Exactly solvable models of 2D dilaton quantum gravity

Physics Letters B 291 (1992) 19-25 North-Holland

PHYSICS LETTERS B

Exactly solvable models of 2D dilaton quantum gravity A l e k s a n d a r Mikovi6 1,2 Department of Physics, Queen Mary and Westfield College, Mile End Road, London El 4NS, UK Received 24 June 1992

We study canonical quantization of a class of 2D dilaton gravity models, which contains the model proposed by Callan, Giddings, Harvey and Strominger. A set of non-canonical phase space variables is found, forming an SL (2, ~) × U ( 1 ) current algebra, such that the constraints become quadratic in these new variables. In the case when the spatial manifold is compact, the corresponding quantum theory can be solved exactly, since it reduces to a problem of finding the cohomology of a free-field Virasoro algebra. In the non-compact case, which is relevant for 2D black holes, this construction is likely to break down, since the most general field configuration cannot be expanded into Fourier modes. Strategy for circumventing this problem is discussed.

S=-~ f d2xx~-ge-2'~(R+4gU~OuqbOu~+c)

1. Introduction

M

Recently there has been a lot of interest in two-dimensional renormalisable models of gravity coupled to scalar fields. These are relevant for non-critical string theory [ 1,2 ], as well as toy models for describing the formation and evaporation of black holes [3 ]. As shown in ref. [4 ], the most general form of the action for such a model is S=-

f d2xx~[½g

~" 0uO0.O+½c~R~+ V(O)] ,

M

(1.1) where M is a 2D manifold, gu, is a metric on M, ~ is a scalar field (dilaton), a is a constant (background charge) and R is the 2D curvature scalar. We will label the time coordinate x ° = t and the space coordinate x Z=x, while the corresponding derivatives will be denoted as • and ', respectively. The form ( 1.1 ) can be always achieved after suitable field redefinitions [ 4 ]. For example, the CGHS action

Work supported by the U K Science and Engineering Research Council. b E-mail address: [email protected],

(1.2) takes the form (1.1), with V(O)=]ce ~m, after the following field redefinitions: 1 e_:~ q~= 4 a '

1__[_ o0/~c, g"~--' 4 a ~ v o~,..

(1.3)

Depending on the form of the potential V(¢) and whether the spatial section of the 2D manifold M is compact or non-compact, one can get models describing a non-critical string theory or 2D black holes. Oneqoop perturbative analysis of ( 1.1 ) has been carried out in ref. [ 4 ], where it was pointed out that the analysis simplifies in the case when V ( 0 ) = A e ~, where A and fl are constants. Canonical quantization methods are more successful in exploring the non-perturbative nature of quantum gravity in four space-time dimensions [ 5 ] then the standard path-integral methods. This may naturally lead one to apply the same methods in the case of 2D gravity, more specifically, to the theory defined by the action ( 1.1 ). The canonical analysis in the case fl= 0 and compact spatial manifold has been already carried out by the author [6], where it was demonstrated that the corresponding quantum theory is exactly solvable. This was achieved by using

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

]9

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non-canonical phase space variables, forming an SL (2, R) × U ( 1 ) K a c - M o o d y algebra, which transformed the constraints into quadratic polynomials of the new variables. By using the free-field realization of the SL(2, ~) currents, the constraints became a free-field realization of the Virasoro algebra, whose cohomology is known [ 7 ]. In this paper we show that the same can be done in the case f i e 0 and the spatial manifold is compact. Adding conformaly coupled matter does not change this result, which means that the physical Hilbert space of the CGHS model on a circle can be obtained by solving the cohomology of a Virasoro algebra realised from N + 2 free scalar fields, with background charges, where N is the central charge of the conformally coupled matter. Since one can very easily construct, level by level, the physical Hilbert space for such systems (for N = 0 there is a complete solution [ 9 ] ), the model is exactly solvable. Unfortunately, this construction breaks down in the non-compact case, due to the absence of well defined Fourier modes for a most general field configuration. Namely, if one wants to include the black hole solutions into the quantum theory, then one has to allow field configurations which blow up either at x = _ oo or at some finite x. Such configurations cannot be expanded into Fourier series. Therefore one needs an alternative way of defining the quantum theory, which is discussed in the conclusions.

17 September 1992

Derivation of the canonical form of the action ( 1.1 ) is simplified by introducing the lapse function Y ( x , t) and the shift vector n(x, t) [5]. Then the metric gu~ takes the form

goo=-y2+n2g,

gm=ng,

gl~=g,

(2.1)

where g(x, t) is a metric on Z. After introducing the canonical momenta for g and 0 as O5o P = O~'

all n= 06'

(2.2)

where Y' is the lagrangian density of ( 1.1 ), then up to surface terms, the action becomes

S= f dtdx(pg+~--~gGo-nGl),

(2.3)

where 2 G o ( X ) = - ~2g 2p2 - ggpn+½(~')2+gV(~)

- - -o~g' -0'+a0", 2g

GI (x) =nCp ' - 2 p ' g - p g ' .

(2.4)

The constraints Go and Gj form a closed Poisson bracket algebra,

{Go(x), Go(y)} = - a ' ( x - y ) [G, (x) + G, (y) ], {G, (x), Go(y)} = - 6 ' ( x - y ) [ G 0 ( x ) + Go(y)] , {G~ (x), G~ ( y ) } = - ~ ' ( x - y ) [G~ (x) + Gl ( y ) ] , (2.5)

2. Canonical analysis The canonical formulation of ( 1.1 ) requires that the 2D manifold M has a topology o r e × R, where Z is the spatial manifold and R is the real line corresponding to the time direction. Z can be either a circle S 1 or a real line. The compact spatial topology is relevant for string theory, while the non-compact spatial topology is relevant for 2D black holes, although we will argue at the end of the paper that the black hole solutions are possible even in the compact case. The compact case is simpler for analysis, due to the absence of the "surface" terms. In the non-compact case, one can assume appropriate boundary conditions at x = +_or, such that boundary terms do not appear. However, in a most general case they will be present. 20

where the fundamental Poisson brackets are defined as

g(y)}=6(x-y), {n(x), #(y)}=a(x-y). {p(x),

(2.6)

G~ generates the spatial diffeomorphisms, while Go generates the time translations of Z, in full analogy with the 3 + 1 gravity case. Note that the algebra (2.5) is isomorphic to two commuting copies of the 1D diffeomorphism algebra, which can be seen by defining the constraints as T_+ = ½(Go + G, ) .

(2.7)

Introduction of the conformally coupled scalar matter changes the cation ( 1.1 ) by

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PHYSICS LETTERS B

P

Sm~_ I

j d2xx/__ggU~

0,0' 0,0,,

(2.8)

17 September 1992

we are going to look for the analogs of the SL(2, E) variables introduced in ref. [ 6 ]. We define

M

(l+0/~)j+=_x/Sr

1

2

1

t

+ A

g

where i = 1, ..., N. The constraints change as

7~'

2

Go-~Go+~ni +~(Oi) , Gl -*Gl + 7~io'i ,

(2.9)

where n~ are the canonically conjugate momenta for 0i. Since we are dealing with a reparametrization invariant system, the hamiltonian vanishes on the constraint surface (i.e. it is proportional to the constraints). Therefore the dynamics is determined by the constraints only. Since Go and G~ are irreducible, there will be (2 + N) - 2 = N local physical degrees of freedom. When N=0, there are only finitely many global physical degrees of freedom (zero modes of g and 0), and one is dealing with a topological field theory. When N ¢ 0, these global degrees of freedom will be present, together with the local ones. In the quantum theory, this classical counting can be spoiled by the anomalies. However, when the anomalies are absent, this counting should still hold, as the subsequent analysis will show.

(l+0/fl)J°=(l+0/fl)gP+~

2 g l'

0/2 J - = ~7~ g,

1 _n

( I +0/fl)'/2 PD = ]~'~ ( -- 20/gg'+ ( l +

0/fl)O') (3.3)

The (J", PD) variables satisfy an SL(2, R ) ® U ( 1 ) current algebra

{J"(x),J~(y)} =fabcJC(x)~(x--y) -- ½t22tlabo' ( x - - y ) , {PD(x), PD(Y)} = - 3 ' ( x - y ) ,

(3.4)

where Ol 2

a2= - 1+aft'

(3.5)

and f"be = 2~"haqac with 3. SL(2, ~ ) ® U(I) variables

r/ab= We now specialize to the case V(0) =A e pc. As in the case fl=0 [6], the variables (g, p, 0, n) are not convenient for quantization, since Go is a non-polynomial function of these variables. First we performed a canonical transformation in order to get rid of the exponential in 0 term: g= e-~,

0=~,

n='~+flpg.

(3.1)

,

(3.6)

and {J, PD} = 0. Instead of using the canonical variables (n~, ~,), we introduce the left/right moving currents

Pg= ~

p=ea~,

-- 1 0

1

(n,+O;),

1 /~,= ~r~ ( h i - O ; ) ,

satisfying

The constraints now become

{e,(x),/'j(y) }= -6,j6' (x-y),

2 2 G 0 ( x ) = - ~-~ (1 +0/fl)g2p2_ _~gpn

{P,(x),/~(y)} = 6,ja' ( x - y ) ,

0/g' +½(l +0/fl)(O')2+Ag - ~ g O'+0/(k",

G~ (x) = nc/)' - 2p' g - p g ' ,

(3.2)

where we have dropped the tildes. As in the fl= 0 case,

(3.7)

(3.8)

and {P, P} = 0. Now one can show that the energymomentum tensor associated to the algebra (3.4) via the Sugawara construction, together with the matter contribution

Y = T , + Tm,

(3.9) 21

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17 September 1992

Y'~[j,m)=O,

1 Tg = - ~ rlabJ~f ' - ( jo ) , ,

n>~l

J~ }j, rn) =J~lJ, rn) , Tm

= 21P D2+

~

, PD + ~1P i2,

(3.9 cont'd)

while ]P,) is the U( 1 ) vacuum, ce~ ]pra) =0,

satisfies ~ = T+ on the constraint surface. Therefore the constraints become

(4.4)

n>~l (4.5)

OL~ IPM ) = P , [Pl ) •

The quantum constraints are defined as

J+(x)-2=0, 5£(x) = Tg(x) + Tin(x) = 0 ,

(3.10)

where 2 = A / x / 2 ( 1 + a f t ) .

1 L,,-k+2

" ~. . . . J ."" .- i n. J .° +. ½ .• "oF rl~b .J . .

oF"

m

+ inQice~,

(4.6)

where 4. Quantum theory S(x) The quantum theory can be now constructed by following the approach of ref. [6]. We promote J's and P's into hermitian operators, satisfying [ja(x), j b ( y ) ] = if~,, ' J q ~ ( x - y ) - i . ½krlaq~' ( x - y ) [P,(x),Pj(y)]=-i,~jS'(x-y),

l=i, D.

,

(4.1)

We introduce a new constant k, which is different from 2~ra 2 due to ordering ambiguities. It will be determined from the requirement of anomaly cancelation. Since the constraints are independent of the P~ variables, the complete physical Hilbert space will be a tensor product of a P Hilbert space with the physical Hilbert space of (J, P,) variables. In the case of Z = S ~, we construct the kinematical Hilbert space as a Fock space built on the vacuum state anhillated by the positive Fourier modes of J and &. If we define the Fourier modes as 1 J ~ ( x ) = ~ ~,, e i..... J~,

P,(x)= ~ 1

~ e i.x ce., I

(4.2)

then (4.1) becomes [ J ~a, J mb ]

1 ab = i f ~ cjC+m + ~krl n~,+,,,,

[c~, a ~ ] =6,jn(~,,+,,,.

(4.3)

The Fock space vacuum is defined as IJ, m ) ® IP,), where IJ, m ) is the SL(2, D) vacuum,

l

~, e in x L n

and the normal ordering is with respect to the vacuum states (4.4), (4.5). Note that the anomaly appears in the quantum algebra of the constraints (4.6), proportional to the central charge of gravity plus matter system c=

3k -6k+N+I+12Q~, k+2

(4.7)

where QD=X/-~ ~. When anomaly appears in the constraint algebra, then one has to use the GuptaBleuler quantization procedure, or the BRST quantization, which is more suitable in this case. The BRST charge 0 can be constructed as Q_=co(Lo-a)+

E c_,Ln+ Z c+,(J+-2~.,o) n#O

n

+...,

(4.8)

where cn and c~+ are the Fourier modes of the ghosts corresponding to the constraints (3.10). The dots correspond to the terms proportional to the ghost momenta, such that (~2=0, and a is the intercept [8]. The nilpotency condition requires vanishing of the total central charge, which includes the ghost contributions c-26-2=0,

(4.9)

and the intercept must satisfy [6] a = l + ~ k _ t ~Ob.

(4.10)

Evaluation of the cohomology of the BRST charge 22

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(4.8) simplifies if one employs the Wakimoto construction for the SL(2, ~) algebra (4.3) [9]. As in the fl= 0 case [ 6 ], we introduce three new variables fl(x), y(x) and PL(X) such that J+(x)=fl(X)

17 September 1992

L , = ½ ~ : a , _ m a , , : +inQ.a~,

(5.1)

I;7

where Xa= (X L, Xt),

X'Y=qabXay b,

,

~/ab= diag ( -- 1, 1..... 1 ) , J°(x) =-:fl(x)7(x):--klPL(X), J- (X) = :fl(~7)}'2(X): +

2k, y(X)PL(X

Q, = (QL, QD, 0, ..., 0 ) .

(5.2)

) +k2?' (x) ,

(4.11)

The BRST charge is then given by the usual expression

0 = ~ GL_,+½ ~ (m-n):CmC, b . . . . : -Coa.

where

n

tn,n

(5.3)

[fl(x), y(y)] = - i 6 ( x - y ) ,

[PL(X), &(y) ] =i6' ( x - y ) ,

(4.12)

with the other commutators being zero. Then the expressions ( 4.11 ) satisfy the SL ( 2, ~ ) algebra (4.1) if

k,=~2),

k2=-k,

(4.13)

where the normal ordering in (4.11 ) is with respect to the Fourier modes offland 7. The scalar constraint now becomes QL p , +

,g= :~'~: -~ :/'~: + ~ + ~ :P~: =0,

QD

~ ~:e~: + ~7~P5 (4.14)

where QL=kl-

The normal ordering is with respect to the vacuum [vac>=lp>®lO> a , Ivac> =cn Ivac> = b , Ivac> =0,

(4.15)

Note that the transformation (4.11 ) is also defined classically, with k~ = a / x / ~ and k2 = - a 2, satisfying the algebra (3.4). If we define B(x) =/~(x) - 2 and F(x) = y(x), then the J+ constraint implies that B=0, and consequently we can drop the canonical pair (B, F) from the theory. Therefore we are left with PL, PD and P~ variables, obeying only one constraint

Q2=-Q~_+Q~=2-~N,

(4.16) 5. BRST cohomology Now one needs to study the BRST cohomology of a Virasoro algebra

a=~N,

(5.5)

which are the conditions for the absence of anomalies. As expected, the first condition in (5.5) is equivalent to (4.9) if QL takes the value (4.15 ). The zero ghost number cohomology is determined by the usual Gupta-Bleuler conditions

n>~O,

(5.6)

where [~u> belongs to the a-modes Fock space F ( a ) . Since the anomalies are absent, due to (5.5), then one can expect that the classical counting of the physical degrees of freedom should hold. Hence only the zero modes of gravity plus dilaton sector should survive, together with N transverse local degrees of freedom, corresponding to the P, modes. This can be verified for the first few levels. Clearly, the ground state IP) is a solution of (5.6) if p2

cog__ __~pLqI 2 QD n , "]-~Pi ~QL _ _ L .p2,. D..t_l_D2 -}----..~rD | 2 -~0 .

(5.4)

where IP> is the a-modes vacuum (aolp> =pip> ), while I0 > is the ghost vacuum, satisfying bo I0 > = 0 (the other possibility CoI0 > = 0 gives symmetric resuits). Nilpotency of Q implies

(L,-ag~,o)[~,)=O, 1 2kl "

n>_-l,

2 2 --PL+Pl =~N,

(5.7)

which looks like the tachyionic ground state of the usual string theory. At the first excited level, the states are of the form I~'>=~'a ~lp>,

(5.8)

which are physical if P2= ~2N-2,

p'~=0,

(5.9) 23

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wherep=p+iQ. The norm of the state (5.8) is )~12=~a*~ a ,

(5.10)

which for the physical states becomes I ~] 2 .~. s l J ~ 7 ~ j

= ( l - 1/~o 1/~. i]12~ 2 Ill ' 2 + (~0 - ]/Oo p' Ps2)'*~j

( PiP, x*~ -

kl/0o I2 ~ , ~ l

) +c.c.

,

(5.11)

where Xe=Xo and XD=Xt. By going into a special frame po=m, p~ =p,=O for N <24 or po=pj = p , p , = 0 for N = 2 4 or p o = p ~ = 0 , pZ=m2 for N > 2 4 , where rn 2= 1 2 - I~NI, it is easy to see that the hermitian matrix S lJ always has N positive eigenvalues and one zero eigenvalue. Therefore this confirms our conjecture that only the transverse modes are physical. This is different from the usual string theory ( Q " = 0 ) , where S lJ is positive definite for N = D - 2 ~<24, while for N > 24, S w has negative eigenvalues, corresponding to the negative norm states. Note that the so called "discrete" states [7,10], arise when ~+ = - ~ _ ,

~,=0,

(5.12)

where ~± = ( 1 / x ~ ) (~Xo+~, ), so that 1~12=21~+ 1 2 > 0 .

(5.13)

The physical state condition then implies ~+ (p+ - p _ ) = 0 - , p + = p _ - ,

p+ = - i Q +

orp+ =iQ~ .

(5.14)

However, the meaning of these states is not clear, since they have imaginary, but fixed, momenta. See ref. [6 ] for a more detailed discussion.

6. Conclusions

We have solved non-perturbatively a class of 2D dilaton gravity theories, defined by a potential V(~) = A e po. Note that the f14=0 case is essentially the same as the fl= 0 case analysed in ref. [ 6 ], since a set o f non-canonical variables, forming an SL(2, Y~)® U ( 1 ) algebra, was found. In terms of these variables the constraints look the same in both cases. The 24

17 September 1992

SL(2, E) variables are gauge independent generalizations of the KPZ currents [ 1 ], which were originally found in the chiral gauge for fl= 0 theory. The Wakimoto construction (4.11 ) and the subsequent free-field expression for 5/' (4.16) is the canonical quantization analog of the D D K construction [2]. We have given only a partial analysis of the BRST cohomology for the zero ghost number sector. This analysis supports our conjecture about the physical Hilbert space which was based on the classical counting of the physical degrees of freedom and the absence of the anomalies. We have also concluded that the physical Hilbert space is well defined for any N, which conflicts with the conclusions of refs. [ 11,12 ]. There are several possible explanations for the discrepancy. First, we have not done the full cohomology analysis. Second, their results are semi-classical, and third, the theories may not be the same, since they are working in the path-integral quantization scheme. The key question now is whether this exactly solvable model contains the phenomena o f interest, i.e. formation and evaporation of 2D black holes. The 2D black hole classical solutions appear in the non-compact case, however we will argue now that one can have black hole solutions even in the compact case. As Mann et al. have shown [ 13 ], the original black hole solutions correspond to a one sided colapse of a 1D dust, so that the singularity is at x = _+oo. However, a symmetric collapse produces a black hole solution symmetric with respect to the origin, with the singularity at the origin x = 0 , and the horizon at x = -+xh, x h < o o [ 13]. By restricting this solution to an interval [ - L , L], such that L>xh, we will obtain a black hole solution on a compact interval. A more serious objection to our solvable model as a model of quantum black holes is the fact that we have used the Fourier modes of our fields to define the quantum theory. A function can be expanded into a Fourier series only if it is piecewise continuous on I - L , L]. However, the black hole metric blows up at the horizon, and therefore cannot be expanded into a Fourier series. Hence by using the Fourier modes we are restricting the phase space of our model to the space of piecewise continuous solutions, which does not contain the black hole solutions. Strictly speaking, this means that our model does not describe the phenomenon of interest. However. one can hope that

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the F o u r i e r m o d e s c o n s t r u c t i o n c a n be v i e w e d as some k i n d o f a discrete a p p r o x i m a t i o n to the full theory. Clearly, a f u r t h e r study is necessary. Therefore o n e s h o u l d f i n d a way o f d e f i n i n g the q u a n t u m theory w i t h o u t u s i n g the F o u r i e r m o d e s . I n the c a n o n i c a l q u a n t i z a t i o n approach, o n e way w o u l d be to recast the scalar c o n s t r a i n t 6e i n t o a Schr/Sdinger type e q u a t i o n . T h i s w o u l d require d e f i n i n g a n extrinsic t i m e variable, in analogy with the 4 D q u a n t u m gravity [ 14 ]. If a n y o f the p r o p o s e d m e t h o d s works, t h e n a physical H i l b e r t space c a n be constructed, a n d therefore the q u a n t u m m e c h a n i c s w o u l d stabilize a 2 D black hole, in analogy with the h y d r o g e n atom, which is classically unstable.

References [ 1] A.M. Polyakov, Mod. Phys. Lett. A 2 (1987) 893; V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819. [2] F. David, Mod. Phys. Lett. A 3 (1988) 1651; J. Distler and H. Kawai, Nucl. Phys. B 321 (1989) 509. [ 3 ] C.G. Callan, S.B. Giddings, J.A. Harvey and A. Strominger, Phys. Rev. D 45 (1992) R1005. [ 4 ] J.G. Russo and A.A. Tseytlin, Scalar-tensor quantum gravity in two dimensions, Cambridge and Stanford preprint DAMTP-I-1992/SU-ITP-92-2 (1992).

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[5 ] A. Ashtekar, Nonperturbative canonical quantum gravity ( World Scientific, Singapore, 1991 ); C. Isham, Conceptual and geometrical problems in quantum gravity, Imperial preprint Imperial/TP/90-91 / 14 ( 1991 ). [ 6 ] A. Mikovi6, Canonical quantization of 2d gravity coupled to c < l matter, Queen Mary preprint QMW/PH/91/22 (1992). [7 ] P. Bouwknegt, J. McCarthy and K. Pilch, BRST analysis of physical states for 2D gravity coupled to c < 1 matter, CERN preprint CERN-TH.6162/91 (1991). [8] Z. Horvath, L. Palla and P. Vecserny6s,Intern. J. Mod. Phys. A4 (1989) 5261; T. Kuramoto, Phys. Lett. B 233 (1989 ) 363. [9] M. Wakimoto, Commun. Math. Phys. 104 (1986) 605. [10] B.H. Lian and G.J. Zuckerman, Phys. Lett. B 254 (1991) 417. [ 11 ] A. Strominger, Fadeev-Popov ghosts and 1+ 1 dimensional black hole evaporation, Santa Barbara preprint UCSBTH92-18 (1992). [ 12 ] S.P. de Alwis, Quantization of a theory of 2d dilaton gravity, University of Colorado preprint COLO-HEP-280 ( 1992 ). [13]R.B. Mann, M.S. Morris and S.F. Ross, Properties of asymptotically fiat 2d black holes, University of Waterloo preprint WATPHYS TH-91/04 ( 1991 ). [ 14] K. Kuchar, Quantum gravity 2: A second Oxford Symp., eds. C.J. lsham, R. Penrose and D.W. Sciama (Clarendon, Oxford, 1981 ).

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