Additional Symmetries of Generalized Hierarchies

TL;DR

This paper identifies and analyzes non-isospectral symmetries in generalized integrable hierarchies, revealing their algebraic structure and implications for tau-functions and quantum gravity models.

Contribution

It generalizes known symmetries of integrable hierarchies using Virasoro and Kac--Moody algebra structures, expanding understanding of their algebraic properties.

Findings

Symmetries form a subalgebra of the Virasoro algebra.

Infinitesimal symmetries act on tau-functions as Virasoro currents.

Comments on implications for matrix-model quantum gravity.

Abstract

The non-isospectral symmetries of a general class of integrable hierarchies are found, generalizing the Galilean and scaling symmetries of the Korteweg--de Vries equation and its hierarchy. The symmetries arise in a very natural way from the semi-direct product structure of the Virasoro algebra and the affine Kac--Moody algebra underlying the construction of the hierarchy. In particular, the generators of the symmetries are shown to satisfy a subalgebra of the Virasoro algebra. When a tau-function formalism is available, the infinitesimal symmetries act directly on the tau-functions as moments of Virasoro currents. Some comments are made regarding the r\^ole of the non-isospectral symmetries and the form of the string equations in matrix-model formulations of quantum gravity in two-dimensions and related systems.