V o l u m e 2 3 6 , n u m b e r 2 P H Y S I C S L E T T E R S B 1 5 F e b r u a r y 1 9 9 0 E X A C T L Y S O L V A B L E F I E L D T H E O R I E S O F C L O S E D S T R I N G S E . B R I ~ Z I N a n d V . A . K A Z A K O V 1

EXACTLY SOLVABLE F I E L D T H E O R I E S O F C L O S E D S T R I N G S

E. BRI~ZIN a n d V.A. K A Z A K O V 1

Laboratoire de Physique Statistique, Ddpartement de Physique de l'Ecole Normale Supdrieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France

Received 7 November 1989

Field theories of closed strings are shown to be exactly solvable for a central charge of matter fields c= 1 - 6 / m ( m + 1 ), m= 1, 2, 3, .... The two-point function Z(2, N), in which 2 is the cosmological constant and N - ~ is the string coupling constant, obeys a scaling law Z(2, N) = N -~,,+ 1/2~f( (2c - 2 )N ,,/~m + ~/2)) in the limit in which N -~ goes to zero and 2 goes to a critical value 2 c; we have determined the universal non-linear differential equation satisfied by the function f From this equation it is found that a phase transition takes place for some finite value of the scaling parameter (2c-2)N m/~m+ ~/2); this transition is a "'condensation of handles" on the world sheet, characterized by a divergence of the averaged genus of the world sheets. The cases m = 2, 3 are elaborated in more details, and the case m= 1, which corresponds to the embedding of a bosonic string in - 2 dimensions, is reduced to explicit quadratures.

1. Introduction

O n e of the most tantalizing goals of m o d e r n q u a n - t u m field theory is the n o n - p e r t u r b a t i v e investiga- tion of various string theories [ 1 ]. This goal seems to be almost inaccessible from s t a n d a r d c o o r d i n a t e ap- proaches which involve integrations over the m o d u l i space a n d the fixing of a gauge (a coordinate system) for the internal metric. Even the first few lowest or- ders of the string loop-expansion ( i n the n u m b e r of holes in the world sheet of a string, for open strings, or in the n u m b e r of handles of the world sheet for closed strings) require considerable calculational efforts.

O n the other h a n d in the past few years a coordi- nate-free approach has been developed for q u a n t u m strings [2,3] which is based on a lattice regulariza- tion o f the world sheet in terms of d y n a m i c a l trian- gulations of the c o r r e s p o n d i n g p l a n a r manifolds. T h e integral over the metrics is represented as the sum over all possible t r i a n g u l a t i o n s with a fixed topology [2]. The i n t r o d u c t i o n of m a t t e r fields for these models of t w o - d i m e n s i o n a l q u a n t u m gravity m a y be done explicitly in two ways: ( i ) one introduces ad-

Permanent address: Cybernetics Council and Academy of Sciences, ul. Vavilova 40, SU-117 333 Moscow, USSR.

ditional variables, such as spins, which lie at the ver- tices of the triangulation; exactly solvable examples o f this approach are the bosonic Polyakov string [ 2 - 5], the fermionic string [6,7], O ( n ) - s p i n s a n d ADE face models [8,9 ], (ii) or by m e a n s o f a multicritical p h e n o m e n o n for d y n a m i c a l graphs of mixed type [ 10 ] (i.e. made of triangles, squares, pentagons, etc. ), which can exhibit different types of critical points, thereby reproducing different kinds of m a t t e r fields.

A n a t u r a l framework for the investigation of these models is the 1 I N expansion of various simple field theoretic models i n v o l v i n g N X N matrices [ 11,12 ].

M a n y of the a b o v e - m e n t i o n e d models have been solved exactly from this approach in the p l a n a r a p p r o x i m a t i o n .

Recently one of the authors has investigated by these m e t h o d s a very simple field theory of open strings [ 13] ( 2 D gravity without a n y m a t t e r field).

A non-perturbative p h e n o m e n o n (with respect to the string coupling c o n s t a n t ) was found there, consisting of the " s p o n t a n e o u s tearing" of the world sheet of a string, a n d some new critical exponents were com- puted.

It was clear from the b e g i n n i n g that this m e t h o d of the 1 / N expansion of matrix field theories provided an elegant n o n p e r t u r b a t i v e d e f i n i t i o n of closed strings. F o r instance a basic quantity, such as the

string susceptibility Z(2, N) (the two-point function of the string theory, see refs. [ 2 - 5 ] for definitions), which is a function o f the bare cosmological constant 2 and o f the string coupling constant 1/N, m a y be represented by the formal expansion

Z(2, N ) =

~ N-ZgZg(2 ), (1)

g=0

in which Zg(2) is the susceptibility for a fixed topol- ogy o f given genus g. Near the critical point 2c o f 2, the susceptibility Z has a singular behaviour charac- terized by a critical exponent Ystr(g) o f the type

Zg(2) ~

( 2 c - 2 ) - 7sir(g) • (2) The exponent ~ s t r ( 0 ) has been c o m p u t e d in the lat- tice approach in some particular examples [ 2 - 1 0 ], and later by elegant c o n t i n u u m methods [ 14,15 ] which yield the general formula for ~str(0 ), in the case o f a manifold with the topology o f a sphere coupled to a conformal matter field o f central charge c~< 1.

This c o n t i n u u m approach was then generalized to in- clude any topology [ 16,17 ], and verified on O (n) lattice models [ 8 ]; it yields

Zstr(g, C ) = --

1/m+

( 2 +

1/m)g,

(3) in which the central charge c is

c= 1-6~re(m+

1) . (4)

In general m is a continuous positive parameter.

In this article we shall use the generalized one-ma- trix models [ 1 1 ] in which the multicriticality yields matter fields out o f pure gravity [10], and investi- gate in the scaling limit the singular point 2 near 2c, at which surfaces o f arbitrary genuses acquire a large average size (the invariant area is proportional to 1 /

( 2 c - 2 ) ) . In the scaling limit N--+ or, A= (2c - - 2 ) ~ 0 and x finite in which

X : z I N 2 m / ( 2 m + l ) , ( 5 )

the singular part o f the susceptibility is expressed in terms o f a universal scaling f u n c t i o n f ( x ) as

Z(2, N) = N - 2 / ( 2 m + l)f(x)

.

(6) In the following we shall d e m o n s t r a t e this scaling law and find a non-linear differential equation for the scaling function f, for any m a t t e r central charge c < 1.

We shall investigate in more details (i) the case o f pure gravity ( m = 2 ) , (ii) the Ising case ( m = 3 ) , (iii)

the general structure o f the equation for the scaling function for any m, (iv) the simplest case m = 1, i.e.

c = - 2, or the ( - 2 )-dimensional embedding o f a bo- sonic string, for which the scaling function f m a y be c o m p u t e d easily.

The large-x expansion o f (6) will reproduce the re- sults (2) and ( 3 ), and the singular b e h a v i o u r near xc (a double pole for m = 2) will exhibit non-perturba- rive effects of the

1/N

expansion. This p h e n o m e n o n is related to the instability o f the world sheet with re- spect to the creation of handles (or to the formation of closed strings) in a regime in which N is large but x is finite near Xc.

2. Multicritical models in the scaling limit

In this section we shall use the approach o f ref. [ l 0 ] in which the matter field is generated from pure grav- ity at a multicritical point, and the mathematical f r a m e w o r k and notations o f ref. [ 12].

In a general one-matrix model o f this type a two- point function can be defined as

Zm.,(2, N)=[ l f

dN2M tr M2m tr M 2n

in which M is an

N×N

hermitian matrix, Z is the partition function, and 2 plays the role of the cosmo- logical constant (see ref. [10] ); the potential

V(x)

is given as

V(x)= y~ gpx 2p. (8)

p>~l

As usual in these problems one expresses in the inte- gral ( 7 ) the matrices in terms of their eigenvalues xa, and of the unitary transformations which diagonalize M. It is then convenient to introduce the orthogonal polynomials Pn (x):

f dl2(x)P.(x)Pm(x) =hn~ ....

(9) with respect to the measure

d p ( x ) = d x exp[ - 2 - '

V(x)

] (lO)

(Po(x) = 1 ); these polynomials satisfy the recursion formula

x P . ( x ) = P,,+, (x) + R . P . _ , (x) , ( 11 ) in which

R , = h , / h , _ ~ . (12)

Itzykson and Z u b e r have obtained [ 12 ] a general re- lation for the coefficients R,:

2n U - 2 R , , ~ (p+l)gp+~ Z Rm...R,~p, (13)

p >/0 s t a i r c a s e s

in which the coefficient o f 2 ( p + 1 )gp+ ~ is a sum over the ( 2 p + 1 ) ! / p ! ( p + 1)! paths along a "staircase"

starting from the height ( n - 1 ), ending at height n in ( 2 p + 1 ) unit steps, ( p + 1 ) up, p down. A down step from k to ( k - 1 ) generates a factor Rk; an up step gives a factor one.

By the same technique which leads to 13 ) one can easily obtain the relation

N - - I

( N - ' t r ( m 2 ) ) = 2 Z R , . (14)

0

If we define a particular two-point function

3/(2, N) = 2 2 ~ ( N - 1 t r ( m 2 ) ) , a (15)

we obtain in the scaling limit

Z(2, N) : R N W O ( N - ' ) , (16) in which

R N - R c = N -Uf(N~A) . (17)

We shall now determine R N - Re in the scaling limit, from the functional recursion relation ( 1 3 ) , as the solution o f a universal differential equation for the function f a t the multicritical point.

Let us recall that the usual large-Nlimit, with 2 fixed (i.e. x going to infinity), is obtained from (13) by setting all the R k ( 2 ) = R ( 2 ) ; R is then given by the equation

2 = w ( R ) = 2 o~ ( 2 p + l ) ! ( p ! ) ~ gp+~R p+~ . (18) A multicritical point 2 ¢ = w ( R ¢ ) o f order m, is ob- tained when the coefficients gp which are defined by the potential (8) are such that the conditions

w'(Rc)=W"(R~) . . . w ~ .... ' ) ( R ~ ) = 0 (19) are fulfilled.

F r o m (19), we find the large-N multicritical be- haviour o f Z(2, oc):

Z(;t, ~ ) ~ ( 2 - 2 c ) ~/m , (20) which corresponds to the appearance of a conformal m a t t e r field with central charge c = 1 - 6 ~ r e ( m + 1 ) coupled to gravity.

For N finite we can perform a Taylor expansion o f the coefficients Rk as follows: s e t t i n g x = n/N, eq. ( 13 ) determines R , ( 2 ) as R (2x). For x near one, we have R . ( 2 ) = R ( 2 ) + 2 ~ - R ' ( 2 ) + ~ R" (2)

+ .... (21)

It is now more instructive to discuss simple examples first, before getting to an arbitrary (even) potential.

3. A first example. Pure gravity

We start from a simple quartic potential which will give an ordinary critical point:

V( x ) =g~ x2 + gzx 4 , (22)

from which we get

W ( R ) = 2 g ~ R + 12g2 R 2 . (23)

This potential has a critical point Rc for which the equation W ( R ) = 2 c has Rc as a double root. The equation which determines R N ( 2 ) is here

2 = 2 R N [ g , +2g2(Ru+~ +RN+RN_, ) ] , (24) and in the large-N limit, using the Taylor expansion

(21 ) we obtain the differential equation

2 = w ( R ) + N - 2 4 g 2 ~ . Z R R " ( J . ) + O ( N -4) . (25) We work now in the vicinity of2~, w ( R ) - 2 c is thus proportional to ( R - R ~ ) 2, with the scaling ansatz ( 17 ). It is i m m e d i a t e to verify that 2 - 2 c , w ( R ) - 2 c , N-2R" (2) are of the same order of magnitude and that the terms o f higher order in 1 / N i n (25) are neg- ligible, iff

/ t = ~ , 4 (26)

This is of course in agreement with the result (6) stated in the introduction (see also eq. (16) for the case m = 2, which corresponds to pure gravity since c = 0 .

The scaling f u n c t i o n f ( x ) satisfies (after a simple rescaling) the non-linear differential equation

x = f 2 + f " , (27)

whose solution is a Painlev6 transcendental of the first kind [ 18 ].

This second-order differential equation should be supplemented by two "initial" conditions. The most obvious way to impose these conditions is to use the expansion ( 1 ), in which the coefficients for any fixed genus g can be computed independently by the stan- dard 1/N expansion. Indeed, in the large-x limit the topological (i.e. 1/Nexpansion), we deduce from ( 1 ) and (6)

f(x)=xt/Z(Ao-~Aix-Z+A2x-9/2+ ...) , (28) in which the coefficients A, correspond to genii 0, 1, 2, ...; the first two are our initial conditions and all the others are determined by the universal equation (27). However the non-linear equation (27) has also some non-perturbative solutions for finite values of x. Painlev6's transcendentals are characterized [ 18 ] by their properties that the only " m o v a b l e " singular- ities (i.e. depending upon the initial conditions) are poles. Eq. (27) exhibits a solution of the form f i x ) ,.~ - 6 / ( x - k) 2 , for x in the vicinity of k ,

(29) in which the parameter k has to be determined by the large-x initial conditions. We do not know the value of k for this problem, and whether it is a universal number. However, if we assume that it is indeed a finite positive constant, we must conclude that our model has a singularity at x = k , and we should thus expect some kind of phase transition, such as a con- densation of handles, when x goes below k.

More precisely we can calculate the average genus g near this transition point from the relation

N O 2 2 2 0 < g ) _ _ 20-NZ ( , N )

[ ( 2 _ 2 ~ ) N 4 / 5 _ k ] - 3 , (30) from which we obtain immediately that <g> di-

verges near the critical value x = k with an index z:

< g ) ~ ( x - k ) - z , w i t h z = 2 . (31) One may speculate that this condensation of handles is likely to be generic and thus to take place in more realistic string field theories.

4. The Ising case

We will see now explicitly that a potential with a tricritical point generates an Ising conformal matter field, c = ½ or m = 3, coupled to gravity. We thus take a potential

V(x) = g l x2 "~ g2 x4 +g3 x6 , (32) from which we get

w(R ) =2g~R + 12gzR2 +60g3 R3 , (33)

which will exhibit a tricritical point (w' = w" = 0)

Rc=-gz/15g3, if2g~=5gtg3. (34)

The equation which determines RN(2) is now in the large-N limit, using the Taylor expansion (21 ), 2 = w ( R ) +N-2),2w" ( R ) R " ( 2 ) / 6 + 3 0 N - 2j.2R'2g 3

+ N - 4 R R ( a ) 2 4 ( g 2 + 3 3 g 3 R ) / 3 + O ( N - 6 ) . (35) In the vicinity Of2o w ( R ) - 2 c is now proportional to ( R - R e ) 3. In the scaling limit (17) all the terms that we have kept in (35) are of the same order of magnitude and higher orders in 1/N in (25) are neg- ligible, iff

/ t = 2, v=-67, (36)

in agreement with the result (6) for the case m = 3 , which corresponds to c = ½, the conformal charge at the Ising critical point.

The scaling function f ( x ) satisfies here the non- linear differential equation

x = f 3 + f f " + ( f ' ) 2 / 2 + f ( 4 ) / l O . (37) A similar discussion on the integration of this equa- tion is required; four initial conditions are required here. We do not know whether all the movable sin- gularities of this equation are poles, however one can verify that, remarkably enough, a double pole is again allowed by (37 ); a priori one could expect more than

one constant analogous to the constant k that we found in the case o f pure gravity, and the phase dia- gram could be richer.

5. The general case

The potential

V(x)

is now an even polynomial o f order 2m, and the corresponding

w(R)

has his first ( m - 1 ) derivatives which vanish at Rc. Expanding as before (21) for n large, the general term will con- tain k-factors involving derivatives o f R with a total of 2p differentiations acting upon them (example:

R ( 4 ) : k = l p = 2 ,

( R ' ) 2 : k = 2 p = l ) ;

by definition

2p>k.

In the scaling limit (17) a simple counting o f powers o f N shows (i) that the scaling exponents (6) are indeed correct, (ii) that all the terms such as

p + k = m

have to be kept (for m = 4 , for instance, this leaves us with R (6),

R'R"', (R")2),

(iii) that all the terms such as p + k < m should have a vanishing coef- ficient at multicriticality for consistency. For in- stance R" has p + k = 2 , but its coefficient is w" ( R ) which vanishes at criticality for any i n > 2; for m = 4 one should check that the coefficients o f R (4) and ( R ' ) 2 vanish at Re, etc. The verification o f the con- sistency o f these conjectures for general in is in prog- ress [ 19 ] ; assuming that all this is indeed right, we obtain an equation for the scaling function f o f the following type:

x = f " +,f "fm--2 + a f (2m--2)

+ Z Cp.f(P)f (2m-p-4)

+ ~, Cp,qf(P!f(q)f (:'-p-q-6)

+ .... (38)

p,q>~O

in which the constants

G, Cp,q,

etc., are all fixed by the multicriticality condition. Note that for any in there is a finite number of terms on the RHS o f (38), which are all characterized in the above discussion. Need- less to say that the analysis o f the singularities o f this general equation remains to be done.

6. Field theory of a bosonic string embedded in a ( - 2 ) - d i m e n s i o n a l space

The planar limit o f this theory has been first inves- tigated in ref. [4]. Kostov and Mehta have found an

elegant expression for the free energy, valid to all or- ders o f the topological expansion [20]. This model appears to be essentially equivalent to the simple gaussian unitary ensemble o f N × N r a n d o m matri- ces. For the study o f this model we shall use a repre- sentation o f the generating function for the expecta- tion values o f the powers o f the r a n d o m matrices, in terms o f a double integral [21 ]; N appears there as an explicit arbitrary parameter.

The connection between the two problems m a y be seen as follows: the derivative with respect to the cos- mological constant of the partition function o f the

"( - 2 )-dimensional" bosonic string is given by

0~ (N,)-) = Z

g = 0 n = l G(n,g) N 2g

( 8 >

(39) in which

E(;(n,g)

is the sum over 03-graphs, drawn on a surface o f genus g and made o f n vertices. The vari- ables xi, -~,, i = 1, 2 ... n, are complex grassmannian variables (corresponding therefore to an embedding space o f dimension - 2 ); Z <,j) denotes the sum over the neighbours defined by the triangulation. In ref.

[20] it was found that (39) could be expressed as an integral over N × N hermitian matrices as

OF 02 - !

f dN2M[exp{-(½NtrM2)}]

× t r { f ( M ) [ M - f ( M ) ] } , (40) in which f ( M ) is the solution o f the quadratic equation

- 2 f 2 + j ~ M .

(41)

Therefore, using a contour integral representation for the prefactor o f the exponent in (40), one can ex- press F' (2) as

OF N ~

02 -- 222

dz Q(z, N)

X [ 3 2 z - 1 + ( l - 2 z ) x / l - 4 2 z ] , (42) in which

Q(z, N)

is the average Green function o f the gaussian unitary ensemble:

Q ( z ) = ( N -1

t r ( z - M ) - 1 ) (43)

(the i m a g i n a r y part o f Q is p r o p o r t i o n a l to the den- sity o f eigenvalues in this e n s e m b l e ) .

Since ref. [21 ] might not be readily accessible let us give an i n d i c a t i o n o f the m a i n steps. A simple gaussian integral in t e r m s o f N c o m p l e x c o m m u t i n g variables ua and 2 N g r a s s m a n n i a n variables Va, va, al- lows one to write

(z--m)ab I = --i f l-I

(dUa du* d/7 a

dVa)l,t*aUb

× exp{i[u*(z--M)abUb+Oa(Z--M)abVb]}.

( 4 4 ) F r o m this r e p r e s e n t a t i o n one can p e r f o r m easily the gaussian average o v e r the m a t r i x M, integrate then over the G r a s s m a n n variables f, v and one o b t a i n s then, after a few s i m p l e algebraic steps, a representa- tion for

Q(z)

which is valid for any finite N, in terms o f a d o u b l e integral

Q ( z ) = - i - ~ - \ - f ~ j dx dy(x--iz)Ny N

- - o o 0

× ( l + x y ) exp[-N(½x2+~y2-izy)].

( 4 5 ) The large-N limit is governed by the saddle p o i n t

x0 =Yo = ½ ( i z + ~ ) ( 4 6 )

(the s a d d l e p o i n t Xo = ½ [ i z - ( 4 - z 2 ) 1/2 ] is sub-lead- ing) a n d one finds that in the large-N limit

Q(z)-,

- ixo. W i g n e r ' s semi-circle law is o f course recovered after taking the imaginary part o f F ( z ) . However near the critical p o i n t

Zc=

2 (at which Xo =Yo = i) the two saddle p o i n t s merge and one has to e x p a n d the ex- ponential in the integrand up to cubic order. Defining

Z=Zc+~O, ( 4 7 )

and e x p a n d i n g the i n t e g r a n d as x o = i + a , y o = i + f l

for small a a n d fl, we o b t a i n at leading o r d e r for large N, in the vicinity o f zc

iN f

Q ( z ) = - ~-~

dadflafl

×exp{N[ko(fl-a)+½i(fl3-a3)]}.

( 4 8 ) F r o m this r e p r e s e n t a t i o n we o b t a i n readily the scal- ing form

Q(z) =N

- 1/3f((Z_Zc)N2/3)

in a range o f extension N - 2 / 3 near the critical p o i n t zc; the scaling function is related to an Airy function.

(This result has been known for a long time, from the c o n s i d e r a t i o n o f the a s y m p t o t i c form o f H e r m i t e p o l y n o m i a l s at large o r d e r [ 22 ], but the present der- ivation is s i m p l e r and self-contained.) Substituting this scaling form into ( 4 2 ) it is straightforward to verify that F ' (2) satisfies the scaling ansatz ( 6 ) for m = 1; the calculation o f the scaling function is re- d u c e d to simple quadratures. The singularity struc- ture o f this function will be given elsewhere.

Note added.

After this work had been s u b m i t t e d for p u b l i c a t i o n we learnt that D.J. Gross and A.A. Migdal in Prince- ton and M. Douglas a n d S. Shenker in Rutgers had d o n e s i m u l t a n e o u s l y interesting work along similar lines.

Acknowledgement

One o f us (V.A.) thanks D.V. Boulatov a n d A.A.

Migdal for useful discussions.

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