Modular Frobenius manifolds and their invariant flows

TL;DR

This paper explores the involutive symmetry of Frobenius manifolds, focusing on the action of this symmetry on almost dual structures, periods, and flows, revealing modular properties and invariance in a special class of fixed-point manifolds.

Contribution

It analyzes the involutive symmetry on Frobenius and almost dual Frobenius manifolds, highlighting modular invariance and fixed points, which was not previously understood.

Findings

The involution $I$ acts on almost dual Frobenius manifolds and their structures.

A class of Frobenius manifolds are fixed points of the involution, exhibiting modular properties.

Flows are invariant under $I$ up to a reciprocal transformation.

Abstract

The space of Frobenius manifolds has a natural involutive symmetry on it: there exists a map $I$ which send a Frobenius manifold to another Frobenius manifold. Also, from a Frobenius manifold one may construct a so-called almost dual Frobenius manifold which satisfies almost all of the axioms of a Frobenius manifold. The action of $I$ on the almost dual manifolds is studied, and the action of $I$ on objects such as periods, twisted periods and flows is studied. A distinguished class of Frobenius manifolds sit at the fixed point of this involutive symmetry, and this is made manifest in certain modular properties of the various structures. In particular, up to a simple reciprocal transformation, for this class of modular Frobenius manifolds, the flows are invariant under the action of $I\,.$