Gauge theory of topological gravity in 1+1 dimensions

Volume 228, number 1

PHYSICS LETTERS B

7 September 1989

GAUGE T H E O R Y OF T O P O L O G I C A L GRAVITY IN 1+1 D I M E N S I O N S A.H. CHAMSEDDINE J and D. WYLER Institut ~ r Theoretische Physik, Universitiit Z~irich, Sch6nberggasse 9, CH-8001 Z~irich, Switzerland Received 5 July 1989

An action for two-dimensional gravity based on the gauge symmetries SO( 1, 2) and ISO( 1, 1 ) is presented. The theory is quantized. It is shown to be renormalizable, and the partition function is computed exactly. The action is generalized to include matter without spoiling renormalizability.

Recent investigations suggest that the induced gravity action might give a description of two-dimensional gravity [ 1,2 ]. This action has been quantized and the obtained results agree with those of the Liouville action [ 3-6]. A disadvantage of the induced gravity action is its non-locality. Locality can be recovered at the expense of adding a scalar field to the metric tensor [ 7 ]. On the other hand, Witten, in his work on threedimensional gravity [8,9], showed that this theory can be completely solved when written as a gauge theory. The topological nature of the action and thus its independence of the metric on the spacetime manifold (because the metric must not be introduced a priori) possibly allow to extract non-perturbative information from the theory. In the context of string theories, two-dimensional gravity is important. It is therefore natural to apply Witten's program to such a theory and investigate the connection to earlier mentioned treatments. In this note ~ we present a topological action for two-dimensional gravity, based on the gauge symmetries ISO(1, 1), S O ( I , 2) and SO(2, 1), depending on whether the cosmological constant is zero, positive or negative, respectively [ 11 ]. Here, we treat the SO ( 1, 2 ) case. SO (2, 1 ) is obtained by simply changing the signature of the S O ( I , 2) metric, while a In6nuWigner contraction yields ISO ( 1, 1 ). Supported by the Swiss National Foundation (SNF). ~' Detailed derivations o f t h e results in this note are given in ref. [10].

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The gauge field is defined by the one-form e=e~rAdx ~ ,

(1)

where zA are the generators of SO ( 1, 2) [rA, ZS] =--eASCZ c A, B, C = 0 , 1 , 2,

,

Eol2=l,

(2)

and the group indices A, B .... are raised and lowered with the flat metric t/a8 =diag( 1, - 1, - 1 )

(3)

The field strength of the gauge field e is the twoform F=F~pzadx'~

Adx

p ,

(4)

with

F ~ = 0,~e~ - 0~e~ - eASCe ~B e eC.

(5)

A (metric independent) topological action on the two-dimensional manifold must be an integral over a two-form. To render the only available two-form gauge invariant, we must introduce a scalar field which transforms under the adjoint representation of the gauge group: ¢=¢zzA.

(6)

The obvious candidate for the action is then simply ~2 ~2 A similar action for four-dimensional gravity was given in ref. [121.

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Volume 228, number 1

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I=--fl f Tr(OF)= ~ f OAFApe"Pd2z, M

Jm=flD~O" = 0 . (7)

M

where M is a general two-dimensional spacetime manifold. Contact with the metric theory of gravity is made by writing e~ = ( e .a, e .2 =cod),

0,

=

(0.,

A

0~ =~)

,

2 F~p = (F~p = Tap, F~p =R~p) ,

(8)

and solving the equations of motion

--up-,,, D,~OA OaOA--eABCeaBOC=0

(9)

which follow from ( 7 ). With 0 = T~p and assuming that the Zweibein field e~ can be inverted, we obtain a~ = - e - I ~'a0ye~ e.~,

(10)

where e=det( e~ ). The equation R~=O in (9) implies, using (10), the Liouville equation [ 3 ]

x/h [R(h,~p)+A]=O,

(11)

where h=det(h.p),

(12)

and A is a cosmological constant that appears after rescaling e~ ~ ~ e g . The action (7) then takes the form

l=fl f O[R(h,~p)+A l,,/h

d2x,

(13)

M

which has been considered in connection with the Liouville theory [ 3 ]. The ISO ( 1, 1 ) case is obtained by rescaling e ~ x / ~ e g , qt"~qi"/x//A, and letting A--,0. The action (7) is, however, easier to quantize than the form ( 13 ) as we will show. To separate space and time coordinates, we write M as C × N t, where C is a closed curve that plays the role of the initial boundary. The Poisson brackets can be read offdirectly from the action and are given by {ei' (x, t),

08(Y, t)} =

1

~

5AS(x--y)

,

(14)

and the constraint equation associated with eg is 76

(15)

Quantization is carried out by changing the Poisson brackets to commutators and imposing the constraint ( 15 ) on the physical states. We quantize the theory in the Landau gauge; the corresponding ghost action is

-- t- (CA~aD°~CA-t-U4~aeA) "

(16)

M

a

a b h.p=e.rla~,e p,

7 September 1989

Here, UA is a Lagrange multiplier and CA, 44 are the ghost and antighost fields. The derivative ~ is covariant with respect to a classical solution e A and general coordinate transformations. The ghost fields are quantized according to

{CA(t, x), OoCS(t,y)} =

-i~(x-y)O A

(17)

and the total current is

JA= fl DIcA --~ABCCB~oC c

(18)

The jA generate an anomaly free SO ( 1, 2 ) -~ SL (2R) current algebra. This is easily verified if we assume periodic boundary conditions in x and compute the operator product expansion of jA (X) jB ( y ). Thus the action (7) may be quantized consistently without introducing matter. Alternatively, the theory could be quantized on the reduced phase space of CA and e~ after imposing the constraint ( 15 ). To prove that the action (7) is renormalizable, we note that the "kinetic term" is ¢0e and the interaction is cubic and of the form Cee. With this, it is possible to show that the only non-vanishing graphs are the one loop graphs with only external e-lines [ 10 ]. However, it turns out that the one-loop graphs with internal ghosts exactly cancel the e0 graphs. This can be seen by evaluating the partition function Z(M) = j

DOADeADUADCADCA

×exp(if~,~POAFAp--UA~e~--CA~D~CA). M (19) where M is a spacetime manifold. Since the determinants we obtain from the ghost/ antighost and e~ integrations are equal ~3 (except for ~3 T h e r a t i o o f g h o s t a n d e 0 d e t e r m i n a n t s is the R a y - S i n g e r t c r sion [ 8 , 9 ] , w h i c h is 1 in t w o d i m e n s i o n s [ 1 3 ] .

Volume 228, number 1

PHYSICS LETTERSB

7 September 1989

the occurrence of zero modes), the partition function is simply given by the number of points in the moduli space X of classical solutions (9) over M. When the topology of M is such that the moduli space X is a manifold, the partition function is [ 10]:

where A~A is an infinitesimal gauge parameter. The equations of motion for X~ and Bua a a r e

1! det' ( - ~ ~D~) Z ( M ) = ~ dn Idet'LI

[B~AI (X, t), X ~ ' ( y , t) ] = i 5~6BAd(X__y),

(20)

where dn is a measure on Y , L is the operator ( ~ ~5-'D,) which acts on one-forms. This expression is formal; the measure as well as the possible JV's are presently under study. Because S is non-compact (the gauge group is not ), Z ( M ) diverges. This can be recognized as a large-distance divergence as follows: Let us define a distance d 2 = d"ld.i by

d"'=~(~-~)dx

~,

(21,

Y

where ~, is a large non-contractible loop in M, and A is the energy scale associated with e. For large enough 7, we take e2~=o9_~0; then dZ_~d"da, which is the usual expression. A rescaling eg, --,/Le~ where # is dimensionless, corresponds, according to (7), to a shift fl--,#fl. Because of (20), we see that Z ( M ) goes to ( 1//t) Z ( M ) and diverges for/z~ 0. In this limit (21) yields large distances and the moduli space Y shrinks to the one o f l S O ( 1, 1 ). Following Witten [9], we would like to take this infrared divergence as a signal of a phase transition to the classical region of macroscopic spacetime and riemannian geometry. The action (7) can be generalized to include matter. For this purpose, we introduce a set of scalar boson fields X ~'~,and vector fields Bus,,/t = 0, 1,...D- 1. Here D is an external dimension; X ~4 and Baa A transform under the adjoint representation of SO ( 1, 2 ). The matter interaction, to be added to (7) has the topological form (in M )

a f ~al~(DaBMjA)XI'A d 2 x .

e~aD~BupA=O.

(24)

Quantization of (22) is effected by writing (25)

and the current associated with B~AOagain generates an anomaly free SO (1, 2) current algebra after the ghost fields, required for fixing the gauge freedom (24), are introduced. The full action is again renormalizable [ 10 ], with the partition function given now by the volume of the enlarged moduli space (which includes the X ¢'Aand B~,o,A solutions). We can recast (24) into a more familiar form, following the separation of group indices in eq. (8). Setting x~A= (X ~a, XU2= - - X u) ,

(26)

and solving (24) for X "a, one gets X ~ = E~he~'0~X".

(27)

Inserting this into (24) yields the Klein-Gordon equation in de Sitter space, [3X ~ = haP( 0~ 0r - F~B 0y) X~ = - A X u .

(28)

In this picture, the X ~" may be viewed as the canonical momenta of X u. A similar analysis for B~,,A yields expressions that simplify only in a special gauge and will not be presented. The generalization to a supersymmetric form of the actions (7) and (22) is straightforward [ 10 ]. Further work is needed to characterize fully the moduli space of classical solutions for different topologies of M. But even more important is to understand whether the anticipated phase transition from small to large distances really occurs. We thank M. Awada for discussions on topological aspects of these theories.

(22)

M

Surprisingly, the full action (7) and (22) is invariant under an additional gauge transformation

6fbA = ( a / f l ) e A B c A ~ X ~c,

D~x~A=0,

6B~,~A=D~AuA,

Note added. After completion of this work, we received a paper by Isler and Truggenberger [14] in which the same action is proposed.

(23) 77

Volume 228, number 1

PHYSICS LETTERS B

References [ 1 ] A.M. Polyakov, Mod. Phys. Lett. A 2 ( 1987 ) 899; V.G. Knizhnik, A.M. Polyakov and A.A. Zomolodchikov, Mod. Phys. Lett. A 3 (1988) 819. [2]A.H. Chamseddine and M. Reuter, Nucl. Phys. B 317 (1989) 757. [3] E. D'Hoker and R. Jackiw, Phys. Rev. D 26 (1982) 3517; Phys. Rev. Lett. 50 (1983) 1719. R. Jackiw, in: Quantum theory of gravity, ed. S. Christensen (Adam Hilger, Bristol, 1984) p. 403; C. Teifelboim, in: Quantum theory of gravity, ed. S. Christensen (Adam Hilger, Bristol, 1984) p. 327. [4] T.L. Curtwright and C.B. Thorn, Phys. Rev. Lett. 48 (1982) 1309. [ 5 ] J.L. Gervais and A. Neveu, Nucl. Phys. B 199 ( 1982 ) 59; B 209 (1982) 125. [6 ] F. David, Conformal field theories coupled to 2 D gravity in the conformal gauge, Saclay preprint SPHT/88-132 (1988).

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[ 7 ] A.H. Chamseddine and D. Wyler, in preparation. [8] E. Witten, Nucl. Phys. B 311 (1988) 46. [9 ] E. Witten, Topology-changing amplitudes in 2 + 1 dimensional gravity, Institute for Advanced Study, Princeton, preprint IASSNS-HEP-89/1 (1989). [ 10] A.H. Chamseddine and D. Wyler, Topological gravity in 1 + 1 dimensions, University of Ziirich preprint (1989). [ 11 ] A.H. Chamseddine and P.C. West, Nucl. Phys. B 129 ( 1977 ) 39; S.W. MacDowell and F. Mansouri, Phys. Rev. Lett. 38 (1977) 739; [ 12] A.H. Chamseddine, Ann. Phys. (NY) 113 (1978) 219. [ 13 ] D. Ray and I. Singer, Adv. Math. 7 ( 1971 ) 145; Ann. Math. 98 (1973) 154. [14] K. Isler and C. Truggenberger, A gauge theory of twodimensional quantum gravity, MIT preprint CTP # 1739 (1989).