Degenerate Riemann-Hilbert-Birkhoff problems, semisimplicity, and convergence of WDVV-potentials

TL;DR

This paper provides a new analytical proof for a theorem on integrable deformations of meromorphic connections and demonstrates that semisimple formal Frobenius manifolds are actually convergent, confirming their analytic nature.

Contribution

It introduces a new proof of Sabbah's theorem and proves the convergence of formal semisimple Frobenius manifolds without additional assumptions.

Findings

New analytical proof of Sabbah's theorem.

Semisimple formal Frobenius manifolds are convergent.

Formal solutions of WDVV equations are analytic.

Abstract

In the first part of this paper, we give a new analytical proof of a theorem of C. Sabbah on integrable deformations of meromorphic connections on $\mathbb P^1$ with coalescing irregular singularities of Poincar\'e rank 1, and generalizing a previous result of B. Malgrange. In the second part of this paper, as an application, we prove that any semisimple formal Frobenius manifold (over $\mathbb C$), with unit and Euler field, is the completion of an analytic pointed germ of a Dubrovin-Frobenius manifold. In other words, any formal power series, which provides a quasi-homogenous solution of WDVV equations and defines a semisimple Frobenius algebra at the origin, is actually convergent under no further tameness assumptions.