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

TL;DR

This paper demonstrates that the two-matrix string model is equivalent to a constrained 2+1-dimensional integrable system involving KP and modified KP equations, revealing new algebraic structures and hierarchies.

Contribution

It establishes a novel connection between the two-matrix string model and a constrained 2+1D integrable system, including explicit hierarchy and algebra representations.

Findings

Equivalence of the matrix string model to a constrained 2+1D KP system.

Explicit representation of the hierarchy and associated algebra.

Identification of the generalized KP-KdV hierarchy related to graded SL(3,1).

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.