THE ADE FACE MODELS ON A FLUCTUATING PLANAR LATTICE I.K. KOSTOV*

Nuclear Physics B326 (1989) 583-612 North-Holland, Amsterdam

THE ADE FACE MODELS ON A FLUCTUATING PLANAR LATTICE

I.K. KOSTOV*

Ser{#ce de Pt~vsique Thdorique de Saclav**, F-91191 Gif-sur- Yvette, France

Received 19 April 1989

The A D E two-dimensional interaction-round-a-face statistical models are formulated on a fluctuating planar lattice. The continuum limit of such systems is described by the minimal conformal theories coupled to quantum gravity. All these models can be reformulated in terms of a gas of self-avoiding nonintersecting loops on a random planar graph. This representation allows us to calculate the partition function and the susceptibilities of the order parameters in the case of lattices with spherical topology. The scaling dimensions of the order parameters are shown to form a linear spectrum.

1. Introduction

The discretization of the bosonic string as a randomly triangulated surface [1 3]

suggested a new approach to two-dimensional quantum gravity. The spacetime manifold is represented by a planar lattice so that the functional integration over the metric field corresponds to the average in the ensemble of planar graphs.

This idea was applied by Kazakov [4] to the Ising model on an irregular planar lattice [5]. The random lattice Ising model is again exactly solvable and has the same phase diagram as the Ising model on a regular lattice. However, the critical exponents are different. Actually, they are the same as these for the 3D spherical model [4].

The Ising model has been solved after being reformulated as a large N matrix problem [6]. Recently, the same strategy was applied to its generalizations: the Q-state Potts [7,8] and O(n) [9] models. Some interesting particular cases are considered in refs. [10, 11]. All results are in perfect agreement with the spectra of scaling dimensions extracted from the degenerate representations of the SL(2, N) current algebra [12].

* Address after November 1988: Institute for Nuclear Research and Nuclear Energy, 72 Boulevard Lenin, 1784 Sofia, Bulgaria.

** Laboratoire de l'Institut de Recherche Fondamentale du Commissariat ~t rEnergie Atomique.

0550-3213/89/$03.50~3Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

584 LK. Kostov / ADE face models

In this paper we consider a class of critical interaction-round-a-face (IRF) models related to the six-vertex model [13]. The local degrees of freedom in these models are labeled by the points of the Dynkin diagram of a simply laced Lie algebra. Thus there are two infinite series (A,,Dn+2; n = 1,2 . . . . ) and three exceptional cases (E6, E7, E8). The A D E models have been introduced by Pasquier [14] as a general- ization of the restricted solid-on-solid (RSOS) models of Andrews, Baxter and Forrester [15]. The latter correspond to the A-series. The continuum limit of the A D E models is described by the minimal conformal invariant theories [16].

After having defined the A D E models on an arbitrary irregular lattice we establish the mapping into a six-vertex model or, equivalently, an unrestricted solid-on-solid (SOS) model. The SOS model allows a geometrical interpretation in terms of a gas of nonintersecting loops (domain walls). The continuum limit of the SOS model is that of a gaussian field on a curved manifold. This mapping is therefore the microscopic version of the gaussian field dominance in two-dimen- s i o n a l critical phenomena [17].

In particular, we express the partition functions of the ADE models on the torus in terms of these of the SOS model, generalizing the results of refs. [18,19] obtained for the regular square lattice.

Finally, we use the mapping onto the SOS model in order to develop a method for actually calculating the partition function and the susceptibilities of the order parameters. The method is based on the Dyson-Schwinger equations for the ensemble of planar random lattices with boundary. We present the explicit solution of these equations at the critical point and show how to extract from this solution the scaling dimensions of the order parameters.

The paper is organized as follows. In sect. 2 we define the A D E models on an irregular planar lattice. The structure of the Hilbert space of states for these models is discussed in the next section. In sect. 4 we consider the unrestricted SOS model on an arbitrary planar lattice. After formulating the cluster expansion for the partition function and the two-point correlators in sect. 5, we consider in sect. 6 the mapping of the A D E models onto an underlying unrestricted SOS model. In sect. 7 we define the critical exponents characterizing the singular behaviour of the parti- tion function and the susceptibilities of the order parameters as functions of the cosmological constant 2, coupled to the volume of the planar lattice. The critical point X c is determined by the dominance of planar lattices of infinite volume. The characteristic volume of the random lattice diverges near the critical point as ( X - Xc) 1. The singularity of the partition function in 2, defines a new critical exponent 7str, the so-called string susceptibility exponent. In sect. 8 we write the D y s o n - S c h w i n g e r equations for a piece of the random lattice with the topology of a disc and find their explicit solution at X = X c. This solution allows us to fix the spectrum of the anomalous dimensions of the order parameters provided Ystr is known. Our conclusions are presented in sect. 9.

I.K. Kostov / ADE face models

Fig. 1. A piece of the lattice ft.

585

2. Definition of the models

Let ~ be an arbitrary planar lattice (fig. 1) whose elementary cells are squares;

we will also call them plaquettes. Any such lattice is dual to a q04 planar graph. At each site x of ~ we introduce a fluctuating field variable (local state) o ( x ) , taking its values in some target space ~ . Different IRF models are characterized by their target spaces.

The simplest one is the unrestricted solid-on-solid (SOS) model [13]. In this case the field variable is an arbitrary integer. However, the allowed field configurations are strongly restricted by the condition that for all finks E= ( x y ) of the lattice f¢, I o ( x ) - o ( y ) l = 1. This condition has a simple geometrical meaning. Indeed, the set of all integers 2v can be given a structure of one-dimensional simpficial complex with points o ~ Z and links (o, o + 1). Then each allowed field configuration defines a map ~ ~ preserving the simplicial structure. That is, if ( x y ) is a link of

~, then the image ( o ( x ) o ( y ) ) is a link of ~ . As a consequence, any closed path on is mapped into a closed path on ~ .

More generally, we can assume that the target space is an arbitrary one-dimen- sional simplicial complex ~ , i.e. a collection of points o and links ( o o ' ) . The graph

is determined uniquely by its connectivity matrix

Coo, = 1 if o and o' are connected by a link,

= 0 if not. (1)

Again we define the allowed field configurations as all possible embeddings f¢--+

preserving the simplicial structure.

Following Pasquier [14] we will specify ~ to be a Dynkin diagram of A D E type (fig. 2).

586 LK. Kostov / A D E face models

We postulate that the Boltzmann weight of each field configuration o: f f ~ ~ is a p r o d u c t of local factors associated with the points x and elementary squares [] of ft.

T h e partition function is defined as the sum over all possible embeddings o: if--* 9 :

z(~)= E II w(o(x))Hw[o(G)]. (2)

o : f g ~ x ~ []

The explicit form of the local weight factors for the A D E models is

points: W ( o ) = S o , (3)

plaquettes:

: - B +

+

01 0 2 01 (7 2 01 0 2

N a m e of D i a g r a m -@ Coxeter E x p o n e n t s

the a l g e b r a n u m b e r h m

1 2 3 4 n

An ^{-- -- -- • • n + l} 1 , 2 . . . n

On 0 ~ _ 2 3 n- 2

D ~ : • • 2 ( n - 1 ) 1,3 ... 2 n - 3 , n - I

1 2 3 4 5

E 6 : -- i 6 = -~ 12

1 2 3 4 5 6

E 7 : : : ~ : ; 18

7

1 2 3 4 5 6 7

E 8 • : : : ~ : : 30

8

1.4,5,7, 8,11

1 , 5 , 7 . 9 , 1 1 , 1 3 , 1 7

1.7, 11.13,17, 19, 23, 29

Fig. 2. Dynkin diagrams and their Coxeter exponents.

LK. Kostov / ADE face models 587

where S, is the eigenvector of the connectivity matrix C, eq. (1), corresponding to the largest eigenvalue fl (the Perron-Frobenius vector). Note that all components of this vector are positive. Eqs. (2)-(4) generalize the definition of the A D E models [14] to an arbitrary planar lattice whose elementary cells are squares. The form (3), (4) at the Boltzmann weights corresponds to the critical point for these models.

Therefore there is no parameter like temperature.

Let us remember that the critical Ising and Potts models are described by the D y n k i n diagrams A 3 and D 4 [14].

After having defined the IRF models on an irregular lattice, we can introduce new local degrees of freedom by allowing the lattice N to fluctuate. The correspond- ing partition functions are obtained by taking the sum over all lattices N with fixed Euler number X(f¢) = 2 - 2g:

1

= E (5)

Here k ( ~ ) is the volume of the symmetry group of the graph ~ and the new parameter k coupled to the volume ]~] - ( # plaquettes of f¢) makes sense of the cosmological constant of the discrete two-dimensional universe.

We can think of eq. (5) as the partition function of random surfaces made of plaquettes and embedded in the target space ~ . Each surface can be considered as the evolution sheet of a string of particles fluctuating in ~ . The Hilbert space of states is therefore the linear span of all closed paths in ~ . In order to identify the ground state we have to consider a random path in statistical equilibrium. Thus we are led to the problem of brownian motion on a Dynkin diagram.

3. Random walks on a Dynkin diagram

The propagation kernel Ko (u) of a random particle moving on a one-dimensional simplicial complex ~ is defined by the number of paths of N steps (links) connecting the points o and o'. This quantity is related to the connectivity matrix C of ~ , eq. (1), by

Ko(oL, ) = ( C L) oo'. (6)

If ~ is a Dynkin diagram of A D E type, then the eigenvalues of its connectivity matrix are of the form

fl(m) = 2 cos

~rm/h,

(7)

where h is the Coxeter number and the positive integers m belong to the set N* of

5 8 8 LK. Kostov / ADE face models

Coxeter exponents. The largest eigenvalue is

/~ = 2 cos ~ / h = / L ) " (8) Introducing the normalized eigenvectors S(,,) = { S{°m) }, o ~ N,

Coo,S~m ) = fl(m)S~m), m ~ ~ * , (9)

o t

we can rewrite the propagation kernel (6) in the form

Ko(~ ) = £ S;~)~(~)S;/~). (10)

r n ~ *

The eigenvector S o -S(~) corresponding to the largest eigenvalue (8) gives the probability amplitude to find the random particle at the point o ~ ~ after infinitely many steps. By the Perron-Frobenius theorem all its components S o are positive numbers.

The eigenvectors S~,~), m ~ ~ * , thus correspond to the ground state and the excited states of a quantum particle on the graph ~ . It is convenient to normalize the wave functions of the excited states by that of the vacuum,

X~m) = Si~)/SO, m ~ ~ . (11)

The new wave functions satisfy orthogonality conditions of the form

Y', S2X~,~)X~,,,)= 8ram,. (12)

o E , ~

The interaction of three particles with " m o m e n t a " m, m' and m" is described by the vertex

C,,,,,,,,,,,, = E S2x~,,,,X~,,,,)X°o,,,, ~ . (13)

o

In particular, according to eq. (12),

C,,,, a = am,,,, C,,al = 8,,,. (14)

An equivalent form of writing eq. (13) is

o o ~ r n o

X(m')X(,,,") Y'~Cm',,," X(m), (15)

m

where C,,,,,,,,,,, = C,,,,,,,,Y since our wave functions are real. The vertex (13) in the

LK. Kostov / ADE face models 589 coordinate space reads

C~°°'°" - E S ( m ) a i m ' ) a i m " ) f m m ' m " ° °' °" = ~oo' aoo"(So) -1 (16)

t)l. ~I',/~i" E ~ *

Explicit formulae for the simplest case when ~ is a Dynkin diagram of type A, are given in appendix A.

N o w let us return to the Hilbert space of paths on the graph ~ . The ground state wave function is a product of factors So, one for each point o of the path. If we go to the normalized wave functions (11), then the factors S o will be absorbed in the s u m m a t i o n measure in the space of paths. Thus the origin of the weights (3) associated with the points of the r a n d o m surface is quite clear.

T h e meaning of the plaquette factors (4) is more subtle. Each plaquette represents an elementary step of the evolution of a string of particles fluctuating in ~ . Consider the segment o~ lOi0.i+ I of such a string. The particle os is allowed to j u m p to a n o t h e r position o~'e N such that C,,_,o; = C o _ o ; = 1:

0"i-I (~i÷I

I

o" i

(17)

If the graph ~ has no cycles of length 4, then the elementary move (17) is kinematically possible only if o~ 1 = °i+1. Thus it is natural to associate with the plaquette (o~ 10.,0.s+ la / ) a weight which is the probability for the particle oi to go to the position 0.[ after being scattered on the state 0.~ 1 ⁼ 0.i+1" According to eq. (16) this probability is given by

1

co oCo ° ' ' ' '+' ' S Ca, , o : C o , + , a , , , (18)

°i I

which is a n o t h e r way to write the first term in the rhs of eq. (4) ( O i _ 1 = 0.1, Oi = 0"2, Oi+l = 0.3, 0.[ = 0.4)" The second term in eq. (4) describes the same process but in the cross channel, the time direction going along the diagonal 0.1 - 0"3-

T h e elementary step (17) is not sufficient to describe the evolution of the string.

Since the n u m b e r of particles is not fixed, we have to consider another elementary

590 LK. Kostov / ADE face models

move which consists in adding or subtracting a backtracking piece at some point or:

I

o ~ ~ (19)

We can describe the move (19) as a decay of the particle o~ into two particles, one of which jumps into position

oi'.

This process yields a factor S,-1 which compensates the excessive factor So due to the presence of two particles at the same point o r ~ 9 .

4.

The unrestricted

S O S model

The target space of the SOS model is the infinite one-dimensional lattice Z. The connectivity matrix in this case is

Coo, = 8o,o,+1 + 8o,o,_ 1 . (20)

The eigenvalues of C form a continuum spectrum

Y'~ (~oo,So,( P ) = fl( P )So( P ),

(21)

0 ~

S ° ( P ) = e ~"e°,

fl(e)=2cosTrP,

- I < P ~ < I . (22) The Boltzmann weights in the definition of the partition function (2) are given by

W(o) = L ( e 0 ) ,

(o4D 3)

VV"

:[go,(Po)]-l~o, a3"+-[goz(Po)]-l~o2°4,

(23)

\ O1 v 2 ]

where the momentum P0 belongs to the interval [ - 1,1]. The model is critical for all P0. On the regular lattice the SOS model is argued to renormalize onto a gaussian free field

o(x)

[20] with an action

q7

d : ~-(1 -

IPol)f(vo)2d2x.

⁽²⁴⁾

If we take P0 =

1/h

with h integer, then all Boltzmann weights will be invariant under translations o---> a + 2h. Therefore the field variable o can be measured

I.K. Kostov / A D E face models 591

m o d u l o 2h. In other words, we can take as a target space the circle 7 ] / 2 h 7 ] = 712h instead of 7/. Then only a finite subset of distinct vectors (22) exists

S?k ) = e x p ( i w K o / h ), fl(k) = 2 cos ~ r k / h , k ~ 7/2h- (25) N o t e that the eigenvalues fl(k) are of exactly the same form as these for the A D E D y n k i n diagrams. This will make possible to m a p all A D E models into the SOS model.

T h e vector S ( P o ) entering the Boltzmann weights of the SOS model describes the ground state only if P0 = 0. Therefore we expect to encounter pathologies in this model when Po + 0. We are going to discuss this point in the next section.

Let us c o m p a r e the Hilbert spaces of the SOS model and the A D E models. The partition function of closed paths of length L on the Dynkin diagram ~ is given by

Z(L ~ ) = ~ K ( L ) = ~[~ ( f l ( m ) ) L , ( 2 6 ) o E ~ m E ~ *

( N o t e that L should be an even integer.) Since all Coxeter exponents are positive integers smaller than h, we can insert a projection operator in the definition of the partition function for the closed paths on 7/2h in order to obtain eq. (26).

where

= - - - o ) Koo,

L 2h Y'~ Y'~ c o s - - m ( o , ^(L)

o - o' = even

(27)

h - 1

Koo,^(L) = (CL)oo, = Y'~ e x p [ i ~ r k ( o - a , ) / h ] ( f l ( k ) ) L (28)

k = - h

is the p r o p a g a t i o n kernel of a r a n d o m particle in Z 2h. The factor ½ comes from the double degeneration of the eigenvalues fl(k), 1 ~< k ~< h - 1. In the next section we will establish a formula similar to (27) for the partition functions of the A D E models on a toroidal random lattice.

5. Cluster expansion

T h e partition function (2) can be reformulated as the partition function of a gas of self-avoiding polymers (clusters). The argument is a straightforward generaliza- tion of the one for the regular square lattice [14,19].

Let N be a lattice with the topology of a surface with Euler characteristic X- This means that

# points - # links + # squares = X ( N ) - (29)

592 LK. Kostoo / ADE face models

Fig. 3. A possible cluster configuration on the piece of the lattice shown in fig. 1.

The partition function (2) can be written as a sum of monomials. A monomial corresponds to choosing one of the two terms in the rhs of eq. (4) for each plaquette of ~. We will represent this choice graphically by drawing a line along one of the diagonals of all squares. Thus each monomial is represented as a pattern of clusters on f¢ (fig. 3). The field variable o is constant along each cluster, because of the 8-symbol in the Boltzmann weights (4). The weight of a cluster is equal to S f with k = ( # p o i n t s - #links) = (1 - #circuits).

Following the argument of ref. [19] we associate to each configuration of clusters a graph whose points represent the individual clusters. Two points of the graph are connected by a line if the corresponding clusters can be connected by a link of ft. It is easy to see that the number of independent loops of this graph is equal to the genus g of the surface f¢. (Note that loops of length zero are also allowed; they are associated with clusters containing noncontractible loops.) Thus if ff has the topology of a sphere, all cluster configurations are described by tree graphs. A typical graph for a surface of genus 1 is shown in fig. 4.

We can interpret each cluster as a time slice of the evolution sheet of the string of particles embedded in ft. A configuration of clusters thus determines the local time direction all over the surface ft. Because of the target space 9 being one-dimen- sional, the time slice of the string occupies a single point o of 9 . Thus the graph associated with a cluster configuration is essentially the trajectory of such a point-like string. The branch points of the graph describe splitting and joining of strings.

Evidently, all cluster configurations associated with the same graph contribute the same quantity to the partition function. Each such graph can be considered as a F e y n m a n diagram in the "coordinate" space 9 . The propagator is the connectivity matrix

Coo,.

A vertex with n legs represents a cluster with n - 1 circuits and therefore has to be given a factor (So) 2-n. The Feynman rules in the coordinate

I.K. Kostou / ADE face models

Fig. 4. A typical Feynman graph describing a cluster configuration on a toroidal lattice.

593

o o m

Caa' ~(rn)= 2cos~ m

0 ~ 0

Sa 6rnl

a m m'

I 6ram'

Sa

m ~ m"

Cram,m,,

( a ) (b)

Fig. 5. Feynman rules for the weights of the cluster configurations (a) in the "coordinate" space .@ and (b) the "momentum" space N*.

s p a c e ~ a n d the m o m e n t u m space 9 " (the set o f C o x e t e r e x p o n e n t s ) are given in fig. 5.

T h e t h r e e - p o i n t vertex Cram're" in m o m e n t u m s p a c e is given b y eq. (13). T h e n - p o i n t v e r t e x c a n b e o b t a i n e d b y p e r f o r m i n g a F o u r i e r t r a n s f o r m a t i o n :

° . . . s ° s 2 " ( 3 0 )

Gin, . . . . = ~ S(ml) (mn) a "

R e m a r k a b l y , a n y such vertex c a n b e d e c o m p o s e d i n t o three-vertices a n d the result

5 9 4 LK. Kostoo / ADE face models

does not depend on the way we do it. For example

m~ m 3 m~ nq 3 ITi4 r'n 3

m l m 2 m l m 2

rn 1 rn 2

(31)

Eq. (31) follows immediately from the form of the three- and four-vertices in the coordinate space. It reflects the duality properties of our discrete string.

Returning to the cluster expansion, let us write the explicit formulae in the two simplest cases of spherical and toroidal lattices ~.

(i) Sphere (X = 2).

All cluster configurations are described by tree Feynman graphs. Applying the F e y n m a n rules in fig. 5b we find that such a graph contributes a factor / ~ # p o i n t s .

Therefore the partition function (2) is given by

Z ( f ¢ ) = Z B ~', (32)

c l u s t e r c o n f i g u r a t i o n s o n

where jV" is the number of clusters. The partition function (32) characterizes only the ground state of the model and does not distinguish between the A D E models with the same Coxeter number h.

(ii) Torus (X = 0).

The Feynman graphs in this case contain exactly one loop as the one shown in fig. 4. The contribution of such a graph is

B E (33)

n,l E ~ *

where JV" is the total number of vertices and JV 0 is the number of vertices forming the loop. These vertices correspond to clusters wrapping the torus around one of its noncontractible cycles. The degenerate case L = 0 corresponds to a cluster wrapping the torus ~ around the two noncontractible cycles. Therefore the partition function (2) in the case of toroidal lattice f¢ reads

Z(f¢) = E /3 ~ - ~ ° E B(,.)~°, (34)

c l u s t e r c o n f i g u r a t i o n s m ~ ~ * o n ,~

where .A r is the total number of clusters on ~ and .At 0 is the number of clusters winding around the torus.

The graph expansions (32) and (33) look nicer if we go to the four-coordinated planar graph ~ * dual to f¢. The graph if* consists of points, links and cells (basic

LK. Kostov / ADE face models 595

Fig. 6. A polygon decomposition of the piece of the lattice shown in fig. i corresponding to the cluster configuration of fig. 2.

polygons) which are dual to the plaquettes, links and points of ~. As in ref. [21] we define a polygon decomposition of the lattice ~ * by separating two of the four lines meeting at each vertex:

T h u s there exist 2" polygon decompositions of a q04 planar lattice with n points. A polygon decomposition of ~ * is determined by choosing between one of the diagonals of all plaquettes of ~. Therefore there is one-to-one correspondence between the cluster configurations on f¢ (fig. 3) and the polygon decompositions of f¢* (fig. 6). Eqs. (32) and (33) now can be reformulated as

zsphere( ~ ) = E ~,A/', (36)

polygon decompositions of~*

where ~4¢" is the number of polygons, and

zt°ms( '~ ) = E ~ j'/'-'A/'O E ~(mJV'~ , (37) polygon decompositions rn ~ ~ *

of ~*

where .A r is the total n u m b e r of polygons and .AP 0 is the n u m b e r of polygons winding around the torus. Eqs. (36) and (37) generalize the results obtained in refs.

[18,19] for a flat lattice.

N o w let us formulate the cluster expansion for the order parameters. F r o m now on we will use only the language of polygons. The formulae we are going to obtain

5 9 6 LK. Kostov / ADE face models

generalize the results of ref. [14] to the case of an arbitrary lattice ~ with the topology of a sphere.

The fields playing the role of order parameters in the A D E models are the wave functions of the one-particle excited states (11)

• m(x) = -,°(x) a . ( m ) , m ~ 2 " . (38)

The correlator of two order parameters at the points xt and x 2 of ~ is defined by

= E l~ W(O(X))I-Iw(o([]))@ml(Xl)@m2(X2)" (39)

o-: if--.- ~,@ x ~ ff []

T h e cluster expansion of the rhs of eq. (39) is constructed in the same fashion as that for the partition function. Inserting the field ~m(X) is equivalent to replacing the local weight factor

W(o(x))= So(x)

associated with the point x with S~m~ ).

Therefore the cluster configurations on ~ will be classified by tree F e y n m a n diagrams containing two sources of " m o m e n t u m " m = m t = rn 2. The contribution of such a diagram is a product of factors fl(m) associated with the propagators along the line connecting the two sources, while the other propagators contribute as before factors fl = rid)" In terms of polygon decompositions of ~*, the propagators of a F e y n m a n diagram correspond to polygons and its vertices to spaces between polygons. Therefore the rhs of eq. (39) reads

z ( v ) =

E

B (#,mVB) ,

p o l y g o n d e c o m p o s i t i o n s o f .~ *

(40)

where JV" is the total number of polygons and .Ar~x, is the number of polygons noncontractible to a point on the sphere punctured at x and x'. A path connecting x and x ' intersects such a polygon an odd n u m b e r of times.

A similar representation exists for the case of three fields

~m,(Xk),

k = 1, 2, 3. The cluster configurations are classified by F e y n m a n diagrams containing a vertex

Cm,,,2m3.

The explicit formula in terms of polygons reads

/3 ) 3

= p o l y g o n d e c o m p o s i t i o n s k = 1

o f f~*

(41)

where JV" is the total number of polygons and .At k is the n u m b e r of polygons surrounding the point k, k = 1, 2, 3.

In general, the correlator of n order parameters is expanded as a sum over polygon configurations classified by tree F e y n m a n diagrams with n external sources.

LK. Kostov / ADE face models 597

N o t e that the identity operator is the order p a r a m e t e r corresponding to m = 1. It is easy to see that eq. (40) is obtained as a particular case of eq. (41).

6. Mapping onto the S O S model

T h e construction of the previous section can be repeated for the SOS model, with m i n o r modifications. The corresponding F e y n m a n rules are given in fig. 7. The arrows show the direction of the m o m e n t u m flow. The integration over m o m e n t a is done in the interval [ - 1 , 1].

It is easy to see that, except for the case of toroidal surfaces (X = 0), the contribution of all cluster configurations will be zero. This is because the F e y n m a n rules in fig. 7 are not compatible with the conservation of the momenta. For a F e y n m a n diagram with 1 - 1 5X loops there is an excessive m o m e n t u m - P o X -

In the case of a sphere (X -- 2) we can repair the model by injecting a m o m e n t u m 2 P 0 at some point of the lattice. This is equivalent to replacing at some point Y ~ the factor

W(o(£))

= exp

iTrPoa(Y )

with its complex conjugate.

If we choose P0 = 1 / h and define the field variable a modulo 2h, the modified partition function of the SOS model on the sphere ~ will be given by eq. (36) with B = 2 cos 7r/h. The factor exp 2~iP0a ( X ) modifying the partition function is exactly

o" a ' P

0 0 )

60', 0,,. 1 + 0 0 , 0 - 1 COS 11; P

P

~ 0 Cf ~ 0

e i~rP° ° 0 (P-Po)

a P P'

I O [ P+P')

e i ( 2 - n )~rPo a

P2

P~

Po

( a l (b)

Fig. 7. F e y n m a n rules for the unrestricted SOS model with modified vacuum in the coordinate (a) and m o m e n t u m (b) space.

598 LK. Kostov / A D E face models

the electric charge " a t infinity" which appears in the Coulomb gas construction on the plane [20].

It is natural to expect that the SOS model on an irregular lattice is again renormalized onto a gaussian field (24) modified by a local term depending on the scalar curvature R (x),

d =

¼,rr(1-

[ P 0 [ ) f ( l T o ) 2 d2x -

¼i of d~xR(x),,(x).

⁽⁴²⁾

The second term in the action introduces an electric charge - P o X distributed all over the surface. This charge has to be compensated by an external source in order to have nonvanishing partition function.

Next, consider the SOS model (P0 = l / h , o ~ 2~/7/2h -- Z 2h) on a toroidal lattice N. The cluster expansion of the partition function is given by the formula (37) with N* = Z2h. In order to obtain the partition function of an A D E model we have to implement a projection operator restricting the sum over m to the set of the Coxeter exponents of a Dynkin diagram. This projector is constructed according to the method of ref. [18]. Namely, we define the partition function Znn, (if) as the sum sos over all configurations o: f f ~ Z 2h having discontinuities 2n and 2n' along the two main circles of the torus (n, n ' ~ 7 / / Z h ) . The cluster expansion of Z,,, ( ~ ) is sos expressed in terms of one-loop Feynman diagrams with discontinuity Ao = 2(n, n') along the loop. Here (n, n') denotes the greatest common divisor of n and n'. The contribution of such a Feynman diagram is equal up to a power of/3 = 2 cos ~r/h to the propagation kernel (28). Applying formula (27) we obtain

, Znn,

Z ' e ) ( f f ) = 2 E ~ E cos 2~r (n n') sos

m ~ * n, n ' ~ Z / Z h

(43)

The rhs can be also expressed as a sum over the partition functions of SOS models compactified on circles of different radii [19]. For example, in the case of the A n models we can use the identity

h - ! m

E c o s ( ~ r - - o ] = h ~ ( 2 h ) ( o ) -- ~}(2)(0)

m=l \ h }

(44)

where 3~n)(o) denotes the 8-symbol modulo n.

Inserting eq. (44) in eq. (43) we have

1 SOS

= - (45)

where Z s°s is the partition function of the SOS model compactified on the circle 7/..

LK. Kostov / ADE face models 599

Finally, let us discuss the SOS representation of the correlation functions of order parameters on a spherical surface. In the case of a regular lattice this problem has been solved by Pasquier [14] who generalized the construction of den Nijs [22].

Consider the SOS model with P0 = 1 / h and the variable o compactified on the circle :~. Define again the order parameters as the wave functions of the "excited"

states e iTrm'~/h normalized by the wave function of the " v a c u u m " state ei~°/h:

~ , , ( x ) = 4 °(x) A ( m ) , (46)

2~m) = e i ' ( m - 1 ) " / h , m = O, + 1 . . . . , -t- ( h - 1), h. (47) The operators ~,, and 4;-,, have to be considered as conjugate to each other.

However, they are complex conjugate only in the limit h ~ m. In particular, the identity operator ¢;]xl _= 1 is conjugate with ~, l(X) = e x p [ - 2 ~ r i o ( x ) / h l .

It is easy to verify that the correlation function ( ~ m ( X ) ~ _ m ( X ' ) ) S O s has the same cluster expansion (40) as the two-point correlator (39) in the ADE models. Indeed, the F e y n m a n diagrams classifying the cluster configurations contain two sources of m o m e n t u m m and - m and all propagators along the line between them contribute a factor/3~,,~ = 2 cos ~rm/h. Therefore, for any spherical lattice ~¢

(~O,,(x)eOm(X')) = ( ~ m ( X ) ~ m(X'))SO s. (48)

Putting m = 1 in eq. (48) we reproduce the SOS representation of the partition function of the A D E models.

The equivalence between the A D E models and the SOS model is the discrete counterpart of the gaussian field (or Coulomb gas) construction of the minimal conformal theories [17]. Unfortunately, we do not know how to extend this equivalence to the case of Green functions of more than two fields and surfaces of genus >~ 2. It is not excluded that the general Coulomb gas picture exists only in the continuum limit.

7. Critical behaviour of the A D E models of a fluctuating lattice

In the previous sections we analyzed the properties of the A D E models on an irregular but fixed lattice ~¢. From now we will allow the lattice to fluctuate and define the partition function Z(g)(k,) as a sum over all lattices ~ with the topology of a sphere with g handles, eq. (5). It is a highly nontrivial fact that the continuum limit of such a system is described by introducing a fluctuating gravitational field [12] coupled to the matter field o ( x ) .

Let us start with the partition function. Using the cluster expansion in the form (36) we write

Z ( X ) = ~ , ~ - - X 2 1 ~ * t k (~¢*) 1 ~ /~polygoos, (49)

polygon decompositions of re*

600 LK. Kostov / A D E face models

where the sum goes over all q94 graphs f¢* with the topology of a sphere, k(f¢*) -- k(~¢) is the volume of the symmetry group of ~ * and I~*l is the number of vertices of f¢*.

The partition function (49) can be written as a series

z(x) = Z zNx", (50)

N = I

where the "microcanonical" partition function Z N is the contribution of the planar graph ~ with fixed area (number of vertices) [f¢] = N. One expects the following large N behaviour of the coefficients

ZN:

Z N - A N N - b .

(51)

This implies that the series (50) is convergent up to some critical value ~c =

1/.4

of the cosmological constant ~.

The singularity at ~ = ~c is characterized by the string susceptibility critical e x p o n e n t "~str,

02Z

0~2 (~,c - ~')--~s,r + regular part. (52)

The second derivative Z " ( ~ ) makes sense of the susceptibility of the identity operator. This is the partition function for a random lattice with two marked points on it. The singularity at ~ = kc is due to the infinite volume of the random lattice ~.

The statistical systems on a fluctuating lattice have larger symmetry and therefore simpler p h e n o m e n o l o g y - only gauge invariant questions are allowed. The Green functions of such a theory are automatically integrated over the positions of their points. Therefore, instead of the correlation functions of the matter fields one has to consider the corresponding susceptibilities. The way they depend on the tempera- ture and the cosmological constant allows us to determine the scaling dimensions. In our case the models are considered from the very beginning at the critical tempera- ture. This is why the scaling dimensions of the order parameters will be determined up to an unknown constant. The latter can be fixed by comparing the results with the K P Z formula [12].

Let us define the susceptibility X, of the scalar field ¢ ( x ) as the double sum of the connected two-point correlator ( ~ ( x ) ~ ( y ) ) over the points x and y of the lattice f¢. The critical singularity of the susceptibility Xo at ~ ---, kc is

(53)

where A is the scaling dimension of the field ~. We have assumed here that the

LK. Kostov / ADE face models 601 operator q~ scales as (volume) - a where the characteristic volume of the random graph grows as (~c - ~)-1. Taking A = 0 we have the singularity of the susceptibil- ity of the identity operator (52).

The analysis of Knizhnik, Polyakov and Zamolodchikov [12] implies that the usual Kac spectrum

( r - xs) 2 - ( x - 1) 2 h

A°~= 4x , x = h - 1 ' (54)

of the minimal conformal theories J g ( h - l / h ) with central charge

c = 1 - 6(x - 1)2/x (55)

is converted by the "gravitational dressing" into a linear spectrum

Ar, = ~(ISX-- r[ -- Ix-- 1[) (56)

through the remarkable KPZ formula A -z~ 0 = A(1 - A ) / x . The string susceptibil- ity in these models is [12, 23]

]/str ^{= - -} 1 / ( h - 1). (57) We will show that the order parameters q~m(x), eq. (38), have scaling dimensions (56) with r = s = m

l m - 1

Am 2 h - 1 (58)

8. Planar Dyson-Schwinger equations

In this section we will develop a method for calculating the partition function and the susceptibilities of the order parameters based on the planar Dyson-Schwinger equation [241.

First we will change our geometrical notations to more convenient ones. Consider a polygon decomposition of a cp 4 planar graph N and decompose all its vertices as pairs of q) 3 vertices by introducing a new kind of propagators (dashed lines) as follows:

- - ] . @ ,, _ , , L (59)

l " 1 - '

We get in this way a ¢p3 planar graph 5 ° with twice as many vertices densely covered by nonintersecting self-avoiding loops. All formulae of sect. 5 can now be

602

I.K. Kostov / ADE face models

translated to the language of the gas of dense loops on a random cp 3 planar graph.

Each loop is taken with fugacity/3. For example, the expression for the partition function (49) reads

Z(~k) = E - - ~ k [ c T [ 1 E /~#loops ( 6 0 )

loops on 5 '~

where the first sum goes over all cp 3 planar graphs 5 ° with the topology of a sphere, k(SP) is a symmetry factor, 15ol denotes the number of vertices of the graph 5 p, and the second sum goes over all loop configurations on 5 p covering all its points.

Now let us consider a piece of a random planar graph with n external lines.

Define the partition function Z,(~,) by eq. (60), where 5 p is this time a planar graph with n external legs which are dashed lines, i.e. not occupied by loops. Let us represent the partition function Z , by a blob with n legs. We are going to write an infinite system of algebraical equations for the quantities Z 0 = 1, Z1, Z 2 .. . . . For this purpose we choose an external line and follow it into the blob. Two things can happen: either the line passes through the blob to end as another external line, or it ends on a loop. Summing up the two contributions we get the following equation (we write it in graphical and analytical form):

@ ' - ' ~ " ~ ^{+ - -} ~ ~ ~ . . . Z _ ^, (61a)

@.L_

n - - 2

Zn= E ZkZn-2 k " ~ ~ ZpZq+n 1 (P+q)!)~p+q+l

^. ( 6 1 b )

k=0

p,q=O P!q!

The combinatorial factor

(p + q)/p! q!

counts the number of ways to put p lines from the left and q lines from the right along the loop.

The infinite system (61) can be written in a more compact form in terms of the generating function

F ( x ) = x - l + Z l x

2 + Z 2 x - 3 - 1 - . . . , (62) The first term in eq. (61) gives simply the square of

F(x).

To evaluate the second term we use the formula

E (p + q)! )kP+q+IxPZq

⁼ F( X-1 - x ) . (63)

p,q=O P!q!

LK. Kostov / ADE face models 603

With the help of eq. (63) we write eq. (61) in the form

xF(x)= l + F2(x) + fl f 2~ri(x_y) F ( 1 - y ) F ( Y ) dy

= l + F 2 ( x ) + f l F ( x ) F ( 1 / X - x ) - ~ ~ x ' ,

n = l

(64)

where the real constants C1, C2,... are the coefficients of the positive powers of the Laurent expansion of the function

F ( x ) F ( 1 / h - x).

In order to get rid of these parameters we take the imaginary part of eq. (64) for x on the real axis

I m F ( x ) { [ 2 R e F ( x ) + f l R e F ( l / X - x ) - x ] }

+

I m F ( l / X - x ) R e F ( x )

= 0 . (65) In order to investigate further this equation we need more information about the analytical properties of the function

F(x).

It follows from the definition (62) that the real function

F(x)

behaves at infinity as

1/x.

Therefore its imaginary part should be supported by some compact interval a, b on the real axis. The micro- canonical partition functions Z n can be written as the moments of the density function

O(x)

= Im

F(x)/1r:

z.= fdxo(x)x',

ⁿ = 0 , 1 , 2 . . . (66) Then

F(x) = fdyp(y)/(x-y).

⁽⁶⁷⁾

The functional equation (65) relates the function

F(x)

and its image with respect to the reflection x ~ 1/X - x. If we substitute the integral representation (67) into the original equation (64) we shall see that it makes sense only if the stationary point x = 1 / 2 X of this reflection lies out of the support [a, b] of the density function p. The critical coupling h c is determined by the condition 2b)t c = 1 [9].

N o w it is clear that if a < x < b , then I m F ( x ) 4 : 0 and I m F ( 1 / h - x ) = 0 . Therefore eq. (65) is equivalent to the following two equations for the real and imaginary part of F ( x ) :

Re F ( x ) +

lflReF(l/X - x) = ~x,1

X E

[a, b] ,

(6Sa)

Im

F(x) = O, x ~ [a, b],

(68b)

604 LK. Kostov / ADE face models

which, together with the condition at infinity

r ( x ) - l / x , Ixl

--, ~ , (68c) fix both the function F and the interval [a, b].

At the critical point ), = ?~c determined by the condition 2b), c = 0 the solution of eqs. (68) can be found in a closed form. After a change of variables,

x = b + ( b - a ) u , (69a)

r ( x )

= / ( u ) +

u ( b -

a ) / ( 2 -

fl) + b/(2 + fl),

(698) eqs. (68) take the form

R e f ( u ) + ~ f l R e f ( - u ) = O ,

- l < u < 0 , (70a) I m f ( u ) = 0 , u < - I or u > 0 , (70b)

f ( u ) -

[ ( b - a ) u ] - l - b / ( 2 + f l ) - u ( b - a ) / ( 2 - f l ) , u ~ . (70c) The solution of (70a), (70b) growing linearly at infinity and finite at u = 0 is

f ( u ) =Cu{[u/(1 +

1 - ¢ ~ u 2 ) ] l / h + [ u / ( 1 + ~ - ~ 2 ) ] - - 1 / h } , ] (71)

where h is related to fl by/3 = 2 cos

~r/h,

see eq. (8), and C is some constant. The latter and the cut [a, b] are fixed by eq. (70c):

C = hv~/sin(~r/h),

b = 2f2-cos(~r/2h), b - a = 2v~-h sin(~r/2h). (72)

The density function

O(x)

= Im

F(x)/~r

is respectively

h2-1/h

p ( x ) -- qT"v~(b - a) (b - x) l-lIb

× [ ( ~ / 2 b - a - x + x ~ L a - a ) 2 / h - - ( ~ / 2 b - a - x - f ~ - a ) 2 / h ] .

(73)

It has been found by other means in ref. [25].

Knowing the density function

p(x)

we can easily reconstruct the partition function of the model. In order to get rid of the combinatorial factor k ( 5 p) on the rhs of eq. (60) we take the derivative with respect to ~ which is equivalent to cutting

LK. Kostov / ADE face models

one of the lines of the graph 5: occupied by a loop

605

d Z ( ) ~ ) / d ~ , =

/ / / ~ N \\~,

x o ( x ) y p ( y )

=/3 ~ (P+q)'zP()k)Zq()k))kP+q=/3fabdxfabdy 1 )k(X-l-y)

p,q=t P!q! - "

(74) It is clear that near the critical point ~ - - 1 / 2 b the singularity of the partition function will be determined by the leading singular behaviour of the density function near the point b of its support. By eq. (73) we have

p ( x ) _ ( b _ x ) l X/h, x<b_ , )~=X c. (75) Evidently this is not sufficient to determine the string susceptibility exponent Ystr;

for this purpose we should know the density function also for )~ _% )~c. Nevertheless, the singular behaviour (75) of the density function implies a very useful relation between Ystr and the critical exponent a 0 for the size of an individual loop.

Let ~o be one of the loops on the random lattice 5 '~. Near the critical point the length ]£,o] of the loop diverges together with the volume ]5 '~ ] of the random lattice.

The exponent a 0 is defined by

15°1- ].L#[ 2-~0. (76)

In the special case /3 = 0 (h = 2) it coincides with the specific heat exponent a [9,111.

The scaling relation (76) enables us to replace the characteristic volume of the r a n d o m lattice which is of order (X¢ - X) -1 with the characteristic length of one of the r a n d o m loops which can be made finite even at )~ = X c by modifying the weights of its points.

Consider the integral representation (74) of the derivative of the partition func- tion for X = )~o where the explicit form of the density function is known. We can change by hand the weights of the points along the loop we have cut, from Xc to Xc - ~ to the effect that the pole will be shifted with ,~/b from the point b. With the help of eq. (75) we get by power counting

Z(~)k) - ~2+2(1 1/h) (77)

6 0 6 LK. Kostov / A D E face models

On the other hand, 6 is the inverse characteristic volume of an individual loop.

Inserting expression (76) with ISPl = (X c - X ) -1 into the definition (52) and com- paring the result with eq. (77) we have

(2 - ao)(2 - "~str) = 2 + 2(1 - l / h ) .

(78)

If we assume that "Ystr is given by eq. (57), then the relation (78) implies

Ystr = - 1 / ( h - 1), a 0 = 2 / h . (79) The connection with the notations of refs. [9,11] is v2d = 2 - a0- Here we prefer to use notations without direct reference to the internal fractal dimension d of the r a n d o m lattice.

Concerning the derivation of eq. (78) we would like to make the following remark.

The partition function Z ( 6 ) is that of a loop gas where all loops but one are critical.

The parameter 8 measures the deviation from criticality of this single loop. There- fore the derivative d Z / d X in eq. (74) and the derivative of Z ( 8 ) with respect to 8 cannot be compared directly. The quantity d " Z / d 8 ~ is the partition function of a loop with n marked points in presence of critical vacuum loops. It can be thought of as the totally connected n-point correlation function of a special operator ~2, integrated over the positions of its points; the operator ~b2(x ) creates two lines starting at the point x (this interpretation is due to Duplantier). If A 2 is the scaling dimension of @2, then

d3 n

~(1--ao)(nA2--n+ 2--3'str) .

(80)

Eq. (80) implies besides eq. (78) the relation

2 - a o = 1 / ( 1 - k 2 ) .

(8a)

The reader can find more details in refs. [9,11, 26].

Up to this point all results concerning the gas of loops on a random graph are known. They have been obtained in refs. [9,11] using the equivalence with a random matrix problem. Here we rederived them in order to illustrate the method of planar D y s o n - S c h w i n g e r equations which turns out to be a more general and powerful mathematical tool. Below we are going to demonstrate this by calculating the susceptibilities of the order parameters (38) in the A D E models.

Let X (m) be the susceptibility of the order parameter ~m" It can be calculated using the cluster expansion formula (40) averaged over all lattices f¢*. As we did when calculating the partition function, we replace the ~04 planar graph ~ * decomposed into polygons with a cp 3 planar graph 5 p containing two kinds of propagators (continuum and dashed lines) as it is indicated by eq. (59). The

LK. Kostov / ADE face models 607 continuum lines form a pattern of dense nonintersecting loops on the planar graph 5 °. Thus for a fixed geometry the sum over all possible mosaics of the two kinds of propagators can be given the meaning of the partition function of a gas of self-avoiding nonintersecting loops (polymers) on the lattice 5 °. In terms of the gas of loops the susceptibility

X (m)

is given by the sum

= E k(so) E E a x'xRY'" ^" "(m) ' 5" x, x' loopson 5"

(82)

where the first sum goes over all ¢p3 planar graphs 5 ° with the topology of a sphere and x, x ' are points of the dual graph (faces of 5O). The loops enclosing only one of these points are taken with a factor

tic,,)= 2cos(~rm/h);

the rest are taken with weights fi = 2 cos(~r/h). Their numbers are denoted by ~4(~ x, and JV'-JV~x, , respec- tively. As before we kill the symmetry factor k ( s o ) by taking the derivative with respect to X. This is equivalent to cutting one of the loops on 5O to obtain a planar graph with two external legs. Exhibiting all the dashed lines starting from the cut loop we represent

OX~')/OX

in the form analogous to eq. (76). Denote by Z,, Z * and Z,** the partition functions for planar random lattices with 0, 1 and 2 marked faces, correspondingly. The marked faces are the locations of the order parameters.

In the loop-gas representation of Z,* and Z * * the loops enclosing exactly one marked face are taken with weights tic-,)=

2c°sOrm/h)"

Then we have

d X<""(Yt)

~ (fl<m)ZpZ*+fiZpZq*) ( p + q ) !

(83)

dX

p , q = l

p!qT

The Dyson-Schwinger equations for Z,* are obtained again by tracing the possible evolutions of an external line:

-

®:.

^{+ - - . + ,}

(84a) or, analytically,

n - 2 ~ ( p -t- ] ) !

Z , * = 2 Y'~

ZkZ,*2_k + P!q! (flZpZ~+,_i + fl(m)ZpZq+,_l).

(84b)

k = 0 p,q=O

In terms of the generating function

F(x) = x 1

+ Z ~ x - 2 4_ Zffx-3 - - t - • • • , eq. (84)

6 0 8

reads

LK. Kostov / ADE face models

x F * ( x ) = 1 + 2 F ( x ) F * ( x ) + f l F * ( x ) F ( 1 / X - x )

+ fl(,,,)F(x)F*(1/X- x) - ~, C.x", (85)

n ~ l

where

C,

are real numbers. Taking the imaginary part of eq. (85) we get 0 = Im r * ( x ) [ - x + 2 R e F ( x ) +

f l R e F ( 1 / X -

x ) ]

+ Im

r ( x ) [ 2

Re F * ( x ) +

fl(m)Rer*(1/X -

x ) ]

+ f l I m F * \ ~ - (

1 - x ) R e F ( x ) + f l < m ) I m F ( 1 - ~ - x ) R e F ( x ) .

(86)

If we assume that F and F* have the same cut from a to b, then for a ~ x ~< b the second factor in the first term is zero, according to eq. (68a). Next, due to eq. (68b), the second factor in the second term should vanish. Therefore, eq. (86) is equivalent to

R e F * ( x ) + c o s ( ~ r m / h ) R e r * ( 1 / X - x ) = O , x ~ [ a , b ] ,

(87a) am r * ( x ) = 0,

x f ~ [ a , b ] .

(87b) The solution we are looking for should vanish at infinity as

1 / x

and be integrable at x -- b. These conditions are satisfied by

[./0

^{+ + l+m'}

2sin(~rm/h )~/1 _ u 2 ,

(88)

where

u = (x - b)/(b - a).

Reasoning in the same way as before we write the Dyson-Schwinger equations for Z * *

n - 2 o o

Z * * = 2 Y'~

(ZkZ**z_k+ Z~Zff z_k)+ 2 ~ (flZ;*Zq+fl~m)ZpZq),

(89)

k = 0 p,q=O

or, in terms of generating functions

xF**(x) = 1 + 2 r ( x ) r * * ( x ) + 2 F * ( x ) V * ( x ) + flF(x)F**(1/X - x)

+ f l r * * ( x ) r ( 1 / X - x) + B~,~)F*(x)r*(1/X- x) + ~, C,x",

(90)

n>~l

I.K. Kostov / ADE face models 6 0 9

where C, are real constants. Taking the imaginary part of eq. (90) and applying eqs.

(70) and (87) we reduce eq. (90) to

ReF**(x)+cos(Tr/h)ReF**(1/X-x)=O, x ~ [ a , b ] ,

(91a) I m F * * ( x ) = 0,

xq~ [a,b].

(91b) It follows f r o m eq. (91) that the singularity of

F**(x)

at x = b does not depend on m. Therefore the contribution of the second term on the rhs of eq. (83) is not essential for the critical behaviour of the susceptibility

x(m)(X).

Introducing the density function

p*(x)=

I m

F*(x)/~r

we rewrite the first term on the rhs of eq. (85) as

_ c

cbdxO*(x)dyo*(y)

d X t ' ) / d X = fi(m) l Ja i - ~ - ( - x ' - ~ y ) + . . . . (92)

As before, we take X = Xc and "regularize" the denominator by replacing Xc with

?'c - 8 which is equivalent to modifying the weights of the points along the cut loop.

Eq. (88) implies

p * ( x ) - ( b - x ) l+m/h

and we obtain from eq. (92) by power counting

x ( m ) ( ~ ) _ ~2+ 2(-l +m/h) : ~(2--a0)(ZA(m)--Ystr) " (93) H e n c e the scaling dimensions of the order parameters are

l m - 1

A ( m ) - - 2 h - 1 ' m = 1,2 . . . (94) T h e first o p e r a t o r in the series is the identity operator with A ~1) = 0.

9. Conclusion

In this p a p e r we have given a method for exactly solving the critical SOS-type models (the A D E models) on a fluctuating lattice. Our analysis consists of two steps. First we have established a geometrical interpretation of the A D E models in terms of a gas of nonintersecting loops (domain walls) densely covering the planar lattice. N e x t we have written an infinite series of algebraical equations for the partition function of a piece of a r a n d o m lattice with the topology of a disc.

Introducing a variable x coupled to the perimeter of the disc we have observed an interesting analytic structure of the corresponding generating functions. Knowing their exact f o r m only at the critical point X c we were not able to predict the string susceptibility exponent Yst~- Nevertheless the scaling dimensions of the order param- eters are determined uniquely provided Ystr is known.

610 LK. Kostov / ADE face models

An interesting geometrical interpretation of the susceptibility X(ml(Yt), eq. (82), is that of the two-point correlator in the momentum space for spherical random surface embedded in a Dynkin diagram with Coxeter number h. In the limit h ---, oo (/3 ~ 2) the embedding space of the random surface becomes the infinite one- dimensional lattice. In other words, the unrestricted SOS model on a fluctuating lattice yields a discretization of the one-dimensional bosonic string. The spectrum of m o m e n t a in this limit becomes continuum: P - -

+_m/h

covers the whole interval [ - 1 , 1]. F r o m eq. (93) we get

x ' m ' ( x ) -- ( X c - X) m/ h-l - (Xc -- X) jpj (9S)

Exactly the same result has been obtained using another, quite different discretiza- tion of the D = 1 bosonic string [27].

Many problems concerning the ADE models still remain to be solved. The most immediate ones are:

(1) To find the generating functions satisfying eqs. (68) and (87) for ~ < Xc- This would allow us to calculate the string susceptibility and the correlation length critical exponents. The problem has been solved in the two extremal cases/3 = +_ 2 (not mentioning the trivial case/3 = 0). The derivative of the solution with respect to x can be expressed in terms of standard elliptic functions [7, 28].

(2) To generalize the method to the noncritical A D E models. Technically this can be performed by introducing an orientation of the loops in the loop-gas representa- tion and supplying clockwise and anticlockwise oriented loops with different weights.

(3) To generalize the method for planar graphs of arbitrary topology. Of special interest are the subleading exponents for the partition function on a toroidal fluctuating lattice. They could give information for the operator content of the corresponding quantum field theory.

We hope that the models we have considered can serve as a playing ground for developing ideas and techniques which will allow us to attack some real physical problems. The most challenging one is the possible realization of the old idea of 't H o o f t [29] that the chromodynamical string could be formed as the result of condensation of high-order planar diagrams. The planar perturbation series can be thought of as a special statistical model defined on a random lattice (the abstract planar graph).

It is a pleasure to thank the members of the Service de Physique Th~orique at Saclay for their stimulating hospitality. I also thank E. Brrzin, F. David, B.

Duplantier, M. Gaudin, V. Pasquier and J.-B. Zuber for many helpful discussions and especially V. Pasquier for a careful reading of the manuscript.