Additional symmetries of the extended bigraded Toda hierarchy

TL;DR

This paper constructs and explicitly describes new symmetries of the extended bigraded Toda hierarchy, revealing a Virasoro subalgebra action on its tau-function, which generalizes previous results.

Contribution

It introduces additional symmetries for the EBTH and details their explicit action on key hierarchy components, extending the understanding of its algebraic structure.

Findings

Identified new symmetries acting on the Lax operator and tau-function

Established a Virasoro subalgebra structure within the symmetries

Generalized previous symmetry results of Dubrovin and Zhang

Abstract

The extended bigraded Toda hierarchy (EBTH) is an integrable system satisfied by the total descendant potential of $\mathbb{CP}^1$ with two orbifold points. We construct additional symmetries of the EBTH and describe explicitly their action on the Lax operator, wave operators, and tau-function of the hierarchy. In particular, we obtain infinitesimal symmetries of the EBTH that act on the tau-function as a subalgebra of the Virasoro algebra, generalizing those of Dubrovin and Zhang.