics. This has been generalized in a variety of ways to different species of black holes in higher dimensions w5–8x.

1 2

E-mail: [email protected] E-mail: [email protected]

What is a common element of all implementations of this program to date is the construction of integrable field theories on some boundary of spacetime. For instance, in 3d, the boundary theory is a version of the 2d WZW model w9x. Characteristically for integrable field theories, the WZW theories contain an infinite number of conserved charges, or equivalently, an infinite dimensional Kac-Moody algebra obeyed by the currents. The association of this algebra with conformal symmetry leads to the characterization of this phenomenon as an example of the AdSrCFT correspondence w10x. The starting point for the construction of the boundary theories is usually Einstein gravity or some generalization thereof. However, the initial work in 3d used the close correspondence of Einstein gravity in that dimensionality with Chern-Simons theory. The reason for this is the fact that in the latter it is comparatively straightforward to disentangle gauge from dynamical degrees of freedom. An important question to ask at this point is if this situation is a consequence of Einstein gravity in

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 2 7 1 - 9

J. Gegenberg, G. Kunstatterr Physics Letters B 478 (2000) 327–332

328

2 q 1 dimensions, or is it more generic? Banados et al. w11x have analyzed higher dimensional ChernSimons theories on manifolds with boundary and found evidence that the situation might be generic. In particular, they showed that the algebras Žin diverse spacetime dimensions. of charges on the boundary are generalizations of the Kac-Moody algebra that they call WZW4 algebras. The latter are known in the mathematical literature as toroidal Lie algebras w12x. They generalize the Kac-Moody algebra in the sense that while the Kac-Moody algebra is the loop algebra with a central charge, the n-toroidal Lie algebras replace the loop with an n-torus. The WZW4 theory Žsometimes known as the Nair-Schiff or Kahler Chern-Simons theory. is a four ¨ dimensional non-linear field theory which is a natural generalization of the 2d WZW model. Some of its properties have been developed in the papers of Nair, Nair and Schiff, and Losev et al. w13x. For example it is classically and perhaps quantum mechanically integrable and one-loop renormalizable. It was shown by Inami et al. w14x that the associated current algebra is the 2-toroidal Lie algebra. The same authors have also formulated a 2 n-dimensional generalization of the WZW4 theory w15x. Banados et al. w11x do not explicitly derive the WZW4 boundary theory from 5d Chern-Simons. They construct the algebra of gauge charges on the boundary under the assumption that the gauge group is the direct product of some non-Abelian group G and the Abelian group UŽ1.. The purpose of this Letter is to fill in this last important step: we will show that with boundary conditions compatible with the geometry of 5d AdS spacetime in the asymptotic region, the induced surface theory is precisely the WZW4 model. We consider the Chern-Simons theory on a five dimensional manifold with boundary. It turns out that the best starting point is the six dimensional functional cubic in the curvature w11x: S6 w Aˆx s

HM ² F Ž Aˆ . n F Ž Aˆ . n F Ž Aˆ . :. 6

Ž 1.

In the above the integral is over a 6-manifold M 6 , the inner product ² : is an invariant trilinear form in the algebra of some group G and which satisfies the ‘cyclic property’ ² ABC : s ² CAB :. Finally, Aˆ is the connection 1-form over M 6 with respect to the ˆ Aˆx group G, with F Ž Aˆ . [ dAˆq Aˆn Aˆs dAˆq 12 w A,

the corresponding curvature. The integral S6 is a topological invariant associated with the principal bundle with group G over M 6 . In the following we assume that the wedge product n of forms is understood and will not be explicitly written. One can show, see e.g. w11x, that the 6-form in the integrand above is locally exact: it is the exterior derivative of a 5-form SCS called the Chern-Simons form. The 2 n q 1 dimensional generalization of this form was added as a topological mass term to the usual YangMills lagrangian and analyzed by Mickelsson w16x. In this Letter, we restrict to the case where the gauge group G ( SO Ž4,2. = UŽ1.. In this we are following the lead of Banados et. al. w11x 3. The reason for this is that supergravity in a five dimensional spacetime with a negative cosmological constant leads us to the super anti-deSitter group, whose bosonic sector is precisely SO Ž4,2. = UŽ1.. Moreover we consider the invariant trilinear form to be given by:

² Ja Jb Jc :s Tr Ž Ja Jb Jc . ; ² Ja Jb J1 :s² Jb Ja J1 :s Tr Ž Ja Jb . ; ² Ja J1 J1 :s² J1 J1 J1 :s 0.

Ž 2.

In the above, the Ja are the generators of SO Ž4,2., while J1 is the generator of UŽ1.. The connection 1-form decomposes as Aˆs A q a s A a Ja q a1 J1. Consider the following functional of A,a on a 5-manifold M 5 : SCS w A,a x s SCS w A x q 3

HM Tr Ž AdA q 5

2 3

A3 . v ,

Ž 3. where SCS w A x [

HM Tr  AŽ dA . 5

2

q 35 A5 q 32 A3 dA 4 .

Ž 4.

with v s da the UŽ1. field strength. It is straightforward to show that the exterior derivative of the integrand in Eq. Ž3. is ²Ž F Ž Aˆ.. 3 :, and hence that SCS w A,ax is the Chern-Simons form for SO Ž4,2. = UŽ1. for the postulated trilinear form.

3 Interestingly, the group G( SUŽ2.=UŽ1. was considered in the mid-1980’s by Mickelsson w17x in the context of topologically massive Yang-Mills theory.

J. Gegenberg, G. Kunstatterr Physics Letters B 478 (2000) 327–332

Variation of the action Eq. Ž3. yields:

d SCS w A,a x s 3

2

HM Tr Ž F d A . q 6HTr Ž Fd A . v 5

q 3 Tr Ž F 2 . d a q d SB ,

H

Ž 5.

where F s dA q A2 and the last term is a boundary variation:

d SB s y

EE M

5

Tr Ž AdA q dAA q 32 A3 q 3 A v . d A

q3 Ž AdA q 23 A3 . d a .

Ž 6.

329

the spatial components of the non-Abelian field strength are zero and the spatial part of the Abelian one form is non-degenerate Ži.e. invertible.. The equations of motion then guarantee that the spacetime components of the field strength obey the generic conditions as well. The symplectic form is found by the varying the kinetic term in the action under arbitrary variations of the gauge potential in the generic sector of the constraint surface. Since the spatial part A of the connection is flat, we can write A s hy1 dh where h is an element of SO Ž4,2.. In terms of h we find:

If M 5 has a non-trivial boundary M 4 [ E M 5, then we refer to fields defined on M 5 as bulk modes, and those on M 4 as boundary modes. The field equations for the bulk modes of the theory are:

d SCS s dt v Tr D Ž hy1 h˙ . D Ž hy1d h .

Tr  Ž F n F q 2 F n v . Ja 4 s 0,

Ž 7.

Tr Ž F n F . s g a b F a n F b s 0.

Ž 8.

where D˜ denotes the covariant derivative along the boundary ES of the spatial slice S . The last line in the above was obtained by integrating by parts and using the fact that DD A F s 0. Eq. Ž9. defines the symplectic form and shows that the physical modes ‘‘live’’ on the boundary of S , in complete analogy with 3-D Chern-Simons gravity w18x. In terms of the Lie algebra valued charges:

Unlike the three dimensional case, the above action describes a theory that does have dynamical degrees of freedom w11x. However there is a ‘generic’ sector of the phase space wherein there are no dynamical degrees of freedom in the bulk of the manifold, although there are propagating modes on the boundary w11x. This sector is described by solutions of the form F s 0 and detŽ vmn . / 0 but otherwise arbitrary. Such solutions are generic in the sense of w11x. Note that the presence of the boundary variation d SB makes the variational principle of the action SCS w A,ax ill defined as given. It is therefore necessary to add a compensating surface term to the action whose variation precisely cancels d SB . This compensating surface term is determined by the choice of boundary conditions and plays a crucial role in determining the boundary dynamics. Before deriving the boundary action, we outline the proof that the dynamics of the physical modes in the generic sector of the theory does indeed take place on M 4 . What is required is to derive the symplectic form on the Hamiltonian constraint surface and show that it is a total divergence. Following the procedure in w11,18x, one first does a standard Hamiltonian decomposition of the fields into spatial and time components. In the Hamiltonian context, the generic sector consists of gauge fields such that

H HS

H EESv Tr

s y dt

D˜ Ž hy1 h˙ . Ž hy1d h . ,

Q [ J0 s 12 e i jkv i j hy1E k h,

Ž 9.

Ž 10 .

we get

˙˜ H HESTr Ž QL

d SCS s dt

y1

d Q. ,

Ž 11 .

where L˜ s 12 e i jkv i j D˜ k , is assumed to be invertible Ži.e. we exclude zero modes.. Using Eq. Ž11. the Poisson bracket for these charges is:

 Q a Ž x . ,Q b Ž y . 4 s 12 v i j Ž x . e i jk D˜ ka b Ž x . d 3 Ž x y y . s 12 Ž f a c d g b c Q d q e i jkv i j E k g a b . d 3 Ž x y y . ,

Ž 12 . where the f a b c are of the SO Ž4,2. structure constants. The algebra Eq. Ž12. is the 2-toroidal Lie algebra, that is, the current algebra for the WZW4 theory w14x. It is a non-trivial central extension of the Lie algebra of SOŽ4,2.. It generalizes the affine Kac-Moody algebra in that the latter is the centrally

J. Gegenberg, G. Kunstatterr Physics Letters B 478 (2000) 327–332

330

extended loop algebra with one central charge; while the 2-toroidal algebra is the centrally extended algebra of maps from a 2-torus to the Žfinite dimensional. Lie algebra in question. The current algebra Eq. Ž12. agrees with the one found by Mickelsson in the mid-1980’s in his analysis of topologically massive Yang-Mills theories in 2 n q 1 dimensions w16,17x. We will now determine the surface term whose variation cancels the unwanted boundary term d SB in Eq. Ž6. for boundary conditions consistent with an asymptotic region of 5d AdS spacetime. In the case of 3d Chern-Simons theory, this procedure results in the WZW model on the 2d boundary w9x. We start by constructing the AdS5 geometry following Witten w19x. Consider the quadratic form: f Ž u,Õ, x i . [ uÕ y hi j x i x j ,

Ž 13 .

where u,Õ, x i with i, j, . . . s 1,2,3,4 are coordinates in R 6 with the ultrarelativistic metric ds62 s ydudÕq hi j dx i dx j ,

Ž 14 .

with hi j the Minkowski metric with signature Žy,q,q,q .. Now consider the quadric given by the solution of f Ž u,Õ, x i . s 1, with points with an overall sign change identified. In a coordinate patch with Õ ) 0, we can solve for u s Ž1 q hi j x i x j .rÕ and the metric Eq. Ž14. reduces, after completing the square, to the AdS5 metric ds52 s

1 Õ

ž

dÕ 2 q hi j dx i y 2

xi Õ

dÕ

/ž

dx j y

xj Õ

/

dÕ .

Ž 15 . From the above, we see that in a region of large fixed Õ, the leading order terms in the frame and spin-connection fields are simply e Õ s 0, e i s dx i , w i j s 0 and w i Õ s e i. The corresponding SOŽ4,2. gauge potential defined in terms of the geometrical fields is therefore flat: a

As

w b ye b

a

0

e s dx i 0 0

ydx j 0 ydx j

d SB s y3

HM Tr Ž A d A v q Ž AdA q 4

A3 . d a . .

We also assume that v is a fixed two-form on the boundary so that the variation d a of the UŽ1. gauge potential must be of the form d a s d l, for some scalar l. Integrating by parts and using the fact that F Ž A. s 0 we find that the second term in Eq. Ž17. vanishes. Now we have assumed that M 5 admits a closed invertible 2-form v and hence the restriction of v to the boundary M 4 is closed. From the flat SO Ž4,2. connection, one may construct a flat frame-field compatible with a ŽLorentzian. spin-connection, which upon Wick rotation, provides a Euclidean structure. The flat Euclidean geometry enables the construction of a complex structure on M 4 w20x. Hence, using this complex structure, we can write:

d SB s y3

HM Tr Ž A d A y A d A . i

4

j

j

i

=v i j dz i n d z j n d z i n dz j .

Ž 18 .

In the above the z i ,i, j, . . . s 1,2 are complex coordinates on M 4 . The above expression cannot be integrated for arbitrary connection A on the boundary. One way to proceed is to assume further boundary conditions on the connection. If we assume, analogously to Carlip for the 2 q 1 dimensional case w21x, that A i is fixed on the boundary, then the second term in Eq. Ž18. vanishes identically and the first term can be trivially integrated. With these boundary conditions, the complete action, including required boundary term is:

HM Tr Ž AdA q 5

Ž 16 .

The first condition we impose on the boundary is that the connection A is flat. In this case we can

2 3

Ž 17 .

S s SCS w A x q 3

0 dx i . 0

simplify the boundary variation by replacing AdA by yA3. Moreover, we have verified that Tr Ž A3d A. s 0 for all Lie algebra valued A and d A. The boundary variation reduces to:

i

i

2 3

A3 . v

EM Tr Ž A dz n A d z . n v .

q3

4

i

i

Ž 19 .

Recalling that on shell F w A x s 0, we can parametrize the gauge potential as A s hy1 dh. What we are left

J. Gegenberg, G. Kunstatterr Physics Letters B 478 (2000) 327–332

with is the gauge dependent part of the action, namely: y1

Sw hx sy

HM Tr Ž h 5

q3

HM Tr 4

3

dh . v

Ž hy1E h . Ž hy1 E h . v ,

Ž 20 .

This is precisely the WZW4 action w22,14x 4 . To summarize, we have examined the 5d ChernSimons theory with gauge group SO Ž4,2. = UŽ1. on a manifold with boundary. We have looked at the sector of phase space wherein the SO Ž4,2. connection is flat and the UŽ1. connection is fixed but not flat. These conditions are obeyed by a connection constructed from the frame-fields and spin-connection of anti-deSitter spacetime. We have shown that the procedure first applied by Carlip to 2 q 1 Chern-Simons theory can also be applied in the present case. The result is that non-trivial field theory induced on the boundary, namely the WZW4 theory, whose action is given in Eq. Ž20.. This theory has the 2-toroidal Lie algebra ŽEq. Ž12.. as the algebra of observables. It seems likely that our analysis can be extended in a straightforward manner to the Chern-Simons action generalized to 2 n q 1 dimensions. The algebra of boundary charges in the generalized case Žthe ‘‘WZW 2 n algebra’’. has already been constructed by Banados et al. w11x. ŽSee also Mickelsson w16,17x.. We are currently in the process of checking whether the boundary theory in this case is the corresponding ‘‘WZW 2 n model’’ of w15x. Finally, we would like to mention that there exists a somewhat different and more natural way to arrive at the complete action, with boundary terms, in Eq. Ž19. 5. In particular, it is not necessary to impose the strong boundary condition that A i be fixed on the boundary. Instead, one notes that the variation of Eq. Ž19. yields the equations of motion for the bulk with no boundary terms if and only if the connection on

4 The term Ž1r10.HM 5 Tr Ž hy1 dh. 5 from SCS in Eq. Ž4. is the winding number of maps from M 5 to 5-spheres in the group manifold SOŽ4,2.. This can be easily shown to be identically zero for the group SO Ž4,2. in five dimensions. 5 We are grateful to S. Carlip for pointing this out to us.

331

the boundary is flat and expressible locally as A s hy1 dh where the h obeys the WZW 4 field equations. In this view, one can think of the full action as containing both the 5-D Chern Simons theory for the bulk modes, and the WZW 4 theory for gauge modes on the boundary. It is important to note in this context that the gauge modes on the boundary are completely decoupled from the bulk modes. This alternative derivation of Eq. Ž19. and its consequences will be discussed in a future publication.

Acknowledgements The authors would like to think Y. Billig, M. Paranjape and M. Visser for helpful comments. We also acknowledge the partial support of the Natural Sciences and Engineering Research Council of Canada.

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