The Inversion Symmetry of the WDVV Equations and Tau Functions

TL;DR

This paper explores the relationship between solutions of the WDVV equations connected by inversion symmetry, revealing how their associated integrable hierarchies and tau functions are related through reciprocal and Legendre transformations.

Contribution

It demonstrates the connection between inversion symmetry in WDVV solutions and transformations of integrable hierarchies and tau functions, providing new insights into their structural relationships.

Findings

Principal hierarchies are related by reciprocal transformations.

Tau functions are connected via Legendre transformations.

Relationships between Virasoro constraints and topological deformations are established.

Abstract

For two solutions of the WDVV equations that are related by the inversion symmetry, we show that the associated principal hierarchies of integrable systems are related by a reciprocal transformation, and the tau functions of the hierarchies are related by a Legendre type transformation. We also consider relationships between the Virasoro constraints and the topological deformations of the principal hierarchies.