Hamiltonian Structures of the Multi-Boson KP Hierarchies, Abelianization and Lattice Formulation

TL;DR

This paper introduces a new bosonic reduction of the KP hierarchy that generalizes classical transformations, revealing deep links between Hamiltonian structures, Miura transformations, and discrete integrable models.

Contribution

It extends the classical Miura transformation and Kupershmidt-Wilson theorem from mKdV to KP hierarchy, connecting Hamiltonian structures with lattice models and free field representations.

Findings

New bosonization form of multi-boson KP hierarchy

Relationship between Hamiltonian structures and lattice models

Representation of nonlinear w-infinity algebra in free fields

Abstract

We present a new form of the multi-boson reduction of KP hierarchy with Lax operator written in terms of boson fields abelianizing the second Hamiltonian structure. This extends the classical Miura transformation and the Kupershmidt-Wilson theorem from the (m)KdV to the KP case. A remarkable relationship is uncovered between the higher Hamiltonian structures and the corresponding Miura transformations of KP hierarchy, on one hand, and the discrete integrable models living on {\em refinements} of the original lattice connected with the underlying multi-matrix models, on the other hand. For the second KP Hamiltonian structure, worked out in details, this amounts to finding a series of representations of the nonlinear $\hWinf$ algebra in terms of arbitrary finite number of canonical pairs of free fields.