- Email: [email protected]
A note on Einstein gravity on AdS3 and boundary conformal field theory1
5 November 1998
Physics Letters B 439 Ž1998. 262–266
A note on Einstein gravity on AdS 3 and boundary conformal field theory J. Navarro-Salas 2 , P. Navarro
1
3
Departamento de Fısica Teorica and IFIC, Centro Mixto UniÕersidad de Valencia-CSIC. Facultad de Fısica, UniÕersidad de Valencia, ´ ´ ´ Burjassot-46100, Valencia, Spain Received 10 July 1998 Editor: L. Alvarez-Gaume´
Abstract We find a simple relation between the first subleading terms in the asymptotic expansion of the metric field in AdS 3 , obeying the Brown-Henneaux boundary conditions, and the stress tensor of the underlying Liouville theory on the boundary. We can also provide an more explicit relation between the bulk metric and the boundary conformal field theory when it is described in terms of a free field with a background charge. q 1998 Elsevier Science B.V. All rights reserved. PACS: 04.60.Kz; 04.60.Ds Keywords: AdS gravity; Conformal symmetry
A crucial property of 2 q 1 dimensional gravity with a negative cosmological constant L s y 1 , l2
described by the action 1 2 Ss d3 x y g R q 2 , 16p G l
H '
ž
/
Ž 1.
is that the asymptotic symmetry of the theory is the conformal algebra with a central charge given by w1x 3l cs . Ž 2. 2G Recently Strominger w2x has used this result to derive 1 Work partially supported by the Comision ´ Interministerial de Ciencia y Tecnologıa ´ and DGICYT. 2 [email protected] 3 [email protected]
the Bekenstein-Hawking area formula for the 3d black holes w3x by counting the number of states of the conformal field theory at infinity with central charge via Cardy’s formula w4x. Other approaches w5x describe excitations associated to a conformal field theory at the horizon or at any value of the black hole radius w6x. Taking into account that Ž1. can be reformulated w7,8x as a Chern-Simons gauge theory with gauge group SLŽ2, R . m SLŽ2, R ., and using WZW reduction, one can argue that the asymptotic dynamics of Ž1. is described by Liouville theory w9x. The aim of this letter is to further elucidate the relation between the gravity theory and the underlying conformal field theory. Following w1,2x we assume the following asymptotic behaviour of the metric
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 0 3 2 - 6
J. NaÕarro-Salas, P. NaÕarror Physics Letters B 439 (1998) 262–266
gqys y
r2 2
q gqy Ž xq, xy . q O
g ""s g "" Ž xq, xy . q O
g " r Ž xq, xy .
g" rs
gr r s
r
l2 q
r2
1
gr r Ž xq, xy . r4
ž / r
ž / ž / ž / r
Ž 3.
1
r5
zys 2Tyq
l2 r
E 2 Tyq O 2 y
l2 r
E 2 Tqq O 2 q
1
ž / ž / r4 1
r4
z r s yr Ž Eq Tqq Ey Ty . q O
,
Ey gqqs 0 ,
z rs
a" r
a
1
r
r
ž / ž /
qO
4
r5
1
qO
r2
6 l
2
,
,
2 2
r l
g" rs0 ,
so the physical degrees of freedom are described by two chiral functions g "" Ž x ". : ds 2 s
1 4l
2
1
r3
dr 2 y r 2 dxq dxyq gqq Ž dxq . 2
1
ž / r
2
,
Ž 18 .
gqqs 2Gl Ž Ml q J . ,
/ ž /
gr r q O
r2
Ž 9.
Ž 11 .
gqyy
l2
q gyy Ž dxy . q O
,
Ž 12 .
Ž 19 . Ž 20 .
gyys 2Gl Ž Ml y J . .
It is interesting to note that, when either gqqs 0 or gyys 0, the omitted terms in the expansion Ž18. vanish Žin the appropriate gauge. and the corresponding BTZ solutions describe extremal black hole geometries. Hence ds 2 s
ž
Ž 16 . Ž 17 .
gr r s 0 ,
Ž 10 .
,
8 sy
In addition one can make, using the gauge transformations Ž10. and Ž11., the following consistent gauge choice
For instance, the standard BTZ black hole solutions can be brought, via gauge transformations, to the form Ž18. with
where a " and a r are arbitrary functions of xq and xy, it is not difficult to see that the variables g "", gqyy 41l 2 gr r are the only gauge invariant quantities. Moreover, the equations of motion imply that 0sRq
Ž 14 . Ž 15 .
Ž 8.
where the functions T "Ž x ". verify the conditions E " T .s 0. Those diffeomorphisms with T "s 0 should be considered as ‘‘gauge transformations’’. Therefore, if one consider the diffeomorphisms
z "s
Ž 13 .
Ž 5. Ž 6.
,
r
gr r s 0 .
The remaining equations of motion Rmn y 12 gmn R s y l12 gmn lead to the equations
Ž 7.
1
4l2
Eq gyys 0 ,
,
ž /
1
Ž 4.
where x "' tl " u , and u and r are the angular and radial coordinates. We have introduced explicitly the first subleading terms in the asymptotic expansion of the metric field and we want to investigate how they are related to the conformal field theory on the boundary. The infinitesimal diffeomorphisms z a Ž r,t, u . preserving Ž3. – Ž6. are of the form w1,2x
zqs 2Tqq
and this requires that
gqyy
,
r4
qO
,
,
1
qO
3
1
263
l2 r2
dr 2 y r 2 dxq dxyq gqq Ž dxq .
2
Ž 21 .
is an exact solution. We want now to identify the fields g "" in terms of appropriate conformal fields on the boundary. To this end, let us determine the conformal transformation properties of these fields. The action of the diffeomorphisms Ž7-9. on the metric Ž18. induces the following transformation law
d T "g ""s 2 Ž T "E " g ""q 2g "" E " T " . 3 y l 2E " T" .
Ž 22 .
J. NaÕarro-Salas, P. NaÕarror Physics Letters B 439 (1998) 262–266
264
It is then clear that the variables g "" are proportional, up to a constant yc 24 , to the stress tensor components Q "" of the underlying conformal field 3l theory with central charge c s 2G
Q ""s
1 4 lG
g ""q
l 16G
.
Ž 23 .
The above relation can also be confirmed by working out the conserved charges J w j x given in Ref. w1x
J w j x A lim
Hd f r™` 1
q l
ž
jH r
½
l r
j Hq
q j , rH
/Ž
r3 l3
ž
j H gr r y
gff y r
2
l2 r2
. q2j
f
/
pfr
5
,
in terms of the metric Ž18.. One obtains J w j x A d f Tq Ž 4gqqq l 2 . q Ty Ž 4gyyq l 2 . 4 ,
H
Ž 25 . which corresponds to the conserved charges associated to the stress tensor Ž23. of a conformal field 3l theory on a cylinder with central charge c s 2G . c Note that the shift term 24 in Ž23. arises from changing variables from the sphere to the cylinder z s e t r lqi u . The Fourier components L nŽ L n . of Qqq ŽQyy . obey the Virasoro algebra c 12
Ž n3 y n . dn ,y m , Ž 26 .
i L n , L m 4 s Ž n y m . L nqm q
c 12
Ž n3 y n . dn ,y m , Ž 27 .
Ln , Lm 4 s 0 , 3l with central charge c s 2G .
ds 2 s yr 2 dxq dxyq
l2 r2
dr 2 ,
Ž 28 .
Ž 29 .
and e 2 f is a positive function depending on the coordinates x ", induces a metric on the boundary r™` dsb s ye 2 f dxq dxy ,
Ž 24 .
i L n , L m 4 s Ž n y m . L nqm q
Our aim now is to provide a more explicit description of the expansion Ž3. – Ž6.. A simple and natural way to do this is to realize that the conformal symmetry also arises from a conformal analysis of 2f infinity. The conformally related metric ds 2 s er 2 ds 2 2 where ds is the elementary black hole solution with MsJs0
Ž 30 .
The 2d conformal symmetry Ž7-8. acts naturally on the metric Ž30. preserving the conformal structure. This is in fact a particular case of the more general AdS dq 1rCFTd correspondence suggested in Ref. w10x and elaborated in Refs. w11,12x. In Ref. w13x the Weyl anomaly for conformal field theories described via the supergravity action in d s 2,4,6 has been calculated regularising the gravitational action in a generally covariant way. In the case d s 2 one recovers the well-known trace anomaly ²Taa : s 24cp R. However, it was pointed out in Ref. w14x that in quantising a 2 d conformal field one can alternatively preserve the Weyl symmetry and partially break diffeomorphism invariance. Therefore, the trace anomaly disappear but one encounters that the stress tensor does not obey the covariant conservation law: c ² =mT mn : s y E RŽ yg g ab . . Ž 31 . 48p n
'
In conformal coordinates this equation implies that ²T mn : transforms according to the Virasoro anomaly w15x. Our approach is related to this second viewpoint because, as we have already mentioned, the diffeomorphism generated by L n ,n / 0," 1 should not be considered as ‘‘pure gauge transformations’’ due to the Virasoro anomaly. Therefore it is natural in our scheme to pick up a trivial metric on the boundary to represent the unique conformal equivalence class. However, due to the breakdown of general covariance in the boundary, we cannot choose a particular form of the flat metric, but instead we have to consider the general form of a flat metric. All this means that f should be a free field. Going
J. NaÕarro-Salas, P. NaÕarror Physics Letters B 439 (1998) 262–266
back to the bulk part of AdS 3 we see immediately that the metric ds 2 s ye 2 f r 2 dxq dxyq
l2
dr 2 2
r
Ž 32 .
induces the boundary metric Ž30. and satisfy Einstein ’s equations. However Ž32. does not satisfy the asymptotic conditions Ž3. – Ž6.. To recover these conditions we have to transform Ž32. in an appropriated way. A redefinition of the radial coordinate r ™ reyf brings the metric into the form 2
2
q
y
ds s yr dx dx q 2l2 y r
l2 r2
dr
2
2
q l 2 Ž Eq f . Ž dxq . q l 2 Ž Ey f . Ž dxy . q 2 l 2 Ž Eq f . Ž Ey f . dxqdxy .
2
Ž 33 .
The above metric fulfils the conditions Ž3., Ž4., Ž6., but to satisfy the requirement Ž5. it is necessary to do 2 a second transformation x "™ x "q rl 2 E. f . We then get ds 2 sy r 2 dxq dxy q
ž
q 4
l4 r4
l2 r2
dr 2
Eq fEy f q O
1
ž // r6
2
ž ž ž
dr 2
ž // ž // ž //
q 2 l Ž Eq fEy f y Eq Ey f . q O q l 2 Ž Ž Eq f . 2 y Eq2 f . q O q l 2 Ž Ž Ey f . 2 y Ey2 f . q O 1
qO
ž / r3
q
dx dr q O
1
ž / r3
1
1
r2 1
r2 y
dx dr ,
r2
g ""s l
Ž Ž E "f .
2
2 yE" f
.,
gqys l 2 Ž Eq fEy f y Eq Ey f . , 4
gr r s 4 l Ž Eq fEy f . ,
df s l 2 2 Eq f Tqq 2 Ey f Tyq Eq Tqq Ey Ty 4 ,
Ž 38 . Redefining now the scalar field f s tensor Q "" takes the form
Q ""s 12
Ž E "f .
2
y
(
l 2G
(
2 E" f q
l 2G
f the stress l
16G
q
y
Ž 39 .
Ž 40 .
where the background charge is given by Q s
(
(
l 2G
2G l
q2 . In a semiclassical description l 4 G the central charge Ž40. reproduces the classical value Ž2.. It is also worthwhile to remark that when f is a chiral field there is not subleading terms in Ž34. and one recovers solutions of the form Ž21.. If we choose a boundary metric of constant curvature l to represent the given conformal equivalence class
Ž 41 .
where f L obeys a Liouville equation Ey Eq f L s l8 e 2 f L , the extension of the metric to the bulk part is more involved than Ž32.. Moreover, the first gauge invariant subleading terms in the asymptotic expansion are
dx dx
Ž dxq . Ž dxy .
,
c s 1 q 3Q 2 ,
dsb s ye 2 f L dxq dxy ,
2
2
Ž 34 .
where the omitted terms can be computed in a recursive way. Therefore we have 2
The transformation law Ž22. can be reproduced if f transforms as
which is similar to that arising in Liouville theory. However, it is well-known w16,17x that a Backlund ¨ transformation convert the Liouville theory into an improved free field theory. It is just this free field theory which emerges in this approach. The quantum stress tensor Ž39. gives rise to a central charge
2
Ž Eq f dxq dr q Ey f dxy dr . 2
265
Ž 35 . Ž 36 . Ž 37 .
but, as we have already mentioned, only the terms g "", gqyy 41l 2 gr r are gauge invariant.
2 g ""s l 2 Ž Ž E " f L . y E " fL . ,
gqyy
1 4l
2
gr r s yl 2 Eq Ey f L y
ž
Ž 42 . l 8
e2fL .
/
Ž 43 .
So we also recover the usual expression for the stress tensor of Liouville theory. Therefore, and according to this, the freedom in picking a metric on the conformal structure seems to be related to canonical transformations of the boundary theory. The boundary metric Ž32. is the most natural one to relate the bulk gravity theory to the boundary conformal field theory.
J. NaÕarro-Salas, P. NaÕarror Physics Letters B 439 (1998) 262–266
266
In this paper we have provided an explicit relation between the ‘‘would-be gauge’’ degrees of freedom of the gravity theory and the underlying boundary conformal field theory. To do this we have made use of the holographic correspondence AdSrCFT of w12x for the theory Ž1.. However, instead of breaking Weyl invariance on the boundary as in Refs. w12,13x, we sacrifice partially diffeomorphism invariance to make contact with the asymptotic boundary conditions of w1,2x. This way we have provided an explicit relation between the bulk metric on AdS 3 , verifying the boundary conditions of w1,2x and an improved free field theory on the boundary. This result raises a question also pointed out by Carlip w18x concerning the derivation of the BTZ black hole entropy of Ref. w2x The Cardy’s formula for the asymptotic density of states of a conformal field theory with central charge c
(
log r Ž D , D . ; 2p
cD 6
(
q 2p
cD 6
,
Ž 44 .
where D and D are the eigenvalues of the two Virasoro generators L0 and L 0 , assumes that the lowest eigenvalues Ž D0 , D0 . of L0 and L0 vanish. In general, the above formula is still essentially valid if one replace the central charge ‘c’ by the so-called effective central charge ceff s c y 24 D0 w19x. However, the minimum value of L0 Ž L0 . is not zero for the improved free field f , in fact ceff s 1 as one should expect by a direct counting of states. Moreover, for normalizable Žmacroscopic. states w20x of Liouville theory we also have ceff s 1. To properly account for the black hole entropy one should include states of imaginary momentum for the improved free field, or microscopic states in the terminology of Liouville theory. In a recent paper w21x it is claimed that stringy degrees of freedom account for the full density of states giving rise the Bekenstein-Hawking entropy of BTZ black holes. Quantization of BTZ black holes in the presence of external fields producing Hawking radiation has been given in Ref. w22x.
Acknowledgements P. Navarro acknowledges the Ministerio de Educacion ´ y Cultura for a FPU fellowship.
References w1x J.D. Brown, M. Henneaux, Commun. Math. Phys. 104 Ž1986. 207. w2x A. Strominger, Black Hole Entropy from Near Horizon Microstates, hep-thr9712251. w3x M. Banados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69 ˜ Ž1992. 1849. w4x J.A. Cardy, Nucl. Phys. B270 Ž1986. 186. w5x S. Carlip, Phys. Rev D51 Ž1995. 632. w6x M. Banados, T. Brotz, M. Ortiz, Boundary Dynamics and the ˜ Statistical Mechanics of the 2q1 Dimensional Black Hole, hep-thr9802076. w7x A. Achucarro, P. Townsend, Phys. Lett. B180 Ž1986. 89. w8x E. Witten, Nucl. Phys. B311 Ž1986. 46. w9x O. Coussaert, M. Henneaux, P. van Driel, Class. Quant. Grav. 12 Ž1995. 2961. w10x J.M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, hep-thr9711200. w11x S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge Theory Correlators from Non-Critical String Theory, hepthr9804083. w12x E. Witten, Anti-de Sitter Space and Holography, hepthr9802150. w13x M. Henningson, K. Skenderis, The Holographic Weyl Anomaly, hep-thr9806087. w14x R. Jackiw, Another View on Massless Matter-Gravity Fields in Two Dimensions, hep-thr9501016. w15x J. Navarro-Salas, M. Navarro, C.F. Talavera, Phys. Lett. B356 Ž1995. 217. w16x E. Braaten, T. Curtright, C. Thorne, Phys. Lett. B118 Ž1982. 115. w17x E. D’Hoker, R. Jackiw, Phys. Rev. D26 Ž1982. 3517. w18x S. Carlip, What We Don’t Know about BTZ Black Hole Entropy, hep-thr9806026. w19x D. Kutasov, N. Seiberg, Nucl. Phys. B358 Ž1991. 600. w20x N. Seiberg, Progr. Theor. Phys. Suppl. 102 Ž1990. 319. w21x A. Giveon, D. Kutasov, N. Seiberg, Comments on String Theory on AdS 3 , hep-thr9806194. w22x R. Emparan, I. Sachs, Quantization of AdS 3 Black Holes in External Fields, hep-thr9806122.
Physics Letters B 439 Ž1998. 262–266
A note on Einstein gravity on AdS 3 and boundary conformal field theory J. Navarro-Salas 2 , P. Navarro
1
3
Departamento de Fısica Teorica and IFIC, Centro Mixto UniÕersidad de Valencia-CSIC. Facultad de Fısica, UniÕersidad de Valencia, ´ ´ ´ Burjassot-46100, Valencia, Spain Received 10 July 1998 Editor: L. Alvarez-Gaume´
Abstract We find a simple relation between the first subleading terms in the asymptotic expansion of the metric field in AdS 3 , obeying the Brown-Henneaux boundary conditions, and the stress tensor of the underlying Liouville theory on the boundary. We can also provide an more explicit relation between the bulk metric and the boundary conformal field theory when it is described in terms of a free field with a background charge. q 1998 Elsevier Science B.V. All rights reserved. PACS: 04.60.Kz; 04.60.Ds Keywords: AdS gravity; Conformal symmetry
A crucial property of 2 q 1 dimensional gravity with a negative cosmological constant L s y 1 , l2
described by the action 1 2 Ss d3 x y g R q 2 , 16p G l
H '
ž
/
Ž 1.
is that the asymptotic symmetry of the theory is the conformal algebra with a central charge given by w1x 3l cs . Ž 2. 2G Recently Strominger w2x has used this result to derive 1 Work partially supported by the Comision ´ Interministerial de Ciencia y Tecnologıa ´ and DGICYT. 2 [email protected] 3 [email protected]
the Bekenstein-Hawking area formula for the 3d black holes w3x by counting the number of states of the conformal field theory at infinity with central charge via Cardy’s formula w4x. Other approaches w5x describe excitations associated to a conformal field theory at the horizon or at any value of the black hole radius w6x. Taking into account that Ž1. can be reformulated w7,8x as a Chern-Simons gauge theory with gauge group SLŽ2, R . m SLŽ2, R ., and using WZW reduction, one can argue that the asymptotic dynamics of Ž1. is described by Liouville theory w9x. The aim of this letter is to further elucidate the relation between the gravity theory and the underlying conformal field theory. Following w1,2x we assume the following asymptotic behaviour of the metric
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 0 3 2 - 6
J. NaÕarro-Salas, P. NaÕarror Physics Letters B 439 (1998) 262–266
gqys y
r2 2
q gqy Ž xq, xy . q O
g ""s g "" Ž xq, xy . q O
g " r Ž xq, xy .
g" rs
gr r s
r
l2 q
r2
1
gr r Ž xq, xy . r4
ž / r
ž / ž / ž / r
Ž 3.
1
r5
zys 2Tyq
l2 r
E 2 Tyq O 2 y
l2 r
E 2 Tqq O 2 q
1
ž / ž / r4 1
r4
z r s yr Ž Eq Tqq Ey Ty . q O
,
Ey gqqs 0 ,
z rs
a" r
a
1
r
r
ž / ž /
qO
4
r5
1
qO
r2
6 l
2
,
,
2 2
r l
g" rs0 ,
so the physical degrees of freedom are described by two chiral functions g "" Ž x ". : ds 2 s
1 4l
2
1
r3
dr 2 y r 2 dxq dxyq gqq Ž dxq . 2
1
ž / r
2
,
Ž 18 .
gqqs 2Gl Ž Ml q J . ,
/ ž /
gr r q O
r2
Ž 9.
Ž 11 .
gqyy
l2
q gyy Ž dxy . q O
,
Ž 12 .
Ž 19 . Ž 20 .
gyys 2Gl Ž Ml y J . .
It is interesting to note that, when either gqqs 0 or gyys 0, the omitted terms in the expansion Ž18. vanish Žin the appropriate gauge. and the corresponding BTZ solutions describe extremal black hole geometries. Hence ds 2 s
ž
Ž 16 . Ž 17 .
gr r s 0 ,
Ž 10 .
,
8 sy
In addition one can make, using the gauge transformations Ž10. and Ž11., the following consistent gauge choice
For instance, the standard BTZ black hole solutions can be brought, via gauge transformations, to the form Ž18. with
where a " and a r are arbitrary functions of xq and xy, it is not difficult to see that the variables g "", gqyy 41l 2 gr r are the only gauge invariant quantities. Moreover, the equations of motion imply that 0sRq
Ž 14 . Ž 15 .
Ž 8.
where the functions T "Ž x ". verify the conditions E " T .s 0. Those diffeomorphisms with T "s 0 should be considered as ‘‘gauge transformations’’. Therefore, if one consider the diffeomorphisms
z "s
Ž 13 .
Ž 5. Ž 6.
,
r
gr r s 0 .
The remaining equations of motion Rmn y 12 gmn R s y l12 gmn lead to the equations
Ž 7.
1
4l2
Eq gyys 0 ,
,
ž /
1
Ž 4.
where x "' tl " u , and u and r are the angular and radial coordinates. We have introduced explicitly the first subleading terms in the asymptotic expansion of the metric field and we want to investigate how they are related to the conformal field theory on the boundary. The infinitesimal diffeomorphisms z a Ž r,t, u . preserving Ž3. – Ž6. are of the form w1,2x
zqs 2Tqq
and this requires that
gqyy
,
r4
qO
,
,
1
qO
3
1
263
l2 r2
dr 2 y r 2 dxq dxyq gqq Ž dxq .
2
Ž 21 .
is an exact solution. We want now to identify the fields g "" in terms of appropriate conformal fields on the boundary. To this end, let us determine the conformal transformation properties of these fields. The action of the diffeomorphisms Ž7-9. on the metric Ž18. induces the following transformation law
d T "g ""s 2 Ž T "E " g ""q 2g "" E " T " . 3 y l 2E " T" .
Ž 22 .
J. NaÕarro-Salas, P. NaÕarror Physics Letters B 439 (1998) 262–266
264
It is then clear that the variables g "" are proportional, up to a constant yc 24 , to the stress tensor components Q "" of the underlying conformal field 3l theory with central charge c s 2G
Q ""s
1 4 lG
g ""q
l 16G
.
Ž 23 .
The above relation can also be confirmed by working out the conserved charges J w j x given in Ref. w1x
J w j x A lim
Hd f r™` 1
q l
ž
jH r
½
l r
j Hq
q j , rH
/Ž
r3 l3
ž
j H gr r y
gff y r
2
l2 r2
. q2j
f
/
pfr
5
,
in terms of the metric Ž18.. One obtains J w j x A d f Tq Ž 4gqqq l 2 . q Ty Ž 4gyyq l 2 . 4 ,
H
Ž 25 . which corresponds to the conserved charges associated to the stress tensor Ž23. of a conformal field 3l theory on a cylinder with central charge c s 2G . c Note that the shift term 24 in Ž23. arises from changing variables from the sphere to the cylinder z s e t r lqi u . The Fourier components L nŽ L n . of Qqq ŽQyy . obey the Virasoro algebra c 12
Ž n3 y n . dn ,y m , Ž 26 .
i L n , L m 4 s Ž n y m . L nqm q
c 12
Ž n3 y n . dn ,y m , Ž 27 .
Ln , Lm 4 s 0 , 3l with central charge c s 2G .
ds 2 s yr 2 dxq dxyq
l2 r2
dr 2 ,
Ž 28 .
Ž 29 .
and e 2 f is a positive function depending on the coordinates x ", induces a metric on the boundary r™` dsb s ye 2 f dxq dxy ,
Ž 24 .
i L n , L m 4 s Ž n y m . L nqm q
Our aim now is to provide a more explicit description of the expansion Ž3. – Ž6.. A simple and natural way to do this is to realize that the conformal symmetry also arises from a conformal analysis of 2f infinity. The conformally related metric ds 2 s er 2 ds 2 2 where ds is the elementary black hole solution with MsJs0
Ž 30 .
The 2d conformal symmetry Ž7-8. acts naturally on the metric Ž30. preserving the conformal structure. This is in fact a particular case of the more general AdS dq 1rCFTd correspondence suggested in Ref. w10x and elaborated in Refs. w11,12x. In Ref. w13x the Weyl anomaly for conformal field theories described via the supergravity action in d s 2,4,6 has been calculated regularising the gravitational action in a generally covariant way. In the case d s 2 one recovers the well-known trace anomaly ²Taa : s 24cp R. However, it was pointed out in Ref. w14x that in quantising a 2 d conformal field one can alternatively preserve the Weyl symmetry and partially break diffeomorphism invariance. Therefore, the trace anomaly disappear but one encounters that the stress tensor does not obey the covariant conservation law: c ² =mT mn : s y E RŽ yg g ab . . Ž 31 . 48p n
'
In conformal coordinates this equation implies that ²T mn : transforms according to the Virasoro anomaly w15x. Our approach is related to this second viewpoint because, as we have already mentioned, the diffeomorphism generated by L n ,n / 0," 1 should not be considered as ‘‘pure gauge transformations’’ due to the Virasoro anomaly. Therefore it is natural in our scheme to pick up a trivial metric on the boundary to represent the unique conformal equivalence class. However, due to the breakdown of general covariance in the boundary, we cannot choose a particular form of the flat metric, but instead we have to consider the general form of a flat metric. All this means that f should be a free field. Going
J. NaÕarro-Salas, P. NaÕarror Physics Letters B 439 (1998) 262–266
back to the bulk part of AdS 3 we see immediately that the metric ds 2 s ye 2 f r 2 dxq dxyq
l2
dr 2 2
r
Ž 32 .
induces the boundary metric Ž30. and satisfy Einstein ’s equations. However Ž32. does not satisfy the asymptotic conditions Ž3. – Ž6.. To recover these conditions we have to transform Ž32. in an appropriated way. A redefinition of the radial coordinate r ™ reyf brings the metric into the form 2
2
q
y
ds s yr dx dx q 2l2 y r
l2 r2
dr
2
2
q l 2 Ž Eq f . Ž dxq . q l 2 Ž Ey f . Ž dxy . q 2 l 2 Ž Eq f . Ž Ey f . dxqdxy .
2
Ž 33 .
The above metric fulfils the conditions Ž3., Ž4., Ž6., but to satisfy the requirement Ž5. it is necessary to do 2 a second transformation x "™ x "q rl 2 E. f . We then get ds 2 sy r 2 dxq dxy q
ž
q 4
l4 r4
l2 r2
dr 2
Eq fEy f q O
1
ž // r6
2
ž ž ž
dr 2
ž // ž // ž //
q 2 l Ž Eq fEy f y Eq Ey f . q O q l 2 Ž Ž Eq f . 2 y Eq2 f . q O q l 2 Ž Ž Ey f . 2 y Ey2 f . q O 1
qO
ž / r3
q
dx dr q O
1
ž / r3
1
1
r2 1
r2 y
dx dr ,
r2
g ""s l
Ž Ž E "f .
2
2 yE" f
.,
gqys l 2 Ž Eq fEy f y Eq Ey f . , 4
gr r s 4 l Ž Eq fEy f . ,
df s l 2 2 Eq f Tqq 2 Ey f Tyq Eq Tqq Ey Ty 4 ,
Ž 38 . Redefining now the scalar field f s tensor Q "" takes the form
Q ""s 12
Ž E "f .
2
y
(
l 2G
(
2 E" f q
l 2G
f the stress l
16G
q
y
Ž 39 .
Ž 40 .
where the background charge is given by Q s
(
(
l 2G
2G l
q2 . In a semiclassical description l 4 G the central charge Ž40. reproduces the classical value Ž2.. It is also worthwhile to remark that when f is a chiral field there is not subleading terms in Ž34. and one recovers solutions of the form Ž21.. If we choose a boundary metric of constant curvature l to represent the given conformal equivalence class
Ž 41 .
where f L obeys a Liouville equation Ey Eq f L s l8 e 2 f L , the extension of the metric to the bulk part is more involved than Ž32.. Moreover, the first gauge invariant subleading terms in the asymptotic expansion are
dx dx
Ž dxq . Ž dxy .
,
c s 1 q 3Q 2 ,
dsb s ye 2 f L dxq dxy ,
2
2
Ž 34 .
where the omitted terms can be computed in a recursive way. Therefore we have 2
The transformation law Ž22. can be reproduced if f transforms as
which is similar to that arising in Liouville theory. However, it is well-known w16,17x that a Backlund ¨ transformation convert the Liouville theory into an improved free field theory. It is just this free field theory which emerges in this approach. The quantum stress tensor Ž39. gives rise to a central charge
2
Ž Eq f dxq dr q Ey f dxy dr . 2
265
Ž 35 . Ž 36 . Ž 37 .
but, as we have already mentioned, only the terms g "", gqyy 41l 2 gr r are gauge invariant.
2 g ""s l 2 Ž Ž E " f L . y E " fL . ,
gqyy
1 4l
2
gr r s yl 2 Eq Ey f L y
ž
Ž 42 . l 8
e2fL .
/
Ž 43 .
So we also recover the usual expression for the stress tensor of Liouville theory. Therefore, and according to this, the freedom in picking a metric on the conformal structure seems to be related to canonical transformations of the boundary theory. The boundary metric Ž32. is the most natural one to relate the bulk gravity theory to the boundary conformal field theory.
J. NaÕarro-Salas, P. NaÕarror Physics Letters B 439 (1998) 262–266
266
In this paper we have provided an explicit relation between the ‘‘would-be gauge’’ degrees of freedom of the gravity theory and the underlying boundary conformal field theory. To do this we have made use of the holographic correspondence AdSrCFT of w12x for the theory Ž1.. However, instead of breaking Weyl invariance on the boundary as in Refs. w12,13x, we sacrifice partially diffeomorphism invariance to make contact with the asymptotic boundary conditions of w1,2x. This way we have provided an explicit relation between the bulk metric on AdS 3 , verifying the boundary conditions of w1,2x and an improved free field theory on the boundary. This result raises a question also pointed out by Carlip w18x concerning the derivation of the BTZ black hole entropy of Ref. w2x The Cardy’s formula for the asymptotic density of states of a conformal field theory with central charge c
(
log r Ž D , D . ; 2p
cD 6
(
q 2p
cD 6
,
Ž 44 .
where D and D are the eigenvalues of the two Virasoro generators L0 and L 0 , assumes that the lowest eigenvalues Ž D0 , D0 . of L0 and L0 vanish. In general, the above formula is still essentially valid if one replace the central charge ‘c’ by the so-called effective central charge ceff s c y 24 D0 w19x. However, the minimum value of L0 Ž L0 . is not zero for the improved free field f , in fact ceff s 1 as one should expect by a direct counting of states. Moreover, for normalizable Žmacroscopic. states w20x of Liouville theory we also have ceff s 1. To properly account for the black hole entropy one should include states of imaginary momentum for the improved free field, or microscopic states in the terminology of Liouville theory. In a recent paper w21x it is claimed that stringy degrees of freedom account for the full density of states giving rise the Bekenstein-Hawking entropy of BTZ black holes. Quantization of BTZ black holes in the presence of external fields producing Hawking radiation has been given in Ref. w22x.
Acknowledgements P. Navarro acknowledges the Ministerio de Educacion ´ y Cultura for a FPU fellowship.
References w1x J.D. Brown, M. Henneaux, Commun. Math. Phys. 104 Ž1986. 207. w2x A. Strominger, Black Hole Entropy from Near Horizon Microstates, hep-thr9712251. w3x M. Banados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69 ˜ Ž1992. 1849. w4x J.A. Cardy, Nucl. Phys. B270 Ž1986. 186. w5x S. Carlip, Phys. Rev D51 Ž1995. 632. w6x M. Banados, T. Brotz, M. Ortiz, Boundary Dynamics and the ˜ Statistical Mechanics of the 2q1 Dimensional Black Hole, hep-thr9802076. w7x A. Achucarro, P. Townsend, Phys. Lett. B180 Ž1986. 89. w8x E. Witten, Nucl. Phys. B311 Ž1986. 46. w9x O. Coussaert, M. Henneaux, P. van Driel, Class. Quant. Grav. 12 Ž1995. 2961. w10x J.M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, hep-thr9711200. w11x S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge Theory Correlators from Non-Critical String Theory, hepthr9804083. w12x E. Witten, Anti-de Sitter Space and Holography, hepthr9802150. w13x M. Henningson, K. Skenderis, The Holographic Weyl Anomaly, hep-thr9806087. w14x R. Jackiw, Another View on Massless Matter-Gravity Fields in Two Dimensions, hep-thr9501016. w15x J. Navarro-Salas, M. Navarro, C.F. Talavera, Phys. Lett. B356 Ž1995. 217. w16x E. Braaten, T. Curtright, C. Thorne, Phys. Lett. B118 Ž1982. 115. w17x E. D’Hoker, R. Jackiw, Phys. Rev. D26 Ž1982. 3517. w18x S. Carlip, What We Don’t Know about BTZ Black Hole Entropy, hep-thr9806026. w19x D. Kutasov, N. Seiberg, Nucl. Phys. B358 Ž1991. 600. w20x N. Seiberg, Progr. Theor. Phys. Suppl. 102 Ž1990. 319. w21x A. Giveon, D. Kutasov, N. Seiberg, Comments on String Theory on AdS 3 , hep-thr9806194. w22x R. Emparan, I. Sachs, Quantization of AdS 3 Black Holes in External Fields, hep-thr9806122.