Introduction The conventional approach to two-dimensional gravity [1] and to string theory [2] is perturbative with respect to fluctuations of the topology

(1)

North-Holland

A NONPERTURBATIVE TREATMENT OF TWO-DIMENSIONAL QUANTUM GRAVITY

David J. GROSS and Alexander A. MIGDAL

Joseph Henry Laboratories, Princeton Unicersity, Princeton, NJ 08544, USA Received 11 January 1990

We propose a nonperturbative definition of two-dimensional quantum gravity, based on a double scaling limit of the random matrix model . We develop an operator formalism for utilizing the method of orthogonal polynomials that allows us to solve the matrix models to all orders in the genus expansion . Using this formalism we derive an exact differential equation for the partition function of two-dimensional gravity as a function of the string coupling constant that governs the genus expansion of two-dimensional surfaces, and discuss its properties and consequences . We construct and discuss the correlation functions of an infinite set of pointlike and loop operators to all orders in the genus expansion .

1. Introduction

The conventional approach to two-dimensional gravity [1] and to string theory [2]

is perturbative with respect to fluctuations of the topology. One sums over two-dimensional geometries by first performing the functional integral for fixed topology (genus = number of handles) and then summing over genus. However, this sum is very badly behaved . The higher terms grow as factorials of the genus, and the positivity of these terms renders the series non Borel summable [3]. This situation is made worse by the absence of a nonperturbative framework for the theory. Such a framework (for example, a useful formulation of second quantized string theory) should be capable of reproducing the topological series as an asymptotic expansion, valid in the perturbative domain; but it should also provide a physical picture and a mathematical framework valid for strong coupling. From what we already know about gravity and strings, we expect dramatic phenomena in this region.

From the point of view of string theory it is essential that we develop nonperturbative methods for treating the theory if we are to make contact with the real world. At the perturbative level of string theory there are many too many possible worlds, i .e. classical vacua about which consistent perturbative expansions can be made. All of them have undesired features, such as unbroken supersymmetry and

*Research supported in part by NSF grant PHY80-19754 . 0550-3213/90/$03 .50 © Elsevier Science Publishers B.V.

(North-Holland)

(2)

334 D.J. Gross, A.A. Migdal / Two-dimensional quantum gravity

massless dilatons. One must hope that nonperturbative physics will lift the degen- eracy and break the unwanted symmetries. This is strongly suggested by the divergence and non Borel summability of perturbation theory which can be taken as an indication of the nonperturbative instability of the classical vacua [3].

One of the main motivations for studying two-dimensional gravity coupled to simple matter is that this provides a toy model for string theory. There are, of course, other motivations. These theories are the simplest examples of quantum gravity, and they can be used to study the critical behavior of random surfaces (phase boundaries). In those cases, however, one is mostly interested in the theory on the sphere or on the torus. Except for the study of wormhole physics, one has little motivation to sum over surfaces of arbitrary genus. In string theory the sum over random surfaces is just a representation of the perturbative semiclassical expansion of the theory, and one is certainly interested in the full sum and beyond.

In particular the minimal models of matter with central charge less than one might be especially simple since they have only a finite number of degrees of freedom [4]. These theories have been studied at length as conformal field theories on a fixed space-time background, and more recently when coupled to gravity [5-7].

In the last few years, a completely different approach to gravity and string theories has been pursued [8]*. The geometry of the world-sheet of the string (or two-dimensional space in the case of pure gravity) is approximated by a dense Feynman graph of the same topology, in the limit where the number of vertices becomes infinite . The topology is selected by means of the 1/N expansion of an SUN invariant hermitian matrix model, while an infinite number of vertices can be produced by adjusting the coupling constant to equal a critical value at which the loop expansion of the matrix model begins to diverge . The sum over all Feynman graphs of given genus and given number of vertices can be regarded as a discrete version of the functional integral over metric tensors. Remarkably, in many cases these discrete models can be handled with greater ease than their continuous analogs. Much work has been done for dimension less than one and spherical topology, with results that are in complete agreement with those of the continuum approach [101.

Recently, significant progress was achieved in this field by three independent groups [11-13], who managed to sum the complete topological expansion in terms of the solution of a certain ordinary differential equation. The nonperturbative effects, which were heretofore unapproachable by standard methods, are now manifest.

The aim of the present paper is to give a systematic derivation of the results presented in our short note [11]. We also present some new results, which extend

*See also ref. [91 where the string interpretation of the random matrix models was first suggested and the topological expansion of the corresponding loop equations was constructed .

(3)

and generalize the previous ones. The main new results are the nonperturbative evaluation of the correlation functions, and a more detailed analysis of pure gravity.

The outline of this paper is as follows . In sect. 2 we review the representation of the sum over surfaces by matrix models, discuss the nature of the double scaling limit. Sect. 3 develops the method of orthogonal polynomials. Here we introduce a powerful operator technique that will be used to solve the models, and write out the basic equations of the theory. In sect. 4 the multicritical behaviour of the one matrix model on the sphere is explored in detail, and the correlation functions of the operators of the theory are constructed . In sect. 5 a complete nonperturbative solution of the model is derived. In the following section, sect. 6, the case of pure gravity (k = 2) is studied in detail . Sect. 7 is devoted to an analysis of the correlation functions of operators to all orders in the genus expansion.

2. Matrix models of random surfaces The partition function of the random hermitian matrix model,

ZN(ß)

=

1d0 exp [ -ß Tr U(O)j, (2 .1)

can be regarded as a discrete version of the sum over two-dimensional surfaces .,/', of genus G and area A . The connection is established by considering the perturbative expansion of the logarithm of the partition function, which generates connected graphs . It is well known that the 1/N expansion of such a matrix model is equivalent to the topological expansion of the sum over Feynman graphs. This expansion has the form

A

In ZN (ß) = regular terms + Y_Nzc1-G>\

N) (2.2)

ß

where the factor F,[U] is given by the products of the vertex weights corresponding to the cubic and higher order terms in U((P), divided by the order of the symmetry group of the graph. The irrelevant regular terms, a + b In ,ß, arise when the matrix field (P is resealed to renormalize to ? N^p2 the quadratic term in ßU((p) = i(p2 + . . . . The vertex weights renormalize, under this resealing, as Ut - U,N(N/ß)(t-2)/2 . For the graph with No vertices, N, links and N2 faces one obtains the overall power of 1/N by noting that, for fixed ß/N, each vertex contributes N, each link (i.e. propagator) contributes 1/N and each face (i.e. index loop) contributes N. The overall power of 1/N follows from Euler's theorem, No - N, + N2 = 20 - G), which yields the power of N in eq. (2.2).

(4)

33 6 D.J. Gross, A.A . Migdal / Two-dimensional quantum gravity

The total area of the graph can be defined as A=2-vertices(` - 2). This has a simple geometrical interpretation in terms of the dual graph where the original vertices represent faces. Namely, a vertex of order l corresponds to a face made out of l - 2 adjacent equilateral triangles of area z each. Thus, a cubic vertex represents one triangle, a quartic vertex represents a rhombus made of two triangles, etc.

In the case of cubic graphs, generated by the potential U(O)=^{Z,~2 -} 3~3, ^the dual graph is a triangulation of the surface and the above definition of the area coincides with one half the total number of triangles. An intrinsic metric on such a triangulated surface can be defined by assigning a unit length to each edge, so that the distance between pair of points is given by the minimal number of steps along the edges.

It is intuitively obvious that by summing over all Feynman graphs of a given genus and a given area, generated by a reasonable U(O), one has a reasonable discrete definition of the geometrical sum over random surfaces of fixed area and genus, which should, in the limit of infinite order, reduce to the continuum definition of this sum.

To make this correspondence more precise we can introduce a metric in space of all triangulations with a given number of points, by which one can measure the distance between triangulations. A triangulation is defined by giving the points and the adjacency matrix Sid =(1, if i and j are connected; 0, otherwise). Define the distance between two triangulations G, G', as ^{d '(G, G')}= z Eij (Sij - Sii)2_, where S, S' are the adjacency matrices for the triangulations. In particular, triangulations differing by aflip of one link are separated by a unit distance in the space of all triangulations.

A generic triangulation of a given topology can be built by a sequence of random flips from some initial triangulation of the same topology. This process of random flips is being used [10] to simulate random surfaces on the computer. One may regard this as a discrete version of the stochastic quantization of two-dimensional quantum gravity, where the variations of the metric tensor gaß are given by infinitesimal random moves, equidistant with respect to the functional norm

d2(g, g + 8g) = Js 8g,ß 8g .ß.

This procedure can be generalized to an arbitrary potential, by introducing extra local moves. In case, say, of the cubic plus quartic potential, one would have both rhombuses and triangles, and one would introduce the local move that cuts a rhombus into two equilateral triangles. The inverse move corresponds to gluing two neighboring triangles into a rhombus.

Let us now discuss the general scaling properties of the matrix models. The continuum limit is achieved by carefully adjusting ß so that the perturbative expansion diverges, since then the sum will be dominated by the terms with an infinite number of triangles. For fixed area this means that the area of the basic building blocks is vanishingly small. For a given genus the behavior of the

(5)

expansion in powers of N/ß is geometric and as one lowers ß eventually one arrives at the critical point. It is convenient to normalize the potential U(O) so, that this will occur at ß/N= 1 for any fixed genus. The fact that this is possible to achieve is not a priori obvious, but will become clear below.

With this normalization In _ZN(ß), for genus G, will have a singularity of the form

In _ZN(/6) =regular terms +/3 Z(' -G)(

0 -

¹⁾

whereas the connected graphs of fixed genus and area will grow as some power of the area

[Y-F`fU]IA,(;

^f' cc Ay([ul-2 . The critical index y(,.[U] depends linearly on the genus

(2 .3)

(2 .4)

yr, [ U ] - 2= (I - G)(yo[ U ] -2) . (2 .5) This was first observed in matrix models by Kostov and Mehta [14] and later confirmed in the continuum theory [6,7]. This allows us to take the double limit, wherein both /3 - oc and ß/N - 1 - 0, keeping fixed

II ß

9. ^ring= I ßZ N - 1 (2.6)

In this case the genus G contribution is weighted by the factor ^g2G, as in standard string perturbation theory.

The critical index yo is highly universal, at least to arbitrary order in the genus expansion of the matrix models. The index yo[U] in particular, does not depend continuously upon parameters of U(O), but rather takes a discrete set of values as we fine tune k of these. When we go beyond the topological expansion, as we shall

see below, there are hidden continuous nonuniversal parameters.

The minimal nontrivial model has k= 2, including adjusting the overall normalization. For example, a purely cubic or quartic potential U((P) is of this type. In this case there are square root singularities in the specific heat (also called string susceptibility) as a function of ß, corresponding to yo = - i . Since in this case the sum over surfaces is one with positive weights there is no question that the k = 2 model corresponds to pure gravity with no coupled matter. It is therefore no surprise to get a value of y~ in agreement with the continuum treatment of pure two-dimensional gravity [5].

(6)

33 8 D.J. Cross, A.A . Migdal / Two-dimensional quantum grauity

Kazakov observed [15] that one may get yo = -1/k by fine tuning k parameters in U(O). This is the same value, that according to previous work, both on continuous [5] and discrete [10] models, corresponds to the coupling to gravity of minimal conformal blocks of matter fields with central charge C = 1 - 6/k(k + 1).

Kazakov therefore suggested that these multicritical one matrix models could represent these unitary theories of matter coupled to gravity [15].

However, as it was realized recently [16], the thermodynamic singularity does not define the model uniquely. In fact there are serious reasons to believe that Kazakov's multicritical models correspond to nonunitary conformal models with central charge C = 1 - 3(2k - 3)2/(2k - 1)*. In particular, the k = 3 case, according to Staudacher [16], corresponds to the Lee-Yang edge singularity with C ^{= - Sz}

[18] . Staudacher proved this relation on the sphere. In recent papers the Ising model on a random surface, which can be formulated as a two-matrix model [19]

and treated by the methods developed here, was solved [20-22]. It was shown that starting from the one loop level the k = 3 multicritical one matrix model and the Ising model on a random surface are not the same. In addition we solved the two matrix model for the Ising system in an imaginary magnetic field explicitly and proved Staudacher's conjecture for arbitrary genus [20] .

Thus, the multicritical one matrix models do not correspond to the coupling of unitary conformal matter to gravity. This is not too surprising since to get these models it is necessary to adjust the couplings in U(0) in such a way as to produce negative weights in the sum over surfaces for some triangulations. The only case where all the weights are positive is k = 2, which we are convinced does represent pure two-dimensional gravity.

However, independent of their relation to conformal field theories, the multicritical models are worth studying by themselves. There are indications [23] that they are disguised topological two-dimensional gravitational models [24]. We shall therefore consider the one-matrix model in its most general form.

To be specific, we shall take the double limit: N - -,,BIN --* 1 and adjust k parameters in the potential U((P), after which the following scaling law will hold : 1nZN(6)= regular terms -F(t), t=(ß-N)O-i1(zk+> ) . (2 .7) We shall prove that the specific heat, f(t) = F(t), obeys the following differential equation:

*This conjecture was first made by M. Douglas [17] .

(2 .8)

(7)

The operator K was introduced by Gelfand and Dikiï in their study of higher order KdV equations [25]. The nonlocal terms involving 1/7, in this expression cancel (see appendix B). In fact K' - 1 yields the higher KdV equations, K . 1 = f, ^{k2 .} 1 = '! (3f2 etc. !

Eq . (2.8) is universal, depending only on the single parameter k. The simplest case, k = 2, corresponding to pure gravity, yields the Painlev6 equation, t = f 2 - 3f, while the case of general k, yields a (2k - 2)th order differential equation, differing from higher KdV equation by only one term t. The above mentioned violation of perturbative universality corresponds to the ambiguity in the Cauchy data for this ODE. We find that half of the free parameters are fixed by requiring that the asymptotic behavior at infinity correspond to spherical topology, i.e.

f ___~ t Ilk . The remaining k - 1 free parameters violate perturbative universality.

3. The method of orthogonal polynomials

Let us recall some aspects of large N technology [26,27]. The one-matrix problem is particularly simple and can be easily solved, for an arbitrary potential U((P), by eliminating the angular matrices from the integrals. Consider the partition function. The Haar measure, DO, can be factored as Dn H i d0i 4N(`!'i)~

where D is the unitary matrix that diagonalizes 0, O, are the eigenvalues of (P, and 4N(Oi) is the Vandermonde determinant

4N= 1Si<j-<Nri (`l'j - Oi)

ZN( P) a

f][1

d~bi4,exp~- ~ßU(oi)) .

-1 i-1

Since the potential depends only on the eigenvalues 0i, we can explicitly integrate out the matrices .fl . We obtain in this case, as indeed we would for the evaluation of any correlation function of "invariant operators" that depend only on the eigenvalues of 0 (i.e . traces of products of 0), an N-dimensional integral over 0c,

(3 .2)

It is instructive to reinterpret this expression as the partition function of a one-dimensional "Coulomb" gas at temperature 1//3. We have N particles, of charge e2 = 1/ß, which sit at position 4i in an external potential U(h). This interpretation of the probability distribution of random matrices was first introduced by Dyson [28], for the case of unitary matrices. At extremely low temperature, 1/(3 - 1/N, and extremely weak charges the attractive forces arising from the potential balance the "Coulomb" repulsion in a finite volume. In this limit the

(8)

340 D.J. Gross, A.A . Migdal / Two-dimensional quantum gravity

Dyson gas freezes, becoming a crystal with each charge i located at a fixed coordinate Oi, thus minimizing the total energy -e2^y^;#j In10_i-cßj^l ⁺Y; U(O).

This frozen state of the Dyson gas corresponds to the planar limit, when we let N -> oc but ß/N =A 1. In this limit the only surfaces that survive are those with the topology of the sphere. Higher-genus surfaces are suppressed by powers of N2 ; for genus G, by (1/N^z^{) ° .} In the standard large N treatment of the one matrix problem one solves the model by minimizing this energy to determine the distribution of eigenvalues. Having determined the density of eigenvalues any function of these can be calculated. This is the classical limit of string theory.

However, we are interested in the contribution of surfaces with arbitrary topology, which correspond to the interacting string theory. How can they be picked out and summed? We shall see that by careful adjustment of^(3/None can enhance the contribution of surfaces with an arbitrary number of handles. The point is that the topological suppression factor is actually given by inverse powers of N Z(ß/N- 1)p, where p is a positive constant that depends on the parameters of the potential. Thus, if we take N to infinity keeping fixed the string coupling constant, gt^ring^--[N a(/3/N- 1)p]^-I, then a surface of genus G will contribute with weight (gt^ring)(;*

Now the eigenvalues are not frozen at the positions of equilibrium, they fluctuate. From the point of view of the Dyson gas this situation corresponds to melting of the crystal near the edges of the one-dimensional volume, where the density vanishes for the critical range of parameters. The charges at the edges fluctuate, so that this problem is a genuine statistical mechanical one. Nonetheless, due to the remarkable properties of logarithmic interaction, it can be exactly solved.

The most powerful method that has been developed for the solution of large N problems is based on the use of orthogonal polynomials [27,29]. It is especially appropriate for our purposes since it enables us to solve the models explicitly for finite N, and thus carefully approach the scaling limit. Let us summarize the basic elements of this method. We introduce the orthogonal polynomials, Pn(O), with respect to the weight dA((~) = exp[ -ßU(0)] d (b, which satisfy

f

do~ exp [ -ßU(b)] P,( o~ )PJ o~ ) =^{Sn,m hn .} (3 .3) These have been normalized so that the coefficient of^xn in Pn equals one, namely

(9)

These orthogonal polynomials satisfy a two-term recursion relation for any U(O),

XP.(x) =P (x) +S P.(x) +R P -1(x) . (3 .5)

For symmetric potentials, U(cß) = U(-0), the polynomials have a definite parity, P(-x)=(-1)Pn(x), and S vanishes. The coefficients R are related to the

normalization constants, h . By using the recursion relation for the P, h ,=

f

d0exp [ - PU(0)1 P, OP,

= f

d <~ exp [-PU(O) ] (P +z+S +1P +1+R +1Pn)P

= R + l h, . (3 .6)

The reason that we introduce these orthogonal polynomials is that the Vander- monde determinant can be written as a sum of products of such polynomials. With the above normalization

d(~i) =det 10 ^-'l =det IPf-1(Oi)I . (3 .7) This enables us to calculate the partition function, as well as arbitrary correlation functions, in terms of R and S. To calculate the partition function expand the Vandermonde determinant, using eq. (3.7), and then perform the integrals over Oi using the orthogonality of the P . This yields

N-1 N-1

ZN = N ! 1-1 _{hi =}_N_!hó1-1 _{RN -` .} (3 .8)

1=0 i=1

In a similar fashion, correlation functions can be calculated in terms of the R's.

The simplest example is given by the orthogonal polynomials themselves, which can be regarded as generating functions for connected correlation functions of tr(P"). To this end note that Pn(W) can be expressed as

1 r,

f

1-1 dg(xi) 32(xi) Il (À -x1) (3 .9)

Zn ¹ 1=1

This formula is easily established by proving that it defines orthogonal polynomials with respect to the measure dl-t(x) with the correct normalization . Now the right-hand side of eq. (3.9) is simply the expectation value of det CAI-4~ I in the

(10)

model with n by n matrices

Pn(

~) - (aetpr - y_{)n .}

(1) (3 .10)

Therefore, the coefficients pni> in Pn, which can be calculated in terms of^Rn and Sn, are equal to the coefficient of A -i in

1 (p i Iexp - tr F_ -j A _n

We find it particularly valuable, at this point, to introduce an operator formalism. We introduce a space of functions of ~, with scalar product

(AIB> --

_ f

dOexp [ -ßU(O)] A(O) B(fi) . (3 .11) An orthonormal basis of vectors in this space, which we denote by In), is given by the nonnalized polynomials, --__Pn(W)l hn . The recursion relation for the orthogonal polynomials yields a representation of the variable 0 in this basis, by the tridiagonal operator ~, with matrix elements

(nI i~ Im> =Sn,m+1 Rn +Sm,n+1 Rm +&",lnsn . (3 .12) With the aid of this operator formalism it is easy to derive explicit expressions for the correlation functions of arbitrary products of invariant operators for arbitrary N. These will prove very useful later in deriving the nonperturbative form of the correlation functions. Consider the generating function of connected corre-

lation functions of U(N) invariant operators

/ P hi

exp[G(lal,l-t25 . . .,UP)] --- ,rlexp(-tr[Fi(1) ])_i-1 N (3 .13)

where Fi is an arbitrary function of the matrix (P. Now, use eq. (3 .7) to express the determinants, 4,(Oi ), in terms of the P.'s, and rewrite J d,~(~)91n(~)F(~)9,n(cß)

as (nI^F(~)Im ), to obtain

P

exp[G(g 92, . . .,tt,P) ] =Det (n1exp( Lhi Fi(~))I M>I , ^{(3 .14)}

i-1 N

where the subscript N reminds us that the matrix is of order N by N. For

(11)

example, the one-point function is given by

where IIN is the projection operator IIN = En-, In)(nI.

Similarly, the connected two point function can be shown to be given by (trF[O]trG[451)connected=Tr[IINF(~)(I-HN)G(~)], (3 .16) where I - IIN = En , N l n)(n l . Similar explicit expressions for the connected n- point functions are easily derived from the generating functional (3.14).

The parameters R., S., govern the recursion relation for the orthogonal polynomials, determine the structure of the operator ~ and can be used to extract all of the physics from the model. They can be determined from the following relations that play the role of the equations of motion. To derive these consider the equation J dA(O) = 0 and J dA(O) 9n_,(d/dc~)9n ⁼ n hn _,/hn=n/ ^Rn , which follow trivially from the orthonormality of the Integrate by parts to derive

(n-11U'(~)In>= ß Rn , (n1U'(~)In>=0 . ^{(3 .17)}

All the above formalism is valid for anyN. Ultimately we will take N - 00 and it will be useful to write these equations in a form appropriate for this limit. We introduce the continuous variables x= n/ß, X = N/ß ^{and write} ^Rn as R(x). We also introduce an operator l, whose eigenvectors are In) [i .e. lln)=(n1/ß)1n)], and the conjugate angular coordinate 0. Note that l= -(i/ß)^{d/d0 and}

0 =

(i/,ß)d/dl. In this formalism 1//3 plays the role of Planck's constant. The operator can be written as

It is sometimes useful to rewrite eq. (3.17) in a different form. First, note that eq . (3 .17) is equivalent to the statement that the diagonal matrix elements of e -`B U'(~) and x/ R(l) are equal, and that the diagonal matrix element of U'(~) vanishes. Then multiply the n by n matrix elements by

XG1

^h^{n - I ) on}

the right (left) and bring the I/ h.-, through to derive x5, 1

= I

^,_e²d0^Tr ^-Qe

U,I

^e-

(trF[O]) = Tr[F(~)IIN] , (3 .l5)

= e",/R(l) + V^/R(î)e -`B ^{+ S(i)}^. (3 .18)

o- = 0,1 . (3 .19)

(12)

344 D.J. Cross, A.A . Migdal / Two-dimensional quantum gravity

4. The matrix model on the sphere

Let us first consider the limit ß - N - oc, but ß/N > 1. In this limit only surfaces with spherical topology survive. This is the lowest order of perturbation theory - the classical limit of string theory. In this case we can drop the derivative terms in eq. (3 .19) and the equations become purely algebraic. They can be interpreted as giving the extrema of (z = e-'9)

( R )

,(l(R,S) --_ -xS +, dz

i U z+ z +S .

(4 .1)

We have analysed in some detail the general case of asymmetric potentials, which might produce the most general multicritical behavior. However, we were unable to find explicit asymmetric potentials which gave results that differ from the symmetric case . We therefore will restrict ourselves to the case of even potentials.

For these we can solve the equation

aR n(R,S) =~Z

d Lz U'~z+

R

+S) =0, (4.2)

by setting S = 0. The second equation can be written as

as

where we have introduced the function W(R),

,f2(R,S) - -x+W(R) =0, (4 .3)

W(R)=~ a-U'Iz+ R I, U(O)=foiduW(u(1-u) 02) . (4 .4) In terms of the power series expansion of the potential, the relation between these

functions is

U(O) - Y-U2kh2k,

k W(R)

= F_ (

_{U2k Rk .} ^(4.5)

k ki (kk) 1

)'

The possible types of critical behavior can be deduced directly from eq. (4.3), after which one may reconstruct the potential U((~) that produces this behavior, and then reinterpret the results from the point of view of the Dyson gas or in terms of random graphs.

Recall that we are interested in adjusting /3, or X =^N//3, so that the perturbative expansion of the free energy diverges, and picking out its singular part. To see

(13)

when this occurs rewrite eq . (3.8) as

N-1 l

ZN( ,ß)=N!hóexp(~ (N-n)1nR ) aexp(ß 2JXdx(X-x)1nR(x)) .

n-1 0

Thus, we see that the scaling laws arise from the singular behaviour of R near the point x = 1, when _ß equals its critical value N. This occurs when 1 - W(R) and k - 1 of its derivatives vanish at, say, R = J*. In other words**

In this case the integral in expression (4.6) will behave as appropriate at this point to introduce scaling variables. We variables x and X,

as well as R,

t= (1 -x)R2kl(2k+l)= (0-n)p^-l1(2k+1) T -- (1 - X) R^{2k l(2k+1)} = (p -N)P-1/(2k+1)

f(t) = (1 -R)p2/(2k+1)

(4 .6)

ß2(l -X)2+llk . It is first rescale the

(4.10)

*We can always scale 0, and thus R,so that the critical point is at R = 1,which will correspond to O=+2.

* * For even k this yields a potential that is unbounded from below; however, as we have noted above, this does not affect the universal critical behavior, as we can cutoff the potential for large (b and the net effect will be exponentially small terms that do not survive the scaling limit.

W= Wk(R) = 1- (1 -R) k . (4 .7)

These W's correspond to the following potentials:

Ul(O)= [ 21<p2] , U2(p) = [02- 1z041

_,) k-I 0kzk n!(k-1)!

] . (4.8)

(n

For the kth model the solution of eq. (4.3) is quite trivial, namely

R = 1 - (1 -x)' lk . (4 .9)

(14)

The free energy, F = - InZN(ß), is then a function of the scaling variable t and is given by

F(t) = ftdt'(t-t')f(t'), (4 .12)

0

where we have added to F(t) terms that are regular and therefore do not contribute to the critical behavior of large surfaces. For the kth multicritical point the specific heat f(t) = f(t) = tIlk, and

k²

F(t) = (k + 1)(2k + 1)t2+t1 k . (4.13) The scaling variable t is, of course, proportional to the (renormalized) cosmologi- cal constant, and thus conjugate to the area. It is also related, when we sum over surfaces of arbitrary genus, to the string coupling constant, which we recall is defined as

2 - t^-(2+Ilk)

gstring ' (4 .14)

The above scaling behavior of the connected sum of genus-zero surfaces corresponds to the string anomalous dimension,

(4.15)

Now let us discuss the correlation functions of the operators of the matrix model. The most general U(N) invariant operators that we can construct are traces of powers of 0. These can be calculated directly by the methods outlined in sect. 3, after which linear combinations with simple scaling properties can be found. This analysis will be pursued in sect. 7 after we have developed the nonperturbative solution of the models. However, a more direct approach, which is sufficient for the spherical limit, is the following. Consider adding to the potential which corresponds to the kth multicritical model, whose W function is W^k(R), perturbations that add to W^k scaling deviations. In other words we add to the potential U^k

(O

operators, 0,,, whose effect will be to produce a W of the form

W(R) = Wk(R) -

r

l-L ra~k)

(ß)[WI,(R) - 11 (4.16)

The ß dependent factor a;k)(ß), to be determined below, is introduced so as to ensure the correct scaling properties .

We shall now show that these operators have simple scaling properties and evaluate their correlation functions. The coefficients ^{lt, i} in front of the operators

(15)

D.J. Gross, A.A. Migdal / Two-dimensional quantum gravity 347 have the meaning of chemical potentials or sources . The derivatives an(In ZN )/aAI. . .dAn can be interpreted as connected correlation functions of these operators. So to calculate their correlation functions we evaluate InZN(ß ; Al, A2, . . . ). To this end consider the perturbed equation x = W(R), where ^W(R) is given by eq. (4.16). This can be rewritten as

1-x=(1-R)k-

r

l-t iaik)(ß)(1- R) l' (4 .17) i

We must solve for R as a function of x and the _Ai's, and then calculate F(t, A,).

This can be done explicitly by means of the Lagrange method, which is outlined in appendix A.

Let us rescale eq. (4.17) so as to write it in terms of the scaling variables. Then

where we see that we must choose a(k)(,ß) = ß2(k-1j)l(2k+1) ^so as to get the correct scaling. Using the method outlined in appendix A we can now solve for f as a function of t and the Ai's and integrate twice, with respect to t, to obtain the free energy. We obtain

a a 1 ^~ ^a ^p-3

0 Wi-O

t = fk- Yhifl, 'i

t(2'+I-k)lk

1 I' [( -y+1-k)Ik1

_ t[1 + 1 -(p -2)k]l k

k F[{ -y+1-(p-2)k}/k] '

where I = Y_1 li, and negative powers of derivatives stand for integrals.

Specifically

k

o )-_ - tl+(11+1)lk

(11+1)(1,+k+1) '

(0102) - _1,+12+11 t(I1_ +12+I)lk

1 _-1+(1

1+1

2

^{+1 3}^+1)lk

0102o3)= - t

k ,

( 01020304)=-

k2

1 ( 1 1+12+13+14+1 - k) t - 2+(11 +1 2 +13 +14+l)lk

(4 .18)

(4 .19)

(4.20)

(16)

348 D.J. Gross, A.A.Mtgdal / Two-dimensional quantum gravity

What do these scaling operators correspond to in terms of matrix model operators? Formally they can be written, using eq. (4.4) to transfer from the W function to the potential, as the operators [15]

0

1

(~)

Pal

P)

Trfo1 du

[1 _ n(1 -

u)p2j 1 . (4.2l)

However, this expression, unlike that for the multicritical potentials given in eq.

(4.8), appears to diverge. The divergent piece, however, is a (0 independent) constant and will not contribute to the scaling behavior of O, . In the following sections we shall develop more powerful methods of treating such operators, but it is useful to show how the direct evaluation of the matrix elements of the above formal expression, using the methods of sect. 3, yields the correlation functions.

Consider for simplicity the one-point function of the operator O,, which according to eq. (3 .15) can be written as

element of ^(fi2k as

(k)(ß) I du

1 -n)I

k 1

(01> =

PaL L [-u( (

^{l )(n}

in>

f

o ü k=0 k . (4 .22)

Now, using eq. (3.18) we can, in the spherical limit, write the diagonal matrix

(nl zkln>

= f

²^7rdB f e re R( n ) + R( n ) e +tel zk 2'rr

r f(t' 1+1

(O1 >=- fodt' l+1 ,

(4.23)

Performing the ^t integral, we obtain En-, Y-k(l/k)k-'(-R)k. Writing the sum over n in the limit as N as

fó

dx and expressing ^R in terms of ^fand the scaling variable, it is easy to see that the singular part of (O,) is

(4 .24)

which reduces, since ^f(t)= t^ilk, to the expression given in eq. (4.21)*.

The operator O,, for l = 0, 1, . . . , -, has dimension d, = ^Ilk, since(1 - R)k scales as 1 -x and hence has dimension one for the kth multicritical model. This is the local dimension of the operator, before it is integrated over the surface. The global

*Note that the expression (4.21) must be regarded in aformal sense in which only the singular contribution of the operator is kept. This is especially clear for the case 1= 0.

(17)

dimension is dt - 1 . Indeed the correlation functions on the sphere scale as 0 op> ,t[2+l/k]+Er[d1 -1]

where the factor ^t2+irk comes from the sum over surfaces and the factor t^d^f^-1 from the insertion of Ot .

The lowest dimension operator, O, is of particular interest. This is the operator that corresponds to 5W= 1, and thus its effect on the equation for W is just to shift t. Since -t is conjugate to the area of the random surface we can think of this operator as a punctureoperator, P, that has the effect of picking out a marked point on the surface. The correlation function of k puncture operators should simply yield a factor of (-Area)^k, since each point can be any where on the surface. Thus,

O() -P dtd -Area .

If we check eq. (4.21) we see that this is indeed the case,

(4.25)

d ⁿ

~(0

o)ni -(di ) F(t)= « -Area)") . (4.26) This equation should be true, not just on the sphere, but to all orders in the loop expansion. We shall verify this below.

The other particularly simple operator is the operator Ok . For the kth multicritical theory the insertion of this operator simply shifts the coefficient of the potential itself. Thus the n-point functions of Ok are proportional to F(t), with coefficients that follow immediately from the fact that Ok modifies eq. (4.18) to read ^{t =fk}(1 -^Ak).

What about the matrix model operators Tr(p 2t ? It follows from eq. (4 .23) that their correlation functions are dominated, for fixed 1 in the scaling limit, by the puncture operator P, since Rt- (1 -f)t -If. This is not surprising since the geometrical meaning of Tr^(p21 is that it corresponds to inserting a 21th order vertex in the graphs, or in the dual graph a loop of length 21. In the continuum limit this is equivalent, for any finite 1, to the insertion of a puncture on the surface. To construct operators of definite dimension it is necessary to take finite linear combinations of these, as in eq. (4.21).

However, if 1 -> co we are then inserting a loop of finite length on the surface and the operator is no longer dominated by a pointlike puncture. In fact, if we scale 1 appropriately as /3 - N -> oc then this operator will correspond to the partition function of a random sphere with a finite boundary. Consider evaluating

(18)

the expectation value of (Tr ,p2M) = En R M(n)(m) for large M. Note that R = 1 - .fß-2/(2k+1) so that if we scale

in the scaling limit, then

M=Le2(2k+1) ---, 00

(Tr(z^e)2`y)_ 1

f

dt e-Lf(`) . 7rL t

We shall discuss all the operators and their correlation functions in greater detail in sect. 7, where they will be written down to all orders in the genus expansion.

5. The nonperturbative sum

Let us now return to the general scaling limit where we will sum the complete topological expansion . We first note that the integral in eq. (3.19) is dominated, for large ß, by the region of small 9 - 1/ß. Furthermore, we already know that in the scaling limit R - 1. We can therefore expand for small B and R - 1,

ete/2~e- rB/2-2-H=2-92- [1-R(x)] =2+ 1 d2

/32 dx2

We can think of H as a Schrödinger operator, x as a continuous variable which we now take to play the role of coordinate and B as the momentum. The potential is given by 1 - R(x) and is to be determined by the equation

(x1U'(2 - H)Ix> =ßx, (5 .2)

where the extra factor of ß arises from the change of normalization of the states (in) - lx)).This problem, of determining the potential given matrix elements of some function of the hamiltonian, is analogous to the inverse scattering problem.

Fortunately we are not interested in general U' but only those that produce the appropriate multicritical behavior. We have already seen that the multicritical points are generated by special forms of U(O), corresponding to W(R) - 1 - (1 -R)k for the kth multicritical theory. The following trick allows us to explore these theories with ease. Define the singular potential (B is the beta function)

U,(0)=B(', 2 - v)(2-~)_ v+1/2+(0--->

(4 .27)

(4 .28)

- [1 -R(x)] . (5 .1)

(5 .3) Here vis a negative exponent, which we will later analytically continue to positive

(19)

integers. One may easily check, using eq. (4.4), that this potential corresponds to

W= W(R) =(1-R)''[1+0(1-R)], (5 .4)

in the sense of analytic continuation from negative v . The advantage of this potential is that eq . (3.19) reduces to the form ßx = - 2 vB(Z, z - v)(x 1H -1/2 1X ) . Now consider the scaling region. Replace R(x) by 1 -f(t)p -21(2k+ ')and x by 1-t/3 -2k/(2k+1) Then H scales as H-[-(d/d t)2+f(t)],B-2(k-i)1(2k+1). We therefore derive the final form of our equation

t= -2vB(-'Z- v)(t 1 fi °^-1121t)

d 2 1 112

_ -2vB(, ,2 -v)(tl ~- ( dt) +f(t)

I '

^it) ^{(5 .5)}

This equation can be solved perturbatively in a semiclassical expansion, which corresponds, since h - 1/ß, to the genus expansion. The leading term, the WKB approximation, should correspond to our previous result for the sphere. To see this note that in the leading semiclassical approximation we replace 1Y by its classical value, p2 + f(t), to derive

t = -2vB (2,2-v)ƒ _{x _Lp2+f(t}277- _Il2 t (5 .6) as expected .

In order to determine f to all orders in the genus expansion we need a useful expression for the diagonal matrix elements of the Schr6dinger operator, H, or equivalently the diagonal matrix elements of the resolvent of H,

R(t,t' ;~) - (tIt= +Hlt,) .1 (5 .7) This function, especially in the limit of integer v of interest to us, was studied by Gelfand and Dikiï [25]. In their work it was shown that the diagonal elements of the resolvent satisfy a simple nonlinear differential equation, and in particular that the coefficients of the expansion in half-integer powers of 116, can be generated by the powers of the nonlocal operator (7, = d/dt),

(20)

We briefly review their results in appendix B, where we show that

Although the operator ^K is nonlocal its powers yield, when acting on a constant function, only local powers of ^f(t) and its derivatives . Thus,

Ro =- z~' R = -¹ ^{4J ,}^'-f

Rz= T'6(3.Îz_ .Î") R;= -~4[1Of3-10ff"-5(f,)z+f(4'],

R4=56 [35570f(f')z-70fZf +21(f°)2 +2gf'f , +14ff(4) -f<6~],

Now we are in a position to solve eq. (5.5). First, express the power of H in terms of the resolvent,

or, using eq. (5.9),

dcj 1

t = 2vB(Z,2 - _{v) ~ 2Tri}

wv-1/2(ti -c) + ^H 10 .

Then expand the resolvent in inverse half integer powers of -a) +f~, where

f0

an arbitrary constant that we will later take to zero. We then perform the integration (for v negative), the result being

r(z) _t r(l - v)

t = -2 1,(-v) rfo 1.(l +'- )Rt(f- .Îo)₂

k! k

t= (2- 1) (K[f(t),o,l)-l .

(5 .9)

(5 .10)

(5 .12) Now we can continue this equation to positive integer ^{v = k,} in which case the above sum consists of a finite number of terms. Furthermore we can exploit the freedom in choosing f0 to set f~ = 0. In that case only one term survives and we derive

t=2 r(Z)r(k+1) (-1)k+1Rk(f(t)), (5 .13) T(k+ )z

(5.14)

(21)

This is the final form of the differential equation for the specific heat for the kth multicritical point of the one matrix model. It can easily be explicitly written for any particular value of k, by expanding eq . (5 .14). Since K is a second-order differential operator it is clear that the equation will contain up to 2k - 2

To reproduce the genus expansion for the specific heat one expands f(t) in a power series in gs«,g = t-(2+Ilk) f= t Ilk(1 - yt Ztg2t) and determines the coefficients perturbatively from the differential equation. To find the specific heat we need more than just this asymptotic expansion, we need the complete solution of the differential equations. This requires boundary conditions, which as we shall see below, introduces new parameters into the theory that cannot be determined perturbatively.

6. Study of pure gravity

In the previous section we derived differential equations for the specific heat.

These equations summarize the relations between successive orders of perturbation theory, can be used to generate the perturbative asymptotic expansion and may serve to provide a nonperturbative definition of the theory. In this section we shall discuss, in some detail, the properties of the simplest and most physical of the models - the case of pure gravity, k = 2.

The specific heat for pure gravity satisfies the Painlevé equation, f= 3(f2 - t).

So to solve the theory nonperturbatively we must solve this equation. First let us consider the analytic properties of the solutions of this equation - the Painlevé transcedents. These are known to be meromorphic functions of t whose singularities are movable double poles with residue equal to 2 with our normalization.

There are always an infinite number of such double poles throughout the whole complex t plane. Some are on the real axis and the others arise in complex conjugate pairs. These poles correspond to double zeroes of the partition function, derivatives. The general term is of the form const.(f)aO~(f)°1(f")Q2 . . . (f(2k-2))a2k-2 where ao + zai

The first few + 2a2 + Za 3 + . . . ka2k -2 k .= equations are

k=1 : t =f, k=2 : t=f2 _ If,

k=3 : t=f3_FF - 2(f,) 2 + TO) (4)'

k=4 : t =ƒ4 -2f( ƒ, ) 2 -2f 2f + 3' l f ) + Sf,fm + _ !3s (6)

k = 5 : . . . . (5.15)

(22)

354 D.J. Gross, A.A. Migdal / Two-dimensional quantum gravity since near a pole t; of f(t)

= F(t)

Z=e-F<r>-(t-t;)Z .

Therefore, we may introduce an entire function Q(t) --_ Z(t) , which has simple zeroes at the poles of f(t).

The solution of the Painlevé equation depends on two parameters which can be specified in a variety of ways. One way is to define f(t) in terms of Cauchy data, f(t

o

) and f'(t

o

), at some particular point t

o

. However, a more useful parametriza- tion is the specification of the position, u, of a pole of f and the first nontrivial term, a(t - u)4 in the Laurent expansion of f(t) in the vicinity of u,

f(t) 2

_ , + 3 U(t-u)Z+I(t-u)3+a(t-u)4 + . . . . (6 .2)

(^t-u)` ^To' ²

This expansion of f converges in some region in the vicinity of u. However, in general it will not converge everywhere in the complex t plane since f(t) has other poles. This can be circumvented by defining a new entire function P(t) = Q(t)/

(t - u), in terms of which f can be calculated,

2 P"P-(P') 2

f(t)= (t-u)z -2 Pz (6 .3)

P(t) can then be calculated in terms of a series in t - u which converges everywhere in the finite complex t plane. One simply uses the Painlevé equation to generate the Laurent expansion of f(t), in terms of u and a, and then uses eq.

(6.3) to determine the Taylor series expansion of P(t). The two-parameter ambiguity in determining P(t) from eq. (6 .3) corresponds to adding the irrelevant terms c,+ C

Z

t to the free energy. A convenient way of removing the ambiguity is to choose P(u) = 1, P'(u)

_ - 5

a which kills the (t - u)6 term of the Taylor expansion for P(t). This allows us to shift the next pole, at t = v, to infinity by starting with large a, where v - u and then continuing to acr;t;ca,(u)where v - -.

Now let us return to the issue of determining which of this two-parameter family of solutions of the Painlevé equation corresponds to the solution of our model.

One requirement appears to be clear: we should reproduce the correct asymptotic expansion at large t, i.e. small g_[ring. This is not automatic since the general solution of the Painlevé equation has an infinite number of poles on the real t axis which accumulate at infinity. This would spoil the asymptotic expansion for large t, f(t) ->

f - 24

t -2 - . . . . This can be avoided by defining f(t) to have, as above, a pole at t = u, and adjusting a so as to push the next pole to + -. This procedure is equivalent to fine-tuning the Cauchy data so as to kill the exponentially growing

(23)

terms at t = - [30] . The advantage of the former procedure is that it provides us with a solution that is defined in the whole complex plane.

The existence of a new free parameter in the theory is annoying. It might be a real feature of the theory; after all the original perturbation series is not Borel summable and therefore certainly requires nonperturbative input to fix it uniquely.

It might be the case also that there are true nonperturbative parameters in this theory, as there are in QCD where instanton effects depend on a parameter ethat cannot be seen in perturbation theory and which lead to non Borel summability.

On the other hand we know of no topological reason that suggests such a B parameter in this case.

It might be that the requirement of unitarity or factorization (a difficult concept in quantum gravity) would determine this parameter. If so what would be the natural guess for the free parameter? Some hints are suggested by the fact that in order to calculate the nonperturbative correlation functions, as below, we required the knowledge of the spectrum and Green functions of the hamiltonian Hthat we introduced. These depend, of course, on the potential f(t), apparently for all values of t from ^-oc to -. If f(t) has a rightmost pole, say at t=u, then H has a unique self-adjoint extension in the region u < t < oo [the residue of the poles is strong enough, i.e. greater than , to prevent tunneling to the region t < u . All the wave functions vanish at t - u + 0 as (t - u)2]. The extension of H to negative t does not contradict any principle, but looks unnatural (negative t means imaginary string coupling). From that point of view the natural choice is u = 0 which incidentally also simplifies the terms of the Taylor expansion for P(t). However, we could not find any convincing argument supporting this guess. There is a symmetric solution, which correspond to a = u = 0, for which Q(t) is an entire function of is = (/3 - N)SN- 1 . This solution has the nice property of being analytic in t3 and N but unfortunately it has poles that condense on the positive real axis, so that it does not approach the topological expansion even for large positive t.

The meaning of such nonperturbative solutions is not clear to us.

Is it possible to find a solution with no poles on the real axis? Even if this were possible (though we could not find such solution numerically), there would still be complex poles. What do these mean? If we could arrange the parameters so that there were no poles to the right of some point u, then we could define, nonperturbatively, the Laplace transform of the connected partition function with one puncture

P(V) dtt

F' (t)e-i,

whose interpretation is the sum over all surfaces of arbitrary genus and (scaled) area V=AN-'~', pinned at one point*.

* Note that this definition does not depend on the normalization of t.

(6 .4)

(24)

This function is certainly well defined order by order in the genus expansion. It is of some interest even there since it is only in terms of the fixed area sums that we expect positivity. The expansion of the partition function itself need not be positive since it may be dominated, for low genus, by regular terms, that are sensitive only to small area contributions, and which we discarded. It is important therefore to check whether the fixed area sums are positive. In appendix C we discuss the evaluation of the fixed area expansion for general k, where we note that when k > 2 the terms turn negative for large enough genus - a sure indication of the nonunitarity of these models. For k = 2 the positivity to all orders is an immediate consequence of the Painlevé equation . Expand f(t) -->

- ~~;-o Z^lt^-51^/2 and use the Painlevé equation to derive recursion relations 251 2 - 1

Zo- -1, ^ZI+1 24 Zl+ 2

L

Z.Zl+1-m' (6.5)

m=1

The corresponding expansion ofp(,2/) is

_C/5(1- 1)/2 P(~)

1L

Zlr(z(1-1)) Note that all terms in this expansion are explicitly positive.

The coefficients Zt increase with genus, 1, as (21)!, which is why the original perturbation expansion was highly divergent. The Laplace transform, however, has an extra gamma function in the denominator, which grows as ('fl)! . Thus the resulting sum is convergent! Does this mean that we could define the theory by this expansion? Unfortunately this has the same ambiguities that would be encoun- tered in attempting to Borel-sum the series, since the nonperturbative sum of p grows too fast. Indeed, for large V, p(V) - exp(,V5), which is not Laplace transformable. However, we might try to rotate the contour to a new one on which ,V5 would be strictly negative. The result would be complex; however, one might define the theory by taking the real part, which is equivalent to averaging over the two allowed complex conjugate contours at angles

± s

r, with respect to the real axis. This prescription has the great advantage of being universal, however, we clearly lack a physical principle that would single it out .

7. Correlation functions

(6 .6)

In this section we shall discuss and evaluate correlation functions of the operators discussed in sect. 5 to all orders in the genus expansion.

Let us consider first the pointlike operators, O;, i = 0, 1, . . . , cc given in eq. (4.21).

We saw above that we must extract from this operator the singular piece in the

(25)

scaling limit before evaluating its correlation functions. In the previous sections we have learned how to do this for essentially the same type of operator. We note that the singular part of eq. (4.21) arise from the integration near t = z, where the integrand behaves as {1 - [1 - (t - ^,','-)Z ^' z ]}. Using the representation = 2 - H (plus a similar contribution near _ -2, where _ -2+H), and evaluating the singular part of eq. (4.21), we derive a formula for O,,

01= 2ßai1k)(0)B(z,- Z-1)H 1+i /z

where we are to understand this formula, as before, as defined for negative 1 and then continued to integer 1.

In sect. 5 we used the formalism of Gelfand and Dikiï to evaluate the half- integer powers of the Schrödinger operator. We can take over this formulism here.

Note that the factor 8aik)(ß) is precisely what we need to get the correct scaling for the trace of 01. Thus, the one-point function of 01 is simply given by

1!

(Or)= (21+1)!! ~dt'

K[f(t')]1+ ' .

¹ ^(7.2)

where

f(t)

in this formula is the exact specific heat of the kth model, and K[f(t)]

is given by eq. (5.8).

The above formula is exact to all orders in the genus expansion . To compare with the formulas presented above for the sphere all we have to do is keep the leading term in K1+1 - 1, i.e. the term [(21 + 1)!!/(l + 1)!]f1+ ', to derive

1

(0¹⁾I^sphere _{= - 1 + 1}

f

dt'f(t')1+' + regular terms, (7.3) which agrees with eq. (4.24), and when we insert into this formula the expression for f(t') on the sphere, f(t') = (t')llk and perform the integral we rederive the form of (O,) given in eq. (4.21). This is precisely the leading WKB approximation to the matrix element of O,, as we have seen in the discussion following eq. (5.5).

Now we can see how the one-point functions of specific operators behave to all orders in perturbation theory. Consider first the puncture operator, O))= P. Since K 1 [f(t)] - 1 =f(t) we immediately derive that (00) _ - fó f(t')=F(t), ^{as we de-}

duced previously [see eq. (4.26)].

Similarly the operator Ok has a simple one-point function, since all it does is rescale t. This we can see from the above formula since, K k[ f(t)] . 1 = [(2k- 1)!!/k!]t, this being the basic equation that we used to determine f(t), it

(26)

358 D.J. Gross, A.A . Migdal / Two-dimensional quantum graoity

follows that

k ! Kk⁺1 [f(t)] -1= 1

K-t= 1 If(t)t+ ƒ

(t», (7 .4)

(2k + 1) !!

2k + 1 2k+ 1 v.

and therefore, using ^{f = F,}

1 d

Cok>_ - 2k + 1 t dt F(t) . (7 .5)

These are two particularly simple operators, whose exact one-point functions are given explicitly in terms of derivatives of F(t). The other pointlike operators have slightly more complicated expressions, but all can be evaluated explicitly in terms of f(t), its derivatives and its integrals. For example, the one-point function of 0 1 is given by

Similar explicit expressions for (Ot ) follow from eq . (7.2).

Next let us calculate the matrix elements of the finite loop operators, Tr(2'-~)2M.

In the scaling limit this gets contributions from 0 - ±[2-H(f(t))p-2/(2k+')], so that if we scale M as Lß z/tzk+'1 , asbefore, we derive*

It is very amusing that this expression, for the loop wave function, involves the matrix elements of the time evolution operator for the hamiltonian ^H, where the length of the loop, L, plays the role of time!

The leading WKB expression yields the result derived above for the sphere. If we desire we can write the diagonal matrix elements of e - 'tr in terms of f(t) and its derivatives, by writing

dz e -L^z

e-ttï _

- ~2rri z-H'

and making use of the results of Gelfand and Dikii. Alternatively, if we were given the potential f(t), we could solve the Schr6dinger equation for the eigenvalues, 6 ,

*This relation was independently derived by Douglas and Shenker [31].

(27)

of^H, which are discrete since f(t) ^fas t ~. Then

CTr(a <t )2ni \ -2~z Lrl n(tl^{)I Z}e (7.Ó)

n

The remarkable feature that emerges from these equations is the nonperturbative quantization of the eigenvalues of Oi = 2 - 2/(2k+>>. From the point of view of the Dyson gas this quantization corresponds to the freezing of the coordinates near the edge of the one-dimensional volume. The continuum spectrum of eigenvalues, i.e. the continuous charge density that we find in each order of the genus expansion, turns out to be discrete when calculated nonperturbatively.

The charges crystallize at fixed positions given by the energy spectrum of above hamiltonian, which is discrete as a consequence of the barrier at the Painlevé pole.

The full content of the theory is contained in the expressions for all the correlation functions. Using the methods that we have developed all of these can be written down in terms of the resolvent of ^H, and in principle calculated once we have solved the differential equation for f(t). So far we have only discussed correlation functions that are determined by the diagonal matrix elements of ^Hor of e -". However, the higher-point functions require the knowledge of the nondiagonal matrix elements of the resolvent.

Let us determine the two-point function of our operators to all orders. Now we make use of eq . (3.16) and go through the same procedure as above . The main difference being the fact that the sum over states in eq . (3.16) involves nondiagonal matrix elements and the presence of the projection operators. The resulting expression for the two-point function of the pointlike operators 01 is (we drop the powers of ß which, as we have seen, will all cancel for the correlation functions of the O1 's)

(0,0,_) = -4B( ;)B(', - ; -l2~

r -

xf

dt, fdt2(t2IH'I +i12 lt,)(ti lfil,+1 /2It2 ) . (7 .9) One may explicitly verify that in the leading WKB approximation we recover the two-point function derived previously on the sphere. In this limit only the region t, ---> t - 0, t2 __> t + 0 contributes to the integral . However, in order to evaluate the correlation functions nonperturbatively we need to know the spectrum and eigen- functions of ^H, which depend on the potential f(t) for the whole range of t from

down to the first pole at u.

Note that the t, integration, which originally ran from -~, is now cut off at the rightmost pole, u, since there is no tunneling through the barrier.

(28)

We would like to thank J. Distler, A. Polyakov, E. Witten, and A.

Zamolodchikov for discussions; and J. Zanetti for help in analytic computations.

THE LAGRANGE METHOD

expression as

Note added

After this paper was typed we received some new papers related to this work.

Banks, Douglas, Seiberg and Shenker have studied the correlation functions and derived results similar to ours [32]. In another very interesting paper Douglas claims to derive a differential equation satisfied by the general minimal model coupled to two-dimensional gravity [33].

Appendix A

We wish to solve the algebraic equation

x=y^k- YCrY`, (A .1)

for y as a function of x and the C;'s. Define F(y) =x, z =^yk and 1(z) = E; C

i

^z'Ik.

Then eq. (A.1) is equivalent to

dF( Y) Y _ dz ^[1-_y,(z)^]^{z '/k} _{(A .2)} Y(x)=~ 27rí F(y) -x - ~2-rri z-x-_(z) '

Now expand the denominator in powers of 1 and integrate by parts to write this

x 1 dz 1

Y(x)

_ ~l k

^p

~2.

rri (z _ x)n

_^p_{( z )}^zllk_

The contour integral can now be done and the result is

n ~ Y(x ;C C2, .)=

1

L,(ax)

~( rCrx'lk) xllk_

n=

(A .3)

(A .4)

The derivatives of y(x ; C) with respect to the C; yield the perturbation of the equation x^{= yk} by the terms C; y'. In particular the terms relevent for the

(29)

THE DIAGONAL RESOLVENT

R'(x)= [Rx+Ry] Jx=y,

calculation of the correlation functions in sect. 4 are

Y(x'C C ¹1^,12, . . . -

The resolvent satisfies the equation

Appendix B act, . .

aCIPY(x) c;=o ( ax ) X^{EC1, +_}

_ _ -k)Ik

k

In this appendix we shall review the classic work of Gelfand and Dikiï [25]

concerning the diagonal matrix elements of the resolvent of the Schrddinger operator,

R(x, Y ;

e) _

(xl -(d/dx)2+u(x) +eIY>,

(A.5)

R(x,x ;e) = RI[u] .

(B .1)

1

[ -17x2 +u(x)+e]R =[ -17y +u(y)+elR=8(x-y), (B .2) from which it follows that c(~) = RRXY - RX RY is a constant independent of x and y, and that Limx^_,7, [ Rx - Ry ] = 1 . Using the asymptotic properties of R for large x it follows that c = 0. The derivatives of R(x) = R(x, y; 6) Ix=y are given by

R"(x) _ [Rxx+ 2Rxy +Ryy ], =y = [{u(x) +~}R+{u(y) +~}R+2Rxy]x=y . (B.3) Putting this together, we deduce that R(x, ~) satisfies the differential equation

-2RR" +(R')2+4[u(x) + fl R 2 = [Rx-Ry]x=y-4c(~) =1 . (B.4)

(30)

If we differentiate this once more we derive a linear equation for R(x),

-R"'+4[u(x) +6]R'+2u'(x)R=0 . (B .5) We can use this to derive recursion relations for the coefficients R1[u]

Ri+1 - áR111 - uRi - zu,R,,1 (B.6)

or, multiplying this equation by 1/7X, we derive

R₁₊₁ ₌ -2K¹ . R1 = [ -2K~^I ¹⁺¹ _

1 d z 1

K[u(x),7x] --- - 2 ( dx +u(x) + pu(x)17X . (B .7)

X

Appendix C

GENUS EXPANSION OF THE SPECIFIC HEAT

Let us calculate the contribution to the specific heat f(t) = F"(t)from surfaces of higher genus. A convenient form of the differential equation is given by eq.

(5 .5), expressed in terms of the diagonal resolvent

2>

can be rewritten as R= 'z

V[1 +

2RR" - (R,)2

1 and iteratively ex- panded in inverse powers of^(t^{Ilk - t)),} if one is given the corresponding expansion of the potential

t - t'l'(1 - ~Z1t-1(2+ Ilk) ).

1

(C .3)

The resulting to integrals reduce to beta functions after which the coefficients become rational functions of v, which can then be replaced by k. Comparing the coefficients in front of various powers of t we obtain a system of equations for the unknown parameters Z1 in the above ansatz for f(t).

The calculations were performed using the MathematicaTM package. One may calculate up to genus 10 or so without too much difficulty. We report only the

1 - t = 2vB(-' Z -OVdtv R(t,t ;-w) .

(C.1) 27ri

The Gelfand and Dikii equation, eq. (B.4),

-2RR" +(R')2+4(f-to)R2=1, (C.2)