Legendre transforms for type $A_{n}$ and $B_{n}$ $\vee$-systems

TL;DR

This paper explores how Legendre transformations act on specific rational solutions of the WDVV equations associated with type A and B systems, revealing connections to new and existing trigonometric solutions.

Contribution

It explicitly computes the effects of Legendre transformations on $A_n$ and $B_n$ rational solutions, linking them to known and novel trigonometric solutions.

Findings

Legendre transformations relate rational and trigonometric solutions of WDVV equations.

Explicit formulas for transformed solutions of types A and B.

New connections between rational and trigonometric Frobenius manifold solutions.

Abstract

The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations have a rich structure related to the theory of Frobenius manifolds, with many known families of solutions. A Legendre transformation is a symmetry of the WDVV equations, introduced by Dubrovin. We explicitly compute the results of a Legendre transformation applied to $A_n$- and $B_n$-type multi-parameter rational solutions, relating them to known and new trigonometric solutions.