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

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 Poincaré 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.

38 pages, 2 figures; v2. corrected typos, more detailed proof of Lemma 5.18; v3. corrected typos; v4. final version

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• Borel $(\alpha ,\beta )$-multitransforms and Quantum Leray-Hirsch: integral representations of solutions of quantum differential equations for ${\mathbb{P}}^1$-bundles
• On the universality of integrable deformations of solutions of degenerate Riemann-Hilbert-Birkhoff problems
• Cyclic stratum of Frobenius manifolds, Borel-Laplace $(\mathbf{\alpha },\mathbf{\beta })$-multitransforms, and integral representations of solutions of Quantum Differential Equations