Flat meromorphic connections of Frobenius manifolds with tt*-structure

TL;DR

This paper studies the geometric structures of Frobenius manifolds with CDV-structures, focusing on flat meromorphic connections and conditions for their formal isomorphisms to converge, especially in semi-simple cases.

Contribution

It establishes a formal isomorphism between two natural holomorphic bundles with connections for semi-simple CDV-structures and provides explicit convergence criteria when the super-symmetric index vanishes.

Findings

Existence of a formal isomorphism between bundles for semi-simple CDV-structures.

Convergence of the formal isomorphism when the super-symmetric index Q is zero.

Explicit conditions for convergence in the case of dimension two.

Abstract

The base space of a semi-universal unfolding of a hypersurface singularity carries a rich geometric structure, which was axiomatized as a CDV-structure by C. Hertling. For any CDV-structure on a Frobenius manifold M, the pull-back of the (1,0)-tangent bundle of M to the product of M by the complex line carries two natural holomorphic structures equipped with flat meromorphic connections. We show that, for any semi-simple CDV-structure, there is a formal isomorphism between these two bundles compatible with connections. Moreover, if we assume that the super-symmetric index Q vanishes, we give a necessary and sufficient condition for such a formal isomorphism to be convergent, and we make it explicit for dim M = 2.