Nuclear Physics B (Proc. Suppl.) 26 (1992) 93-110North-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 todiscussion of two dimensional gravity, dynamical triangulation, matrix models and Regge calculus . We also discsthe possibility of constructing the theory of quantum gravity as an ordinary field theory in four dimensions.

1 . Introduction

There are many conceptual problems in thetheory of quantum gravity, such as the originof the space-time and the measurement of thewave function of the universe . We also do notknow a priori whether topology changes shouldbe 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 conceptualproblems . In these three years, since the equiv-alence of the dynamical triangulation and quan-tum gravity is established, we have experienced asubstantial progress in understanding dynamicalproperties of quantum gravity . In this talk, wesummarize the present status of quantum grav-ity on the lattice and random surfaces . In section2, we discuss two-dimensional quantum gravityfrom various points of view . In section 3, we con-sider random surfaces and discuss the difficultyofdefining random surfaces in higher dimensions .Finally, in section 4, we investigate the possibil-ity of defining four-dimensional quantum gravityas an ordinary local field theory.

2.1 . Continuum theory

0920-5632/'92/'$05 .00 0 1992- Elsevier Science Publishers BY

Allrights reserved.

2. Two-dimensional quantum gravity

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

and a scale invariant matter field with centralcharge c defined on the manifold . We then con-sider the partition function of this system withthe total volume of the. manifold fixed to A:

Z(A)

voDDff ZM[.9]6(

d2z~- A), (1)1( )

where ZM[g] is the partition function of the mat-ter field for the background metric g,�,(z), andvol(Diff) stands for the formal volume of thespace of diffeomorphisms.

In the conformal gauge, gauge slices are parametrized 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 modeand 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 DlO is induced fromthe following metric in the functional space of0:

ilb0 l 1 1 =

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 d2x

gr

Vygpvgxobgpx bgvo => Dg .

(5)

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

and Dog is the translation invariant measurewhich 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-25 SLUM

.b(

d2x/7g : eao :i -A),

(11)

where ¢ can be regarded as a free scalar field de-fined on a curved space with metric g,,� . If wetake this point of view, we should renormalize

H. Kawai/ Quantum gravity andrandom surfaces

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

jd2x-,Fg : e ° `O :i,

(12)

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

eaO i -

lim

(e.Î davp(y),O(y)P(y)~ab~~l (y-z)

.e-112 fd2yd'zp(y)P(z) K log d2#(y,z)) . (13)

Here, di (y, z) is the geodesic distance between yand z, and rc log d9 (y, z) is the singular part ofthe 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 underthe transformation given by

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

(15)(x)

-+ OW - 0(x),

(16)

because the original theory (1) is defined in termsof 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) itselfshould 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) .

(19)12

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,

SLW 0] = d2x,r(ieva00av0+ A0), (9)

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

-~

-I- 1 logA.

(20)a

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) .

(25)12

The exponent b is a monotonically decreasingfunction of c when c < 1, and becomes complexwhen c is greater than 1 . The latter indicates aninstability 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 maintainthe 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 ofgauge invariance, it is not easy to express it on alattice because infinitesimal motions it requires

H. Kawai/Quantumgravity andrandom surfaces

are not allowed on a discretized space-time. Wecan reinterpret, however, the diffeomorphism in-variance in the following way: First we cut outtwo disks from the manifold on which fluctuat-ing fields are defined. If the theory has the dif-feomorphism invariance, the field fluctuations onthese two disks should have the same structure,because any field configuration on one disk canbe isomorphically mapped to that on the otherdisk . In other words, the diffeomorphism invariT.nce in quantum theory can be regarded as theproperty that field fluctuations on any two disksare isomorphic . If we use a fixed discretizationof the manifold, it is not easy to guarantee thisproperty, because the two disks in general havedifferent lattice structures . On the other hand,this property is automatically satisfied if we sumover all possible ways of discretization, becauseif one lattice structure appears on one disk, thenthe same lattice structure should appear also onthe other disk . Ordinary field theories have theproperty of universality, that is, once two theo-ries have the same symmetry, then they have thesame continuum limit unless extra fine tuningsare introduced among the coupling constants ofthe theory. If we assume that universality alsoholds for the case of quantum gravity, details ofdiscretization should not matter for constructionof 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 afixed shape and size for triangles. In this case,the partition function Z(A) defined by (1) canbe discretized as

Zreg(A)

95

= E

Z n(G; 0 = Qc)6n(G),A-

(26)G:triangulation

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

D00 --> V00, (21)

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

ƒd2 xXrg : ea0 :k-> A d2xxrg : eaß :k, (23)

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

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

where the value of b coincides with that of thecontinuum 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 haveto check the Green's functions in order to provethe real equivalence of these two theories . For thecontinuum theory several groups calculated theGreen's function on the sphere up to four-pointfunctions[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 techniquesfor the continuum theory is not completely de-veloped, there seems to be little doubt in theequivalence of the dynamical triangulation andthe 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 thevarying metric in terms of length variables ofbonds . Then the cosmological constant term andthe Einstein action are expressed as

dDx~

=

E volume of simplex,

(28)simplex

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

H. Kawai/Quantum gravity andrandom smfaces

(29)

where 1i is the length of the bond i, and 9i is thedeficit angle around the bond i . Although thisformulation gives a clear geometrical meaning tothe action, it is not clear a priori what measurefor the li's corresponds to the reparametrizationinvariant measure Dg . Therefore we have to tryvarious measures and examine whether they givethe correct answer . In ref[9], a Monte Carlo sim-ulation was performed for two-dimensional puregravity with the Regge calculus . Using lattices ofa size 42 -642 , the authors found that the scaleinvariant measure dfi/$i reproduces the resultsof 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 theexponent -3.5 and -1, respectively. This goodagreement indicates that the Regge calculus shouldbe taken more seriously, although the symmetryproperty is not very transparent in this formula-tion .

,2.3 . Instability for c > 1

(31)

As we have seen in eq . (19), the cosmologicalconstant term becomes complex when c becomesgreater than 1 . This phenomenon can be under-stood as an instability of the space-time againstthe formation of pinches . In the dynamical trian-gulation picture, Zreg(A) in (27) represents thenumber 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 thearea of the two spheres separated by the pinch(see Fig.1), we can estimate a lower bound forthe contribution of such pinched spheres to (32)as follows :

2A/sconst .

Z(A') A' - Z(A - A') (A - A')dA'A/3

A,-°° const .icAA-2e+1,

(33)

Here, we have used the fact that making apinched sphere is equivalent to making two punc-tured spheres, and that the partition function fora punctured sphere of area A' is proportional toZ(A')A' . By comparing (32) and (33), we findthat the space-time becomes unstable against theformation of pinches for b smaller than 2 . Onthe other hand, eq.(25) shows that b decreasesmonotonically to 2, when c increases from -ooto 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 cis greater than 1, the space-time has infinitelymany pinches, which means that it takes theshape 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 nottoo small a domain on the manifold, and com-pare its volume with its "typical size" . If theysatisfy the relation,

dF is called the intrinsic fractal dimension of thespace-time. Obviously, the value of dF dependson the definition of the typical size . One extremechoice is to consider the geodesic diameter of thedomain . 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 latticewhich can be arrived within L steps starting froma fixed site P. Monte Carlo simulations have beenperformed for two-dimensional quantum gravitywith c = 0[11] and with c = -2[12] . The resultsare controversial : The authors of reQ11] claim adoubly logarithmic behavior,

log R(L) - co + ci log L + c2(log L)2 ,

(36)

which indicates dF = oo, while the authors ofref.[12] claim that R(L) shows a fractal struc-ture,

log R(L) ~ co + dF log L,

dF = 3.55 .

(37)

We can not directly compare these two resultssince they are for different models. However, theanalysis of [12] indicates that one has to considera very large lattice (N ti 5 x 106) in order tosee the fractal structure. Therefore the result ofreQ11] might still suffer from finite size effects,since their lattice size is only N - 1 x 105 . Onereason 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 geodesicdiameter, we can introduce a smeared size bytaking the average length of various paths in the

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

Quenched field on the fluctuating space-timeAnother way to examine the fluctuation of

the space-time is to introduce quenched matterfields . For example, we can introduce the Isingmodel and consider the quenched average of thefree energy :

F(h, t, A) -_ f DM(g)F(h't ; {g}) .f DP(g) (38)

Here, Dp(g) is the measure for two-dimensionalquantum gravity coupled to the scale invariantmatter with central charge c,

DP(g) = voVBiff ZM[9]$(d2x~-A)'

(39)( )

and F(h, t; {g}) is the free energy of the Isingmodel defined on the curved space-time withmetric g,,,, . In (38) h and t are the magneticfield and the temperature, respectively ; that is,we consider the action for the Ising model givenby

S = S* + hƒd2x~er + t

d 2x~e,

(40)

where S* is the fixed point Hamiltonian, ando-(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,and an analysis similar 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 dimensionsde and dE are given by

di = 2(1 - O'),

(42)api = 25-c-

(25-c)(1+12di'-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=a

(44)1,

i=e '

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

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

D-d,'D-2do

y =D-d,

'

we can calculate various critical exponents .(Here, a, ft and y stand for the critical exponentsfor the heat capacity, the spontaneous magneti-zation and the susceptibility, respectively. Notethat a in (45) has nothing to do with a in (19) .)In ref.[14] a Monte Carlo simulation was madefor 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 takesthe 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 localfield theories . As is well known, such a procedureis 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 . Fornon-critical strings with c < 1, however, we cangive a non-perturbative definition by using theequivalence of the dynamical triangulation andthe matrix model. For simplicity, we considerhere the case of pure gravity (c = 0) .

Let us consider the following one-matrix model:

S = N(1tro2 -a tro3) ~

(48)2 3where ¢ is an NxN hermitian matrix . An anal-ysis of Feynman diagrams shows that the 1/Nexpansion of the free energy of this theory can beregard as a gene.ating function of the partitionfunction of two-dimensional quantum gravity forvarious topologies . In fact, we have

F = log (

doe-S)

= ENXZX(A)x

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

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

ZX(A) =

1:

A#of triangles

triangulations

= 1: ÁAZreg(A) .

(50)A

Using (27) we find that the series in (50) hasthe convergence radius A, = 1/rc and its criticalbehavior 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 thedouble scaling limit, N -+ oo and A -+ oo with(Á, - A) - N2/ b kept fixed, then all topologiesequally contribute to (49), and yet informationfor each topology can be extracted by examiningthe various power behavior in t. The authors ofref.[6] obtained a kind of the Schwinger-Dysonequation for (49) by using the orthogonal poly-nomial method:

f2+1d2f =t3 dt2

'(54)

where f = d2F/dt2 . This equation is expectedto contain full non-perturbative effects, althoughthe way to fix the integration constant is not yetclear. This method is applicable to any (p, q) con-formal field coupled to two-dimensional quantumgravity. In fact, it was shown that the two-matrixmodels with various polynomial potentials forthe hermitian matrices A and B defined by

S = Ntr(2A2 + ~B2 + pAB

+V(A) + U(B))

(55)

reproduce all the (p, q) conformal field theoriescoupled to two-dimensional quantum gravity.[15]These beautiful results might give a clue to anon-perturbative description of superstring the-ories, if one could succeed in introducing localscale invariance and worldsheet supersymmetryinto 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 mostfundamental concepts in the ordinary field the-ories . In fact, if one knows the set of all localoperators, and the operator product expansionamong them, the theory is completely specified .On the other hand, in the presence of quantumgravity, operator product expansion can not besimply formulated, because the notion of dis-tance between two points is not well defined . Itis still possible, however, to discuss whether thetwo 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 whathappens when the two local operators coincide .As we discuss below, this is indeed true at leastin 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 wecan construct a complete set of local operatorsby expanding the loop amplitudes with respectto the loop length . More precisely, take a two-

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

9(k)(£1~ . . .,tk)D9

-tf d2 -- ..fg-1topologies

Vol(Diff)e

k

6( /

g,,�dzPdxv - fi),

(56)i-1

where t is the cosmological constant . These loopamplitudes can be expressed as the double scal-ing limit of the connected Green's functions ofthe 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,

(57)

w2mk

where fi

=

2N-1 /5mi (i

=

1, 2, - - -, k), and<

wn , * - wnk

>, stands for the connectedGreen's function of the loop operators wn =

tr(on )/N for the ¢4 hermitian N x N matrixmodel defined by

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

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

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

8vr2i 5/2 1/2- tVir

(~i8~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, cní,* . .,nk% 1

. . .~kí=o nk=o

(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+3I'(nl + 3/2)

r(nk + 3/2)

02ní+1 . . . 02nk+Z >ci (62)

102

where, for the later convenience, we have labeledthe operators by odd integers and introduced thenormalization factors shown above . In this waywe have constructed an infinite set of operatorsfrom 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

3=0m-1

-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

£ESWn t >, (63)

where S runs over all subsets of {1_,. .., k} andS is the complement of S. The diagrammaticmeaning of (63) is shown in Fig.3 : If one re-moves the square attached to the boundary oflength m+1, the length of the boundary becomesrra+3 (left-hand side of (63)) . There are, however,the exceptional cases in which the boundary isdirectly connected to one of the other bound-aries or to itself. These cases correspond to theright-hand side of (63) . Therefore the Schwinger-Dyson equations are a set of identities indicatingwhat happens under the deformation of a bound-ary. Taking the double scaling limit of (63) andconsidering the definition of the local operators

(61) and (62), we obtain

t < OPOni . . . onk >,

k

-3 < Op+40n, " . . Onk

_E nj < On, . . . Oni-lOp+nk-ldni+1 . . .j=1

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

< orr SC{1,2,. . .,k}

jeS

>c

< Op-1-r

Ont > c .

(64)1ES

The geometrical meaning of (64) is depicted inFig.4 for the case of k = 1 and surfaces withgenus 1 . These equations show what happens ifthe two operators 0, and Op-1, coincide . Inthis sense we can regard eq.(64) as a prototypeof ¢he operator product expansion for quantumgravity. We emphasize here that the Schwinger-Dyson equation (63) can be derived by a purelygeometrical analysis as in Fig.3, which can alsobe 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, operatorproduct expansion-like relations are expected forvarious local operators .Equation (64) turns out to have a simple struc-

ture if we introduce the generating function ofthe connected Green's functions defined by

F(x1, X3 i 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 :

LneV = 0

(n > -1),

(66)

n

94 Zr

where Ln 's are Virasoro generators defined by

2Ln = 2~

E

pgxpxq +

pxpôÓgp+q=-2n p-q=-2n

3

~~(Yr arp-1- r

® ~n

+ 1

a

a

+ 1E

bn,o.

(67)2p+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 quantumgravity. 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 and xL. = 0 otherwise . The samemethod applies to two-matrix models, and it waschecked that the (p, q) conformal field coupledto two-dimensional quantum gravity is describeduniversally for all values of q by aformal vacuumcondition of the Wp algebra.[17,16]

H. Kawai/Quantum:graváry and random surfaces

2

r

. (in

Fig. 4. Continuum Schwinger-Dyson equation

,2 .7. Other progress in two-dimensionalquantum gravity

103

(i) The BRST cohomology for the Liouville-matter system was analyzed in [18] . It was foundthat even the physical state carries ghost num-bers . This is expected, however, because the de-grees of freedom for two-dimensional quantumgravity coupled to c < 1 matter is negative . At afirst glance, matrix models seem to contain moreoperators than are expected by the BRST coho-mology. However, those operators which do notcorrespond to the BRST cohomology are shownto 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 backgroundsource of the other operators . (See the third ref-erence of [16])

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

104

terpreted the Schwinger-Dyson equation for theloop 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 Virasoroconstraint has been discovered .[23]

(iv) The W-gravity was constructed by ex-tending the worldsheet Virasoro algebra to theW-algebra.[24] The geometrical meaning of thegeneralization of the diffeomorphism invarianceis yet to be clarified .

(v) Two-dimensional quantum gravity coupledto 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 modeiscorresponding to the topological gravities wereconstructed in [28] based on the idea of trian-gulation of the moduli space. They are closelyrelated to the open-string field.

(vii) The Liouville theory was analyzed forgeneral values of c using the quantum group, anda possibility to go beyond c = 1 was pointed outin[29] .

3 . Random surfaces

3.1 . Models of random surfaces

As is well known, spin systems or scalar fieldsare closely related to the problem of randomwalks. In particular, the universality class of thefree 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 seemto 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 freerandom surfaces may be a natural starting pointfor treating gauge systems analytically. It is alsoexpected that theory of random surfaces wouldhelp us to understand certain three dimensionalstatistical 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 theusual lattice gauge theory with the Gaussianmeasure:

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

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

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

F = log (

dAn,pe-S)n,p

2: AA(s)NX(s),S:closed surface

(69)

where A(S) is the number of plaquettes con-tained in the closed surface S, and X(S) is theEuler number of S. Although (68) is not well-defined for a finite N, it generates a well-defined1/N expansion.

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

S =

d2~~9abaa X uabXis+ i1

d2 ~. (70)

These two approaches are essentially equiva-lent . In fact, if we consider squares instead oftriangles 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 cannot represent smooth surfaces . This is explic-itly checked by a Monte Carlo simulation for theWeingarten's model for a large N.[33] The par-tition function is found to behave as

Z(A) A-*00 rcAA-bo-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~ themodel 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 curvatureterm into the worldsheet action .[35] For simplic-ity, if we consider three dimensional target space,the worldsheet action is given by

d2 ~9abôaXóbX + a

d2~~

+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 thesurface in the target space . It is obvious thatfor strong enough Pc the surface is almost flat,because the extrinsic curvature term forces eachsurface element to align in one direction . On theother hand, if is = 0, the surface is in branchedpolymer phase. Therefore it is natural to expecta 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 triangulatedlattice:

S = 2E(X, - X,)2 + leE(1- COBBij), (74)ij

ij

where eij stands for the angle between the twotriangles 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#c c = 0.885 f 0.01 . A similar phase structure wasalso 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 measurethe central charge at this fixed point. If the cen-tral charge is less than 1, the model is consistentwith the general analysis of Section 2-3, but itshould violate positivity. (This is because thereis a renormalization group flow from ic = x, to#c = 0, and therefore the c-theorem asserts thatthe central charge for x = x, is greater than 3,the central charge for ic = 0, unless positivityis violated .) However, if we do not care aboutpositivity, we can always decrease the total cen-tral charge by introducing non-unitary matterson 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 theother 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 greaterthan 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 targetspace and the worldsheet area:

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

< XI`XIU >

(75)

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

S=

87r

d%~g°baaXj`abX

j`+a

d2~~,(76)

the Hausdorff dimension is calculated in the con-formal gauge with the use of several additionalassumptions [39] ; the extent of the surface be-haves as

-DV1-D -F

25-D 1ogA+ui1-D

for D < 1,

(77)= (log A)2 + vi log A + v2

for D = 1,

(78)

where the left-hand side stands for the meansquare distance defined by1_AZ < ƒ d2~i

g( i)

C1Zg2

g(~2)(Xo(~i) - X" (6))2 >, (79)

and U1, vi, v2 are non-universal constants . MonteCalro simulations for D = 1 were performed bothfor dynamical triangulation [40] and for Regge

H. Kawai/Quantua: gravity and raisdom surfaces

calculus[9] . In [40] it was found that the coef-ficient of (log A)2 in (78) is very close to 1, whilethe authors of [9] observed that the coefficientis almost 0 . Since the predictions (77-78) arestill rather ambiguous, it is hard to concludesomething definite from these simulations at thepresent stage .

4 . Quantum gravity in d > 2

4.1 . Qualitative picture of quantum gravity ind>2

It is well known that the Einstein gravity isformally 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 thecase of the non-linear model: Since the Einsteinaction is a total derivative in two dimensions, thegraviton propagator has an exta pole in 2 + c di-mensions . This extra pole invalidates the simplepower counting for d = 2, and thus causes a greatdifficulty when one tries to apply the conven-tional idea of the e-expansion around the lowercritical dimension .

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

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

If the ft-function behaves as naively expectedfrom asymptotic freedom (see Fig.5), we have acritical value Gó for Go. For Go < Gó, the ef-fective Newtonian constant Geff tends to van-ish in the infrared region, and we expect smalllarge-distance fluctuations . On the other hand,for Go > Go*, Geff becomes large in the infraredregion, and we have severe large-distance fluc-tuations . As we shall discuss in the next sub-section, numerical analyses seem to support thephase structure discussed here . If this is true andthe phase transition is of second-order, we canachieve 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 beexpressed as

d3x.Vrg = N3 ,

(82)

d3xVg-R = cNl - 6N3,

where c = 27r/ arccos(1/3), and No, N1,N2 andN3 are the numbers of vertices, links, trianglesand tetrahedra, respectively, contained in the tri-angulated manifold . Only two of four Ni's areindependent, 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 factthat each triangle is shared by two tetrahedra,and the second means that the Euler number ofany three dimensional closed manifold vanishes .Using these relations, we can express the actionin terms of No and N3 as

S = Aod 3xf-

d3x~RGo(ao -_

6Gnc)N3

Gn No .

(86)

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

E

ecNolGobN3~V .triangulations of S3

(87)

In the second reference of [42], for example, aMonte Carlo simulation was made for N3 =8000, and a clear phase transition was foundaround c/Go se 40 . This phase transition, how-ever, seems to be first order, which suggests thenecessity of adding another type of interactionsin 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 calculusThe authors of [43] performed Monte Carlo

simulations of Regge calculus for three and fourdimensions, and they found a second-order phasetransition, which is consistent with the resultfrom the c-expansion .

5 . Conclusion

At present we have two possible approachesto quantum gravity. One is the theory of super-strings and the other is based on the ordinarylocal field theory. The main difficulty of super-strings lies in the fact that they have too manyclassical vacua and we need a non-perturbativetreatment in order to make physical predictions .The techniques developed for the matrix modelin the last two or three years will help us tostudy this problem greatly if one could find away to implement eyl invariance and world-sheet supersymmetry . On the other hand, the re-cent progress in the lattice gravity have openeda possibility of describing quantum gravity as anordinary local field theory. One of the the mosturgent problems is to examine whether a non-trivial fixed point exists in four dimensions ornot . It seems that both approaches, dynamicaltriangulation and Regge calculus, are worth pur-suing to answer the problem . These approachesare quite suitable for Monte Carlo simulations,and we expect that they will produce a lot ofnew physics in the field of quantum gravity justas they have done in the lattice gauge theory.

Acknowledgements

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

H. Kawai/Quantum gravity midrandom surfaces

ment and patient reading of the manuscript . Heis also indebted to J. Ambjorn, C . Baillie, T.Filk, H. Hamber, N. Kawamoto, A . Migdal andH. Neuberger for fruitful discussions during theSymposium . This work is supported in part bythe International Scientific Research Program ofthe Japanese Ministry of Education, Science andCulture .

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