Introduction There are many conceptual problems in the theory of quantum gravity, such as the origin of the space-time and the measurement of the wave function of the universe

Nuclear Physics B (Proc. Suppl.) 26 (1992) 93-110 North-Holland

QUANTUM GRAVITY AND RANDOM SURFACES H . Kawai

Department of Physics, University of Tokyo, Tokyo 119, Japan

We review recent progress in lattice quantum gravity and random surfaces with a particular emphasis given to discussion of two dimensional gravity, dynamical triangulation, matrix models and Regge calculus. We also discs the possibility of constructing the theory of quantum gravity as an ordinary field theory in four dimensions.

1. Introduction

There are many conceptual problems in the theory of quantum gravity, such as the origin of the space-time and the measurement of the wave function of the universe. We also do not know a priori whether topology changes should be taken into account. These problems, how- ever, are hard to consider without understand- ing the dynamics of fluctuating metrics. There- fore, it seems better to investigate the dynam- ics first, and then to think about the conceptual problems. In these three years, since the equiv- alence of the dynamical triangulation and quan- tum gravity is established, we have experienced a substantial progress in understanding dynamical properties of quantum gravity. In this talk, we summarize the present status of quantum grav- ity on the lattice and random surfaces. In section 2, we discuss two-dimensional quantum gravity from various points of view. In section 3, we con- sider random surfaces and discuss the difficulty ofdefining random surfaces in higher dimensions.

Finally, in section 4, we investigate the possibil- ity of defining four-dimensional quantum gravity as an ordinary local field theory.

2.1. Continuum theory

0920-5632/'92/'$05.00 0 1992- Elsevier Science Publishers BY All rights reserved.

2. Two-dimensional quantum gravity

The continuum theory of two-dimensional qu- antum gravity can be solved either in the light cone gauge [1] or in the conformal gauge.[2,3] Let us consider a two-dimensional closed orientable manifold and introduce the metric field glw(z)

and a scale invariant matter field with central charge c defined on the manifold. We then con- sider the partition function of this system with

the total volume^ofthe. manifold fixed to A:

Z(A) vo DDff1( )^ZM[.9]6( d2z~- A), (1) whereZM[g] is the partition function of the mat- ter field for the background metric g,,(z), and vol(Diff) stands for the formal volume of the space of diffeomorphisms.

In the conformal gauge, gauge slices are para metrized by a scalar field ¢(z) and a finite num- ber of parameters r:

g, (x) = 9,(r; x)e'O(x), ⁽²⁾ where ¢(z) and r are called the conformal mode and the moduli, respectively. In this gauge, the

partition function Z(A) is rewritten as Z(A) = jdrj*VjOAFp[je'O1Zm[§eO1

( d2x-^%/7geO - A),

Here, ®Fp is the Faddeev-Popov determinant, and the functional measure D^lOis induced from the following metric in the functional space of0:

ilb0l11 = d2x~e#(bo)2 D1~~ (4)

This is because the measure Dg in eq.(1) is in- duced from the reparametrization invariant met- ric in the functional space of g,,v(x) defined by

116g112

= f

yg

gpx

gvo

The conformal mode 0 can be easily factored out in eq.(3), using the following identities:

and Dog is the translation invariant measure which is induced from the functional metric de- fined by

lebO112 = d2x_ 7(bO)2 =* DOO. (10) Substituting (6)-(8) to (3), we obtain

Z(A) = dr DoO®Fp[9]ZM[9]e^-WC-25SLUM .b( d2x/7^g : eao_{:i -A),} (11) where ¢ can be regarded as a free scalar field de- fined on a curved space with metric g,, . If we take this point of view, we should renormalize

H. Kawai/ Quantumgravity andrandom surfaces

the cosmological constant term in an appropri- ate manner. In eq.(11), we have introduced an ansatz that the renormalized cosmological con- stant term can be expressed as

jd2^{x-,Fg : e}°`O :i, (12)

where the normal ordering with respect to the background metric 9,,v is defined by

eaO i - lim ^(e.Îdavp(y),O(y) P(y)~ab~~l (y-z)

.e-112_fd2yd'zp(y)P(z)Klog d2#(y,z)) . (13)

Here, di (y,z) is the geodesic distance between y and z, and rc logd9 (y,z) is the singular part of the two-point Green's function of 0 with

The constant a in (11) is fixed by the require- ment that the theory should be invariant under the transformation given by

9~v(x) -+ gttv(x)e~(x)^, (15)

(x) -+ OW - 0(x), ⁽¹⁶⁾

because the original theory (1) is defined in terms of^g,,,,(x) and not in terms of j.v and ⁰ sepa- rately. It is easy to check that the combination

DoO®FP[g]ZM[g]e

is indeed invariant under (15) and (16) . There- fore the cosmological constant term (12) itself should also be invariant, which leads to the fol- lowing equation for a:

a + 6 a2 = 1.c-25

limit c --" -oo, we obtain

(14)

(17)

(18) Requiring that a should go to 1 in the classical

a= 25-c- (1-c)(25-c) .12 (19)

AFP[jeO] = ®FP[9]e- MSL[i ;ml 6 Zm[9eO] = Zm[9]e +8*sL[i;~] (7)

D10 = DoOeT-ksL[9 ;01, (8)

where SL is the Liouville action,

SL W 0] = d2

x,r(ieva0

We thus find that the partition function of two-dimensional quantum gravity (1) can be ex- pressed in terms of a free field as (11) with a given by (19). From these expressions, the A de- pendence of Z(A) can be exactly calculated by considering a constant shift of 0 given by

-~ -I- ¹alog A. (20)

It is easy to see the following changes under (20) :

where X is the Euler number of the two dimen- sional manifold. Substituting (21)-(23) into (11), we obtain

Z(A) = Z(A = 1) - A-b''^{-i , (24)} where b is given by

b - 25 - c + (1- c)(25 - c) .12 (25) The exponent b is a monotonically decreasing function of c when c < 1, and becomes complex when c is greater than 1 . The latter indicates an instability as we shall discuss in subsection 2.3 .

2.2. Discretization

Dynamical triangulation [4]

A:~ in the case of lattice gauge theory, the ba- sic problem of discretization is how to maintain the symmetries of the theory under considera- tion. In the theory of quantum gravity, the fun- damental symmetry of the system is the diffeo- morphism invariance. Although it is a kind of gauge invariance, it is not easy to express it on a lattice because infinitesimal motions it requires

H. Kawai/Quantumgravity andrandom surfaces

are not allowed on a discretized space-time. We can reinterpret, however, the diffeomorphism in- variance in the following way: First we cut out two disks from the manifold on which fluctuat- ing fields are defined. If the theory has the dif- feomorphism invariance, the field fluctuations on these two disks should have the same structure, because any field configuration on one disk can be isomorphically mapped to that on the other disk. In other words, the diffeomorphism invari T.nce in quantum theory can be regarded as the property that field fluctuations on any two disks are isomorphic. If we use a fixed discretization of the manifold, it is not easy to guarantee this property, because the two disks in general have different lattice structures. On the other hand, this property is automatically satisfied if we sum over all possible ways of discretization, because if one lattice structure appears on one disk, then the same lattice structure should appear also on the other disk. Ordinary field theories have the property of universality, that is, once two theo- ries have the same symmetry, then they have the same continuum limit unless extra fine tunings are introduced among the coupling constants of the theory. If we assume that universality also holds for the case of quantum gravity, details of discretization should not matter for construction of a continuum theory of quantum gravity pro- vided that the property discussed above is satis- fied.

One of the simplest choice of discretization is to consider all possible triangulations using a fixed shape and size for triangles. In this case, the partition function Z(A) defined by (1) can be discretized as

Zreg(A)

95

=

E

G:triangulation

Here, Z,n(G; Q = #^c) stands for the matter parti-

D00 --> V00, (21)

St[9; 0] -} St[9; 0] + logA - 47rX,a (22)

ƒd2xXrg :ea0 :k->A d2xxrg :eaß :k, (23)

tion function defined on the triangulation G with the coupling Q fixed to its critical value Q,, and n(G) is the number of triangles contained in the triangulation G. Using the matrix model tech- nique, one can evaluate (26) for a variety of min- imal conformal matter fields with c < 1 . One finds that

Zregreg rc`~A- ;r*-1, (27)

where the value of b coincides with that of the continuum theory given by (2,b) .

So far we have considered the partition func- tion, and checked that the dynamical triangula- tion indeed reproduces the results of the contin- uum theory. Strictly speaking, however, we have to check the Green's functions in order to prove the real equivalence of these two theories. For the continuum theory several groups calculated the Green's function on the sphere up to four-point functions[5] . On the other hand, with the dynam- ical triangulation one can calculate Green's func- tions for any number of points for any genus[6], and the results all agree with those of the contin- uum theory. Although calculational techniques for the continuum theory is not completely de- veloped, there seems to be little doubt in the equivalence of the dynamical triangulation and the continuum theory.

Regge calculus [7,8]

Another possibility of discretizing quantum gravity is the Regge calculus[7] . In this formu- lation, we fix the triangulation and express the varying metric in terms of length variables of bonds. Then the cosmological constant term and the Einstein action are expressed as

dDx~

= E

simplex

dD x ,,fg-R = 40i, i:bond

H. Kawai/Quantumgravity andrandom smfaces

(29)

where 1i is the length of the bond i, and 9i is the deficit angle around the bond i. Although this formulation gives a clear geometrical meaning to the action, it is not clear a priori what measure for the li's corresponds to the reparametrization invariant measure Dg . Therefore we have to try various measures and examine whether they give the correct answer. In ref[9], a Monte Carlo sim- ulation was performed for two-dimensional pure gravity with the Regge calculus. Using lattices of a size 42 -642 , the authors found that the scale invariant measure dfi/$i reproduces the results of the continuum theory:

Z(A) - A-3.ss±0.22 for sphere, (30)

A-1 .025:0.022 for torus

which is compared with the exact value of the exponent -3.5 and -1, respectively. This good agreement indicates that the Regge calculus should be taken more seriously, although the symmetry property is not very transparent in this formula- tion .

,2.3. Instability for c > 1

(31)

As we have seen in eq. (19), the cosmological constant term becomes complex when c becomes greater than 1. This phenomenon can be under- stood as an instability of the space-time against the formation of pinches. In the dynamical trian- gulation picture, Zreg(A) in (27) represents the number of configurations consisting of A trian- gles. For example, if we consider a sphere (X=2), the partition function

Z(A) a^{%o KAA-h-1} (32)

corresponds to the canonical ensemble of trian- gulations of the sphere consisting of A triangles.

Let us consider how many configurations belong- ing to this canonical ensemble have the shape of

A

Fig. 1 . Sphere with a pinch

A -A'

a pinched sphere . Letting A' and A - A' be the area of the two spheres separated by the pinch (see Fig.1), we can estimate a lower bound for the contribution of such pinched spheres to (32) as follows:

const . 2A/sZ(A') A' - Z(A - A') (A - A')dA'

A/3

A,-°° const .icAA-2e+1, (33)

Here, we have used the fact that making a pinched sphere is equivalent to making two punc- tured spheres, and that the partition function for a punctured sphere of area A' is proportional to Z(A')A' . By comparing (32) and (33), we find that the space-time becomes unstable against the formation of pinches for b smaller than 2. On the other hand, eq.(25) shows that b decreases monotonically to 2, when c increases from -oo to 1. Therefore, it is natural to regard the com- plex value of b and a for c > 1 as an indica- tion of such an instability. In other words, if c is greater than 1, the space-time has infinitely many pinches, which means that it takes the shape of a branched polymer.

,2.4. Fractal structures of space-time Intrinsic fractal dimension

One way to see how wildly the space-time met- ric fluctuates is to consider the intrinsic fractal

H. Kawai/Quantum gravity andrandom surfaces

dimension dF .[10] We consider a small but not too small a domain on the manifold, and com- pare its volume with its "typical size" . If they satisfy the relation,

dFis called the intrinsic fractal dimension of the space-time. Obviously, the value of ^dF depends on the definition of the typical size. One extreme choice is to consider the geodesic diameter of the domain. In the dynamical triangulation picture, this extreme choice can be expressed as

R(L) L ...°° LdF ,

97

(volume) ti ("typical size"^)dip, (34)

(35) where R(L) is the number of siteq on the lattice which can be arrived within L steps starting from a fixed site P. Monte Carlo simulations have been performed for two-dimensional quantum gravity with c = 0[11] and with c = -2[12] . The results are controversial: The authors of reQ11] claim a doubly logarithmic behavior,

log R(L) - co + ci log L +^c2(log L)2, ⁽³⁶⁾ which indicates dF = oo, while the authors of ref.[12] claim that R(L) shows a fractal struc- ture,

logR(L) ~ co + dF log L, dF = 3.55. (37) We can not directly compare these two results since they are for different models. However, the analysis of [12] indicates that one has to consider a very large lattice (N ti 5 x 106) in order to see the fractal structure. Therefore the result of reQ11] might still suffer from finite size effects, since their lattice size is only N - 1 x 105. One reason why we need such a large lattice to mea- sure ^dF seems to be the use of an extreme defi- nition of the typical size. Instead of the geodesic diameter, we can introduce a smeared size by taking the average length of various paths in the

domain. Then the approach to the asymptotics is expected to be faster.[13]

Quenched field on the fluctuating space-time Another way to examine the fluctuation of the space-time is to introduce quenched matter fields. For example, we can introduce the Ising model and consider the quenched average of the free energy :

F(h, t,A) -_

f

f

Here, Dp(g) is the measure for two-dimensional quantum gravity coupled to the scale invariant matter with central charge c,

DP(g) = voVBiffd2x~- A)' (39)( )^ZM[9]$(

and F(h, t; {g}) is the free energy of the Ising model defined on the curved space-time with metric g,,,, . In ⁽³⁸⁾ h and t are the magnetic field and the temperature, respectively; that is, we consider the action for the Ising model given

by

S = S* + h

ƒd2x~er + t d2x~e, (40) where S* is the fixed point Hamiltonian, and o-(x) and e(x) are the spin and the energy op- erators, respectively.

The free energy of the gravitational spin glass (38) can be expressed in the conformal gauge, andan analysissimilar to that leading to eq.(24) gives the following scaling equations:

F(h,_{t, A) =}^F(A-(2-d«)h,A-(2-d.)e;A2A) . (41 )

The quenched gravitationally dressed dimensions

deand dE are given by

di = 2(1 - O'),a (42)

pi

c)

12 '

H. Kawai /Quantum gravity and randons surfaces

(43)

where do is the scale dimension of the corre- sponding operators ,

do = {1/8, i=a1, i=e ' (44)

and a is given by eq.(19) . By using the general relations (D = 2 in this case)

a= D - 2d, D-dE ' D-d,'do D-2do

y = D-d, '

we can calculate various critical exponents.

(Here, a, ft and y stand for the critical exponents for the heat capacity, the spontaneous magneti- zation and the susceptibility, respectively. Note that a in (45) has nothing to do with a in (19) .) In ref.[14] a Monte Carlo simulation was made for the case of c = 0 by taking one sample back- ground configuration . The results are

a ti 0, 6 - 0.149 - 0.207, y , 1 .75 - 2.5, (46) while the values obtained from (45) are

a = -0.868 . - -,P =0.386 - - -, y = 2 .096 - .(47) The discrepancy should disappear, if one takes the average over various background configura- tions or employs a lattice of a much larger size.

2.5. Summing up topologies

In the usual framework ofstring theories, scat- tering amplitudes are calculated for each topol- ogy of the worldsheet separately, and their rel- ative weights are fixed by requiring unitarity.

This procedure corresponds to calculating Feyn- man diagrams order by order in ordinary local field theories. As is well known, such a procedure is not sufficient when non-perturbative effects

are important. In the case of superstrings, non- perturbative effects play an essential role to ob- tain the true vacuum from infinitely many clas- sical vacua. So far, there is no satisfactory non- perturbative formulation of superstrings . For non-critical strings with c < 1, however, we can give a non-perturbative definition by using the equivalence of the dynamical triangulation and the matrix model. For simplicity, we consider here the case of pure gravity (c = 0) .

Let us consider the following one-matrix model:

S = N(1tro2 -

a

2 3

where ¢ is an N x N hermitian matrix. An anal- ysis of Feynman diagrams shows that the 1/N expansion of the free energy of this theory can be regard as a gene.ating function of the partition function of two-dimensional quantum gravity for various topologies. In fact, we have

F = log ( doe-S)

= E

x

= N2Z2(A) + N°Za(A) + N-2Z_2(A) + . . . ,

where Zx(A) is the partition function of the dynamically triangulated two-dimensional quan- tum gravity on the manifold of Euler number X:

ZX(A)

= 1:

triangulations

= 1:

A

Using (27) we find that the series in (50) has the convergence radius A, = 1/rc and its critical behavior is given by

Zx(A) a-, (A C - A)6.2,

where cx is a constant . By substituting (51) into

H. Kawai/ Quantum: gravity

(49)

(51)

This equation indicates that if we consider the double scaling limit, N -+ oo and A -+ oo with (Á, - A) - N2^/b kept fixed, then all topologies equally contribute to (49), and yet information for each topology can be extracted by examining the various power behavior in t. The authors of ref.[6] obtained a kind of the Schwinger-Dyson equation for (49) by using the orthogonal poly- nomial method:

f2+1d2f =t3 dt2 ' (54)

where f = d2F/dt2 . This equation is expected to contain full non-perturbative effects, although the way to fix the integration constant is not yet clear. This method is applicable to any (p, q) con- formal field coupled to two-dimensional quantum gravity. In fact, it was shown that the two-matrix models with various polynomial potentials for the hermitian matrices A and B defined by S = Ntr

(2

+V(A) + U(B)) (55)

reproduce all the (p, q) conformal field theories coupled to two-dimensional quantum gravity.[15]

These beautiful results might give a clue to a non-perturbative description of superstring the- ories, if one could succeed in introducing local scale invariance and worldsheet supersymmetry into the matrix models.

ndranduni surfaces 99

(49), we have

F ⁿⁱ⁶ cXNX(A, x

= cxtb-f, (52)

x

where t is defined by

t=(A -A)-Ni . (53)

Fig. 2. Loop amplitude

H. Kawai/Quantum gravity and randon: surfaces

2

.6. Operator product expansion in quantum gravity

Operator product expansion is one of the most fundamental concepts in the ordinary field the- ories. In fact, if one knows the set of all local operators, and the operator product expansion among them, the theory is completely specified.

On the other hand, in the presence of quantum gravity, operator product expansion can not be simply formulated, because the notion of dis- tance between two points is not well defined. It is still possible, however, to discuss whether the two points of operator insertions coincide or not . Therefore, if something similar to operator prod- uct expansion exists in the case of quantum grav- ity, it should be a set of relations telling us what happens when the two local operators coincide.

As we discuss below, this is indeed true at least in the case of two-dimensional quantum gravity, and such operator product expansion-like rela- tions completely specify the theory.

We consider for simplicity the case of two- dimensional pure gravity (c = 0) . In this case we can construct a complete set of local operators by expanding the loop amplitudes with respect to the loop length . More precisely, take a two-

dimensional manifold M with k marked bound- aries bl ,b2, - - -, bk (see fig.2) We then perform the path integral over the metric with the length of each boundary bi kept fixed to 1i

9(k)(£1~ . . .,tk)

D9 -t

f

1

topologies Vol(Diff)e

k

6( / g,,dzPdxv - fi), (56)

i-1

where t is the cosmological constant. These loop amplitudes can be expressed as the double scal- ing limit of the connected Green's functions of the one-matrix model. For example, if we con- sider the 04 matrix modal, we have

9(i(~1, .  ^{Jk) =}

(2N k/2

Nimoo ( (2-~)2(m,~.. ...~mk) ^w2m1

-(nonuniversal parts)

1,

w2mk

where fi = 2N-1^/5mi (i = 1, 2, - - -, k), and

< wn , * - wnk >, stands for the connected Green's function of the loop operators wn =

tr(on )/N for the ^¢4 hermitian N x N matrix model defined by

5' = Ntr(2102 - A04).4 (58)

The reason why the non-universal parts should be subtracted in (57) is that Feynman diagrams with a small number of squares dominate for sur- faces with topologies of a disk (k = 1) and a cylinder (k = 2). The explicit forms for the non- universal parts are given by

By examining the matrix model (58), one can show that the loop amplitudes can be expanded

8vr2i _5/2-

t

Vir (~i

8~i ) for disk, (59)

1 _(91£2)-1/2

for cylinder. (60) Ir V1 +^t2)

LHS

= RHS

k

H. Kawai/ Quantum:gravity an :anyni surfaces 101

ní+1/2 nk+1/2 . . .

r,

(61)

We then identify the coefficients cní,. ..,nk with

s

Fig. 3. Schwinger-Dyson equation for the q54 matrix model

remove the bridges

by half-odd integer powers of li's : the connected Green's functions of local opera- tors 01,d3,X75, - .:

9(k)(I1' . . . ilk)

00 Cn i,. ..,nk

_ 2ní+3

. . . 2nk+3

I'(nl +3/2)

r(nk +

02ní+1 . . . 02nk+Z >ci (62)

102

where, for the later convenience, we have labeled the operators by odd integers and introduced the normalization factors shown above . In this way we have constructed an infinite set of operators from the loop amplitudes.

The next step is to rewrite the Schwinger- Dyson equation in terms of these operators.

The Schwinger-Dyson equation for the 04 ma- trix model is expressed as

< wM+lwn, . . .w >e

m-13=0

-A < _wm+.wn,

nj < wn, " . . wni_iZhni+m-1Wni+i . . . . . .wnk >

< Wj Wm-j+lwn, " . . wnk

E <W.,

=0 SC{1,. . .,k} rESw . < WM-j-1

H. Kawai/ Quantumgravüy and random surfaces

. .wnk

£ESWnt >, (63)

where S runs over all subsets of {1

_ , ..., k} and

S is the complement of S. The diagrammatic meaning of (63) is shown in Fig.3: If one re- moves the square attached to the boundary of length m+1, the length of the boundary becomes rra+3 (left-hand side of (63)) . There are, however, the exceptional cases in which the boundary is directly connected to one of the other bound- aries or to itself. These cases correspond to the right-hand side of (63) . Therefore the Schwinger- Dyson equations are a set of identities indicating what happens under the deformation of a bound- ary. Taking the double scaling limit of (63) and considering the definition of the local operators

(61) and (62), we obtain t < OPOni . . ._{onk >,}

k

-

3

j=1

-~.~ E < OrOp-l-rOn, . . Onk

< or r SC{1,2,.. .,k} jeS

>c

< Op-1-r Ont >c ^. (64) 1ES

The geometrical meaning of (64) is depicted in Fig.4 for the case of k = 1 and surfaces with genus 1. These equations show what happens if the two operators 0, and Op-1, coincide. In this sense we can regard eq.(64) as a prototype of ¢he operator product expansion for quantum gravity. We emphasize here that the Schwinger- Dyson equation (63) can be derived by a purely geometrical analysis as in Fig.3, which can also be carried out in higher dimensions if we in- troduce boundaries of various topologies. There- fore, if the continuum limit of discretized quan- tum gravity exists in higher dimensions, operator product expansion-like relations are expected for various local operators.

Equation (64) turns out to have a simple struc- ture if we introduce the generating function of the connected Green's functions defined by

F(x1,X3i X5,- . .) =

< e-("- t)01-x303-(xa+8/15)05-zzpr-. .. > .(65) Then the continuum Schwinger-Dyson equation (64) is expressed as a formal Virasoro constraint :

Lne V = 0 (n > -1), (66)

n

9 4

Zr

where Ln's are Virasoro generators defined by

2Ln = 2

~ E

3

~~ (Yr arp-1-_r

® ~n

+ 1

a a

E

2_p+q=2n ôxp ôxq 8

It is shown that eF/2 satisfying (66) is a r- function of the KdV hierarchy, and that if ^eF/2 is such a r-function the full set of equations (66) reduces to ^L_1eF/2 = 0. Furthermore, eq.(66) gives a universal description of various confor- mal fields coupled to two-dimensional quantum gravity. In fact, the free energy of the (2,q) con- formed field coupled to two-dimensional quan- tum gravity can be obtained by setting xl =

t, 02+q =const andxL. = 0 otherwise . The same method applies to two-matrix models, and it was checked that the (p,q) conformal field coupled to two-dimensional quantum gravity is described universally for all values of q by aformal vacuum condition of the Wp algebra.[17,16]

H. Kawai/Quantum:graváry and random surfaces

2 r .

Fig. 4. Continuum Schwinger-Dyson equation

,2.7. Other progress in two-dimensional quantum gravity

103

(i) The BRST cohomology for the Liouville- matter system was analyzed in [18] . It was found that even the physical state carries ghost num- bers. This is expected, however, because the de- grees of freedom for two-dimensional quantum gravity coupled to c < 1 matter is negative. At a first glance, matrix models seem to contain more operators than are expected by the BRST coho- mology. However, those operators which do not correspond to the BRST cohomology are shown to be redundant in the sense that they are to- tal derivatives with respect to the matrix 0.[19]

In other words, such boundary operators are ab- sorbed into the redefinition of the background source of the other operators. (See the third ref-

erence of [16])

(ii) Some of the Virasoro constraints (66) were reproduced from the continuum Liouville theory, and are understood as Ward-Takahashi identities for the string field.[20] In [21] the authors in-

104

terpreted the Schwinger-Dyson equation for the loop amplitudes as the Wheeler-DeWitt equa- tion.

(iii) The generalization to the case of two- dimensional supergravity is rather straightfor- ward in the continuum theory.[22] However, cor- responding lattice models are not yet known.

Only a plausible generalization of the Virasoro constraint has been discovered .[23]

(iv) The W-gravity was constructed by ex- tending the worldsheet Virasoro algebra to the W-algebra.[24] The geometrical meaning of the generalization of the diffeomorphism invariance

is yet to be clarified .

(v) Two-dimensional quantum gravity coupled to c = 1 matter turned out to have rich structure.

A string field was constructed in [25], and a W,,.- structure was discovered in [26] .

(vi) The topological gravity coupled to mini- mal topological matter was shown to be equiva- lent to the (p,1) conformal field coupled to two- dimensional quantum gravity.[27] Matrix modeis corresponding to the topological gravities were constructed in [28] based on the idea of trian- gulation of the moduli space. They are closely related to the open-string field.

(vii) The Liouville theory was analyzed for general values of c using the quantum group, and a possibility to go beyond c = 1 was pointed out in[29].

3 . Random surfaces

3.1. Models of random surfaces

As is well known, spin systems or scalar fields are closely related to the problem of random walks. In particular, the universality class of the free field corresponds to the free random walk, for which universality is guaranteed by the cen-

H. Kawai/ Quanngravity mid rcuulom surfaces

tral limit theorem. Similarly gauge systems seem to be related to the problem of random surfaces.

This is expected from the strong coupling ex- pansion of lattice gauge theory or from large- N fishnet diagrams.[30] This suggests that free random surfaces may be a natural starting point for treating gauge systems analytically. It is also expected that theory of random surfaces would help us to understand certain three dimensional statistical models.[31]

There are two typical models for the free ran- dom surface :

(i) random surfaces on a regular lattice. [32]

The Weingarten's model is obtained by replac- ing the Hear measure for SU(N) matrices in the usual lattice gauge theory with the Gaussian measure:

S = N(L: An,IAAn,p_n,p

An~pAn+p,vAnn,v) (68)

n p,v

where An,p's areNx N complex matrices . If we regard the second term in (68) as a perturbation to the first term, the perturbation series corre- spond to summing up free random surfaces. The free energy is given by

F = log ( dAn,pe-S)

2:

S:closed surface (69)

where A(S) is the number of plaquettes con- tained in the closed surface S, and X(S) is the Euler number of S. Although (68) is not well- defined for a finite N, it generates a well-defined

1/Nexpansion.

(ii) dynamical triangulation of the Polyakt string. An alternative approach to the free ran- dom surface is to apply dynamical triangulation to the Polyakov string,

S = d^2~~9^abaa X^uabXis_{+ i1} d2 ^~. (70) These two approaches are essentially equiva- lent. In fact, if we consider squares instead of triangles in (ii) and discretize X" in an appro- priate way, the model of (ii) is reduced to (i) .

3.2. Problem with D > 1

The Polyakov string in the D-dimensional tar- get space is nothing but a two-dimensional quan- tum gravity coupled to D scalar fields, that is, c = D. Therefore, as is discussed in Section 2- 3, it is in the branched polymer phase, and can not represent smooth surfaces. This is explic- itly checked by a Monte Carlo simulation for the Weingarten's model for a large N.[33] The par- tition function is found to behave as

Z(A) ^A-*00 rcAA-b^o-1 (for sphere), (71) with the exponent

bo = 1 .4 f 0.2 for D = 2,

S=

1 .5 f 0.2 for D = 3, (72)

which is clearly less than 2, indicating tha~ the model is in the branched polymer phase . An ex- act proof of this statement was made later in

[34] .

3.3. Random surfaces with extrinsic curvature One possible way to avoid the blanched poly- mer phase is to introduce an extrinsic curvature term into the worldsheet action.[35] For simplic- ity, if we consider three dimensional target space, the worldsheet action is given by

d2 ^~9abôa^XóbX

+ a

+ic d2~-29abaanabn, (73)

H. Kawai /Quantum gravity and random surfaces 105

where the last term represents the extrinsic cur- vature term, and n is the normal vector to the surface in the target space. It is obvious that for strong enough Pc the surface is almost flat, because the extrinsic curvature term forces each surface element to align in one direction. On the other hand, if is = 0, the surface is in branched polymer phase. Therefore it is natural to expect a phase transition at some critical value of rc.

The dynamical triangulation version of (73) was proposed in [36]. They considered the fol- lowing action on the dynamically triangulated lattice:

S

= ₂ ^{E(X, -}

^E(1-

ij ij

where

eij

triangles connected by the bon:1 < i, j > . Monte Carlo simulations were performed on the regular triangular lattice in [37], and the au- thors observed a second-order phase transition at

#cc = 0.885 f 0.01. A similar phase structure was also observed in the case of the dynamical trian- gulation.[38] These results suggest that the sys- tem can be regarded as a two-dimensional quan- tum gravity coupled to the scale invariant mat- ter that corresponds to the fixed point found in [3:']. In this context, it is important to measure the central charge at this fixed point. If the cen- tral charge is less than 1, the model is consistent with the general analysis of Section 2-3, but it should violate positivity. (This is because there is a renormalization group flow from ic = x, to

#c = 0, and therefore the c-theorem asserts that the central charge for x = x, is greater than 3, the central charge for ic = 0, unless positivity is violated .) However, if we do not care about positivity, we can always decrease the total cen- tral charge by introducing non-unitary matters on the worldsheet in addition to the original tar- get space coordinate. Therefore, there is no rea-

son to insist on the special form (73) . On the other hand, if we require that the theory be pos- itive definite, we need some miraculous cancel- lation in order to suppress branched polymers, since the central charge in this case is greater

than 3.

3.4. Hausdorf dimensions

The Hausdorff dimensions ^dH of random sur- faces is defined by the power-like relation be- tween the extent of the surface in the target space and the worldsheet area:

< X'` X" >^{A ,00A2/dH}

< XI`XIU >

(75) For the simple random surface with D < 1 de- fined by

S=

87r d%~g°baaXj<sub>`abX</sub>j`+a d2~~,(76) the Hausdorff dimension is calculated in the con- formal gauge with the use of several additional assumptions [39]; the extent of the surface be- haves as

-D V^{1-D -F} 25-D 1ogA+ui

1-D

for D < 1, (77)

= (log A)2⁺vi log A +^v2

for D = 1, (78)

where the left-hand side stands for the mean square distance defined by

AZ < ƒ d21_ ^{~i g( i)}

C1Zg2 g(~2)(Xo(~i) -X" (6))2 ^>, (79) andU1,vi,^v2 are non-universal constants. Monte Calro simulations for D = 1 were performed both for dynamical triangulation[40] and for Regge

H. Kawai/Quantua: gravity and raisdomsurfaces

calculus[9]. In [40] it was found that the coef- ficient of (log A)2 in (78) is very close to 1, while the authors of [9] observed that the coefficient is almost 0. Since the predictions (77-78) are still rather ambiguous, it is hard to conclude something definite from these simulations at the present stage.

4 . Quantum gravity in d > 2

4.1 . Qualitative picture of quantum gravity in d>2

It is well known that the Einstein gravity is formally asymptotically free in two dimensions.

Therefore it is natural to attempt to apply the c- expansion around two dimensions.[41] However, there is an essential difference compared to the case of the non-linear model: Since the Einstein action is a total derivative in two dimensions, the graviton propagator has an exta pole in 2 + c di- mensions. This extra pole invalidates the simple power counting for d = 2, and thus causes a great difficulty when one tries to apply the conven- tional idea of the e-expansion around the lower critical dimension.

Although nobody has succeeded in construct- ing a satisfactory c-expansion due to the above mentioned difficulty, there is evidence enough to believe that the two-dimensional quantum grav- ity is the c --" 0 limit of the renormalization group fixed point of 2 + e dimensional quan- tum gravity. In the case of the two-dimensional quantum gravity, the conformal mode is stable if c < 25, and the space-time is stable against pinch formation if c < 1 as was discussed in sec- tion 2-3. This suggests that the 2+c dimensional gravity is stable if e is not too large. Therefore we can naturally believe that the integral,

is well-defined for d-dimensional quantum grav- ity.

If the ft-function behaves as naively expected from asymptotic freedom (see Fig.5), we have a critical value Gó for Go. For Go < Gó, the ef- fective Newtonian constant Geff tends to van- ish in the infrared region, and we expect small large-distance fluctuations. On the other hand, for Go > Go*, Geff becomes large in the infrared region, and we have severe large-distance fluc- tuations. As we shall discuss in the next sub- section, numerical analyses seem to support the phase structure discussed here. If this is true and the phase transition is of second-order, we can achieve a field-theoretical description of quan- tumgravity by taking the continuum limit, Go = Gó - 0, as in the case of ordinary field theories.

4.2. Numerical analyses

(i) Dynamical triangulation [42]

In the dynamical triangulation picture ofthree- dimensional quantum gravity, the action can be expressed as

d3x.Vrg = N3_{, (82)}

d3xVg-R = cNl - 6N3,

where c = 27r/ arccos(1/3), and No, N1, N2 and N3 are the numbers of vertices, links, triangles and tetrahedra, respectively, contained in the tri- angulated manifold. Only two of four Ni's are independent, because of the following relations:

H. Kawai /Quantum gravity and random surfaces

(83)

P(Go)

Fig. 5. fl-function for d > 2 dimensional gravity 107

N3-N2+Ni-No=0, (85)

The first equation is the consequence of the fact that each triangle is shared by two tetrahedra, and the second means that the Euler number of any three dimensional closed manifold vanishes.

Using these relations, we can express the action in terms of No and N3 as

S = Ao

d 3xf-

G

(ao -_ 6Gn c

)N3 Gn No . (86)

The path integral (80) on a three-dimensional sphere then becomes

E

triangulations ofS3

(87)

In the second reference of [42], for example, a Monte Carlo simulation was made for N3 =

8000, and a clear phase transition was found around c/Go se 40. This phase transition, how- ever, seems to be first order, which suggests the necessity of adding another type of interactions in order to obtain awell-defined continuum limit.

A similar first-order phase transition was ob- served also for four-dimensional quantum grav- ity.[42]

vol(Diff) ( d2XVF V),-- (80) with S the Einstein action

S = - 16 Go ddx~R. (81)

(ii) Regge calculus

The authors of [43] performed Monte Carlo simulations of Regge calculus for three and four dimensions, and they found a second-order phase transition, which is consistent with the result from the c-expansion.

5. Conclusion

At present we have two possible approaches to quantum gravity. One is the theory of super- strings and the other is based on the ordinary local field theory. The main difficulty of super- strings lies in the fact that they have too many classical vacua and we need a non-perturbative treatment in order to make physical predictions.

The techniques developed for the matrix model in the last two or three years will help us to study this problem greatly if one could find a way to implement eyl invariance and world- sheet supersymmetry. On the other hand, the re- cent progress in the lattice gravity have opened a possibility of describing quantum gravity as an ordinary local field theory. One of the the most urgent problems is to examine whether a non- trivial fixed point exists in four dimensions or not. It seems that both approaches, dynamical triangulation and Regge calculus, are worth pur- suing to answer the problem. These approaches are quite suitable for Monte Carlo simulations, and we expect that they will produce a lot of new physics in the field of quantum gravity just as they have done in the lattice gauge theory.

Acknowledgements

The author is grateful to the organizers of Lattice 91, especially to Prof. M. Fukugita and Prof. A. Ukawa, for their persistent encourage.-

H. Kawai/Quantumgravity midrandom surfaces

ment and patient reading of the manuscript. He is also indebted to J. Ambjorn, C . Baillie, T.

Filk, H. Hamber, N. Kawamoto, A . Migdal and H. Neuberger for fruitful discussions during the Symposium. This work is supported in part by the International Scientific Research Program of the Japanese Ministry of Education, Science and Culture.

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