Department of Physics and Astronomy, Rutgers Unit'ersity, Piscataway, NJ 08855-0849, USA

Nuclear Physics B362 (1991) 665-709 North-Holland

F R O M L O O P S TO STATES IN T W O - D I M E N S I O N A L Q U A N T U M G R A V I T Y *

Gregory MOORE

Department of Physics and Astronomy, Rutgers Unit'ersity, Piscataway, NJ 08855-0849, USA and

Department of Physics, Yale Unicersity, New Haren, CT06511, USA

Nathan SEIBERG**

Department of Physics and Astronomy, Rutgers Unirersity, Piscataway, NJ 08855-0849, USA

Matthias STAUDACHER

Department of Physics and Astronomy, Rutgers Unicersity, Piscataway, NJ 08855-0849, USA Received 2 April 1991

We investigate macroscopic loop amplitudes (at genus zero) using the matrix model and the Liouville theories of two-dimensional quantum gravity. Some loop amplitudes, interpreted as wave functions of local operators, satisfy a linear differential equation known as the Wheeler- deWitt equation. Moreover, from the properties of the loop amplitudes an inner-product space structure on the space of wave functions emerges naturally. In the course of our analysis we resolve several apparent discrepancies between the matrix model and Liouville theory. Macro- scopic loops provide a natural ultraviolet cutoff on the theory, rendering universal analytic terms in the coupling constants. They contain more information than the local operators and should be regarded as fundamental.

I. Introduction and conclusion

T h e s t u d y o f t w o - d i m e n s i o n a l r a n d o m s u r f a c e s h a s m a n y a p p l i c a t i o n s . O n e a p p l i c a t i o n is t o q u a n t u m g r a v i t y . M a j o r i s s u e s o f p r i n c i p l e in q u a n t u m g r a v i t y (e.g.

t h e n a t u r e o f t h e H i l b e r t s p a c e a n d t h e f a c t o r i z a t i o n o f a m p l i t u d e s , m e a s u r e m e n t t h e o r y , t h e i m p o r t a n c e o f t h e s i g n a t u r e o f s p a c e - t i m e , e t c . ) a r e n o t u n d e r s t o o d . T h e t h e o r y o f r a n d o m s u r f a c e s is a u s e f u l t o y m o d e l f o r q u a n t u m g r a v i t y s i n c e , in

* This work was supported by DOE grants DE-AC02-76ER03075 and DE-FG05-90ER40559 and by a Presidential Young Investigator Award.

** On leave of absence from the Department of Physics, Weizmann Institute of Science, Rehovot 76100, Israel.

0550-3213/91/$03.50© 1991 - Elsevier Science Publishers B.V. (North-Holland)

66¢~ (;. ,:~h;ore ~': a/. / /~vo dimvn~iona/ q m l n l u m ,~,ra~'i:~;

some cases, it is solvable and is not plagued by the technical difficulties of its higher-dimensional counterparts. (Of course, it is possible that it consequently misses some crucial aspects of these counterparts.)

One source of the difficulties in the interpretation of quantum gravity is the ill-definedness of time, a result of general covariance. When we attempt to quantize gravity in a canonical formalism, general covariance leads to a constraint on physical wave functions: They should be annihilated by the generator of time reparametrization. This condition is known as the W h e e l e r - d e W i t t equation.

Unlike ordinary constraints in field theory (e.g. Gauss' law in A 0 = 0 gauge), the W d W constraint is i m p l e m e n t e d weakly. Consider the path-integral representation of the propagator. The Lagrange multiplier, A , , which implements Gauss' law is integrated from - ac to + oc. T h e Lagrange multiplier for the W d W constraint - the elapsed time - is integrated only from - z c to 0. (This point has been stressed in ref. [1].) Therefore, we can specify space coordinate invariant boundary conditions on the functional integral which do not necessarily satisfy the W d W constraint and find non-zero answers. This p h e n o m e n o n is familiar in (0 + 1)-dimensional gravity and we will meet it below in I + 1 dimensions. Interpreting ordinary field theory as (0 + 1)-dimensional gravity on the world line, this is the origin of off-shell physics.

Thus, we may hope that our understanding in two dimensions will help clarify, the subject of off-shell string theory.

With the above issues in mind we will examine macroscopic loop amplitudes in two-dimensional gravity. In particular we will show that, when appropriate, they satisfy the W d W constraints. Moreover, we will show how one can introduce an inner-product space structure on the "space of wave functions" of the theory. The introduction of this structure allows us to express matrix model macroscopic loop amplitudes as transitions between states, and paves the way for an understanding of factorization properties in (1 + 1)-dimensional gravity.

Some readers might find these results surprising since the WdW equation is a linear differential equation. On the other hand, in matrix model theory, it is well known that loop amplitudes obey non-linear equations [2-4]. This fact has even led to suggestions that in q u a n t u m gravity the superposition principle must be aban- doned. We disagree with these proposals. Once the wave functions of the theory are properly identified, there is no need to abandon the superposition principle in two-dimensional q u a n t u m gravity (at least, at genus zero). The Liouville interpre- tation of the nonlinear equation of the matrix model is an interesting question we do not address. It is clearly related to string interactions, and thus takes us out of the single string Hilbert space.

We will need several technical results in order to address the issues discussed above. The tools we will use involve both the matrix model [5-9] and the continuum formulation [10-12] which, in conformal gauge, reduces to the Liouville theory [13-18]. Despite recent progress [19] exact calculations in Liouville are still very complicated. On the other hand, the matrix model provides us with an

G. Moore et al. / Two-dimensional quantum grat'ity 667

extremely efficient calculational tool. T h e r e f o r e we need a complete and precise dictionary between Liouville and matrix model correlation functions. This is the first technical issue addressed in this paper. Some difficulties of the matrix m o d e l / L i o u v i l l e translation are noted in sect. 2. The difficulties can, in part, be traced to an issue of contact terms. This allows us to find a partial resolution of the difficulties. Using the WdW equation as a guide we unravel a tangle of mistaken identities and complete the resolution of the difficulties in subsects. 4.3 and 4.4. It is a nontrivial fact that, when correctly understood, Liouville theory and the matrix model are in complete agreement.

We also will need to develop some technology for handling macroscopic loop amplitudes. We study some general properties of these amplitudes in sect. 3.

Guided by semiclassical reasoning we investigate in detail the relation between macroscopic loop operators and sums of local operators. In particular we will see that macroscopic loop amplitudes contain more information than correlation functions of local operators. With these insights, we obtain general formulae for loop amplitudes in subsect. 4.1. Then, after we have understood better the relation to Liouville theory in subsects. 4.3 and 4.4, we are in a position to note some intriguing factorization properties in subsect. 4.5. Having settled these technical problems we will finally be in a position to discuss the inner product space structure in sect. 5. Most of our analysis is phrased in terms of the one hermitian matrix model in its one-cut phase, or, from the continuum point of view, in terms of Liouville coupled to perturbations of the ( 2 , 2 m - 1) minimal conformal field theories. In sect. 6 we show that the ideas discussed in the paper can be applied to a wide variety of models of two-dimensional gravity.

A n o t h e r application of the theory of random surfaces we have not yet men- tioned is to string theory. In subsect. 5.4 we remark on some string-theoretic implications of our work.

2. Contact terms and the Liouville/matrix model dictionary

According to the standard dictionary, the mth multicritical point of the matrix model corresponds to the coupling of the Liouville theory to the conformal field theory ( 2 , 2 m - 1) [20-23]. Moreover, it is generally accepted that we have the correspondence

~ e"i~LdPl m - l - j , j = 0 . . . m - - 2 , (2.1)

where aj = ½y(m - j ) , q~,,,, are the primary fields of the conformal field theory in the standard parametrization of the Kac table, and o) is the matrix model operator whose correlation functions are described by KdV flow. Eq. (2.1) means that the Liouville path integral is related to the matrix-model free energy at the mth

6 6 8 (;. Moore et al. / Two-dimensional quantum grat'iO'

multicritical point by

zmm(tj)

Z , . ( r , ) . t, = ~ i . ( 2 . 2 )

where " L " stands for "Liouvilte", the t i couple to %,

= f ,,, ( 2 . 3 )

¢ is the Liouville field, S[¢] is the Liouville action, and

m - - 2

Si.~ = ~ zif, e"i'~4 ~, ^{. . . .} _ l _ i + . . . (2.4)

i = 2 -

According to eq. (2.2), if, at the mth multicritical point we tune all tj = 0 except for t m_ 2, the resulting model is expected to be a conformal field theory coupled to gravity. If so, the three-point functions of the theory should vanish in accordance with the fusion rules. In particular this implies that, up to analytic terms in Tin_ 2 ^{= ] . Z ,} the cosmological constant, we have ( % ) = 0 for j 4: m - 2, and (o)~r k) 6jk. Explicit computations in the matrix model shows these expectations to be false [24]. Similar paradoxes occur in the computations with the q-matrix model at its p t h multicritical point if one accepts the dictionary

where

9

~rk(~. ) ~4 f , e-k.,~ ~ , . , , , (2.5)

ak. r p + q - ( k q + r ) k = [ - ~ ] - n

y 2q

[ pn' ]

r = n ' p - q [ ~ - . (2.6)

In particular, the two-point functions are not diagonal.

It has been conjectured that these paradoxical results are related to contact terms arising when two operators are at coincident points [25] (this is indeed the case in the topological interpretation of these models [4, 26]). Since contact terms play a key role in the following let us pause to recall the basic idea behind them.

In quantum gravity (and string theory) we consider operators like (2.1) and (2.5) which involve integrals over space-time (or the world-sheet). Therefore, to com- pute correlation functions we must give a definition to the local correlation functions at coincident points: ( . . . ( / ) I ( Z ) t ~ 2 ( Z ) . . . ) . (More generally, we must define correlation functions at the boundaries of moduli space.) Typically such expressions are singular and cannot be obtained as a limit of ( . . . q~(w)q~2(z)... ) as w --* z. If we do find prescriptions to define correlation functions at coincident

G. Moore et al. / Two-dimensional quantum grauity 669 points, two such prescriptions will, in general, differ by distributions s u p p o r t e d at w = z (more generally, by distributions supported on the boundary of moduli space). These ambiguous 8-function-supported changes in the G r e e n functions are known as contact terms. A useful way to think about a change of contact terms is that it is equivalent to an analytic redefinition of coupling constants [27]. Suppose that couplings h i multiply operators f ~ i so that correlation functions are com- puted according to

(2.7)

If we m a k e an analytic change of couplings h i --* iz i +A~k/zi/z k + ... and define correlation functions of new operators ~ by taking derivatives O/31zJ:

(ex.[ j'O ] H 0 + (ex.[ j'°,])

^(2.8)

then we see that locally the qt-correlators differ from the q~-correlators by delta functions. With a little thought one can see that conversely every modification of the G r e e n functions by additions of delta functions (subject to some criteria of physical reasonableness, e.g., locality) can by summarized by an analytic redefini- tion of coupling constants*.

In the examples studied in this p a p e r the original frame of operators u~(G,) and couplings t , ( a ) , the KdV frame, will be transformed to a new frame of operators

~,(~¢~) and couplings % , , , which we refer to as the "conformal field theory frame".

O n e important property of the KdV frame is that the operators ~r are scaling fields. In order to preserve this feature, the change of variables to the C F T frame must be compatible with the dimensions. T o g e t h e r with the analyticity of the transformation the form of the allowed transformations is severely restricted**.

We now consider three examples of analytic o p e r a t o r mixing.

Example 1: Pure gravity. As an extreme example of analytic redefinitions of couplings we show that at the m = 2 fixed point there is, up to analytic redefini- tion, only one nontrivial coupling. T h e specific heat for the m = 2 fixed point is obtained from the string equation

y ' t i u i +/./2 ⁼ to ' (2.9)

* In supersymmetric theories the situation is more intricate [28].

** S. Shenker has pointed out to us that a similar ambiguity in the identification of the scaling field was noticed in Wegner's original paper in the case where the critical exponents are rational.

67()

and thus has the form

(;. Mo¢)re ct al. / l i v o - d i m e n ~ u m a l q u a n t u m g r a t ' i t y

n i

Let us split the sum in eq. (2.10) into two terms )d' and v,, defined according to whether E i n i ( i / 2 - 1) is an integer or a half-integer. It follows that if we define

7" i = t i , i ~ 0 ,

tx = "r,, = to(1 ⁺

~.~,)2

^(2.11)

then the specific heat is of the form

u = f ~ - + analytic. (2.12)

T h a t we can do this is not surprising, although it contrasts sharply with the result of ref. [29] where an infinite set of B R S T cohomology classes in the Liouville × ghost system was found. We expect infinitely many of the states in ref. [29] to correspond to redundant operators in the Liouville theory. Eq. (2.12) is one way of making that statement precise. This example can probably be considerably general- ized: an analytic redefinition of couplings should show that, up to analytic terms in the free energy, the only physical couplings in the p, q models are those coupling to the m a t t e r operators in the Kac table. Moreover, such a redefinition should lead to fusion rules in a g r e e m e n t with conformal field theory*. We will see special cases of this in example 3 and in sect. 4 below.

Example 2: The boundary operator [30]. H e r e we briefly summarize ref. [30] as an example of analytic o p e r a t o r mixing. Although we work at genus zero through- out the paper, in this example there is no difficulty in extending our argument to all orders. In the string equation

E t i ( j + '2)Ri = 0, (2.13)

the coupling t m_ ~ in the m t h multicritical point is redundant, both on the lattice and in the continuum. Using analytic o p e r a t o r redefinitions we may eliminate it using the identity

R k [ u + p ] = R k [ u ] + y" a k . i p k - J R i [ u ] , (2.14)

j < k

* There are two possible directions one might follow to extend this example to other models. First, it could be useful to apply the methods of singularity theory. Second, our change of variables t i ~ r i is probably closely related to the Virasoro constraints on the matrix model partition function [3,4]. It would be worth understanding this connection better.

G. Moore et al. / Two-dimensional q u a n t u m gracity

where p is a constant. In particular ak, k_ ~ = - ( k - ½), so defining

671

1

P - - I t i n - I , m - ~ .

= tj + E F, ?O k,/k

(

k>j S ~ - ~ \ m - - z ]

(2.15)

we find that the operator coupling to p completely decouples beyond the one-point function. That is, computing at fixed ~'i we have (O/Op)"u = 0 for n > 1. The operator coupling to p is the boundary operator, and does nothing but measure the length of macroscopic loops.

E x a m p l e 3: The Ising m o d e l . As a third example we show how analytic redefini- tions of couplings in the Ising model can resolve some paradoxes. The dimension of the temperature, t, is 3 where the dimension of the cosmological constant, x, is 1

always taken to be 1. The correlation functions on the sphere are determined by solving [21-23, 31]

U 3 + tU 2 = X . (2.16)

Using our rules above there is a one-parameter family, labeled by a, of redefini- tions

T = t , tx = x + a t 3 . (2.17)

For a = - 2 eq. (2.16) becomes

_ I T 2 ( u I T

( u + l r ) ^, _(2.18)

The one-point function of the energy operator ( e ) = 1 z ~/z is analytic in /z and hence does not correspond to macroscopic surfaces (below we will study such analytic terms in/~ in detail). Moreover, it is clear from eq. (2.18) that (e n) = 0 for n odd precisely as expected from the continuum Liouville approach. Since the magnetic field enters the string equation quadratically, all of the Ising fusion rules (at the level of selection rules) are now satisfied. A similar redefinition in the unitary discrete series coupled to gravity can resolve some paradoxes presented in ref. [32] where the correlation functions in this theory were computed from the KdV [9] point of view. For example, it is easy to check that using analytic (in /~) redefinitions of the operators the two-point function can be made diagonal, as expected from the continuum approach. We suspect (but did not check explicitly) that as in the example of the Ising model, a similar redefinition can make the correlation functions compatible with the fusion rules of ref. [33].

672 (;. Moolz" et al. / l~'o-dime*l,s'iotlal quantum grat'itv

In general, finding the analytic redefinitions of operators with good physical properties is complicated. One of the technical results of this paper is an identifi- cation of the matrix model couplings ti(/x) corresponding to the Liouville theory coupled to a conformal field theory. That is, the theory defined by "r i = 0 for j 4= m - 2, %, 2 =/x. This is given in eqs. (4.22) and (4.23) below. Since, from the point of view of string theory the tj define a background for string propagation we refer to such a choice of couplings as a "conformal background." We will see in subsect. 4.3 below that the conformal background is most easily derived by examining macroscopic loop amplitudes. In sects. 3 and 4 we discuss macroscopic loop amplitudes, from the point of view of Liouville theory and of the matrix model.

3. L o o p s in Liouville

3.1. A C T I O N A N D B O U N D A R Y C O N D I T I O N S

We first discuss Liouville theory on manifolds with boundary along the lines of ref. [18]. The action is

+ 8 ~ ~ / ~ + ~ , . ~ / ~ d £ + ,u [ e ~ * + P-770~ e~*/2dg, (3.1)

_ 8 T r y 2 J,.: 4vr`/ ~.,'

where /~ is the extrinsic curvature of the boundary, /x and p are the volume and boundary cosmological constants. Classically, Q = 2 / - / where 3, is the Liouville coupling constant*.

From S we obtain the usual Liouville equation of motion:

1 tx

- Aq~ - ~ e r'p + E °~ i6~2'( z - z i ) = 0 (3.2)

(we have included sources of curvature for later convenience). We may set the boundary term to zero in the variational principle by choosing Dirichlet boundary conditions 6~ la_,- = 0, or Neumann boundary conditions:

~t0,,/p~ + ~: + p_ e v~/2 = 0, (3.3)

On 2

where the first term is the normal derivative.

* We will write the formulas in terms of Q in such a way that the generalization to the q u a n t u m case is straightforward.

G. Moore et al. / Two-dimensional quantum gracity 673 3.2. C L A S S I C A L T H E O R Y

At the classical level we should solve the constant negative curvature equation.

Recall that in the absence of a boundary a solution exists only when

X = ~ a , + Q ( 2 h - 2 ) / 2 (3.4)

i

is positive (c~ i are sources of curvature and h is the n u m b e r of handles). Constrain- ing the area of the surface the equation of motion states that the metric has constant curvature. T h e nature of the surface and hence the nature of associated q u a n t u m states depends crucially on the sign of X. T h e solution has negative curvature for X > 0 and positive curvature for X < 0. A similar story is true in the presence of boundaries. It is no longer necessary to constrain the area, since, if there is at least one boundary there is always a constant negative curvature metric.

T h e r e f o r e we henceforth restrict attention to the case with a single boundary. T h e nature of the surface, and hence the nature of the associated q u a n t u m states

I I

depends crucially on the sign of Y = X + ~Q = Y ' . a i - ~Qx. We must consider several cases.

Case let: Fixed tz, Y > O. If Y > 0 there exists a classical solution with constant negative curvature as the loop is shrunk to a point. In this case a small loop of length l behaves like a local source of curvature Q / 2 (this is the origin of this term in the definition of Y). In other words, the loop behaves like a puncture, and the surface has the shape shown in fig 1.

Case 2ct: Fixed Iz, Y < O. If Y < 0 there is no classical solution with constant negative curvature when the loop is replaced by a puncture. As l becomes small the surface looks like fig. 2 and the loop cannot be thought of as a local

Fig. 1. A large constant negative curvature sur- face with a small loop. T h e spikes originate from sources of curvature, e.g. from operator insertions.

1 > 1

Fig. 2. A constant negative curvature cone with two loops of decreasing length sliding towards the

apex.

674 G. Moore et al. / Two-dimensional q u a n t u m grat'ity

Fig. 3. A large c o n s t a n t positive curvature surface with a small loop.

d i s t u r b a n c e to t h e surface. F o r very small l t h e c o s m o l o g i c a l c o n s t a n t in t h e e q u a t i o n o f m o t i o n is n e g l i g i b l e (of o r d e r tzl 2) a n d t h e r e f o r e t h e s u r f a c e is a l m o s t flat. W e c a n u n d e r s t a n d this c a s e b e t t e r if w e c o n s t r a i n t h e a r e a to b e A . T h e r e a r e t h e n two r e l e v a n t d o m a i n s :

Case 3 % Y<O, F i x e d A > > l 2 . If w e c o n s t r a i n t h e a r e a A w h e n Y < 0 a n d e x a m i n e t h e c a s e o f small 12/A t h e classical s o l u t i o n has positive c o n s t a n t c u r v a t u r e a n d looks like fig 3. In this c a s e t h e small-/ limit is s m o o t h a n d t h e l o o p b e c o m e s as a p u n c t u r e with c u r v a t u r e Q / 2 .

Case 4% Y < O, Fixed A << 12. W h e n 12//t is l a r g e o r o f o r d e r o n e t h e s u r f a c e has c o n s t a n t n e g a t i v e c u r v a t u r e a n d t h e l o o p c a n n o t b e t h o u g h t o f as a local d i s t u r b a n c e . W e t h e r e f o r e always o b t a i n a s u r f a c e o f c o n s t a n t n e g a t i v e c u r v a t u r e as A ~ 0 .

W e e n d this s u b s e c t i o n by i l l u s t r a t i n g t h e s e r e m a r k s with s o m e e x a m p l e s o f c o n s t a n t c u r v a t u r e m e t r i c s . U p to c o o r d i n a t e t r a n s f o r m a t i o n t h e s o l u t i o n o f (3.2) with a s o u r c e o f c u r v a t u r e o f s t r e n g t h 1 - a is

w h e r e

16a 2 s c

g = e ~''p - - - ( 3 . 5 )

(1 _ ' - o '

t x l 2 sc = i z l 2 + 6 4 , n - 2 a 2 •

A s l ~ 0 t h e c h a n g e o f v a r i a b l e s to t h e s t a n d a r d c o n s t a n t c u r v a t u r e p u n c t u r e d d i s k g o e s like w ~ ~ / 2 z , t h u s i l l u s t r a t i n g fig. 2. Similarly, u p to c o o r d i n a t e t r a n s f o r m a - tion, t h e s o l u t i o n with fixed a r e a a n d b o u n d a r y l e n g t h is

g = ey~ - Y ( 3 . 6 )

(1 - - X ] Z [ 2 ) 2 '

w h e r e y = ( 2 A l l ) 2 a n d x = 1 - 4 r r A / l 2. A s an i l l u s t r a t i o n o f c a s e 3 d n o t e t h a t as l - + 0, x c h a n g e s sign, giving a positive c u r v a t u r e m e t r i c . C h a n g i n g v a r i a b l e s to

G. Moore et aL / Two-dimensional quantum gracity 675 W = ( f A / l ) z we see that the surface approaches the Riemann sphere as the boundary Izl = 1 approaches the pole at w = ~.

3.3. QUANTUM THEORY

Recall that, at the quantum level the expression for Q is renormalized Q = 2 / 7 + y. Every matter operator is "dressed" by e ~+ and the coefficients a i lead to curvature sources as in the classical theory. The probability measure for a surface of area A, with h handles and insertions of operators is Z ( A ) d A = ( d A / A ) A X / v z ( 1 ) [12, 15]. Therefore, in the partition function at fixed p,,

Z ( t x ) = fo~e-uA Z ( A ) d A (3.7)

for X > 0 the large-A part of the integral dominates and the typical surface has negative curvature. The fixed cosmological constant amplitude behaves l i k e / ~ - x / ~ . Thus, when negative curvature surfaces dominate the path integral the result is convergent and gives a negative, nonanalytic power of /~. For X < 0 the typical surface has constant positive curvature and the integral over the area diverges at small A. Regularizing this integral, the dominant configurations have very small A. Their contribution is analytic in Iz and not universal. The large-A surfaces c o n t r i b u t e / z - x / ~ (times log p. when - X is an integer). We now generalize this to the case with boundaries.

First, we need the quantum analog of Y. Consider an amplitude with one loop of size 1. The small-/behavior of Z(/~, l) is controlled by ~" = X + am, where a,~ is the curvature associated with the lowest dimension operator ¢¢,, which can couple to the loop (typically the lowest dimension operator in the theory). The term Q / 2 in the classical theory is replaced by a m because this is the maximum curvature that can be localized in a point in the quantum theory. Since there is always a constant negative curvature metric on such a surface, there are no small area divergences in the definition of Z ( ~ , l). Thus, the amplitude is well defined and universal. As in the classical analysis we consider several different cases.

Case lU": Fixed tz, y > 0. When y > 0 the small-/limit of the typical surface is smooth. Replacing the loop by the leading local operator G,, we find there is still a classical solution and hence there are no divergences in the l ~ 0 behavior of the path integral. By scaling we then find that the amplitude behaves like I~Q-Z~'")/vtz-Y/v (we will derive this explicit formula below). Notice that the amplitude is not analytic in /z. Higher-order corrections in the small-/ expansion correspond to replacing the loop by other local operators.

If there is more than one loop, the situation is similar. In this case, shrinking all the loops but one is smooth and the shrunk loops can be replaced by local operators. T h e reason is that there is a classical solution with constant negative

676 G. Moore et al. / Two-dimensional quantum gra~'iW

curvature whenever there is at least one macroscopic loop. Thus, in these cases we can safely replace a loop by a local operator expansion.

Case 2~": Fixed/.t, y < 0. When there is only one macroscopic loop and y < (I there is no classical solution without the loop*. Therefore, Z ( t z , / ) diverges /'or small 1. The typical surface is small and looks like fig. 2. Note that since the amplitude is finite at nonzero l, the loop length plays the role of an ultrauiolet cutoff on the theory. Indeed, by the shift q~ --, ~ + ( 2 / y ) l o g l, we may scale l out of the Liouville path integral except for the cosmological constant term

f # e 7~ ~ f #l ze ~ . (3.8)

In Liouville theory it does not make sense to expand in the cosmological constant:

there is no sense in which the interaction is small. However, in this case, having used our freedom to shift the zero mode of q~ we see that an expansion in txl 2 around free field theory could make sense, and should give the leading contribu- tions in a small-/ expansion. In particular, by scaling the leading term is l(Q + 2 X / y ) and the negative powers of 1 multiply analytic terms in ~. The expansion in

~12 is sensible at least for the first few orders.

Surfaces of the type shown in both fig. 3 and fig. 2 contribute to

~ c

Z ( i z , I ) ⁼ f , d A Z ( A , l ) e -"A (3.9)

m the large- and small-A region of integration, respectively. We can understand the nature of these contributions by analyzing each case as follows.

Case 3~": y < O, Fixed A >> 12. If we constrain the area to be large relative to l z the typical surface looks like fig. 3. In this case the presence of the loop is a small perturbation on the geometry of the surface so we expect that the loop can be replaced by a local operator expansion. As opposed to case 1 q" the large surface has constant positive curvature because, replacing the loop by the leading local operator ~'m, the resulting parameter X is negative. By scaling one finds that the contribution of such surfaces to (3.7) is l(Q - 2 a m / y ) A v/~- ~, leading to nonana- lytic terms in p.**.

Case 4q": y < O, Fixed A << 12. For small A the loop is not a small perturbation on the geometry of the surface and we cannot expect to replace it by a sum of local operators. Nevertheless, for nonzero loop length there is always a classical solution so the path integral Z(/x,l) discussed in case 2 qu is finite and well defined. In particular, in the expression (3.7) the integral over A converges at small A. As

* Note, in particular, that this occurs in the situation with one operator inserted on a disk, a situation we study in great detail below.

** If - y / y E 2+ then the amplitude has a factor of log p..

G. Moore et al. / Two-dimensional quantum gracity 677 discussed above, the short-distance cutoff is provided by the length l and the small-A part gives analytic terms in /z. T h e moral of the story is that even the analytic terms in /z can be universal since the loop length provides a physical ultraviolet cutoff on the theory*.

We end this subsection by illustrating these remarks with some examples of the behavior of partition functions on the disk. Using the solutions (3.6), (3.5) one can calculate the action (3.1) and verify the behavior we have predicted for Z ( A , l) in the leading semiclassical approximation on the disk. In this approximation one finds that Z ( A , l) has the form AXl y e -#-/A. From the semiclassical expansion in Liouville theory we would expect that the exact disk amplitude with fixed area A and p e r i m e t e r l has the s a m e general form Z ( A , I ) = A X l y e - / : / A , the exponents x, y merely being expressed as an expansion in 1 / c . Accepting this, we can derive the exact values of x, y as follows. For simplicity assume the lowest dimension o p e r a t o r which couples to the loop is the unit operator. Applying the scaling arguments used to derive the K P Z formulae [12,15] we find that Z ( A , I ) =

A-3/2-Q/2vf(l/~-A).

Now, as we have discussed, the limit as l -~ 0 should give the disk with an insertion of the cosmological constant, thus the leading t e r m as l ~ 0 must have the form U A Z ( A ) , where Z is the partition function for a closed surface. Again by ref. [15] the A - d e p e n d e n c e of Z ( A ) is known so we find

Z ( A , l) = l-3+O/Z'A - Q / e e -#-/A . (3.10) We will verify from matrix model calculations below that (3.10) is indeed exact.

O n e easily checks that (3.10) is compatible with the previous discussion. At large A the exponential is negligible, and the amplitude behaves according to the scaling AXl ~' predicted by the insertion of a local operator. As A shrinks to zero we pass from fig. 3 to fig. 2. T h e exponential in eq. (3.10) provides an explicit ultraviolet cutoff, which is present as long as I v~ 0, and cures the small-A divergences in the Laplace transform to fixed cosmological constant.

3.4. L i o u v i i l e wave f u n c t i o n s

Understanding the functional integral over a manifold with a boundary is an important step towards a construction of the space of states of a theory. As is standard in q u a n t u m field theory, and often used in conformal and topological field theories, the functional integral with Dirichlet boundary conditions on the fields defines a vector in a state space associated to a boundary. This makes the

* This point of view resolves a paradox about topological field theory. Since this is somewhat outside our main line of development we explain this in appendix A. Another application of this remark is that in the Ising model, the one-point function of the energy operator is computable and nonzero.

The boundary, which provides the UV cutoff leading to a universal analytic term breaks chirality and leads to a nonzero answer.

678 (;. Moore et al. / Two-dimensional quantum grarity

factorization and gluing properties of amplitudes obvious. We would like to do something similar in the case of gravity. Here, our wavc functionals, Z[~b(~r), M(o,),c(~r)] d e p e n d on the Liouville mode, the matter fields and the ghosts on the boundary.

By general covariance the wave functionals Z of canonical quantization satisfy some constraints. They should be in the B R S T cohomology of the left- and right-moving B R S T operators Q and Q, i.e. they should satisfy the m o m e n t u m constraints associated with space diffeomorphisms and the hamiltonian constraint associated with time reparametrizations. The latter is known as the W d W equa- tion.

As stressed in ref. [1], the m o m e n t u m constraints should be i m p l e m e n t e d as ordinary constraints since their Lagrange multiplier is integrated between - z c and + ~ . This is not the case for the hamiltonian constraint. The range of integration of its Lagrange multiplier is b o u n d e d so Z does not necessarily satisfy this constraint. This p h e n o m e n o n can equivalently be understood as a contact term arising when an operator, a handle or a n o t h e r loop touches the boundary*. For example, the disk amplitude with one insertion of an o p e r a t o r c ~" (the wave function of this operator) should satisfy the W d W equation but the disk with two insertions should not. More generally, whenever our amplitude involves integration over moduli, the region of integration is such that the W d W equation is not i m p l e m e n t e d on the boundaries. This issue of the range of integration over thc moduli makes the factorization and gluing properties of quantum gravity more subtle than in a theory which does not have integrated moduli. In the context of the critical string this p h e n o m e n o n has been discussed, for example, in ref. [34].

In the matrix model, we do not know how to study Z as a functional of all its arguments. First, the role and the origin of the matter is mysterious. In the one-matrix model at its third critical point the m a t t e r can be identified as in ref.

[20]. In the p-matrix model we can think of the matter as being the label of the matrix. However, in general we cannot specify M(~r) on the boundary. T h e ghosts are even more elusive and are not easy to identify in a completely gauge invariant description like the matrix model. Furthermore, we do not know how to specify

~h(~r). T h e only observables we know how to compute are macroscopic loops. They are related to Z through

( w(l)...) = Z(l ....

) =

f tD4aDMDcla(~er'~/2-1)J( M,

c , O ) Z [ ~ ( ~ ) , M ( ~ ) , c ( ~ ) ] ,

(3.11)

where

f(M, c, 4~)

is some "wave function" for matter, ghosts and Liouville, 1 is the

* We thank T. Banks and S. Shenker for stressing this interpretation of the violation of the WdW equation.

G. Moore et al. / Two-dimensional quantum gracity 679 length of the boundary, and the ellipsis stands for an unspecified insertion of operators a n d / o r loops. (In the p-matrix model we have p different loops and correspondingly p f's.)

This wave function Z ( I .... ) is similar to the minisuperspace wave function discussed in the quantum gravity literature. By general covariance physical wave functions are functions on superspace, which in pure gravity is the set of spatial metrics modulo spatial diffeomorphisms. In the minisuperspace approximation one simplifies the problem by reducing further to a one-dimensional quotient of superspace defined by the spatial volume. In pure gravity in two dimensions superspace and minisuperspace are identical since the length completely specifies the spatial geometry up to spatial diffeomorphisms. Given this equivalence, one might expect the minisuperspace approximation to be exact. In particular, from our remarks regarding integration over moduli, one might hope that the amplitude for a disk with a single operator ee inserted, i.e. the wave function associated to the operator ~', Z ( l , ee) =- $e(l), will satisfy the minisuperspace WdW equation.

We have not proven from the path integral that minisuperspace is exact.

Nevertheless we can already present some evidence in favor of exactness. In the conformal backgrounds of sect. 2 the wave functions factorize between matter and gravitational degrees of freedom:

= d t matter dt gravity (3.12)

Therefore the (minisuperspace) WdW equation simplifies drastically and becomes Bessel's differential equation

[l: t 2

^{- l} + 4 / z / z + v ²

1

0,¢ ^gravity = 0 , (3.13)

w h e r e v 2 is related to the undressed matter conformal dimension A°(6~) by

1,2 L [ Q2 )] ~2 (0 t Q ) 2

= y 2 [ 8 ( 1 - - A ° ( ~ ' ) = -- (3.14)

and a is the Liouville charge associated with the dressing. In sect. 2 we argued that the formula (3.10) for Z ( A , I ) is exact. Given that, one can take the Laplace transform, using eq. (B.7) of appendix B* to obtain a Bessel function,

Z ( I z , l ) = l - 2 ( f ~ ) c 2 / ~ ' - ' K o / ~ _ , ( 2 v ~ l ) "

(3.15)

From eq. (3.15) we can obtain the wave functions of the cosmological constant and

* Several pertinent facts about Bessel functions are gathered in appendix B.

68[) (;. Moore et al. / Two-dimensional quantum grat'itv

the boundary operator, and we can verify that both of these indeed satisfy (3.13).

In sect. 4 we will prove that the minisuperspace approximation is exact using the calculational techniques of the matrix model.

4. Loops in matrix models

4.1. n-LOOP AMPLITUDES AT GENUS ZERO

In this section we c o m p u t e wave functions and loop amplitudes from the matrix model formalism elaborated in refs. [8, 35].

As is well known, in the representation of matrix model integrals as sums over surfaces, the o p e r a t o r w m m ( l ) = ( 1 / l ) t r ~b l/" creates a hole of boundary length / in units of the lattice spacing a. The extra factor of 1/l is a symmetry factor needed to take into account the fact that we are working with u n m a r k e d loops. In the double-scaling limit, a --* 0, N -~ ~, correlation functions of this o p e r a t o r may be expressed in terms of integrals of the kernel

<xle ~C*ly ), (4.1)

where Q = - u 2 d 2 / d x 2 +U(X,K) is the Schr6dinger o p e r a t o r associated to the model, • is the topological coupling, and U ( X , K ) = X ~ / " + G ( K 2 / X 2 ) . In the double-scaling limit, the amplitude for one macroscopic loop, to be c o m p a r e d with the Liouville quantity Z(/z, l) discussed in sect. 3 is given by [8, 35]

d x ( xte-lC*lx ) .

(4.2)

z m m ( I ; t i ) - 1 ,,,

In the following we will use the notation w(l) = lwmm(l).

We now c o m p u t e n-loop amplitudes at genus zero. We begin with one loop.

Using the C a m p b e l l - B a k e r - H a u s d o r f f formula, and inserting a complete set of eigenfunctions

1

{ p l x ) - e ip'/K , (4.3)

we obtain the genus-zero answer

f d x e -/u(x: t')

( w(1)) - KI,/~ ,,,

~¢11/2 dy y" j t j y j - l e-i~,

i=- I

El ~" j t j u J - ' / 2 ~ b i - , ( i ) ,

j = l

(4.4)

G. Moore et al. / Two-dimensional quantum grat'ity

where

~bj( x) - j ! x - j -

'/2(1

+ x + x2/2! + . . . +xJ/j!)e -x , [ - ul,

and we have used the string equation

681

(4.5)

tju j ⁼ 0. (4.6)

j>~O

Surprisingly, the genus-zero two-loop formula is much simpler. The expression to all orders is [35]

d x e-Z'~-)ly)(y I

( w ( l , ) w ( 1 2 ) >= f + f- "dv<xl

-- /(I

(4.7)

so the genus-zero approximation is

e-"'+L')" f +

dxf-l,,dyexp[_Cx_y)-,(,, +12)/(4K21,12)]

tvC ,tSK - , , , _

- - e-U(/u +/2)

=2/~u/~ l , + l 2 (4.8)

Note that the small-/ behavior of these amplitudes is in accord with the predictions of the Liouviile theory discussed in sect. 3. In particular, the two-loop amplitude is smooth as li ~ 0, while the one-loop amplitude is divergent as 1 ~ 0.

Indeed, using the string equation we have

( w ( l ) ) = - ~ j!tjl - j - ' / 2 +

1 O ( I ' / 2 ) . (4.9)

K j~>O

(Henceforth we will set x = 1.) In accordance with the previous discussion we interpret the nonanalytic terms in t i in the small-/expansion as arising from the insertions of local operators in the theory. It is in this sense that we may use the formula

1,,+ i/2

"w(t) = E ( - 1 ) "+' . = o n! ~" " (4.10)

The operators* (5, are the standard operators of two-dimensional gravity, in

* Note that o-. corresponds to O/at. with t . being defined in eq. (4.6). O u r normalization therefore differs from the " G e l f a n d - D i k i r ' normalization.

(~S2 ( ; . :~10121'(' ~'I ~11. / Jll(2-dilllCtl~lttll¢l/ ¢lllOlllllltl t,'l'~l/ il~'

particular their correlation functions arc computed from KdV flow [35]. The expansion (4.10)is correct in correlation functions if we ignore divergent terms in 1. However, as noted before, if there is at leasl one other hole on the surface then there is no divergence as /--, 0 and then (4.10) is exact. This simple observation allows us to compute a formula for the genus-zero n-macroscopic loop amplitude.

Consider, for example the case n = 3, which we can write as

1,1,+ i 2

= E

n :(I n !

-1 it +

E ( - 1 ) " " II ^{I / 2} ;j (W(12)W(I~)}

n ~ () t! [ Otn

-2V/1213 e "¢"-~'" E ( - 1 ) " " li'+ ,,,2 at,

,,=~ n? at,,

Oll

= 2.77~/~/~/2l~ e ,,,/, +/_.*, ~, (4.11) c,t,, ~

where in the last line we may obtain the /-dependence immediately since the amplitude must be totally symmetric in l~,12,1 ~. The same argument may be applied recursively to obtain the n-loop formula. T h e argument giving the last line of (4.11) implies that the n-loop amplitude is, up to a factor of ~ I l i t / 2 a function only of the sum of the loop lengths E - £ l i. Thus we can obtain the general formula immediately by specializing to the case where all but one loop is micro- scopic*. In this case the amplitude is also proportional to (cr[~'-~w(Z)). Hence, defining ~ ( l ) = w ( l ) / ~ we obtain the final result for the genus-zero n-loop formula:

/H

^{g,(l,) = - -} ( ( # ( Z ) ) ) . (4.12) It is remarkable that this amplitude only depends on the sum of the loop lengths.

This fact deserves to be understood better.

4.2. W A V E F U N C T I O N S F R O M T H E M A T R I X M O D E L

By shrinking one of the loops we obtain the amplitude for one macroscopic loop and one local operator, (o-jw(l)). In subsect. 3.4 above we identified a similar quantity Z(I, dP) in the Liouville theory and argued that it should be thought of as

* A m o r e formal p r o o f of (4.12) proceeds by using KdV flow and induction in n.

G. Moore et al. / Two-dimensional quantum gravity 683 the wave function associated to a microscopic operator. Since wave functions are half-densities, what we call the wave function is ambiguous up to a function of l until we specify the measure. The proper measure for the wave function of the operator G, which is a half-density (dO) I/2, is obtained by computing [18]

~t~,( l )

=( (~'lwmm( [)) =( (~W( I)) .

(4.13)

By expanding (4.8) we obtain the wave functions in terms of the truncated exponential functions (4.5):

z ; n m ( / ) =(O'jW(I)) = H 'i+l/2~tj(lu)

= j ! l - i - I / 2 + regular. (4.14)

Using the matrix model, it was shown in ref. [18] that the leading behavior of the wave function of o) is l - j - ~/2. This was derived by examining the large-n limit of the coefficients a,,(j) in ~ = t r ( 1 - M ) J + l / 2 = Y ' . , , a , , ( j ) n -l t r M " , which can be interpreted as the wave function of ~.

4.3. CONFORMAL BACKGROUNDS

The wave f u n c t i o n s Z ? lm of sect. 3 do not satisfy simple linear differential equations. As we emphasized in sect. 1, the KdV basis of operators is not necessarily the best basis for comparing Liouville theory with the matrix model. In particular, we mentioned that the dictionary (2.1) leads to paradoxical results for one-, two-, and three-point functions. The origin of these problems is two-fold.

First, we have not correctly identified the matrix model operators ~ coupling to ri"

Furthermore, we have not even correctly identified the matrix-model background corresponding to gravity coupled to a conformal field theory.

We will find the correct conformal background using the W h e e l e r - D e W i t t equation as a guide. Recall from subsect. 3.4 that in a conformal background the wave function factorizes 0~ = Off ~'tte' ® ~ltgravity and the minisuperspace WdW equa- tion reduces to the Bessel equation (3.13). The Bessel equation admits a two- dimensional space of solutions spanned by the modified Bessel functions I t , ( 2 f / z I), where

1 2k

l.(z)

⁼

= ) k ! F ( k + v + 1) (4.15)

Since ,y~,.J/tgravity must decay in the infrared (l--* ~) we learn that in conformal backgrounds the wave functions of scaling operators are expressed in terms of the

684 G. Moore el al. / Two-dimemional q m m m m gra~ tty

modified Bessel function K,.(x):

JT

K , , - 2 s i n v ~ [ l , , ( z ) - l , , ( z ) ] , (4.16)

w h e r e v ¢ 2. In sect. 2 we saw that the c o n f o r m a l b a c k g r o u n d is defined in terms o f the couplings 7~ of (2.4) by r , = 0 for i = ~ r n - 2 and % , - 2 = / z - In such a b a c k g r o u n d the insertion o f ~,,, 2 in correlation functions is the derivative with respect to the physical cosmological constant:

( ~ . . . ) = ( ~ , , , 2 c # . . . ) . (4.17) 0/.t ... ,,

A p p l y i n g this relation to the wave function o f the area o p e r a t o r in the b a c k g r o u n d with % = . . . = r,,, 3 : 0 we expect

() / ~ J n 3 / 2

( 4 , , _ 2 w ( / ) ) = - ~ g ( w ( l ) ) = (~/# ) K m _ 3 / 2 ( ~ f ~ l ) . (4.18)

It is possible to integrate (4.18) with respect to /x with the result that

<w(t)>

⁼

t- (¢-;-)'" ,j2.

¹ K,,,_,/2 ( f ~ - l ) " (4.19)

W e can now express the c o n f o r m a l b a c k g r o u n d in terms o f the K d V c o o r d i n a t e s ti, j = 0 . . . m - 2, at least to first o r d e r in % . . . % - 3 . T h e most general analytic c h a n g e o f variables has the f o r m

t i = r i + E c ~ i , ~ I , ( / ' , , (4.2(i)

n

where, by dimensional analysis,

c °) :/: 0 ,-, 1 - i / 2 = Y]ny(1 - j / 2 ) . (4.21)

In particular, for m even, the most general analytic c h a n g e of variables, to first o r d e r in % . . . % - 3 has the f o r m

t2j = r 2 j + c(2J)l&m/2-) j = O, . . . , 2 m - 1,

1 . . . , ~ r F l

t2j+ ~ = r2j+ I j = (1, - 1 , (4.22)

for m o d d we modify tj for j o d d by powers o f / x . C o m b i n i n g eqs. (4.9), (4.19) and

G. Moore et al. / Two-dimensional quantum gravity 6 8 5

(4.22) one may d e t e r m i n e the c i from the first m singular powers of l. T h e result is

- - ' 7 7 "

C(m-2p)__ ( 1) m+l 2 m - 2 p

( r n - 2 p ) ! p ! F ( p - m + ~ ) " (4.23)

This completes the determination of the conformal background. Note that we have not yet shown that (4.18) and (4.19) are satisfied. We only arranged the t's such that the singular part of these equations is correct. However, using the explicit expressions (4.4) and (4.5) it is possible to show that the regular part is also correct.

In a conformal background we can also use the W d W equation to identify unambiguously the conformal scaling operators ~ since these should diagonalize the action of the W d W o p e r a t o r on the space of wave functions. We expect

( ~ ( u ) w ( l ) ) - u j+ ' / 2 K j + ,/2( u l ) , (4.24) where u = 2V/~ -. T h e transformatin coefficients used in passing from d to ¢r are most easily found by comparing the singular terms in the wave function for l ~ 0.

Using eq. (4.15) one can c o m p a r e (4.14) with (4.24) to obtain

• "/7" [ j / 2 ] b / 2 s

~ = ( - 1 ) J - ~ 2 j + ' / 2 .,=o ! ( j _

2s)!V(s+ , _j)%-2~.

^(4.25)

Note that the transformation is analytic in ~*.

It is a nontrivial fact that once we have fixed the singular powers of l to get (4.25) the remaining powers of l work out to give (4.24). O n e can prove this by explicit computation of the coefficients of l ''+ z/2. A n o t h e r proof is given at the end of subsect. 4.4.

4.4. CORRELATION FUNCTIONS IN CONFORMAL BACKGROUNDS

As a check on our identification of the conformal background we now c o m p u t e the one- and two-point functions of the operators ~ . These calculations finally resolve the paradoxes pointed out in ref. [24] and discussed in sect. 2.

We begin by noting that the wave functions ~O~(l) have a small-/ behavior which agrees nicely with the general predictions of sect. 3. Using the defining relation (4.24) and (4.16) we split the wave function into a sum of two terms. From the expansion (4.15) we see that the divergent terms for l ~ 0 come from

I_j_l/2.

Since u 2 = 4/z these are analytic in the couplings, as expected. T h e nonanalytic terms come from the expansion of I j + l / 2 and, in a small-/ expansion, the first nonanalytic t e r m in /x enters at order I j+ ~/2

* In a conformal background d,,, 1 ⁼ O'B is the boundary operator [30], which satisfies (O'BW(I)) = l(w(l)).

68~ (;. ,'~tOOl't' dl tl/. // fwo-dilHt,tlMOlltll q t l a l t l u m ,qrdl ll 3

O n e - p o i n t functions arc most efficiently c o m p u t e d by using the o p e r a t o r expan- sion of w(l) to c o m p u t e the nonanalytic terms in # in a small-/ expansion of ( w ( l ) ) . To obtain the expansion of w(l) in terms of ~ wc first invert the expansion (4.25) to express ~r in terms of {) as

[j/2l ( 2 j + 1 - 4 s )

cr = 2.iv ~ 2.,^

.

,:,2'+'/2s!F(i-s+~ -)

/~/ O" i ~ ~ ^- • ^(4.26)

Substituting eq. (4.26) into eq. (4.10) yields the expansion

" w ( / ) = 2 ~ ~)i( - 1 ) J ( 2 j + 1 ) / i + ' / 2

lJ+'/2(lu) ,,

^(4.27)

, ~ .

(lille+

^1/2

Notice that we expand in terms of li+ ~/2 and

not Ki+

~/2. This is in accord with the results of sect. 3 w h e r e we learned that only the nonanalytic terms in # are related to an o p e r a t o r expansion.

T h e o n e point functions of &~+

~/2

are now easily c o m p u t e d by c o m b i n i n g eq.

(4.27) with (4.19). Using p r o p e r t i e s (B.6) of the Bessel function we may rewrite eq.

(4.19) as

llnl ~ I / 2

( w ( l ) ) -

2 m ~ ( K m + , / 2 ( 2 ~ l ) - K , , , ~/2(2~/-~l)).

^(4.28)

It follows from eqs. (4.27) and (4.28) that

{,4i ) = O, j 4= m , m - 2,

( t),,,) = 2~r u2,,, + t ( 2 m - l ) ( 2 m + 1)

- 2 r r

('),,, 2) = u 2''' -' (4.29)

( 2 m - 1 ) ( 2 m - 3)

T h e s e e q u a t i o n s are u n d e r s t o o d to hold up to terms analytic in ~. W e thus may write o u r matrix m o d e l / L i o u v i l l e dictionary as follows:

8 i ~ f e,,,< q~. _{J r} , , m - . I - i ' j = 0 , . . m - 2

<-+ (]6 e r ¢ / 2

~ m I " a Z

m + 4 /~ 1

~,,, = ~ ~,,, - u28,,,_2 ~ -OSq~ + - - e re + - - / ~ . (4.30)

m ~ 8 y 4~-y

G. Moore et aL / Two-dimensional quantum gracity 687 In the last line we have written the matrix-model version of the Liouville equation of motion. The linear combination is determined by demanding that the one-point function of the equation of motion is zero. As a nontrivial test, one can then check that on the sphere < ~'5~'~;n(O~m_2)"> = n<(~,n_2)">. Furthermore, we can consider an insertion of the equation of motion ~;,, on the disk. From the above formulae we find

( ~ w ( l ) ) = l ~ ( w( I)) ,

d (4.31)

as expected from the surface term in the equation of motion.

The two-point functions are also easily computed. Using the expansion (4.27) in (4.24) we learn that

{ u2j+'

< ak>=ai'k

^{- -} 2 j + 1 '

(4.32)

up to analytic terms in ~.

In this subsection we have used analytic redefinitions of couplings, of the type discussed in sect. 2. However, as emphasized in sect. 2, redefinitions involving only the relevant couplings are only a small part of the story. We expect that many properties of the macroscopic loop amplitudes discussed below will become much more transparent when the correct notion of analytic redefinitions of loop cou- plings (and loop contact terms) has been found.

We conclude this subsection by noting a peculiar fact about the above redefini- tions, namely, that one can carry out many of the above manipulations in an

arbitrary

background

t i.

To see this, define the operator

2 1

~r~.j--- -- l~- q- U212 q- ( j q'- 12)2 .

A short calculation reveals that

and hence

~ j g ? m =

u2j(j

^--

])Z~_~, _(4.33)

Z i ( / ; u ) - - Z j + Y'~ ( - 1 ) k u 2k

Zi_2k

(4.34)

k=J 2 k ] ~ 2 k J

satisfies eq. (3.13) with /~ ~

u2/4.

In the mth multicritical theory the space of relevant couplings is (m - 1)-dimensional. For a general point in that space the specific heat cannot be simply related to the cosmological constant t m_ 2. However,

688 (;. Moore el al. / Two-dimensiomd qttanttcm grat'it)"

there is a submanifold in which u 2 = t,,,__ 2 + analytic. One can check that in this case the coefficients agree with those found in passing from ~r to c; in (4.25).

Using the properties of Bessel functions one can check that one- and two-point functions vanish modulo

u 2.

Of course, away from a conformal background u 2 is not necessarily analytic in /x. Moreover, we can only interpret eq. (3.13) as the W h e e l e r - D e W i t t equation if the background is such that in the continuum theory only the cosmological constant is turned on. Thus the significance of these last observations is unclear.

4.5. O P E R A T O R F O R M A L I S M

In subsect. 4.4 we have analyzed the operator expansion (4.27) of

w(l)

in some detail. This allows us to develop an " o p e r a t o r formalism" for constructing correla- tion functions of the local operators 4 ' To do this we use the expansion (4.27), subject to the rules of section three, to compute macroscopic loop amplitudes.

We begin with the two-loop amplitude

G(l~, l e) = (w(ll)w(12)).

For l I < 12 we can replace

w(l t)

by a sum of local operators as in (4.27) because one loop remains. Substituting this and using (4.24), we find

c ( l , , l : ) = E

j = 0

( - 1)J(2j + 1)u-~i+

,/2,!i + ~/2(1.)(4.W(/2))

zc

= Y'. ( - l ) J ( 2 j + 1)1i+ l / 2 ( l l U ) g i + l / 2 ( 1 2 u ) . (4.35) j=o

The fact that this infinite sum of Bessel functions is equivalent to (4.8) follows from the G e g e n b a u e r addition formula (see eq. (B.13) of appendix B) which in our case reads

e - u ( l l + / 2 )

( - 1)J(2j +

1)Ij+l/2(l,u)Kj+,/2(12u )

= ~ (4.36)

j=0 /~ + 12

Proceeding to amplitudes for n > 3 loops we first note that from eq. (4.12) it follows that as all l i ~ 0 there is no singularity, so we can freely use eq. (4.27), and consequently the (n > 3)-point functions can be written

( w ( l , ) . . . w ( l , , ) )

= Y'~ I-I [ ( - 1 ) h ( 2 J , +

1) u-j'-I/2l.j,+,/2(ul,)]

( l~-I 4.,). (4.37)

j~ s

We therefore extract correlation functions of the local operators ~ by extracting the coefficients of the "wave functions"

Ii+

J/2 in the expansion (4.37). (Note that, by the reasoning of sect. 2 we could have expected divergent terms in 1 in

G. Moore et a L / Two-dimensional quantum gracity

() ,.)

5". Iv !

i

K v

V

Fig. 4. A cap with an o p e r a t o r inserted. Fig. 5. A tube with I at one end and K at the other.

689

Z

p.,v,p>O

Fig. 6. n macroscopic loops with l ' s on the ends.

(~d'k...w(l))

for large j or k since X would then become negative. The above calculation shows that with our universal cutoff these potentially divergent terms in fact have zero coefficient.)

Eqs. (4.24), (4.35) and (4.37) strongly suggest the following geometrical picture.

We associate (4.24) with fig. 4 depicting the creation of a state by insertion of an operator on a disk. Similarly, eq. (4.35) suggests that we interpret the annulus amplitude as a propagator. We see a sum of states with diagonal propagation, every term in the sum corresponding to the propagation of a different state. The two wave functions

lj+ J/2

and Kj+l/2 are physical wave functions satisfying the WdW equation with two different asymptotics. Since lj < l 2 we should impose good behavior (no divergence) only as 11 ~ 0 and as l 2 --* oc. These asymptotics determines the combination

lj+ l/2(l~u)Kj+ 1/2(12u).

Thus we associate (4.35) with fig. 5*. Moving on to n >t 3 loops, eq. (4.37) can be associated to fig. 6 where we have wavefunctions lj+ 1/2 on the external legs. Geometrical intuition suggests that the correlation function of n local operators ~ should be obtained by "sewing"

pictures like fig. 4 into surfaces like fig. 6. We see from the above formulae that this intuition is reproduced if we define a formal " i n n e r product" in the space of modified Bessel functions by the rule

( I , , Ip) ~ 8,+~. (4.38)

The inner product ( , ) represents the geometrical operation of sewing.

* Remarkably, a l t h o u g h we have different expansions for the p r o p a g a t o r for / I < / 2 and for 12 < Ii, the total sum (4.35) is not singular at / l = 12. This follows from eq. (4.36), which also shows that the singularity is at I I = - I 2.