Some Constraints on Frobenius Manifolds with a TT*-Structure

TL;DR

This paper establishes conditions for Frobenius manifolds to have CDV-structures, demonstrates their existence on semi-simple cases, compares natural connections, and explores the harmonic potential in harmonic Frobenius manifolds.

Contribution

It provides a necessary and sufficient condition for Frobenius manifolds to admit CDV-structures and analyzes their geometric properties.

Findings

Existence of positive definite CDV-structures on semi-simple Frobenius manifolds

Comparison of three natural connections on CDV-structures

Hermitian manifolds of non-trivial CDV-structures are not Kahler

Abstract

The article gives a necessary and sufficient condition for a Frobenius manifold to be a CDV-structure. We show that there exists a positive definite CDV-structure on any semi-simple Frobenius manifold. We also compare three natural connections on a CDV-structure and conclude that the underlying Hermitian manifold of a non-trivial CDV-structure is not a Kahler manifold. Finally, we compute the harmonic potential of a harmonic Frobenius manifolds.