Volume 255, number 2 PHYSICS LETTERS B 7 February 199 1

Two-dimensional gravity, string field theory and spin glasses

I.Ya. Aref’eva and I.V. Volovich Steklov Mathematical Institute, Vavilov 42, GSP-I, SU-I 17 966 Moscow, USSR

Received 5 July 1990

It is noted that an auxiliary matrix model which has appeared in recent works on non-perturbative two-dimensional gravity can

be considered as a lattice version of Witten’s string field theory. We consider the string field theory on a D-dimensional lattice and

obtain that for the simplest interaction it is equivalent to a scalar matrix field model on the lattice with local gauge invariance.

We investigate the matrix model in the large N limit using the quenched prescription for D> 1 and argue the existence of a spin-

glass-like phase in the string theory.

Recently, some interesting results have been ob- tained on the exact solutions for non-perturbative two-dimensional gravity [ 11. In there, the discreti- zation of the string by random graphs [2] and l/N expansion of the random matrix model [ 3 ] have been used.

It is important to note that the sum over surfaces with arbitrary genus was reduced to a model involv- ing an auxiliary NX N matrix BiP As is well known in string theory there are two approaches: the first quantization in which the summation over different topologies is considered, hence it is intrinsically per- turbative, and the second quantization approach, i.e. string field theory (SFT), which is formulated in a non-perturbative way. A coupling constant expan- sion in SFT corresponds to summation over different topologies in the first quantization approach. There- fore there appears a natural question: what is the meaning of the auxiliary NXN matrix @i, from the viewpoint of the string theory?

In this paper we point out that this matrix Dij is nothing but a lattice version of the string field @[x(a) ] in Witten’s formulation of the open string field theory.

Let us show how the Witten open string field the- ory [4] leads to a matrix random model. In covari- ant SFT one deals with a string field @[X((T) ] which is a functional of x( a) as well as the ghost fields. Here the string x( a) is an image of the interval [ 0, rc] into D-dimensional space-time [RD. The string fields form

an algebra with the associative multiplication * and with the integral J which satisfy the conditions

(i) (A*B)*C=A*(B*C),

(ii) J A*B= B*A. j

These operations are defined in terms of string over- laps. Each string has a preferred point, the “mid- point”. The midpoint divides a string x into left and right halves (x,, xn ). The product x*Y of two strings x and y is zero unless xR coincides with y, in space- time, and if so is proportional to (x,, yn). Hence the string field @[xl is considered as a functional @ [ (.q, xR) ] and one has

= @[(XL>YR)l~](YR>XR)l 9YR(O) > s (1)

s s @= @I (xl_, X‘) 1 g&(a) . (2)

The string action has the Chern-Simons form [ 41:

S= j @*Q@+&j @*@*a, (3)

or the @ ‘-interaction [ 5 ] :

S=$g @*CD*@. I (4)

Elsevier Science Publishers B.V. (North-Holland) 197

Volume 255, number 2 PHYSICS LETTERS B 7 February 1991

The action (4) is invariant under gauge trans- course we can introduce some parameter (T to de- formations: scribe the string (8 ) .

@+V*@*V-‘. (5)

The string theory needs a regularization. Usually one uses the discretization of the parameter 0, i.e. o,=ik/2n, i=O, 1, . . . . 2n and denotes as xj=x(ai).

This regularization was considered in ref. [ 61. The string field @[x(a) ] is a function of the 2n coordi- nates, @(x0, . . . . xn_], x,, xn+,, . . . . xZn). The product now has the form

Note that we assume that the string yX has no dis- continuities. The representation (7) means that the

string yX is described by the initial point x and k di- rections of its links

(8)

(@* V(xo, .**, x*,)

Let I,( I) be the space of all strings on the lattice ZD of length I with start in the point XEZ~. It is clear that I,( I) and F,( 1) are isomorphic for any x, YE ZD, and have a finite number of, say N, elements, which can be enumerated in some way.

For our purpose it is convenient to imagine the string of length 21~ 2ka, = s @(x0, . ..) x,-l~xn,Yn-l,-, Yo>

XY/(yo,...,yn-,,x,,x,+~,...,xz,)d~~o...d~yn-,.

(6)

Rule (6) looks like but is not yet the matrix multipli- cation law. To obtain a matrix let us introduce a lat- tice ZD in IRD.

We will work with a hypercubical lattice ZD in [RD with lattice spacing a. A general point in Z* is of the form

x=(x’, . . . . xd-‘) , P=anp, p=O, . . . . d- 1 ,

n integer. Furthermore let e, be the vector of length a in direction ,u. Pairs (x, y) of nearest neighbor lat- tice points x, ycZD are called links. Any link can be given by a point x and a direction ,u, so the corre- sponding pair has the form (x, x+e,).

String is a contour on the lattice. We shall consider a string of length 1 for some fixed I= ka, k integer. To specify the string configuration it is necessary to specify the initial point, say x, as well as links along which the string lies. This can be done if we denote the string links as

(

k-1

(x>iu,)> (X+e,,,~2),...> x+ is, e,,,pk . >

(7)

Eq. (7) leads to a trivial but very important obser- vation. In the continuum case in the string theory, we deal with a path, ~~(a), i.e. a map [0, n]-+iRD, and with path functionals, in contrast with loops and functionals on loops in the contour formulation of Yang-Mills theory [ 7,8 1. Considering the set ( 8 ) we do not get the path but the contour on the lattice. Of

198

&(21)= (

x- 5 eptjh,...rp2k r=l >

, (9)

to be composed of two “halves”. We will call the left half of the string the configuration

and the right half the configuration

yxR(21)=(x,~k+L,...,~Zk).

The point x will be called midpoint.

(11)

The space of right halves I,” (21) is isomorphic to F,(I) and its elements can be numerated in the same way as the elements of I,( 1). At the same time, the space F:“( 1) being the space of left halves, will be nu- merated as if it was the space of strings

(x, -fik, ...> -PI) . (12)

Below we shall use the following notation: any string of length 21 with midpoint x will be denoted as

Y=y(x, a, b) > (13)

where a specifies the right half of the string, i.e. the element of F,( I), and b numerates the left half of the string, i.e. again the element of F,( 1).

From the above description of the string on the lat- tice it is clear that a functional @[y] on the string y= y(x, a, 6) on the lattice is in fact a matrix function

@[r] =(Pnb(x,) , a, b= 1, . . . . N, XEZ~. (14)

Therefore we get a matrix realization of the Witten

Volume 255, number 2 PHYSICS LETTERS B 7 February I!?? 1

algebra where fields are matrices oab(x) depending gauge field interacting with matter fields for the gauge

on the parameter x, the product * is the matrix group SU (N), in the limit N-+co and the spacing of

product, the lattice going to zero.

(@* y?,,(x) = @a=(x) Ku,,(x) >

and the integral is given by

(15) An expression for the quadratic term in SFT with-

out ghosts has the form

@Y&-l)@, (20)

where Lo is the Virasoro generator. In the matrix re- alization it gives an expression

J CD= C tr Q(x) . (16) x

It is clear that the conditions (i) and (ii) are satisfied. We ignore the ghosts at the moment. The ghost de-

pendence would lead to new indices in the matrix field 0(x) and to an extra coordinate xD for the midpoint [we have in mind the bosonized representation of the (b, c) ghost system] _ Note that a matrix realization of the string field algebra including ghosts was con- sidered in ref. [ 9 1.

The action (4) now has the form

S=g 1 tr Q3(x) . X

(17)

It is invariant under local gauge transformations:

@(x)+U(x)@(x)U-‘(x) ) (18)

which are the matrix analogues of ( 5 ). Here V(x) is an Nx N matrix function on the lattice. So we arrive at the surprising conclusion that a lattice regulariza- tion of the string field theory with the action (3) is the scalar matrix theory on the D-dimensional lattice with the action (17) which is invariant under local gauge transformations ( 18). The gauge invariant correlation functions for this model have the form

1

z J es(@) tr cDp’(x,) . . . tr opk(xk) 9@. (19)

However, the above considerations are oversimpli- fied. In fact the presence of ghosts in the SFT is quite important for obtaining a gauge invariant theory be- cause it gives a nilpotent BRST operator Q. It was shown in ref. [ 5 ] that one can derive the action (3) from (4), but for this purpose one should have the BRST operator. We are not discussing carefully the ghost dependence in this paper, however, we will consider the quadratic term in the matrix action and therefore lose local gauge invariance. It seems that in order to preserve the local gauge invariance in the lat- tice formulation of string field theory, one should consider a Wilson-like lattice formulation for the

&=~a~-’ C C [%(x)@~nb(x+e,)

-@aab(X)@ba(X)l. (21)

Here is used the fact that the operator Lo is the oper- ator which moves the string as a whole.

This expression is reminiscent of the gauge invar-

iant kinetic operator (P@, (Lo - 1 )P@) in the Banks- Peskin formulation [ lo] of SFT, with P a projector on the states satisfying L_,@= 0 for n > 0. Note once again that in the case of the string on the lattice we deal with contours which are parametrization-invar- iant, hence in eq. (2 1) we do not have to use the an- alogue of the projector operator. In this context the action (2 1) resembles the Marchal-Ramond [ 111 string field action, where they considered the contour functionals as the string functionals. So the matrix action has the form

s=s, fgl/( 0) ) (22)

where V( @) is a local function depending on 0(x). To recover from the action (22) the usual Vene-

ziano or Koba-Nielsen amplitudes in the continuum limit (i.e. then a-0, ~-+Go, N+cx, and I is fixed) within the perturbative expansion on the coupling constant g, one has to introduce ghosts, in a similar way as has happened in the lattice formulation of gauge fields. A simpler way to incorporate ghosts is to work in a special gauge, namely, the Siegel gauge. In this gauge the kinetic operator contains only the Lz’ operator, Lip’ = Lo + fLfjh.

Now let us discuss the role of the dimension D in the lattice formulation of SFT. One can interpret the case D < 1 in the following sense. Let variables xi take values not on the lattice ZD but on some finite set, for example ( + 1). Such a model has a finite number of degrees of freedom and one can interpret it as having D-c 1. In the case D= 1 one can get in the continuum limit the lagrangian

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Volume 255, number 2 PHYSICS LETTERS B 7 February 1991

Y=tr d(x)‘+ V(Q) . (23)

This model was considered recently in ref. [ 12 1. From the above discussion it is clear that in the

same approximation as (23) one can consider the following lagrangian for D> 1:

U=i tr[a,@(x)12+ V(Q) , (24)

where 0(x) is the NX N matrix model field in D-di- mensional space-time. One does not have local gauge invariance for the lagrangian (22) and (23) but only global invariance and relativistic invariance. To re- ceive a local gauge invariance one needs to consider SU (N) Yang-Mills fields as was noted above.

Note that we are discussing the open string. David [ 2 ] has obtained the following D-dimensional la- grangian for the closed bosonic string:

L=jtr(@ee-d@)+Ltr@3. fi

(25)

It is amusing to note that the same lagrangian (25) was obtained in the context of the p-adic string (for N= 1, p = 2). It has a soliton (instanton) solution for anyD [13].

The problem of constructing the 1 /N expansion of the theory (24) was considered a long time ago. In ref. [ 141 it was shown that the evaluation of the vac- uum energy B in the leading order of the 1 /N expan- sion is equivalent to the calculation of the free energy in the quenched theory with the action

2J% S(P, @) = - A F (Pf-P?)*@i,@,i+ I’(@), (26)

and subsequent integration over all values of the quenched momenta pL, i.e.

(27)

Here I/ is the volume of the system,

’

/i is the ultraviolet cutoff (lattice spacing). As was noted in ref. [ 151 this means that one is dealing with the theory of disordered (amorphous) systems, like for instance spin glasses. It was shown in ref. [ 15 1,

200

using the replica trick [ 161, that a new phase can ap- pear for this system when the parameter

q= s W(p) N2 tr((@>s) ,

CD>

’ = I@exp[-W, 011 d@

Iexp[--Sk 011 d@

is different from zero, but p defined by

p= d&) N-‘I2 tr( O>p

is equal to zero. This phase is an analogue of the spin glass phase. This was obtained for the potential

V(@)=tm2trQ2+$trQ4,

where the renormalized coupling constant g depends on D in the following way:

g= - m2/2 exp( -2/D) , m2-c0.

We have considered in this note the non-perturba- tive approach to string field theory in D-dimensional space-time (D> 1 ), and found an indication of the existence of a new spin-glass-like phase in this theory. A natural next step would be the consideration of the Parisi replica symmetry breaking solution and the ul-

trametric topology [ 17 1.

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