THE STRING DIFFERENCE EQUATION OF THE D=1 MATRIX MODEL AND W1+∞ SYMMETRY OF THE KP HIERARCHY

Abstract

We give a connection between the D=1 matrix model and the generalized KP hierarchy. First, we find a difference equation satisfied by F, the Legendre transformation of the free energy of the D=1 matrix model on a circle of radius R. Then we show that it is a special case of the difference equation of the generalized KP hierarchy with its zero mode identified with the scaling variable of the D=1 string theory. We propose that the massive D=1 matrix model is described by the generalized KP hierarchy, which implies the manifest integrability of D=1 string theory. We also show that the (generalized) KP hierarchy has an underlying W1+∞ symmetry. By reduction, we prove that the generalized KdV hierarchy has a subalgebra of the above symmetry which again forms a W1+∞. We argue that there are no W constraints in D=1 string theory, which is in contrast to D<1 theories, where there are W1+∞ constraints.