ELSEVIER

12 June 1997

PHYSICS LETTERS B

Physics Letters B 402 ( 1997) 270-275

Conformal and non-conformal symmetries in 2D dilaton gravity *

J. Cruz a*1, J. Navarro-Salas av2, M. Navarro b,c,3, C.F. Talaveraa,d*4, a Departamento de Fisica Tedrica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Facultad de F&ccl.

Universidad de Valencia, Burjassot-46100, Valencia, Spain h Institute Carlos I de Fisica Tedrica y Computational, Facultad de Ciencias. Universidad de Granada.

Campus de Fuentenueva, 18002, Granada, Spain c Institute de Matemdticas y Fisica Fundamental, CSIC. Serrano I13-123, 28006 Madrid, Spain

d Departamento de Matemdtica Aplicada, E.T.S.I.I. Universidad Politknica de Valencia, Camino de Vera, 14, 46100-Valencia. Spain

Received 20 March 1997 Editor: L. Alvarez-Gaume

Abstract

We introduce new extra symmetry transformations for generic 2D dilaton-gravity models. These symmetries are non- conformal but special linear combinations of them turn out to be the extra (conformal) symmetries of the CGHS model and the model with an exponential potential. We show that one of the non-conformal extra symmetries can be converted into a conformal one by means of adequate field redefinitions involving the metric and the derivatives of the dilaton. Finally, by expressing the Polyakov-Liouville effective action in terms of an auxiliary invariant metric, we construct one-loop models which maintain the extra symmetry of the classical action. @ 1997 Elsevier Science B.V.

PACS: 04.60+n Keywords: Non-conformal symmetries; Solvable models

One of the main properties of the classical CGHS theory [ 1 ] is the existence of an extra symmetry which allows to solve the theory completely. To account for

the back-reaction effects one has to add the Polyakov-

Liouville term which breaks the extra symmetry. How- ever it is still possible to introduce certain modifica-

tions of the semiclassical CGHS action which maintain the extra symmetry thus restoring the solvability of the

*Work partially supported by the Comisi6n Interministerial de Ciencia y Tecnologia and DGICYT.

’ E-mail: cruz@lie.uv.es. 2 E-mail: jnavarro@lie.uv.es. s E-mail: mnavarro@ugr.es. 4 E-mail: taIavera@lie.uv.es.

one-loop theory [ 2-41. In this paper we investigate the existence of extra symmetries for generic 2D dilaton- gravity models. Firstly, we look for models which, alike the CGHS model, are invariant under an extra

(conformal) symmetry. We shall show that, when the kinetic term has been dropped out by an appropriate

resealing of the metric, these theories are restricted to have a potential of exponential form V = 4A2eP4 where $J is the dilaton field (when /? = 0 one recovers the CGHS model). Secondly, we shall present three (model-dependent) transformations which are sym- metries for generic 2D dilaton gravity models with arbitrary potential V = V( C/J). These symmetry trans- formations generalize that of the CGHS and expo-

0370-2693/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693 (97) 00458-9

J. Cruz et al./Physics Letters B 402 (1997) 270-275 271

nential models in the sense that, although they are non-conformal for a generic potential and involve the space-time derivatives of the fields, a linear combina- tion of them is conformal and equals the extra con- formal symmetry of the CGHS and the exponential model when the potential takes the appropriate form. Therefore, by following parallel lines to those which produce the semiclassical CGHS models, we are able to provide one-loop models for a generic 2D dilaton gravity theory which maintain the extra symmetry.

We shall start our analysis showing a simple mech- anism to reconstruct one-loop corrected theories asso- ciated with the CGHS action restoring the extra clas- sical symmetry. The CGHS model [ 1 ] is given by the action

$=&-/d2xfi

X

i e-24(R+ 4(V4)2 + 4h2) - 4 $(Vfi,?

1 ,

I_

il) This action can also be written by means of a confor- ma1 redefinition of the fields in the form

X RJ+4A2-$e(Vfi)’ . 1 (2) i=l

In this way it is straightforward to see that there is a (conformal) extra symmetry transformation

@pv = 0, (3)

s$=E. (4)

To account for the back-reaction effect one has to add the one-loop Polyakov-Liouville term

Sp=-& s d2xJ--gRO-’ R.

This term breaks the symmetry (3), (4), and there- fore the semiclassical theory is no longer solvable. The simplest way to construct a semiclassical theory invariant under (3)) (4) is to consider the Polyakov- Liouville term with respect to an auxiliary metric g,,,

which is invariant under the symmetry transformation. Introducing also the new field

4 = 6 + ;G(& (6)

where G (4) is an arbitrary function, it is clear that the action

(7)

is invariant under the transformation

&L” = 0, (8)

s$ = E. (9)

Returning now to the primitive fields (gfir, +>, the action

So@, 4) + SP @)* (10)

turns out to be

So(g, (6) + &(g)

- 2f”‘“V,F(d~)Vv4 + F(4)R - 4R], (11)

where F(4) = G(&r$)), and the transformation (8), (9) takes the form

(12)

(13)

The BPP [3] and RST [2] models are obtained by choosing F (4) = 0 and F (4) = $4 respectively.

A natural way to analyze the existence of extra conformal symmetries in generic dilaton gravity theo- ries, generalizing the powerful symmetry of the CGHS model, is based on the non-linear sigma model formu- lation of these theories. In Ref. [ 51 it is given the con- dition for the existence of an extra symmetry transfor- mation in a generic 2D sigma model. If we consider the parametrization of a generic 2D dilaton gravity model in which there is no kinetic term [ 6,7]

272 J. Cruz et d/Physics Letters B 402 (1997) 270-275

in the corresponding 2D sigma model

s= ~Sd’n(-4a,~~_pifv(~)e2~). (15)

The condition for the existence of a symmetry trans-

formation [ 51 becomes simply

d21%V(4) =* dqS2 ’ (16)

with general solution V( 4) = 4A2eP$, where A and p

are constants. The CGHS model is recovered for p = 0. If /3 # 0 we have a model with an exponential po-

tential. This model contains black hole solutions with

temperature proportional to the mass. The symmetry transformation of the model is

Q,” = -&/.LLY, (17)

$4 = E. (18)

There is an auxiliary metric which is invariant under

(17), (18)

g/,, - = eP6g,,, (19)

enabling the construction of an invariant (and solv-

able) semiclassical theory in a similar way as in the

CGHS case. Our aim now is to investigate the symmetries in

2D dilaton gravity models from a different standpoint.

We shall show that suitable modifications of the ex-

tra conformal symmetries we have dealt with up to now produce transformations of the fields which are

symmetries of all the models (including spherically symmetric Einstein gravity) whose Lagrangian can be

brought to the form

In this way, we shall be able to generalize the pro- cedure described above to construct one-loop models with an extra symmetry.

To find out these symmetries our strategy consists in finding the generalized conserved currents firstly. Then, by applying a very useful (sort of reciprocal) version of the Noether theorem which is presented next, we shall find out the symmetries these (Noether) currents are associated with.

For any Lagrangian C = L (9”) and arbitrary trans- formations of the fields &I@, we have

6L = (E - L), SW - V,sP, (21)

where (E - L), = 0 are the Euler-Lagrange equa- tions of motion for the fields V/*, and V,s@ is a total derivative term which appears due to the “integrations

by parts” which are generally required to produce the

equations of motion.

Let now jp be a current, which is made from the fields P and which is conserved on-shell: V, jp ls0i =

0. It is easy to see then that a transformation &Pa is the Noether symmetry associated to jp iff, without using

any of the equations of motion, the following equality

holds as an identity:

(E - L),, 6W’ = V,jP. (22)

In general, and due to semi-invariance, the current jfi

will not be equal to sp.

For the Lagrangian in Eq. ( 14) we have

cT=~{[ R + V’(4)] 84

+ [g/a4 - &LPV (4) - v,vv4] w + 0, [-4 (gP”VOGgp - g”‘“V”Qw) 1

+V”@ g&y - VP4 ..,I } . (23)

Taking into account that G,, = 0 implies 04 = V the equations of motion are equivalent to

R + V’(4) = 0, (24)

v/.&v,4 - &L”v4) = 09 (25)

and then it is not difficult to check that the following currents are conserved,

(26)

(27)

where V,j{ = R. Now we can make use of the Noether theorem to

show that these currents are, in fact, Noether currents

associated with symmetry transformations of the the- ory. The transformations which satisfy (22) for the currents jf and j: are, respectively,

J. Cruz et al./Physics Letters B 402 (1997) 270-275 273

424 = f2.

( gw SLgpv = E2V - - w>* 2 v,wdP

> (W4 (29)

Though neither of these variations, Si or 82, repro- duce the conformal symmetry of the CGHS model

when the potential V is constant, V = 4A2, it is easy to show that a linear combination of them do,

8 = s, - 4/P&. (30)

Therefore the transformation S must be regarded as a

symmetry which generalizes that of the CGHS model.

Observe that both 61 and 82 are area-preserving

(i.e., S1,2fi = -~fig&i,2gPp = 0). It is also interesting to remark at this point that when the fields

are taken to be on-shell the symmetry transformation & can be identified with a diffeomorphism, with in-

finitesimal space-time vector field fp = $$. Moreover the model (23) has the following sym-

metry

aEd = 0,

SE&v = &Luagvu+- i (a,Vd + hv#$) , (31)

for arbitrary constant vector a,. By means of the Noether theorem this symmetry can be easily shown

to give rise to the following conserved current

Jpv = gc”“E,

where

(32)

E= + [(V&2 - J(4)] 9

and

(33)

J’ (4) = v(4) . (34)

‘Ihe conservation law for Jp” implies the space-time independence of the local energy E.

In the present case, as in the CGHS model, the conservation law for the currents jf, jr turns out to

imply the existence of two free fields. It is not difficult

to check that jr and jf - ji satisfy the integrability condition. The corresponding free field equations are:

Ujr = 0, R+Uj2 =O, (35)

where

jl = s dr

2E + J (7) ’ (36)

and

j2 = log(2E + J) = log(V4>*. (37)

At this point it is clear that we have generalized the

conformal symmetry of the CGHS model in the sense

that it could be recovered as a special linear combina-

tion of 61 and 62. We have also showed that the CGHS

model can be seen as a particular case (p = 0) of a family of models (with potential V = 4A*e*Q) having

a conformal symmetry 8,. However, it is easy to see that, for non-constant potentials, no non-trivial linear

combination of 61 and 82 is conformal. This suggests

that it may exist another symmetry 63, which must be independent from 6, and 82, such that a linear com-

bination of 81.62 and 83 gives rise to a conformal

symmetry, at least in the particular case in which the

potential is an exponential of the dilaton.

To find out this symmetry we note that, for the ex-

ponential models, the Noether current associated to the conformal symmetry 6, can be written as

ji = jr + 2pEjr. (38)

However, since E is constant, this current is also con- served for a generic 2D dilaton gravity model. Our

task, therefore, is to work out, for a generic model,

the Noether symmetry 83 which is associated with the

conserved current Ejl , A straightforward application of the Noether theorem yields

(39)

Hence the conformal symmetry of the exponential models, for which V = /3J, is given by

6, = 62 •t 2pS3. (40)

274 J. Cruz et a/./Physics Letters B 402 (1997) 270-275

Therefore, the CGHS model and the models with an exponential potential are special only in the sense that

a) One of the symmetries Si ,62 and 63 is a linear

combination of the other two. b) A linear combination of SI,Sz and 6s is confor-

ma1 and does not involve the space-time derivatives of

the fields. It is interesting to mention that the three symmetries

close down to a non-abelian Lie algebra. The symme-

try 82 is a central generator, but 6, and 63 generate the affine algebra: [ 81,631 = F) 6,.

To complete the program we have outlined we must consider now the construction of an invariant metric

&. Although for a generic 2D dilaton gravity model

the invariant metric will not be unique, in the present

paper, and for the sake of clarity, we shall consider the

simplest choice. Here we shall consider the metric &,, which fulfills

the following requirements: a) It is invariant under 6 = 82 - 4h2Si,

b) gFiy = g,, when V = 4A2, and

c) det g,, = det &,, . The requirements a) and b) guarantee that, when

V = 4h2, our semiclassical model will reduce to the

BPP model [ 31. The requirement c) appears to be a natural one since the symmetry 6 is area preserving.

Bqs. (30) and (30) suggest that an invariant metric

may be of the form

(41)

A=A(&&, B=&&P)

are scalar functions to be determined. It can be written

in the form

- =A -- ( g,v V&V”4 & (V4j2 W414

+ BVpPVv4, (42)

where the new scalars A = A( g,,, 4) and B = B(g,,, 4) multiply quantities which are invariant under S. Therefore A and B must also be invariant. The simplest scalar which is invariant under S is

E* = i ((v+)2 - J(4) +4A*+)

f E + 2A2@ (43)

Therefore, it appears natural to consider that A and B are functions of E*,

The condition c) implies AB = 1. Moreover, since

for V = 4A2 we have (V4)2 = 2Eh, condition b)

requires

A = 2E,,. (44)

Therefore, a metric which fulfills the three require-

ments above is

The inverse metric is given by

-pl_ wP)2g’” g -- =A

1 - - - =A

0c”fp0”~.

(45)

(46)

Moreover (45) defines a one-to-one field transforma-

tion and, since (04)* = SEA the explicit inverse re-

lation of (45) takes the form

2EA _ --

+

where

BA = ; ((8&2 + J(4) - 4A24) .

(47)

(48)

Therefore, the inverse transformation is obtained from

the direct one by (essentially) changing the sign of the potential.

Once obtained the metric &, (45), which is in-

variant under the symmetry (30)) we can immediately construct a one-loop action which preserves this sym- metry. This can be achieved by adding the Polyakov- Liouville action with respect to the invariant metric gpy and conformally coupling the matter fields to the metric &,,. We obtain in this way

.I. Crur et al./Physics Letters B 402 (1997) 270-275 275

Therefore, we have provided a one-loop action for a generic dilaton-gravity model, invariant under the transformation 62 -4A*&, which in terms of the aux- iliary metric takes the form

Sd=E, (50)

6&, = 0. (51)

The action (49) reduces to the BPP model 131 for V = 4h2 and the transformation (50)) (5 1) coincides with the extra symmetry of the CGHS model (3)) (4). In terms of the metric g,, the counterterm, Sp (&,) - Sp (gpy), required to maintain the extra symmetry is in general non-local.

J.N-S. would like to thank J.M. Izquierdo and A. Mikovic for interesting discussions. M.N. is grateful to the Spanish MEC, CSIC and also the IMAPP for a research contract.

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