Two-Matrix String Model as Constrained (2+1)-Dimensional Integrable System

Abstract

We show that the 2-matrix string model corresponds to a coupled system of $2+1$-dimensional KP and modified KP ($\KPm$) integrable equations subject to a specific ``symmetry'' constraint. The latter together with the Miura-Konopelchenko map for $\KPm$ are the continuum incarnation of the matrix string equation. The $\KPm$ Miura and B\"{a}cklund transformations are natural consequences of the underlying lattice structure. The constrained $\KPm$ system is equivalent to a $1+1$-dimensional generalized KP-KdV hierarchy related to graded ${\bf SL(3,1)}$. We provide an explicit representation of this hierarchy, including the associated ${\bf W(2,1)}$-algebra of the second Hamiltonian structure, in terms of free currents.