f dN• A2(2)

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Volume 238, number 2,3,4 PHYSICS LETTERS B 5 April 1990

M I C R O S C O P I C A N D M A C R O S C O P I C L O O P S

I N N O N - P E R T U R B A T I V E T W O D I M E N S I O N A L G R A V I T Y

T o m B A N K S 1, M i c h a e l R. D O U G L A S , N a t h a n S E I B E R G 2 a n d Stephen H. S H E N K E R Department o f Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849, USA

Received 22 December 1989

We study the correlation functions of microscopic loops (local operators) and of macroscopic loops in non-perturbative two dimensional quantum gravity. They are easily calculated using a free fermion formalism. The microscopic loop correlation functions can be expressed in terms of the KdV flows. The specific heat as a function of the scaling fields obeys the generalized KdV equation. The physical interpretation of macroscopic loop correlation functions is discussed.

Two d i m e n s i o n a l q u a n t u m gravity is relevant b o t h for string t h e o r y and as a toy m o d e l o f higher d i m e n - sional q u a n t u m gravity. T h e d e f i n i t i o n o f pure 2 D q u a n t u m gravity a n d q u a n t u m gravity c o u p l e d to m a t t e r in t e r m s o f m a t r i x m o d e l s [ 1 ] is very explicit a n d rigorous. M a t r i x r e a l i z a t i o n s o f pure gravity a n d gravity c o u p l e d to certain m i n i m a l c o n f o r m a l field theories ( a n d their m a s s i v e d e f o r m a t i o n s ) can be solved by the a p p l i c a t i o n o f large-N techniques. Re- cently, an exact expression for the specific heat o f s o m e o f these m o d e l s was f o u n d in the c o n t i n u u m limit [ 2 - 6 ] . In this note, we will show that the correlation functions o f o p e r a t o r s in these m o d e l s can also be easily computed. We distinguish between two kinds o f operators, m i c r o s c o p i c a n d m a c r o s c o p i c loops. By m i c r o s c o p i c loops we m e a n expressions like Tr M p in the m a t r i x m o d e l s with p finite. T h e y c o n t a i n all the i n f o r m a t i o n a b o u t integrals o v e r the surface o f local operators. M a c r o s c o p i c loops are also given by T r M p but p is t a k e n to infinity in the c o n t i n u u m l i m i t in such a way that they c o r r e s p o n d to e x t e n d e d b o u n d - aries on the surface.

We start by d e r i v i n g a free f e r m i o n r e p r e s e n t a t i o n for the c o r r e l a t i o n functions o f a r b i t r a r y loops in theories o f two d i m e n s i o n a l g e o m e t r y based on a single

On leave of absence from the Department of Physics, Univer- sity of California at Santa Cruz, Santa Cruz, CA 95064, USA.

2 On leave of absence from the Department of Physics, Weiz- mann Institute of Science, Rehovot 76100, Israel.

large-N m a t r i x integral. The m e t h o d s we present gen- eralize i m m e d i a t e l y to a r b i t r a r y one d i m e n s i o n a l chains o f matrices o f the type studies in ref. [7]. In the limit when the m a t r i x chain b e c o m e s infinite a n d continuous, our m e t h o d reduces to the f e r m i o n i c description o f large-N m a t r i x q u a n t u m m e c h a n i c s dis- c o v e r e d by Br6zin, Itzykson, Parisi a n d Z u b e r [8 ].

We will indicate the form o f this generalization only briefly here, reserving the details for a lengthier pub- lication [9]. We will study c o r r e l a t i o n functions in the m a t r i x m o d e l o f the form

f [ d M ] e x p [ - t r ~ "(M) ] tr M p'...tr M p"

f [ d M ] e x p [ - t r Y - ( M ) ]

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G e o m e t r i c a l l y these represent sums over r a n d o m surfaces o f a r b i t r a r y genus with b o u n d a r i e s o f lengths Pl, ..., Pn.

I n t r o d u c i n g the d e c o m p o s i t i o n o f the h e r m i t i a n matrices M into u n i t a r y and diagonal matrices, M = U*A U ,

this integral can be written as

f dN2A2(2)

e x p [ - ~ ( 2 i ) ] E2~'...E2 p"

f dN• A2(2)

e x p [ - E ~ ( 2 i ) ] ( 2 ) where A(2) = ]~,<s ( 2 , - 2 s ) is a Van d e r M o n d e de- t e r m i n a n t . This expression m a y be v i e w e d as the ex- p e c t a t i o n value o f a p r o d u c t o f one b o d y o p e r a t o r s in a Slater d e t e r m i n a n t [ 10 ] c o n s t r u c t e d from the first

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Volume 238, n u m b e r 2,3,4 PHYSICS LETTERS B 5 April 1990

N members of the complete orthonormal set of one body wave functions

~'n =Pn(2) exp[ - ½ "t'(2) ] , (3)

f W~Um d 2 = g . . . . (4)

These functions are determined completely by eq. (4) and by the requirement that Pn be a polynomial of order n. A set of recurrence relations determining them explicitly was given by Bessis, Itzykson and Zuber [ 11 ].

For large N we are dealing with a many fermion system and it is convenient to introduce second quantized notation. We define the fermion field

g-'(2)= ~ a n g , ( 2 ) , (5)

n = O

where the an are annihilation operators. The correlation function ( 1 ) can then be written as

( F [ ~*JP' gJ... ~J*JP"~IF> , (6) where I F> is the filled Fermi sea for N fermions, and J is the one body operator of multiplication by 2. In order to take the continuum limit of this equation it is best to work in the orthonormal polynomial basis

[ 11 ], where J is an infinite matrix given by

Jrnn = N ~ m (~m,n+ l "}- ~ n C{m + l,n . (7) It is now easy to write down formulae for the correlation functions in terms of one body operators, by considering how the creation and annihilation operators act on the Fermi sea. For example, the connected one and two point functions are

N--I

( F I ~ J t J P ~ l F ) = t r C J J P = Z (JP)nn, (8) n=0

< FI ~ j m ~ , j v 2 ~1 F> c = tr ~geJ p'

(

1 - 5 p ) J p : ,

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where ~ is the projecnon operator [ 10] on the sub- space of one body wave functions with n ~< N - 1. Be- low we will show how to take the continuum limits of these formulae.

To facilitate computation of higher order connected Green's functions we can use Wick's theorem.

In order to do this it is necessary to write our correlation functions in terms of time ordered products.

This is done by inventing a one body hamiltonian

whose eigenstates are ~u,, and whose eigenvalues arc monotonic functions of n. The details of the hamiltonian are irrelevant because all of the correlation functions of interest are of operators at almost equal times. With respect to any such one body hamiltonian, our Slater determinant is the normalized N fermion ground state. The expectation value of ordinary operator products that we want to compute may be written as time ordered products by assigning the k-th operator from the left a time ka. g is taken to zero at the end of the calculation. We can now use Wick's theorem with the fermion propagator for s ~, t

(El Yg-'(2, t) g-'* (2', s ) I F >

= y, ~ u , ( 2 ) [ S ( s - t ) ( 1 - , ~ )

n m

- S ( t - s ) , ~ l , m ~ , * ( 2 ' ) , ( 1 0 ) where S(t) is an ordinary step function with support on the interval from zero to infinity. Note that we did not have to know anything about the spectrum of the fictitious fermion hamiltonian because all times are taken to zero at the end of the calculation. We must, however, be cautious about one point. The fermions inside each bilinear operator are in ordinary rather than time ordered products. This difficulty is easily remedied by writing

~mO~= ½ ( [ ~m, O7/] +{ ~ , O ~ )

=½[ 7 ~, O ~ ] + ½ t r O (11)

for any one body operator O. The connected Green's functions of the time ordered operators (the com- mutator piece of ( 11 ) ) are given by the usual ring like Feynman diagrams for free fermion bilinears, with the propagator given above. However, the time ordered and ordinary products of two fermion operators differ by a c number. This subtraction thus af- fects only the connected one point function. Thus higher point connected Green's functions of the operators that interest us are given correctly by apply- ing Wick's theorem and ignoring the subtraction.

We now want to note an important property of the expressions we have derived for correlation functions. Consider the connected two point function of eq. (9) when the length of the loops is a small number of lattice spacings. As can be seen from eq. (7), the operator J is "local" in fermion level space. It is a finite difference operator that connects only the

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n_+ 1 levels. Low powers of it have short range in fermion level space. Note that in the two point function j l and jk are sandwiched between orthogonal projec- tors that project out the states above or below the Fermi surface. Only states in the neighborhood of the Fermi surface contribute. For example,

( t r M 2 tF M 2 ) c = R N + I R N + RNRN_I ,

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( t r M 2 tr M 2 tr M2)c

=RN+ I RN( RN+ 2 + R N + I - - R N - - R N - 1 )

+ R N R N _ I ( R N + 1 + R N - - R N _ I --RN_2) . (13) This is important since the universal physics is lo- cated in the immediate neighborhood of the Fermi surface. We see that the continuum theory is sensi- tive only to the states right near the Fermi surface.

The space of levels near the Fermi surface becomes continuous, and operators like J, that are local in Fermi level space, become finite order differential operators.

We have shown that general correlation functions in string models based on single large-N matrix integrals can be written as expectation values of products of fermion bilinears in a free fermion lattice field theory. These considerations are easily extended to matrix-chain models [7]. The new element there is a transfer matrix along the chain. The correlations in these models can be written as expectation values of products of fermion bilinears and transfer matrices in the state I F ) described above. Since this is not an eigenstate of the transfer matrix, an overlap integral must be computed. However, for infinitely long chains, the divergent part of the free energy is independent of the overlap and depends only on the ground state of the transfer matrix. This is a discrete version of the large-N quantum mechanics ofref. [ 8 ] and our fermion formalism converges nicely to theirs in this limit. We note, however, that the matrix chain contains variables that cannot be described in terms of fermions. These are the unitary parts, or angular variables, of the matrices. They disappear from certain correlation functions because of a global U (N) symmetry under which both the transfer matrix and the state I F ) are invariant [ 10]. Other correlators involve the angular variables in an essential way, and are more difficult to study. We will give a detailed

description of our results for matrix chain models in ref. [9].

We now discuss the continuum limit of the correlation functions using the scaling limit introduced in refs. [2-4]. For simplicity we present the formulas for c = 0, m = 2. We follow the notation of ref. [ 3 ].

Introduce a lattice spacing a, a renormalized cosmological constant

fl=(flo--lAc)/a 2,

a renormalized string coupling 2 = a - 5 / 2 e ( e =

1/N),

finite in the continuum limit, and a variable z that describes the universal infinitesimal region near the Fermi surface

( x = n / N ~ l ) , e x p ( - a 2 p ) x = l - a Z z .

The constants Rn are replaced by a function

r(x)

whose universal part is

P(z),

defined by

r-p=aP(z)

where p is a non-universal constant. As shown in ref. [2-4],

P(z)

satisfies the Painlev6 equation of the first kind.

We now consider macroscopic loops. These are operators of the form T r M p with p ~ in the continuum limit so that

l=pa,

the physical length of the loop, is held fixed. It is convenient to write the Jacobi operator as

J= [p+aP(z)

]1/2 exp(e 3x) + e x p ( - e 0 x )

X [p+aP(z)

]1/2. (14)

Introducing the above quantities and expanding to first order in a we find

J=2p1/2 +a[pl/222 O~ +p-l/2P(z)

] , (15) and

tr

JP

= tr

j~/a

l 2 2 _{_p --

= e x p ( ~ l o g ( 2 p ' / 2 ) ) e x p ( ~ ( 2 0 ~ ' P ) ) (16) The first factor is a non-universal boundary energy which we absorb by a multiplicative renormalization. The remainder is universal. We see that the loop of length l is described by the heat kernel exp ( - H l ) of the Schr6dinger operator ~1

H = - - ½ ) ; 2 O z w v ( z ) , V ( z ) = - ½ P ( z ) . (17)

~ This Schr6dinger operator was derived independently by Gross and Migdal [ 4 ] who showed that the multicritical string equations could be determined from its Seeley coefficients and by Douglas and Shenker [ 12] who showed that its heat kernel described the macroscopic loop.

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Volume 238, number 2,3,4 PHYSICS LETTERS B 5 April 1990 Where we have scaled p to one and P(z) obeys the

c = 0, m = 2 string equation

p 2 + l)~2p,, = z . (18)

It is often useful to measure lengths in units o f l/x/%. After rescaling z and P to accomplish this, equations (17), (18) remain unchanged except 2 is replaced by the dimensionless handle counting parameter x = 2~It 5/4.

The changes in scaling necessary to account for the negative dimension operator to which the " c o s m o - logical constant" very probably couples in the general multicritical model are discussed in ref. [ 5 ]. The general string equation is discussed in ref. [ 2 - 4 ] .

Using the fermion formalism and the heat kernel H w e can now write the master formula for the expectation value o f k macroscopic loops with lengths lb /2 . . .

l,:

( w~, w~2...wl, )c

= l-I ~ P * e x p ( - / , g ) . (19)

i = 1 c

As an example we calculate the expectation for one loop

( W t ) = i d z S ( z ) ( z l e x p ( - H l ) l z ) , (20)

and for two loops

( Wh Wt2 )c = i dz d w S ( z ) ( z l e x p ( - H l l )l w)

- - c c

× [ 1 - S ( w ) ] ( w l e x p ( - H l 2 ) [ z ) . (21) The step functions S ( z ) represent the existence o f the Fermi sea as explained above and have support for z > / t .

We now turn to a discussion o f microscopic loops.

The natural operators to examine are the scaling operators [ 13 ] Ok that couple to sources Tk in the tree level equation ~2 for the specific heat as

tt=c2 S2f2-t- c3T 3f 3-l- c4 T4f4-t-...

+ckTkfk+ .... (22)

~z Gross and Migdal [4] give a general formula for the correlation functions of scaling operators at tree level.

where the c, are normalization constants. Such a potential describes the general massive model interpo- lating between the multicritical fixed points [ 13 ].

One way o f isolating such scaling operators is to observe that as we take the lattice spacing to zero, lattice correlators like (8) and ( 9 ) will have expansions in powers o f the lattice spacing whose coefficients are matrix elements o f c o n t i n u u m scaling operators. In particular, these scaling operators can be found by examining the behavior ofcorrelators for boundaries only a few lattice spacings long. The locality o f small powers o f J implies that the matrix elements o f the scaling operators are given by polynomials o f the Painlev6 function and its derivatives. For example, in the scaling limit ( 12 ) and ( 13 ) become

( t r M 2 t r M 2 ) c = 2 p 2 + 4 p a P ( / ~ ) + O ( a 2 ) , (23) ( t r M 2 t r M 2 tr M 2 ) c

= - 82,o 2a3/2P' (#) + O ( a z) . (24) the additive constant 2p 2 in (23) is not universal. It is not present in higher n point functions such as (24).

Such a constant exists also in the one point function.

It appears there because the expectation value depends on the entire Fermi sea and not just on its universal surface. A similar non-universal additive constant could appear in the calculation o f the one point function (20) where we integrate over the entire Fermi sea. However, as is clear from eq. (16), because o f the limit p=l/a-~ov the additive non-universal constant exponentiates and turns into a multiplicative constant. The contribution o f the b o t t o m o f the Fermi sea ( z ~ or) is exponentially suppressed in (20) and therefore it does not shift the answer.

In interpreting the correlation functions one should be careful not to forget the additive constants. These can usually be removed by differentiating a large enough n u m b e r o f times with respect to the cosmological constant/~. This has the effect o f removing all the analytic dependence on/t. An equivalent way to understand this is to study the theory at large (rela- tive to the c u t o f f ) fixed area A. Then the correlation functions on the sphere have the form A r for some constant r. For r~< - 1 the integral overA diverges for small A. This is the origin of the additive constant.

For r < - 1 and not an integer, the universal term is proportional t o / t - r i. For r an integer smaller or

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Volume 238, number 2,3,4 PHYSICS LETTERS B 5 April 1990 equal to - l , the s i t u a t i o n is m o r e c o m p l i c a t e d . In

this case, differentiating the answer - r - 1 times with respect to p we expect to f i n d log # in the answer. F o r example, for the one p o i n t function o f the energy operator in the Ising model, r = - 3. Differentiating twice with respect to the cosmological c o n s t a n t we expect r = - 1 and the universal t e r m is p r o p o r t i o n a l to log #.

Since the exact result o f the lattice calculation is a c o n s t a n t i n d e p e n d e n t o f p, we conclude that the one p o i n t function o f the energy o p e r a t o r in the Ising m o d e l vanishes ~3. We w o u l d like to note, however, that such c o n s t a n t s m a y well have universal physical meaning.

A m o r e elegant a p p r o a c h to c o n s t r u c t i n g the correlation functions is to use the singular p o t e n t i a l s in- t r o d u c e d by G r o s s a n d M i g d a l [4] to pick out the pure scaling operators. T h e y show that the m a t r i x potential c o r r e s p o n d i n g to a v a r i a t i o n o f the field Tk is t r ( 2 - M ) k+~/2, which in the c o n t i n u u m limit is Hk+ ~/2. In the f e r m i o n f o r m a l i s m it is r e p r e s e n t e d by the one b o d y o p e r a t o r 7ttH k+ 1/21/./, T h u s we can write a general f o r m u l a for the n p o i n t c o r r e l a t i o n function o f scaling o p e r a t o r s ( u p to a n o r m a l i z a t i o n )

( Ok, Ok2...Ok. )

~ d l l

~

dln .

(kl+3/2)'"ln(k~+3/2)

. . . . ~i~i l F

/ k /

× l-I ~ * e x p ( - / g H ) tp . ( 2 5 ) i=1

F o r e x a m p l e the one p o i n t function o f the o p e r a t o r conjugate to Tk is given ( f o r m a l l y ) , up to a n o r m a l i - z a t i o n by

(Ok) = i'

dz ( z l H k + l / 2 [ Z )

.

u

^{( 2 6 )}

F o r higher p o i n t functions care m u s t be taken with the d i s t r i b u t i o n s i m p l i c i t in eq. ( 2 5 ) .

Eq. ( 2 6 ) has i m p o r t a n t consequences. In o r d e r to explain t h e m we t e m p o r a r i l y shift n o t a t i o n to con- form to ref. [ 14 ], whose results we use extensively in what follows. R e p l a c e z by x, set 22 = 2, a n d let u = V.

The Schr/3dinger o p e r a t o r b e c o m e s H = - 0~ + u ( x ) .

~3 This is in contrast to the statement in ref. [ 5 ].

I n t r o d u c e the diagonal o f the resolvent to define frac- t i o n a l powers o f H,

c~ R t [ u ( x ) ] ( x [ ( H + f f ) - I I x ) =

l=0 ~l+1/2

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The coefficients R~[u] are p o l y n o m i a l s in u a n d its d e r i v a t i v e s and are the generalized K d V potentials.

Gross and M i g d a l [4] showed that the m u l t i c r i t i c a l string e q u a t i o n s are d e t e r m i n e d by these quantities.

The diagonal o f H k+~/2 is d e t e r m i n e d up to a nor- m a l i z a t i o n by Rk+ i. The string e q u a t i o n for the general massive m o d e l i n t e r p o l a t i n g between m u l t i c r i t i - cal p o i n t s [ 3,4 ] is

x = ~ ( k + l ) T k R k [ u ] ( 2 8 )

k=0

( w h e r e we have now fixed the n o r m a l i z a t i o n s ) , or, using the i d e n t i t y ( ~ / Su ) Rk + ~ = -- ( k + ½ ) Rk,

x = - - ~ Tk~uuRk+,[u]. ( 2 9 )

k=0

We list the first few Rt:

R o = ½ , R I = - I u , R 2 ~.~_ 1 ( 3 u 2 - u " ) ,

83=--~44[lOu3--10Ul.I"--5(U')2-t-U ''' ] . ( 3 0 ) N o t i n g that the specific heat u ~ O Z F a n d that ( O k ) = (O/OTk)Fwhere F is the free energy, we see that eq. ( 2 6 ) for u(T~, T>.., x) can be written, after d i f f e r e n t i a t i n g twice, as

0 0

O~kk U= ~xRk+I

^[U] ^{( 3 1 )}

for every k. These are j u s t the ( g e n e r a l i z e d ) K d V e q u a t i o n s ~4. We see that the specific heat u as a function o f the scaling fields a n d x is j u s t a solution o f the K d V hierarchy. This o b s e r v a t i o n raises an i m p o r t a n t question; if we start at a given m u l t i c r i t i c a l model, and then flow up to a higher one using eq. (31 ) which special solution o f the higher string e q u a t i o n ( i f any ) do we c o m e to ~5?

We can express eq. (31 ) m o r e c o m p a c t l y by intro-

~4 The connection to the free fermion formalism discussed above is quite likely to be made through the grassmannian and its associated r function [ 15 ].

~5 Issues related to this have been considered by Witten [ 16 ] in his topological field theory derivation of low genus correlation functions.

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ducing the vector fields which generate the KdV flows:

~l= ~ RF +l~

_,=0 _{8u~ ) ,}⁸ ₍₃₂₎

where the superscript refers to differentiation with respect to x and the u ~) are considered independent.

Integrability of the KdV hierarchy depends crucially on the fact that these vector fields commute:

[4,, ~m]=O. (33)

We can then write eq. (31 ) as

- - u = ~ k + , ' u . 0 (34)

aTk

Correlation functions of the general massive multicritical model are then given by the simple formula

a~ ( Ok, Ok2...Ok. )

=~k.+,..gk2+,~k,+, "U. (35)

The ordering is unimportant because ofeq. (33). This expression is a polynomial in u and its derivatives, as the matrix expressions imply. It is straightforward to show, using identities following from eq. (33), that the differential equation for the correlation function [41 derived from varying eq. (28) with respect to irk is satisfied by eqs. (33), (34).

We now discuss some of the physics of eqs. (20), (21 ). Since we are dealing with a free fermion theory, all the correlators can be written simply in terms of the heat kernel of H. Let us examine the properties of H. Fig. 1 is a rough sketch of Vfor m = 2 , when the string equation is Painlev6. Recall that the detailed shape depends on the non-perturbative free parameter. There is an infinite sequence of double poles as z--, - o o that asymptotically become periodic. We will call the location of the pth double pole zp. Since the potential approaches +oo at these points as ( z - z p ) -2, the wavefunction must vanish there faster than z-zp. With such a behaviour, the region between each pair of poles is disconnected from the others, i.e. the hamiltonian H is self adjoint once re- stricted to a single region. The region that joins onto perturbation theory is z~ < z < ~ .

In the perturbative region z - - , ~ the potential V~

x~z, and the wavefunction must decay. So there is a well posed eigenvalue problem with discrete spectrum in this region. Referring to eq. (20), we see that

V(Z)

2

Fig. 1.

loops will decay with an infinite number of distinct exponentials of their length due to this discrete spectrum. This is a dramatic and non-perturbative phe- nomenon in 2D quantum gravity. Note that the dis- creteness of the spectrum is a consequence of the first double pole, even if it is not visible in the free energy for "physical"/z> 0. This suggests a physical role for these singularities ~6. We might wonder what hap- pens if the free parameter is adjusted so that the Fermi level/z is in between two poles. We then couple to the part of the spectrum supported entirely between them.

Perhaps this is one of an infinite number of strong coupling phases of 2D quantum gravity. The spectrum of H would be gapped in each new "phase."

In the weak coupling regime we might expect that geometrical intuitions about sums over surfaces of different topologies would give a qualitatively correct picture of the physics of two dimensional quantum gravity. At tree level David [ 1 ] has argued that the behavior of ( W t ) is qualitatively similar to that of a large loop spanned by a surface of constant negative curvature - x / ~ . The expectation value of the area

O u log ( Wi ) ~ 12 for lx/@ << 1 ,

~ l / x / ~ , for/x/~>> 1 . (36)

~6 That the first double pole might have physical consequences was first pointed out by Br6zin and Kazakov [2] who sug- gested that it might describe a "handle condensation" phase transition.

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Volume 238, number 2,3,4 PHYSICS LETTERS B 5 April 1990 So at large l the area is large and for small enough K 2

a dilute o f gas of handles picture of the kind used in wormhole physics might be expected to be valid.

In fact, at asymptotically large l, for any finite ~c, the behavior o f expectation values in two dimensional gravity does not coincide with the dilute wormhole picture. The spectrum o f the hamiltonian is discrete and at asymptotically large l, formula (20) is d o m i n a t e d by the ground state energy. We find

W / = J dz 0 g ( z ) e x p ( - E o l ) . (37)

/z

This looks more like what might be expected from a simple renormalization o f the cosmological constant;

the web of higher genus surfaces seems to behave at large l like a genus zero surface with an effective cosmological constant. Note that the value of this effect constant is not zero.

It should come as no surprise that the dilute wormhole gas is not a valid approximation for large volume universes. There is no cluster expansion for wormholes as there is for ordinary instantons. The contribution o f wormhole interactions (non-quadratic terms in the action for fluctuating couplings) to the logarithm o f the partition function contains cubic and higher powers of the volume, while the dilute gas contribution is quadratic. Even if there is a small parameter (x in the present context) control- ling the wormhole density, the interaction terms dominate at large volumes. O u r exact solution o f the two dimensional problem allows us to see the correct asymptotic behavior.

The above discussion was valid for asymptotically large volumes. As we let l become smaller we see the possibility o f a regime in which wormholes give a correct picture o f the physics. The gap in the Schr6- dinger spectrum is o f order ~c, so if l,,/~ ~c<< 1 we can no longer approximate the macroscopic loop by the contribution of the ground state alone. Thus for small

~c and 1 << 1,~/~ << 1/K we can expect to approximate the Schr6dinger spectrum by a c o n t i n u u m and the result for the loop in this regime can be written as an integral over fluctuating values o f the cosmological constant o f the tree level result ~7

This is a rather weak probe o f the validity of worm-

~7 We thank L. Susskind for discussions of this point.

hole ideas. To be more precise we can investigate the behavior o f the loop expansion order by order. The diagonal matrix element o f the heat kernel that appears in W / c a n be written as a path integral with action

i

S = ~ + ~ - + ~ ~c2"a~,z - (38)

p = l 0

It can be expanded in powers of K by writing x ( t ) = z + ~ c A ( t ) . This generates, in each order o f x, an action which is polynomial in A. The path integral can then be written in terms of Feynman graphs whose l dependence is determined by simple dimensional analysis.

When this analysis is performed in order x2 (genus one) we find a result consistent with wormhole ideas:

a term proportional to l and a term proportional to l 2. The average area is again proportional to l, so the first o f these resembles a renormalization o f the cosmological constant while the second can be inter- preted as a single wormhole contribution. However, at genus two and higher there appear to be contribu- tions which do not fit into a wormhole picture, even when wormhole interactions are included. In particular, at genus g the leading large I behavior appears to be 13g 1 rather than the l zg one would expect from wormholes. This may be an indication that even in this perturbative regime, the contribution o f "fat"

surfaces to the path integral at large I dominates over that of wormhole configurations. Indeed, the arguments (such as they are) that have been adduced to justify restricting attention to wormhole configurations in the path integral over four-geometries, are not obviously applicable in the present context ~8. We caution, however, that our understanding of this is- sue is not complete, and that a wormhole picture of the sum over two-geometries is not completely ruled out in the perturbative regime. What is clear is that for asymptotically large l, all such arguments fail, and the behavior of the loop is controlled by aspects o f the problem that are invisible in the genus expansion.

We do not at present have an intuitive geometrical picture of the origin of these non-perturbative effects.

~8 Carlip and de Alwis [ 17 ] have discussed problems with the dilute wormhole approximation in 2+ 1 dimensions.

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Volume 238, number 2,3,4 PHYSICS LETTERS B 5 April 1990

Note added. A f t e r s u b m i s s i o n o f this n o t e we re- c e i v e d a p a p e r by D. B o u l a t o v w h o uses s i m i l a r free F e r m i fields to w r i t e t h e t r e e level m u l t i p o i n t f u n c - t i o n s for t h e c = 1 t h e o r y .

We w o u l d like to t h a n k J. C o h n , M. D i n e , D.

F r i e d a n , D. G r o s s , E. M a r t i n e c , A. M i g d a l , L.

Susskind a n d E. W i t t e n for v a l u a b l e discussions. T h i s r e s e a r c h was s u p p o r t e d in p a r t b y g r a n t s f r o m t h e D e p a r t m e n t o f E n e r g y a n d t h e N a t i o n a l S c i e n c e F o u n d a t i o n .

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