Virasoro Symmetry of Constrained KP Hierarchies

TL;DR

This paper develops a framework for Virasoro symmetry in constrained KP hierarchies, clarifying compatibility issues and linking to matrix models and Toda-like structures.

Contribution

It explicitly formulates Virasoro symmetries for constrained KP hierarchies and addresses their compatibility with constraints, providing new insights into integrable systems and matrix models.

Findings

Established conditions for commutativity of symmetries and Darboux-Bäcklund transformations.

Derived a new approach to the string-equation constraint in matrix models.

Linked constrained KP hierarchies to Toda-lattice-like structures.

Abstract

Additional non-isospectral symmetries are formulated for the constrained Kadomtsev-Petviashvili (\cKP) integrable hierarchies. The problem of compatibility of additional symmetries with the underlying constraints is solved explicitly for the Virasoro part of the additional symmetry through appropriate modification of the standard additional-symmetry flows for the general (unconstrained) KP hierarchy. We also discuss the special case of \cKP --truncated KP hierarchies, obtained as Darboux-B\"{a}cklund orbits of initial purely differential Lax operators. The latter give rise to Toda-lattice-like structures relevant for discrete (multi-)matrix models. Our construction establishes the condition for commutativity of the additional-symmetry flows with the discrete Darboux-B\"{a}cklund transformations of \cKP hierarchies leading to a new derivation of the string-equation constraint in matrix models.