Generalized Legendre transformations and symmetries of the WDVV equations

TL;DR

This paper generalizes Legendre transformations within the WDVV equations framework, expanding symmetry understanding and linking different classes of solutions, including rational and trigonometric types.

Contribution

It introduces a broader class of Legendre transformations for Frobenius manifolds, extending the symmetry structure of the WDVV equations beyond flat vector fields.

Findings

Generated symmetries between almost-dual Frobenius manifolds.

Established a map between rational and trigonometric solutions.

Extended the concept of Legendre transformations to non-flat vector fields.

Abstract

The Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations, as one would expect from an integrable system, has many symmetries, both continuous and discrete. One class - the so-called Legendre transformations - were introduced by Dubrovin. They are a discrete set of symmetries between the stronger concept of a Frobenius manifold, and are generated by certain flat vector fields. In this paper this construction is generalized to the case where the vector field (called here the Legendre field) is non-flat but satisfies a certain set of defining equations. One application of this more general theory is to generate the induced symmetry between almost-dual Frobenius manifolds whose underlying Frobenius manifolds are related by a Legendre transformation. This also provides a map between rational and trigonometric solutions of the WDVV equations.