Constrained KP Hierarchies: Additional Symmetries, Darboux-B\"{a}cklund Solutions and Relations to Multi-Matrix Models

TL;DR

This paper explores the structure of constrained KP hierarchies, their symmetries, solutions via Darboux-Bäcklund transformations, and their connection to multi-matrix models, advancing understanding of integrable systems and matrix model relations.

Contribution

It introduces a systematic framework for constrained KP hierarchies, their symmetries, and solutions, linking them explicitly to multi-matrix models and generalized Toda lattice structures.

Findings

Darboux-Bäcklund symmetry generates tau-function orbits as Wronskians.

Any DB orbit of cKP_{r,1} defines a generalized 2D Toda lattice.

Explicit relationship established between truncated KP hierarchies and multi-matrix models.

Abstract

This paper provides a systematic description of the interplay between a specific class of reductions denoted as \cKPrm ($r,m \geq 1$) of the primary continuum integrable system -- the Kadomtsev-Petviashvili ({\sf KP}) hierarchy and discrete multi-matrix models. The relevant integrable \cKPrm structure is a generalization of the familiar $r$-reduction of the full {\sf KP} hierarchy to the $SL(r)$ generalized KdV hierarchy ${\sf cKP}_{r,0}$. The important feature of \cKPrm hierarchies is the presence of a discrete symmetry structure generated by successive Darboux-B\"{a}cklund (DB) transformations. This symmetry allows for expressing the relevant tau-functions as Wronskians within a formalism which realizes the tau-functions as DB orbits of simple initial solutions. In particular, it is shown that any DB orbit of a ${\sf cKP}_{r,1}$ defines a generalized 2-dimensional Toda lattice structure. Furthermore, we consider the class of truncated {\sf KP} hierarchies ({\sl i.e.}, those defined via Wilson-Sato dressing operator with a finite truncated pseudo-differential series) and establish explicitly their close relationship with DB orbits of \cKPrm hierarchies. This construction is relevant for finding partition functions of the discrete multi-matrix models. The next important step involves the reformulation of the familiar non-isospectral additional symmetries of the full {\sf KP} hierarchy so that their action on \cKPrm hierarchies becomes consistent with the constraints of the reduction. Moreover, we show that the correct modified additional symmetries are compatible with the discrete DB symmetry on the \cKPrm DB orbits. The above technical arsenal is subsequently applied to obtain complete