Legendre transformations of a class of generalized Frobenius manifolds and the associated integrable hierarchies

TL;DR

This paper explores how Legendre transformations connect generalized Frobenius manifolds and their associated integrable hierarchies, revealing a linear reciprocal relationship between their deformations under certain conditions.

Contribution

It demonstrates that Legendre transformations induce a linear reciprocal relationship between integrable hierarchies and their deformations in generalized Frobenius manifolds.

Findings

Legendre-extended Principal Hierarchies are related by a linear reciprocal transformation.

Topological deformations of these hierarchies are also connected by the same transformation.

The results hold under the semisimplicity condition.

Abstract

For two generalized Frobenius manifolds related by a Legendre-type transformation, we show that the associated integrable hierarchies of hydrodynamic type, which are called the Legendre-extended Principal Hierarchies, are related by a certain linear reciprocal transformation; we also show, under the semisimplicity condition, that the topological deformations of these Legendre-extended Principal Hierarchies are related by the same linear reciprocal transformation.