Reduction of Toda Lattice Hierarchy to Generalized KdV Hierarchies and Two-Matrix Model

TL;DR

This paper develops a unified formalism connecting Toda lattice hierarchies with generalized KdV hierarchies, providing explicit algebraic structures, free-field representations, and demonstrating their relation to two-matrix models.

Contribution

It introduces a unified approach to derive generalized KdV hierarchies from Toda lattices using free currents and describes their algebraic and matrix representations.

Findings

Unified formalism for reduced $SL(M+1,M-k)$-KdV hierarchies.

Explicit free-field representations of $W(M,M-k)$ Poisson algebras.

Equivalence of two-matrix string model to $SL(M+1,1)$-KdV hierarchy.

Abstract

Toda lattice hierarchy and the associated matrix formulation of the $2M$-boson KP hierarchies provide a framework for the Drinfeld-Sokolov reduction scheme realized through Hamiltonian action within the second KP Poisson bracket. By working with free currents, which abelianize the second KP Hamiltonian structure, we are able to obtain an unified formalism for the reduced $SL(M+1,M-k)$-KdV hierarchies interpolating between the ordinary KP and KdV hierarchies. The corresponding Lax operators are given as superdeterminants of graded $SL (M+1,M-k)$ matrices in the diagonal gauge and we describe their bracket structure and field content. In particular, we provide explicit free-field representations of the associated $W(M,M-k)$ Poisson bracket algebras generalizing the familiar nonlinear $W_{M+1}$-algebra. Discrete B\"{a}cklund transformations for $SL(M+1,M-k)$-KdV are generated naturally from lattice translations in the underlying Toda-like hierarchy. As an application we demonstrate the equivalence of the two-matrix string model to the $SL (M+1,1)$-KdV hierarchy.