Dubrovin's duality for $F$-manifolds with eventual identities

TL;DR

This paper explores a duality concept for F-manifolds with special invertible vector fields called eventual identities, characterizing them and showing how they preserve certain geometric structures and enable metric constructions.

Contribution

It introduces a new duality for F-manifolds with eventual identities, characterizes these identities, and demonstrates their compatibility with various geometric structures.

Findings

Characterization of eventual identities on F-manifolds.

Development of a duality preserving Riemannian structures.

Conditions under which harmonic Higgs bundles are preserved.

Abstract

A vector field E on an F-manifold (M, o, e) is an eventual identity if it is invertible and the multiplication X*Y := X o Y o E^{-1} defines a new F-manifold structure on M. We give a characterization of such eventual identities, this being a problem raised by Manin. We develop a duality between F-manifolds with eventual identities and we show that is compatible with the local irreducible decomposition of F-manifolds and preserves the class of Riemannian F-manifolds. We find necessary and sufficient conditions on the eventual identity which insure that harmonic Higgs bundles and DChk-structures are preserved by our duality. We use eventual identities to construct compatible pair of metrics.