Reciprocal $F$-manifolds

TL;DR

This paper studies how reciprocal transformations affect the structure of semisimple F-manifolds with compatible flat connections, introducing the concept of reciprocal F-manifolds and analyzing their properties and solutions.

Contribution

It introduces the notion of reciprocal F-manifolds resulting from reciprocal transformations and examines their properties, especially in the bi-flat case, including preservation of flatness and solution behavior.

Findings

Reciprocal transformations produce new F-manifolds called reciprocal F-manifolds.

Certain reciprocal transformations preserve the flatness of bi-flat F-manifolds.

Analysis of solutions to the augmented Darboux-Egorov system under these transformations.

Abstract

We consider the action of a special class of reciprocal transformation on the principal hierarchy associated to a semisimple $F$-manifold with compatible flat structure $(M,\circ,\nabla,e)$. Under some additional assumptions, the hierarchy obtained applying these reciprocal transformations is also associated to an $F$-manifold with compatible flat structure that we call reciprocal $F$-manifold. We also consider the special case of bi-flat $F$-manifolds $(M,\circ,*,\nabla^{(1)},\nabla^{(2)},e,E)$ and we study reciprocal transformations preserving flatness of both the connections $\nabla^{(1)}$ and $\nabla^{(2)}$ and how they act on corresponding solutions of an augmented Darboux-Egorov system.