Volume 233, number 1,2

PHYSICS LETTERS B

21 December 1989

T H E P A R T I T I O N F U N C T I O N O F T W O - D I M E N S I O N A L Q U A N T U M GRAVITY M.A. A W A D A 1 Physws Department, Impertal College, London SW7 2BZ, UK and A.H C H A M S E D D I N E Instttut fur TheorettschePhystk, Umversttat Zurich, CH-8001 Zurtch, Swttzerland Received 25 September 1989

We propose a can&date parttUon funcUon of two-d~menslonal quantum gravity on an arbitrary R~emann surface wtth boundary The conjecture xsbased on the SO (2,1 ) gauge Chern-S~mons theory m three d~mens~ons We show that when the surface is a &sk, the wave functton coincides wtth the one obtained by quant~zmg the Polyakov reduced action of grawty

I. Introduction Classical Einstein gravity m two d i m e n s i o n ( 2 D ) ms a trivial theory Although there have been various proposals for the action [ 1 ], it is not yet clear whether such actions lead to a consistent q u a n t u m theory o f gravity m d = 2. Recently, Polyakov [ 2 ] succeeded in quantxzmg the " r e d u c e d g r a w t y " action [ 3] m the hght-cone gauge. The theory has an unexpected connection with SL (2, ~ ) current algebra. This SL (2, ~ ) symmetry can be clarified further by studying the oneloop effective action o f q u a n t u m gravity m d = 2 [4,5]. Unfortunately the action o f ref [2] is nonlocal and therefore not attractive from a physical p o i n t o f view. A topological action based on SO (2, 1 ) ~ SL(2, E) gauge m v a r m n c e has been p r o p o s e d [ 6 ] though there is not a clear connection between this model a n d the Polyakov theory, ~t has nevertheless a distinctive feature o f being topological and it is also exactly soluble However, the canonical quantlzation can only be p e r f o r m e d consistently for the surface C X ~ In this p a p e r we would like to give a new and altert Address after 30 September 1989 Physics Department, University of Florida, FL 3261 l, Gameswlle, USA 2 Supported by the Swiss National Foundation (SNF)

natxve proposal for the partition function o f quant u m gravity m d = 2 . O u r conjecture is based on the q u a n t u m theory o f SO (2, 1 ) gauge C h e r n - S l m o n s (CS) theory m d = 3. We wdl prove that when the twod~mens~onal surface Z as a disk, the H a r t l e - H a w k l n g [7] wave functional coincides with the p a r t m o n function o f the induced q u a n t u m gravity o f Polyakov.

2. S0(2, 1) gauge CS theory Recently, Witten [ 8 ] showed m a very interesting work, that the q u a n t u m CS theory for a ( c o m p a c t ) gauge group G is potentially rich for knot theory, and leads to new insights into solvable and statistical models in d = 2. In particular the spaces o f c o n f o r m a l blocks [9] m 1 + 1 & m e n s l o n s are the q u a n t u m Hilbert spaces o b t a i n e d by quant~zlng the CS theory, w~th a c o m p a c t gauge group, o n M 3 = E X ~. Therefore it seems plausible to think o f the SL(2, ~ ) current algebra m the Polyakov theory [2 ] as being the consequence o f quantlzmg the SO (2, 1 ) gauge CS theory on ~ × E We start by considering the CS action for the gauge group SO (2,1 )

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79

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I=~

kf

PHYSICS LETTERS B

from which we can read the Polsson brackets

tr(AdA+~A^A^A),

(1)

{AT(x),Aj°(y)}= k Ev~abd~Z)(x--y) ,

M3

where M3 is an arbitrary three-dimensional ( 3 D ) manifold and A is a one-form gauge field

A=A~T ~ ,

(2)

CJF~=O,

[ T a, T b ] =lC, abCTc,

where

tr( T a T b ) = q ab ,

(3)

and rlab is the group metric. I f M has a boundary, then under a finite gauge transformations

A ~ A v= V - I A V + V - l d V

(4)

the action changes to

I--,IV+I+ -~

t r ( A d V V -~ ) 0M

k ftr(g_ldg) 127r -

3

f tr(AdVV-~)=O,

(6a)

0M

f tr( V - ' d V) s = 0

(6b)

M

An Immediate plausible soluUon of (6) is to set V equal to one on the boundary [ 8 ] However, we will be interested in solutions that do not necessarily restrict Vto be one on 0M. We take M = Z × R with boundary, and we consider the canonical quantIzation of (1) in the temporal gauge Ao = 0 The action ( 1 ) reduces to

ff(o)

~tr A,~Aj E

80

F ~ = O,A; - OjA ~ + ~ab~A,bAjc

(10)

Since (7) is only first order in the time derivative, the associated hamlltonian vanishes This means that the dynamics o f the above system is completely determined by the symplectlc structure on the classical phase space Let ~' be the space o f all connections on Z and (¢ be the gauge group (set o f all maps from Z to SO (2,1 ) ) then the classical phase space is the space ~ / ~ ; The symplectlc structure, for level k on ~ is defined by

S(A~,A2)=~

k is an arbitrary parameter and there are no quantlzatlon conditions on it This is simply because zt3 (SO (2,1 ) ) = zt3(SO (2) ) = 0 If M has no boundary then the action (1) is gauge mvariant since the last term in (5) vamshes as a consequence o f the vanishing of the third cohomology group of M, H 3 (M, ~r3(SO(2,1 ) ) = 0 . Assuming M has a boundary, then gauge mvarlance o f I Implies that

dt

(9)

(5)

M

I=~-~

(8)

with t a n d j being vector Indices on Z The above system is supplemented by the "Gauss law" constraint, which is 8I/8Ao ( 1 )

with T ~ being the SO (2,1 ) generators

_

21 December 1989

,

(7)

k I tr(A~A2),

(11)

z

w h e r e A 1 and A 2 e J .

Very often one proceeds in quantlzing ~ , which is an lnfinite-&menslonal space, and then imposes the constraints, as has been performed in ref [ 10 ] We will adopt such an approach in this paper Alternatively one can impose the constraints and then quantlze In this case the phase space is the modull space o f flat SO(2,1 ) connections on 2; modulo gauge transformations, s~o/N, and it is canomcal to the Telchmuller space o f Z whose dimension IS ( 2 g - - 2 ) × Dim (SO (2,1 ) ) g > 1 represents the genus o f the Riemann surface 2; The Hllbert space H~ is obtained by quantizmg ~ o / f f Since ~o/ff is finite dimensional, a natural notion for quantlzing it would be the holomorphlc quanUzation [ 8 ] In our case we will quantlze 4 by separating the canonical variables into "coordinate", A~ = q a and " m o m e n t a " A ~ =pa. The quantum Hflbert space is Infinite dimensional (since the p~ are unbounded) and it is the space of square lntegrable funcUons of the q". The Hartle-Hawklng wave functional [ 7 ] ~(qlz) is only a function o f the q~ We will determine such a functional from the path integral of the SO (2,1 ) gauge

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CS theory with fixed values o f the restriction Y The path integral must be

ofq a to

T(qtz)

= f dpdqexp[l(.[ tr(AdA+~A3))],

(12)

21 December 1989

by (27c/lk)~)/Sq a when acting on the wave functional ~(qlz) The quantlzatlon con&tlon is the standard commutation relation between p and q. The A ~ equation of motion gives the Gauss' law constraint which quantum mechanically becomes a condition on the physical states

M

F _ + [ ~T'/ph> : 0 where j: is to denote integrations over restricted values of qa=A~ to Z Eq (12) has been proposed in the context o f 3D gravity by Witten. The wave functional T s h o u l d be a half-density m the m o d u h space N = d / ~ of flat SO (2,1 ) connectmns on Z. For this to be true the Hllbert space HM must be embedded into Hz (which is the space of half-densities of N ) Wltten [ 8 ] has argued that this would be the case for manifolds M that are obtained as the interior of a manifold Z embedded into R 3 In this paper we will assume such manifolds M

We wouMhke to propose the Hartle-Hawkmg wave functional T m eq (12) of the SO( 2,1 ) gauge CS theory as the partttlonfunctton of quantum gravity m two dimensions The functional ( 12 ) can be determined completely (this is the significant step in our proposal) on an arbitrary genus g Riemann surface Z As explained in ref [8 ] any flat SO(2,1 ) connectmn on E is determined by 2g holonomy elements (u,, v,) around the cycles (a,, b,) of Z Integrating o v e r p in (12) one can exphcltly evaluate ~u (u,, v,). We will prove our proposal when Y = D, the disk We quantlze using the Schr6dlnger representation and in the gauge A~ = 0 (the mdex 2 refers to the radial directions of the &sk) In this case the wave functmnal 7'is defined on C × E which coincides with the boundary o f D × E. We will show that the wave functional is the partition function of the Chlral WessZ u m l n o - W l t t e n ( C W Z W ) theory of which in turn we prove that it its nothing but the induced quantum gravity of Polyakov

2 2 Gauged CWZW theory We will use the light-cone (LC) coordinates x -+= ( l / v / 2 ) (x°+_x I ) and In the gauge A 2 = 0 the action (1) takes the form in (7) with the replacement of d/dt by d/cbc 2, and the indices l, j run over + and - We will treat A a_ as the "coordinate" qa, and A% as the " m o m e n t a " p~ which are represented

(13)

The infinitesimal SO (2,1 ) gauge transformations are generated by T(e)= ~

tr(eF)-= ~ CXN

(14a) CXN

which satisfy

[ Ta(x), Tb(y) ] =ld~b~T~d(2)(x-y)

(14b)

We can exponentlate (14a) to get the finite SO (2, l ) gauge transformation by taking g = e x p [ ~ ( x ) ] We obtain g2(g) -- exp [ i T ( g ) ] ,

(15a)

which satisfies the group law £2(g, )£2(g2)

=~2(glg2),

g~, g2 eSO(2,1 ) . The £2 acts transformation

on

(15b) AL=qa

by

£2(g)q~ +(g) =qg=gqg-~-dg g-'

the

following (16)

Acting with (16) on ~(qtz) it is straightforward to find the following representation. t2(g) ~ ( q )

k =exp[e(kS(g)+ ~ f

tr(q0+g

g_,))J~(qg),

CX~

(17) where S(g) is an arbitrary function o f g . We can determine it by requiring ,Q(g) to satisfy (15b) when acted on 7j We recall that eq (13) can be rewritten in terms ofg2(g) as Tph(q g) .= $2(g) ~ph (q) ----~ph (q)

( 18 )

The solution for S(g), for gauge transformations connected to the identity, is nothing but the W Z W action [ 10 ] 81

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kf tr(g-lO+gg-~O_g)

S~: (g) = ~

k -T-l~n f t r ( g - l d g ) 3

(19)

D×~q

By definition, and for a fixed value of g, the wave functional must satisfy

~(qg, g).= g2(g) ~ ( g ) .

(20a)

Combining(17) and (19), we deduce from (20a) that

T( qg, g)=exp [1kS_ (g, q)] (P(qg) ,

21 December 1989

the gauge parameter, and the gauge connecuon q This ~mphes that we actually have a reduced gauge lnvarlance We denote the corresponding gauge group by Therefore in gauge fixing (22) we must factor out by the volume of ~ rather than ~ = M A P (D, SO(2,1 )). Eq (24) is equivalent to the con&tlons in eq (6) aswell as J" tr(V-~O+ VV-~O_ V)=O.

(25)

CX~

The most general solution of (6b) and (25) is generated by the following SL (2, • ) matrices.

(20b) V l ( X +, x - ) =

1

'

where

kS~: (g, q) =kS~ (g)

+k -

2~z

f tr(qO+gg -~)

V 2 ( x +, x - ) = ( u ( x +,1 x - )

(26a)

(21)

V3(x+,x- )= (O 1 y ( x +,I x - ) )

cx~

~s the gauged WZW action The physical wave functmnal is obtained by integrating (20) over the gauge group, and using ( 18 ): Tph (q) = ~Yph(qg)

= ~ [ g - l d g ] e x p [ i k S _ ( g , q ) l T ( q g)

01) '

(22)

We leave it to the reader to check that eq. (18) is satisfied In [ 10 ] it has been shown that ( 19 ) is the actmn ofa chlral WZW (CWZW) theory. This is because there as an extra gauge symmetry on the boundary given by

V4(x+,x-)=(tO(X+'lX-)_

~),

' (26b)

where v, u, y, and o9 are arbitrary functions The solution (19) assumes gauge transformations that are connected to the ~dentlty Therefore only the solutions (26a) are relevant m this paper However (26b) will have an important global lmphcation which we will discuss elsewhere. The most general solutions can be obtained using the Gauss product [ 11 ] representations of a general element g of SO (2,1 ): g=(l u

01)(20 0 i )(10

~)

(27)

g(x +, x - ) --*U(x + )g(x +, x - ) V ( x - ) . We have a slmdar s~tuaUon for the action (21 ) we are considering In fact the action (21 ) corresponds to the gauged CWZW theory. Th~s is because of the spectrum of the kinetic term, and the presence of an extra gauge symmetry on the boundary of D:

g(x +, x - ) ~ g ( x +, x - ) V(x +, x - ) , q__.qV= VqV-I + V - I O_ V

(23)

A general solution of (6b) and (2 5 ) in the form (27) determines 2 to have the following form

2=exp[f(x+ ) + h ( x - ) ] .

Unfortunately the general solution (26a) and (28a) is not useful because it corresponds to gauging away all q m (6a) Therefore for a non-trivial connectlon q we will take

Gauge lnvarmnce of (21 ) under (23) implies that S+( V, q) = 0 ,

(24)

which is defined in (21). Eq. (24) as a constraint on 82

(28a)

or

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PHYSICS LETTERSB

,,: ('u

, 8c,

as the most general interesting solutmns of (6b) and (25). Eventually 1{or t2 will lead to the same theory, so we can choose either of them for the remaining analysis. Inserting ~ ' l n t o (6a) and using (2) for the definition of qa we find that (6a) is satisfied if, and only ff ( a = 0 , +, - ) A - " = q = (00 0 - )

(29,

21 December 1989

This action has been studied prevxously in ref [ 11 ] and elaborated further in ref [ 12 ] which gives a geometrical interpretation of the appearance of the SL(2, R) symmetry in induced 2D quantum gravity in terms of the theory of co-adjolnt orbits of the Vlrasoro group The energy-momentum tensor of the classical action in (32 ) can be determined m terms of the SL (2, ) currents [ 13 ], [ 14 ]. T(x)= ~

1

~

Ja(x)Ja(x) -O+J°(x)+O_bc (33)

In fact we can understand the structure of (28c) (respectwely (28b)) by noting that we can gauge away v (respectively u) to zero for a general SO(2, 1 ) element (27) using the V~ (respectively V2) solution In this gauge the action (21) becomes"

kS_ (g, q) _ k ~dx+dx_[(0+ln2)(0_ln2 2n

)

+ u q - 0+ (In u22 ] .

(30)

2 2 The equtvalence of gauge C W Z W theory wtth mduced quantum gravtty

(a runs over 0, +, and - ) The central charge of (33) 1s

c (k)-

3k -6k-2, k+2

(34)

which ~s nothing but the anomaly contrIbutmn of induced 2D quantum gravity discovered by Polyakov [ 2 ] In fact the CWZW action S_ (g) in (32) is nothing but the first term m (30) which ~s the Polyakov action in the conformal gauge. We can rewrite ~t in the LC by performing coordinate transformation [4]. In the LC gauge the metric gives the hne element d s 2 = h + + (dx + ) 2 + d x + d x -

,

We consxder the wave functmnal ~'vh(q) as the path integral over the space of all flat connectmn q -

while m the conformal gauge it ~s given by

Z = f [ d q - ] [g.~ldg] exp[lkS_(g, q)]~o(q g)

The reqmred transformatmns are

Vol(~)

(31) where we have used (22) (~ is the gauge group corresponding to (23) and ~o is the ground state whmh we can normahze to l If we do the q - mtegratmn we obtain a condmon which fixes u m terms of 2. Alternatively we can impose the gauge-fixing con&tlon q - = 0 and compensate for that in (31 ) by introducing a ghost system (b, c) ofconformal dimension ( 1, 0). We thus obtain Z = f [ g - l d g ] dbdcexp(lkS_(g)

'f

- fnn bO-cdx+dx-

)

(32)

ds2=22dzd2.

(35a)

(35b)

z = x +, g = ~ ( x + , x - ) .

(36)

The (35a) is equivalent to (35b) provided we do the following ldenUficatlons h++ = 0 + ~ / 0 _ ~ ,

~,2= 1/0_ ~ .

(37)

Inserting (37) into S_(2, 0) (30) we obtain the Polyakov actmn m the LC gauge for k=c/6. The measure [ g - ~dg] can be easily shown to coincide with [d~l Thus we have proven that for Z = D , the wave functional ~(qlz) integrated over the space of all flat connectmns qlz is the partmon functmn of reduced quantum gravity Further work remains to be done, m partmular, we need to find a scheme of computing (12) for an ar83

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bltrary R ~ e m a n n surface Z, a n d a n o t h e r i n t e r e s t i n g ~ssue ~s the i n c o r p o r a t i o n o f m a t t e r Th~s r e q m r e s the u n d e r s t a n d i n g o f c o u p h n g o f m a t t e r to t h e S O (2,1 ) CS t h e o r y m 3 D [ 8 ] In c o n c l u s i o n we r e m a r k that o u r p r o p o s a l has the f o l l o w i n g d i s t i n c t i v e features (1) T h e w a v e f u n c t i o n a l ~ ( q ) ~ ) is t o p o l o g i c a l as a c o n s e q u e n c e o f the t o p o l o g i c a l n a t u r e o f the threed i m e n s i o n a l S O (2,1 ) CS t h e o r y ( u ) T h e w a v e f u n c t i o n a l is c o n s t r u c t e d w~th no refe r e n c e to a classical t h e o r y m two &mens~ons. (Ul) W h e n c o m p a c t l f i e d to 2D, the SO (2,1 ) CS theory gives the o n e p r o p o s e d in r e f [ 6 ] for 2 D g r a v i t y w h i c h was s h o w n to h a v e m a n y a t t r a c t i v e features

Acknowledgement We w o u l d like to t h a n k J P D e r r e n d l n g e r , G Felder, C J Isham, T. K a k a s a n d R S l l v o t t l for useful & s c u s s l o n s M A. w o u l d lake to t h a n k D. W y l e r for the h o s p l t a h t y e x t e n d e d to h i m at the U m v e r s ~ t y o f Z u r i c h w h e r e this w o r k was d o n e

References [ 1 ] R Jacklw, m Quantum theory of gravity, ed S Chnstensen (Adam Hflger, Bristol, 1984) p 403, C Teltelbolm, m Quantum theory of gravity, ed S Chnstensen (Adam Hflger, Bristol, 1984) p 327

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21 December 1989

[2] A M Polyakov, Mod Phys Lett A2 (1987) 899, V G Kmzhmk, A M Polyakov and A A Zamolodchlkov, Mod Phys Lett A3 (1988)819 [3] G Vllkovlsky, m Quantum theory of gravity, ed S Chnstensen (Adam Hllger, Bristol, 1984) [4] A Chamseddme and M Reuter, Nucl Phys B 317 (1989) 757 [5 ] J Helayel, S Mokhtarl and A Smith, ICTP preprmt (Feb 1989) [6]A Chamseddme and D Weyler, Umverslty of Zurich preprlnts (1989) [7] J Hattie and S Hawking, Phys Rev D 28 (1983) 2960 [ 8 ] E Wltten, Commun Math Phys 121 (1989) 35 I, preprlnts IASSNS-HEP-88/89, 89/1 [9] A M Polyakov,, V G Kmzhnlk and A A Zamolodchlkov, Nucl Phys B260 (1980)333 [10]S Ehtzur, G Moore, A Schwlmmer and N Selberg, IASSNS-HEP-89/20, see also M Bos and V P Nam preprmt 89-0118, Y Hosotam, preprlnt IAS-HEP 89/8, G V Dunne, R Jacklw and C A Trugenberger, preprmt MIT-CTP-1711 (1989), K Gawedzkl, preprmt IHES/P/89/6 [ 11 ] A Alekseev and S Shatashvlh, LOMI preprmt ( 1988 ) [12] M Bershadsky and H Oogun, preprmt IASSNS-HEP-89/ 9O [ 13 ] A Belavm, m Quantum stnng theory, Proc second Yakawa Memorial Symp (Nlshlmomlya, Japan, 1987), eds N Kawamoto and T Kugo, Lecture given at Tamguchl Foundation Symp (Kyoto, 1988) [14]M HalpernandJ Freerlcks, Ann Phys 188 (1988)258