From Additional Symmetries to Linearization of Virasoro Symmetries

TL;DR

This paper explores the symmetries of integrable hierarchies, deriving formulas and showing how certain hierarchies relate to Virasoro symmetries, Frobenius manifolds, and solutions satisfying string equations.

Contribution

It constructs additional symmetries for the two-component BKP hierarchy and establishes the linearization of Virasoro symmetries for Drinfeld-Sokolov hierarchies of type D.

Findings

Derived the Adler-Shiota-van Moerbeke formula for the hierarchy.

Showed Drinfeld-Sokolov type D hierarchies have Virasoro symmetries.

Connected hierarchies to Frobenius manifolds and string equations.

Abstract

We construct the additional symmetries and derive the Adler-Shiota-van Moerbeke formula for the two-component BKP hierarchy. We also show that the Drinfeld-Sokolov hierarchies of type D, which are reduced from the two-component BKP hierarchy, possess symmetries written as the action of a series of linear Virasoro operators on the tau function. It results in that the Drinfeld-Sokolov hierarchies of type D coincide with Dubrovin and Zhang's hierarchies associated to the Frobenius manifolds for Coxeter groups of type D, and that every solution of such a hierarchy together with the string equation is annihilated by certain combinations of the Virasoro operators and the time derivations of the hierarchy.