Meromorphic connections over F-manifolds

TL;DR

This paper reviews the construction of Frobenius manifolds via holomorphic bundles with meromorphic connections, discusses constraints and freedoms in the process, and presents new results including a conjecture and its proof in 2D cases.

Contribution

It introduces a new conjecture on the existence and uniqueness of meromorphic bundles over F-manifolds and proves it in two-dimensional cases.

Findings

Conjecture on meromorphic bundle existence and uniqueness proposed

Proof of the conjecture established for 2-dimensional cases

Discussion of steps, constraints, and freedoms in Frobenius manifold construction

Abstract

This paper review one construction of Frobenius manifolds (and slightly weaker structures). It splits it into several steps and discusses the freedom and the constraints in these steps. The steps pass through holomorphic bundles with meromorphic connections. A conjecture on existence and uniqueness of certain such bundles, a proof of the conjecture in the 2-dimensional cases, and some other new results form a research part of this paper.