Volume 255, number 2 PHYSICS LETTERS B 7 February 199 1
Two-dimensional gravity, string field theory and spin glasses
I.Ya. Arefeva 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 Wittens 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  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 Wittens formulation of the open string field theory.
Let us show how the Witten open string field the- ory  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
(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 ) .
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
(@* 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>
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
(x>iu,)> (X+e,,,~2),...> x+ is, e,,,pk . >
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
x- 5 eptjh,...rp2k r=l >
to be composed of two halves. We will call the left half of the string the configuration
and the right half the configuration
The point x will be called midpoint.
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
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
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
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,)
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 operato