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

EXACTLY SOLVABLE F I E L D T H E O R Y O F D = O C L O S E D AND O P E N S T R I N G S

Ivan K. KOSTOV Service de Physique Thborique 2 de Sac&y, F-91191 Gif-sur- Yvette Cedex, France

Received 3 January 1990

The field theory of interacting closed and open strings is solved using the equivalence with a random matrix model. The parti- tion function of the theory is given in the scaling limit by an universal function of the renormalized cosmological constant A, mass M at the ends of the open string, and couplings G and F for the closed- and open-string interactions, correspondingly. Its second derivative in A obeys an ordinary differential equation of fourth degree.

The discret izat ion of the bosonic string by random planar graphs [ 1,2 ] provides not only an unambigu- ous defini t ion of the string path integral, but also al- lows for exact solutions [2 -5 ]. Most of the exact re- sults have been obta ined using the equivalence of the discret ized string with an N x N matr ix model in its p lanar l imit N - - , ~ . The loop expansion in the string field theory corresponds to the 1 /N expansion of the random matr ix model.

It is a common place that the 1 / N e x p a n s i o n (and therefore the string loop expans ion) diverges. But very recently, six authors [6 -8 ] succeeded to make sense out o f the field theory of closed string assuming that the bare coupling constant 1 /N 2 is renormal ized by a power o f the cut-off.

A similar phenomenon has been previously ob- served by Kazakov [9] who s tudied the open-str ing field theory in the Veneziano l imit ( the world sheet o f the string may have holes but no handles) , embed- ded in zero dimensions. Kazakov observed that a nontr ivial con t inuum l imit of the model can be achieved only i f the bare interact ion constant 7 for open strings tends to zero as certain power of the cut- off.

In this paper we generalize these results by present- ing the solution of the field theory of interacting closed

Permanent address: Institute for Nuclear Research and Nu- clear Energy, 72 Boulevard Lenin, 1784 Sofia, Bulgaria.

2 Laboratoire de l'lnstitut de Recherche Fondamentale du Commissariat/l l'Energie Atomique.

and open strings in D - - 0 embedding dimensions. First we regularize the string path integral by discre- tizing the world-sheet of the string. Then we take the scaling l imit of the equations o f mot ion and find a four th-order differential equation for the string sus- ceptibili ty. This equat ion contains as par t icular cases the Painlev6 equat ion found in ref. [ 6-8 ] for the in- teracting closed string, and the algebraic equation de- scribing the cont inuum l imit of the interact ing open string in the Veneziano l imit [ 9 ].

Let G and F be the couplings for the closed and open strings. Then the world-sheet of the string is al- lowed to form handles and holes with fugacities G and F, correspondingly. The par t i t ion function of the the- ory is a weighted sum over all configurations of the world sheet. Each configurat ion represents a two-di- mensional surface with g handles and h boundar ies (holes) . Geometr ica l ly it is character ized by its area A and length of the boundary L. The par t i t ion func- t ion r e a d s

F = ~ G g - ' F h e x p ( - A A - M L ) , (1) connec ted surfaces

where A is called cosmological constant and M is the mass at the ends of the open string.

The power of G is chosen so that the piece of the par t i t ion function due to the non-interact ing closed string ( g = 1, h = 0 ) is mul t ip l ied by one. I f we re- quire the same for the non-interact ing open string ( g = 0, h = 2 ), then two coupling constants should be related by G = F 2.

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The sum over surfaces in ( 1 ) makes sense only after being regularized. In what follows we shall do this by discretizing the world-sheet of the string. The discre- tization introduces a cut-off a ( = elementary length). We shall see that the renormalized parameters of the theory behave under a rescaling a-~p- ~a as

A~p2A, M-~pM, G---,pSG, F--,pS/2F. (2)

The partition function of the theory in the contin- uum limit a ~ 0 does not depend on the cut-offat all, as it should. It depends on its four arguments only through three dimensionless combinations of them:

F(A, M, G, F)=F(p2A, pM, pSG, pS/ZF) . (3)

The string susceptibility f = 02F/OA 2 obeys as a func- tion ofA a fourth-order differential equation

G( R 'R ' - 2RR" ) = 1 - 4 ( f + M ) R 2 , (4a)

where

2 F R = A - 3fZ + Gf" . (4b)

This equation is the main result of this paper. When F = 0 , eq. (4b) reproduces the Painlev6

equation found in refs. [ 6-8 ] in the case of interact- ing closed strings. In the Veneziano limit G = 0 (no closed-string interaction) all derivatives disappear and the renormalized string susceptibili tyf(A, M, F) can be found as a solution of a fifth degree algebraic equation.

The initial conditions for eq. (4) are partially fixed by the perturbative expansion in G. If we replace A by the dimensionless combinat ion X=AG -2/5, then the perturbative regime is reached in the limit X--, ~ . At finite X however, the solutions of (4) may contain nonperturbative terms that have zero Taylor expan- sion at infinity. The complete fixing of the boundary conditions remains an open problem, as in the case of interacting closed strings [ 6-8 ].

Let us also note that eq. (4) can be easily general- ized to the multicritical models of random surfaces proposed by Kazakov [10]. The renormalized pa- rameters of the theory scale for the m-critical model as

A ~ p " A , M-~pM, G-~p2m+lG , F~p'n+l/2F. (5)

These scaling relations make doubtful the interpre- tation of these multicritical points as string theories

in D = 1 - 6 /m (m + 1 ) embedding dimensions when m > 2. Indeed, the exact solutions of the string field theory in the Veneziano limit in D = 0 [9] and D = - 2 , D = 1 [ 11 ] embedding dimensions imply that A scales as M 2 and perhaps this is the case for any D.

In what follows we present the actual calculations based on a particular discretization of the path inte- gral of the string.

The world-sheet of the string is discretized by a col- lection of plaquettes (elementary squares) glued pairwise along their edges. The length a o f the edges gives the cut-off in our model. The free edges form the boundaries.

A discretized surface is characterized by its area A = # plaquettes, length of the boundary L = ~ free edges, as well a the number h of connected compo- nents o f the boundary and the number g o f handles. The genus g is given by the Euler formula

2 - 2 g - h = ¢~ p laque t tes - # edges + ~¢ vertices. ( 6 )

The partition function of the model reads

F= ~ (e2)g--'Th2AltL, (7) discrele connected surfaces

where ~2 and ~ are the bare couplings for closed and open strings, 2 is the bare cosmological constant and ~t is the bare mass at the ends of the string.

Each discrete surface is dual to a planar graph ~ with two kinds of vertices: 4-coordinated vertices dual to plaquettes and an L-coordinated vertex for each connected boundary of length L (fig. 1 ). The parti- tion function (7) is equal to the sum of all connected vacuum Feynman diagrams composed according to the Feynman rules given in fig. 2. Therefore the RHS of eq. (7) can be evaluated as the perturbative piece of the vacuum energy of a zero-dimensional N × N hermitean matrix field q~gj; i , j= 1, 2 ..... N = 2 ~ -

eg= d~ exp - -

V(0 ) = 102-- 104-~- y ln( 1 - /~202 ) . (9)

Knowing the result for all e=2/N, NeN, one can re-

~1 By planar graph we understand a graph with the structure of a two-dimensional simplicial complex. This structure is im- plied if we imagine the lines of the graph as thin strips.

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i

Fig. 1. A piece of the discretized world sheet containing one hole and the planar graph dual to it.

g}' + &

Fig. 2. Feynman rules for the planar graphs dual to discretizations.

produce the function F in the vicinity of the point e=O.

Due to the U ( N ) symmetry of the integrand the measure dO involves only integration over the eigen- values ~o~, ..., ~ON of the matrix 0

( 2 g ) N ( N - 1 )/2 f lYl ( 1 ) e F= , ~ dq~i exp - - V(q~,)

1 !2!...N~. i=l e

Xd2(~o), (10)

where

N

3(fp) = 1-I (~0i--~0j) . ( 1 1 ) t < J

The integral (10) can be evaluated by the method of orthogonal polynomials [ 12 ]. Let us introduce an infinite set of polynomials P~(q~); n = 1, 2, ..., orthon- ormal with respect to the integration measure d~0 exp[ - (1 /2g) V(~o) ]

f ( I v ( ~ ° ) ) P'(~)P'(~) ( m l n ) =- d~0exp -

(The fact that the measure is not integrable is out of relevance for the formal manipulat ions that follow. )

The determinant ( 1 1 ) is proport ional to the anti- symmetr ized product of Po, ..., P x -

d(q0 = d e t ( ~ - ' ) =e~/Zdet(Pj_l (~oi) ) . (13)

Therefore it is convenient to use the language of quantum mechanics and consider 5%(~01,...,~0N)= A (q~) exp [ -- ( 1/2e) Z~ V(~0~) ] as the ground state of a system of N Fermi particles in the q~-space.

The U(N) - inva r i an t part of the random matrix 0 is given by the coordinate operator ~b: f (~0) -~0f (~) . This operator is represented in the basis of polyno- mials (12) by a symmetr ic tridiagonal matrix

(ml~ln)=,,/~,~,~,,,+~ +~/~c~,,,m+l. (14)

The averages in the discrete string can be expressed in terms of the nontrivial matrix elements Rn. For ex- ample, the string susceptibility Z= ;32F/c322 is equal to the connected correlator of two operators N tr 0 4. A simpler quanti ty with the same scaling behaviour is the connected correlator of two N t r 0 2

a2F 02~_ ~ ( N t r 0 2 N t r 0 2) . . . . .

= Y~ <7"olN®~b21~><~lN@~b2l~o> ~ ~ o i i

= ~ N2(nl~21k>(kl(~21n> n<~N k > N

=N2RN+I(RN+RN+2) . (15)

Therefore the singular part of the string susceptibility near the critical point is that of the function RN(2,/t, Y).

The matrix elements of the operator ~b are deter- mined by the "equat ions of mot ion" [ 12 ]

(nl(oV'(~)ln)=(2n+l)e, n = 0 , 1 , 2 .... ,N, .... (16)

These equations are stationarity conditions for the functional

W[~b]= ~ t r V ( ~ b ) - ~ nln(nl(oln-l>. (17) n = O

The explicit form of the equations of motion (18) for our potential (9) is

=6ran. (12)

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e(n+ ½ ) = ½R.(1 -R, ,_I -R, , -Rn+, )

+ ½R,+I ( 1 - R n -Rn+l -Rn+2 )

1 + y - 7 ( n l 1 _ / / 2 ~ 2 In) , n=0 , 1 ..... N. (18)

The last term is the diagonal element of the propaga- tion kernel of a random motion on the lattice Z with the coordinate depending hopping parameter/~x/FR~. In general its evaluation represents a formidable combinatorial problem. However, in the scaling limit that we are going to take below, the mass of this ran- dom particle tends to zero and the effects of the lat- tice disappear.

The partition function F(2) makes sense up to some critical value 2¢. At 2=2c the entropy due to the fluctuations of the geometry of the world-sheet com- pensates the exponential factor and the partition function diverges due to the contribution of surfaces with infinite area. When 2 approaches 2¢, the parti- tion function is dominated by surfaces of area A ~ (2c -2 ) -1. The critical coupling 2c is the same for surfaces with any topology. Therefore it can be found by taking the large N limit of eq. ( 18 ) for 7 = 0, n = N:

2 = R - 3 R 2, R = l i m Rn. (19) N~o~

The solution of this equation gives the singular be- haviour of the string susceptibility of surfaces with spherical topology

g2F,, (j.) ,~, (/~¢ __~) 1/2 , /~¢ = h • (20) e=y=O

In order to find the critical singularity in Ft, let us al- low y> 0 but keep N--* ~ . Eq. ( 18 ) degenerates again to an algebraic equation for R = limN~o~Rs [ 10 ]

2 = R - 3 R 2 + 7 - 7 ( 1 - 4 # 2 R 2)-1/2 (21)

Keeping 2 close to 2¢ and increasing# we create a sec- ond singularity at ~t = (2R¢) - 1. This singularity is due to surfaces with infinite boundary; the length of the boundary diverges as L~(Ztc-/~) - l , where /z¢= (2R¢) -l , R¢=R (2~) =~. The term due to the bound- aries in (21 ) diverges when g--,/G therefore the bare coupling 7 should tend to zero in the scaling limit.

Below we shall find as a preliminary exercise the partition function of the open Veneziano string in the continuum limit, thus completing the analysis of Kazakov [9 ].

Near the critical point 2=2o, #=/zc, y=0 eq. (21) takes the form

2c - 2 = 3(Rc - R ) 2 + ? [ (/x~ _p2) + (Rc - R ) ] -1/2 (22)

The double scaling limit 2~2¢,/z-/z¢ implies that all three terms of the equation are of the same order of magnitude. This is achieved if we approach the criti- cal point along the trajectory

2 ¢ - 2 = a 2 A , ~=aS/2F,

, u ~ - # 2 = a M , R c - R = a f , (23)

where the parameter a makes sense of an UV cutoff. The equation for the cutoff-independent part f o f the string susceptibility is an algebraical equation of fifth degree

A = 3 f 2 + F ( M + f ) - ' /2 . (24)

The function f (A , M, F) is related to the partition function (7) by

3F = a _ 4 OF (25) e - 2a f= ~ 5 OA---5.

Note that i fe~ a 5/2, then the partition function does not depend on the cutoff a. Later we shall see that this is indeed the case as is expected in a theory of gravity.

The characteristic area of the surface scales as the square of the length of its boundary. The exact for- mulae for the mean area, length and number of holes are, correspondingly,

(A ) =a22 ~--~21ogf~- ~--~logf

=f[ ½F(M+J) --3/2 -- 6jq --1, ( 2 6 a )

0 ( L ) = a/z log f ~ - ~ logf

= F ( M + f ) - 3/2 (A) , (26b)

0 logf=Fo@log f

= 2 ( M + f ) - I / 2 ( A ) . (26c)

These mean values are finite unless

12f(M+f) 3/2F> 1 . (27)

Along the critical line where the inequality (27) is

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saturated, the mean area of the surface diverges but the density of holes ( h ) (A) -1 and the perimeter of an individual hole ( L ) ( h ) - ~ remain finite.

At this point we terminate the discussion of the open string in the Veneziano limit and go to the gen- eral case of interacting closed and open string. In or- der to incorporate the closed-string interaction we have to find the solution of the "equations of mo- tion" ( 18 ) as a function of the small parameter e = 2 / N. Proceeding in the same way as in refs. [6-8 ] we define an analytic function R(x) such that R,,= R(e(n- ½ ) ). Then 95 can be written as an infinite-or- der differential operator acting on the functions o fx

+ t C - - 952=I~exp(-+~-e~'l R~)exp(' Ox) -' 00)]2

=4,,.,[ l ,.,

Inserting this in eq. (19) for n = N a n d neglecting the derivatives of order higher than 2 (this will be justi- fied below) we get

02 2 = R - 3 R 2 - e 2 ~ R

- y ( R l { 1 - 4 / t e R [ l + e 2 ( 0 / a R ) 2 ] } - ' l R ) e . (29)

Next we consider the vicinity of the critical point and change the variable 2 as well as ¢t and y according to eq. (23).

We have up to terms involving higher derivatives

a2A= 3 a 2 f 2 - e e a - 302f/ OA 2

+eS/2F( Ai [aM +af_e2a-4( O/OA )2 ] -i iA )ea-2. (30)

The last term on the RHS can be written as eS/2Fe-lae(A ] [e-2aS(M+f)_ (0/0/1)2]-1 IA).

Therefore the dependence on the cutoff a disappears if we introduce a normalized coupling for closed- string interaction

e2=aSG. (31)

Then all higher derivatives disappear in the limit a - ,0 (each derivative in A is multiplied by a factor e/a 2 = Gx/a) and we arrive at the renormalized equation (4b) with

R(A)= (AI (M+f--G02) -~ fA) . (32)

The resolvent R (A) satisfies the second order differ- ential equation (4a) [13]. Eq. (4) can be inter- preted as the stationarity condition for the functional

W=f dA[JG(Of/OA)2+f3-Ftrln(M+f-GO2f)]

(33)

We intend to analyze eq. (4) in a future publication. Here we only mention that the double-pole singular- itiesf(A ) ~ 2 (A - A * ) -2 of the Painlev6 equation for the closed string [6-8] are allowed by our equation as well. Suppose that there exists a surface of such singularities A* =A (M, F, R). Near the singular sur- face the mean area of the world-sheet will diverge

(A) =a2 ~j Inf=- O l n f ~ ( A - A * ) - ' , (34)

while the densities of the holes and handles will re- main finite

OA* Oil* ( g ) ( A ) - ' - - ( h ) ( A ) - l = - . (35) OR ' OF

We see that the singularity at A* is produced by a kind of phase transition where the renormalized area of the surface blows up. We expect that in the limit G-*0 this singularity will match the singularity of the open string in the Veneziano limit located by eq. (27). The limit A-,A* for the string is the analog of the massless limit for the random particle where the length of its world-line blows up.

In conclusion, let us present the obvious generali- zation of the scaling law (3) for a field theory of closed and open strings in D dimensions ( - o r < D < 1)

F(A, M, G, F)

= F(1)2A, pM, p 2 ( 2 - vstr)G, p 2 - YstrF) ( 36 )

where 7s,~ is the string susceptibility exponent [ 1-3 ]. The exponent y string has been calculated within the conformal-field-theory approach in ref. [ 14 ]

D - l - x / ( D - 1 ) ( D - 2 5 ) ~)str ~" 12 (37)

Eq. (37) has been also checked for various discrete models of matter with central charge C=D coupled to gravity. For a recent review see Kostov [ 5 ].

One can interpret eq. (36) by saying that a hole on

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the wor ld sheet o f the s tr ing has scaling d i m e n s i o n

2zJ=Yst r and that a hand le is c o m p o s e d out o f two

holes. Even though the source ~Uofholes is a non loca l

operator , it is not t rans la t ion invar iant . There fo re the

in te rac t ing str ing is o b t a i n e d f rom the free one by

add ing a source t e r m F f ~ d ( v o l u m e ) to the act ion.

T h e coupl ing cons tan t F has scal ing d i m e n s i o n

2 (1 - - z / )=2 - -Ys t r . T h e d i m e n s i o n o f G ~ F 2 is

4 ( 2 - y s t r ) , and eq. (36 ) follows.

Par t o f this work has been done dur ing the vis i t o f

the au tho r at SISSA, Tries te . The a u t h o r thanks V.

K a z a k o v for sending h i m ref. [6] and ref. [9] p r io r

to pub l i ca t ion and J.-B. Zube r for a cr i t ical r ead ing

o f the manusc r ip t .

References

[ 1 ] F. David, Nucl. Phys. B 257 (1985) 45, 543; V. Kazakov, Phys. Lett. B 150 (1985) 28; J. Ambjorn, B. Durhuus and J. FriShlich, Nucl. Phys. B 257 (1985) 433.

[2] v. Kazakov, I. Kostov and A. Migdal, Phys. Lett. B 157 (1985) 295.

[3] D. Boulatov, V. Kazakov, 1. Kostov and A. Migdal, Nucl. Phys. B 275 (1986) 641; Phys. Lett. B 174 (1986) 87.

[ 4 ] V. Kazakov and A. Migdal, Nucl. Phys. B 311 ( 1988 ) 171 ; I. Kostov, Phys. Lett. B 215 (1988) 499.

[5] I. Kostov and M. Mehta, Phys. Len. B 189 (1987) 118; I. Kostov, Nucl. Phys. B (Proc. Suppl. ) 10 A (1989) 295.

[6] E. Brezin and V. Kazakov, preprint Ecole Normale ENS- LPS 175 (October 1989).

[7]D. Gross and A. Migdal, Princeton University preprint PUPT-1148 (October 1989).

[ 8 ] M. Douglas and S. Shenker, Rutgers University preprint RU- 8>9-34 (October 1989).

[ 9 ] V. Kazakov, Ecole Normale Sup6rieure preprint ( 1989 ). [ 10] V. Kazakov, Niels Bohr Institute preprint 89-25 (1989). [ 11 ] D. Boulatov, Cybernetics Council preprint, Moscow (1989). [ 12 ] C. Itzykson and J.-B. Zuber, J. Math. Phys. 21 ( 1980) 411;

D. Bessis, C. Itzykson and J.B. Zuber, Adv. Appl. Math. 1 (1980) 109.

[13] I. Gel'land and L. Dikii, Usp. Mat. Nauk XXX, No. 5 (1975) 67.

[14] V. Knizhnik, A. Polyakov and A. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819; F. David, Mod. Phys. Lett. A (1988) 1651.

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