13 February 1997

ELSEVIER

PHYSICS LETTERS B

Physics Letters B 393 (1997) 316-320

Combinatorics of solitons in noncritical string theory Masafumi Fukuma G’, Shigeaki Yahikozawa b,2

a Yukawa Institute for Theoretical Physics, Kyoio University, Kyoto 606-01, Japan h Department of Physics, Kyoto University, Kyoto 606-01, Japan

Received 5 November 1996 Editor: M. Dine

Abstract

We study the combinatorics of solitons in D < 2 (or c < 1) string theory. The weights in the summation over multi-

solitons are shown to be automatically determined if we further require that the partition function with soliton background be a 7 function of the KP hierarchy, in addition to the WI+, constraint.

PACS: I 1.25.Pm; I 1.25.Sq Keywords: Noncritical string theory; Nonperturbative techniques; Soliton; Schwinger-Dyson equation; Combinatorics

One of the clues to understanding nonperturbative behavior of strings is that nonperturbative effects in string theory generally have the form e-Co”St/R, where g is the closed string coupling constant [ 11. In fact, these effects naturally appear in D-brane theory [ 2,3] and M-theory [4], and play important roles in their dynamics. However, the origin of such dependence is still not understood well because we do not have the self-contained closed string field theory such that both elementary and solitonic excitations of strings can be treated in a systematic manner. Therefore, it should be of great importance to investigate these nonpertur- bative effects in simpler models such as D < 2 string theory, which can be explicitly constructed and also exactly solved as the double scaling limits of matrix models [ 51.

In our previous paper [ 61, we explicitly constructed soliton operators in the Schwinger-Dyson equation approach to D < 2 (or c < 1) string theory, and

’ E-mail: fukuma@yukawa.kyoto-u.ac.jp. ’ E-mail: yahiko@gauge.scphys.kyoto-u.ac.jp.

investigated the nonperturbative effects due to these solitons. Furthermore, we suggested that fermions should be regarded as the fundamental dynamical variables in (noncritical) string theory, since both ele- mentary strings (macroscopic loops) and solitons are constructed in their bilinear forms. The purpose of the present letter is to study the combinatorics of solitons in noncritical string theory, namely, how to determine the weights in the summation over multi-solitons.

We start our discussion with reinvestigating the cel- ebrated string equations [ 51. For pure gravity (D = 1 (or c = 0) string), it is represented as the PainlevC I equation:

4u(t,g)2+ $3;.(t.g) =t, (1)

where u( t, g) is the connected two-point function3 of cosmological terms (31, U( t,g> = ( 0101 )C, and t

’ In matrix models with even potentials, there arises the “doubling phenomenon” and the connected two-point function would be identified with f(t,g) = 2u(t,g).

0370-2693/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved. PI1 SO370-2693(96)01642-S

M. Fukuma, S. Yahikozawa/Physics Letters B 393 (1997) 316-320 317

and g are, respectively, the (renormalized) cosmolog- ical and string coupling constant. The solution of this equation has the following form of asymptotic genus expansion:

(2) h=O

The nonperturbative corrections [ 5,7,1] to this asymp- totic solution, Au(t,g) = u(t,g) - Upert(t,g), can be evaluated by expanding the string equation around rlpert( t, g). In fact, if we first make a linear approx- imation and also take into account only the leading

term in g, then we obtain AU N &t- 118 e-4&t514/ 5g

Then treating nonlinear terms in the string equation as perturbation, we can calculate higher corrections as 4

where the coefficients a, satisfy the recursion equation (n2 - 1) a, + c;,’ ak an-k = 0, and are solved to be

CE,, = (-g-’ nun. (4)

The first coefficient al = a is one of the constants of integration for the string equation5 and remains un- determined. This parameter is analogous to the theta parameter in QCD which cannot be seen in pertur- bation theory. Thus, this simple analysis shows that, once 1-soliton (n = 1 case in the above equation) ex- ists, namely, a does not vanish, then a series of multi- solitons also must exist and be summed up with def- inite weights, so that the solution satisfies the string equation. In the generic unitary case, (p, 4) = (p, p + I), the same conclusion holds except that the expo- nential factors in Eq. (3) are generalized to the form exp (-n . a t’+“2J’/g), w h ere n resides on the domi- nant integral lattice 6 of some dimension (2 for p = 3 and 4 for p = 4, for example), and the vector (Y =

4 Note that fit- ‘is (and thus Pi4/g) is a dimensionless com- bination of g and t.

5 Another one is fixed by demanding the asymptotic form u$!L ( I) = -d/2. Here the minus sign is chosen so that Au( t, g)

takes a real value. h Here we consider only stable solitons with negative value of

exponents.

(W,LY2,**.) consists of the exponents of I-soliton (i.e. linearized) solutions.

In order to systematically study the above re- sult, we here briefly review our formulation given in Ref. [ 61. We first introduce complex [ plane on which live p pairs of free fermions c, (0, E, (5) (a = 0, l;.. , p - 1) . We then construct p bosons (P~ (5) through bosonization:

&G(5) =: G(S)%(C) :,

which in turn give the fermions as

(5)

E,(l) = Ku : e’+‘fl(5) :,

c,(c) = K, : e-IDacr) : . (6)

Here K, is the cocycle factor and ensures the correct anticommutation relations between different indices a # 6, and all operators are normal ordered with re- spect to SL(2, C) invariant vacuum (vat 1. The string field which describes elementary excitations of strings is identified with the macroscopic-loop operator q (1) [8] which creates a loop boundary of length 1. The Laplace transforms of their (generally disconnected) correlation functions are represented in terms of the boson &o( 5) as

(drpo(L.1) ...a+Jo(5lv)) co

= - s

dll ...dl~e-‘~~l-“‘-‘~~~ (*(II) . ..~(ZN)) 0

where the normal ordering again respects (vat I. In the above expression, the state ( -B/g / is defined by

(8)

where ( v 1 is the vacuum with Z,-twist operator a( l) inserted at the point of infinity, (g 1 = (vat 1 (T( ca) [9]. B( 5) is the background which characterizes the theory, and for the minimal (p, q) case it has the form B( 5) = c:T B, {“if’ with nonvanish- ing BP+,. The contour should surround 5 = 00 p times, since the bosons arp, (5) have the monodromy

318 M. Fukuma, S. Yahikozawa/Physics Letters B 393 (1997) 316-320

as ((~(ap,(e~“‘f) = (alJcpla+il(J’) with [a] 3 a (mod p). Furthermore, &a(J) has the mode ex- pansion under (CT 1 as

and thus ( -B/g ( can be rewritten as

P+Y

-(l/g)CB,a, . n=l

The Schwinger-Dyson equations [ lo-131 are com- pactly expressed by the requirement that the state ( @ ) is a decomposable state satisfying the WI+~ con- straint 7

W,kl@)=O (kbl,n3--k+l). (10)

Here the generators of the WI+~ algebra [ 151 are given by the mode expansion

Wk(l) = c w,“s-“-” IlEZ

In general, a state ( Q, ) is called decomposable if it is written as I@ ) = eH ig), where H is a bi- linear form of fermions and 1 CT) E u(O) I vat ). This is also equivalent to the statement that r(x) = (G I exp{Cz, x,cy,,} I @ ) is a r function of the RF’ hierarchy [ 161.

Connected correlation functions of macroscopic loops are obtained as the cumulants of the correlation functions above, and have the following dependence on the string coupling constant g:

c

=log( -+I : exp {f” &j(l)~rp0(J)l: j @) (-;I @>

=IEc O” g-2;+” p&p& h=o N=l

7 The equivalence between the Schwinger-Dyson equations and the Douglas equations [ 141 is proved in Ref. [ 13 1.

x j(ll>...j(h;N) (~(p0(5i>...a~~o(5~))~') . (12)

Soliton backgrounds are constructed by inserting the soliton operators [ 61

D rrh = (0 # b) (13)

into correlation functions, where the soliton fields &, (5) are defined by

&(0 = G(5) cb(j) (a + 6)

C

_ K& : ,%(~)-Pb(c) : (a < 6) =

+K,,K,, : eC%(4”-Ph(~) : (a > b) . (14)

As shown in Ref. [ 61, the soliton operators commute with the generators of the WI+~ algebra, [Wi, Do,,] = 0 (k 3 1, n E Z), and thus, if the state / @ ) satis- fies the WI+~ constraint, then the multi-soliton state

Da,/,! DL,zt? . . . / @ ) also satisfies the same constraint. Furthermore the nonperturbative effect from 1-soliton background was evaluated and found to have the form e -co”st /g [ 61. Nonperturbative effects for multi-soliton backgrounds are then roughly estimated by adding the exponents and found to construct the same lattice with the one described in the beginning of the present paper.

At first sight these multi-solitons could be summed in a completely arbitrary manner, since each multi- soliton state satisfies the WI+~ constraint. On the other hand, our analysis of the string equation shows that the weights in the summation are fixed except for a few undetermined parameters (see Eqs. (3) and (4) ). This paradox can be solved if we further re- quire that the multi-soliton state be decomposable. In fact, since D,b is a fermion bilinear, the multi-soliton states must have the following form if we impose this decomposability condition:

I Q’, t9) = n e8‘g6Dnb 10) , a+b

(15)

where I CD ) is also a decomposable state satisfying the WI+, constraint, and 8&, (a # b) are arbitrary con- stants. Thus, we find that the generating function for macroscopic loops with soliton backgrounds is gener- ally given by

C, H

M. Fukuma, S. Yahikozawa/Physics Letters B 393 (1997) 316-320 319

=log ( -3 : exp {j” ~j(5)~spo(O} : ( @, 0 ) ( -;I w) ’

(16)

where the parameters 6,b (a # b) remain undeter- mined.

We here make a comment that the nonperturbative effects from multi-solitons can be explicitly calculated for connected correlation functions of lower dimen- sional operators 01, 02 . . . , c?,+,. In fact, these con- nected correlation functions are obtained by taking derivatives with respect to the background sources as

(17)

and log{-B/g1 @,,0) can be evaluated at 6’ = 0 by rewriting it as follows:

b-q1 %+lo,(-ii @)

+ log (,&Lq , where

(18)

(19)

As an example, we consider the (p, q) = (2,3) case (pure gravity), setting the background as B1 = t, Bs = -15/8 and B, = 0 (n + 1, 5). Since contributions from Die can be absorbed into 801 when p = 2, we can restrict our consideration to the case 801 = 0 and 010 = 0. The quantity

(20)

is then calculated as follows. First we evaluate

I’ dl = f’ - exp {( e(Po(O-(Pl(5) _ 1 ),)

. 2Vl

’ dcJ =

f ~exPUcp0(Y) -n(5))

+i( (PO(l) -5p1m)2)c+-~}. (21)

All the leading contributions to the exponent come from spherical topology, and thus we have

+; ((soo(5) - Pd5))2)io) +0(g) . 1 (22)

The first term (denoted TOI (I) in Ref. [ 61) is calcu- lated by integrating the disk amplitude with respect to 5 and found to be [7,6]

(Po(5) - Pi(l) j(O) = 2 (5 - ifi) (r + &)3’2.

(23)

On the other hand, to compute the second term, we have to integrate two-point function (cylinder ampli- tude) twice. Explicit form of the cylinder amplitude can be found in Ref. [ 17,181, and we obtain*

(24)

where R is the infrared cutoff necessary to define the two-point function. In the weak coupling limit (g --+ +0), we can evaluate the integral by the saddle-point method at 5 = fi/2 (see [ 61 for the detailed investi- gation on this point), and find that

(DoI) =PR&t- 51s (?-4x.4 PI 5g+O(‘q) (25)

8 For the general (p, q) = (p, p + I ) case, we have

((Pn(ll) ‘f’b(l2i) ):“I

= log [ cd (a, + fi) ‘lp + co--” (i, - Jr:-t) lit,

-cd (l2 + fi)“” -w-b (l* - &yj

where o = e2*‘/P. This normalization corresponds to B, = t (> 0~.~2p+~=-~~/(p+l~(2p+1)and~,,=O(~~# 1,2p+l).

320 M. Fukuma, S. Yahikozawa/Physics Letters B 393 (1997) 316-320

Here /3 is a definite numerical constant. On the other hand, ( D:i ) vanishes for n 3 2. In fact, we encounter the following expression in the process of evaluation:

X rI 2 (l/g) cy=, ro,ci,,+oc!T”, (li-ii) e (26)

i>,;

which vanishes in the weak coupling limit, since in the p = 2 case we have a single saddle point9, 51 = 52 = . . = ln = d/2. Therefore, we obtain

log+%1 .,+=lo,(-;I Q)

+ log 1 + ,gR &t-518 e-4h’i’/5/: , 1 (27)

where Ba = j? R 0 is the renormalized theta parameter. Thus, taking derivatives with respect to the cosmolog- ical constant t, for example, we get

, + oR fit-S/8e-4&-?iJ/5g

>

-2

The first term represents the asymptotic solution for sphere, z$!Ji( t), and the second term exactly repro- duces Eqs. (3) and (4) with a = 6 8~. Recall that we have neglected contributions from higher topologies.

In conclusion, we have shown that we can obtain correct combinatorics of solitons in our formulation if we require that soliton background be expressed by a decomposable state, namely, the corresponding partition function is a r function of the KP hierarchy. For further study of the dynamics of these solitons, it would be convenient to construct the string field action in terms of fermions. The investigation in this direction is now in progress and will be reported elsewhere.

We would like to thank N. Ishibashi, A. Ishikawa, K. Itoh, H. Itoyama, H. Kawai, T. Kawano, and M. Ni- nomiya for useful discussions. This work is supported

‘) For p > 3, higher terms ( D$, ) (n 2 2) could survive since we have multi saddle points.

in part by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture.

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