Riemann-Hilbert-Birkhoff inverse problem for semisimple flat $F$-manifolds, and convergence of oriented associativity potentials

TL;DR

This paper classifies solutions to oriented associativity equations for flat F-manifolds using monodromy moduli and proves convergence of these solutions under certain conditions, linking complex analysis, algebra, and geometry.

Contribution

It introduces a monodromy local moduli system for classifying formal solutions and establishes convergence criteria for semisimple flat F-manifolds.

Findings

Classification of solutions via monodromy local moduli.

Reconstruction of solutions from Riemann-Hilbert-Birkhoff problems.

Proof of convergence for non-resonant formal flat F-manifolds.

Abstract

In this paper, we address the problem of classification of quasi-homogeneous formal power series providing solutions of the oriented associativity equations. Such a classification is performed by introducing a system of monodromy local moduli on the space of formal germs of homogeneous semisimple flat $F$-manifolds. This system of local moduli is well-defined on the complement of the "strictly doubly resonant" locus, namely a locus of formal germs of flat $F$-manifolds manifesting both coalescences of canonical coordinates at the origin, and resonances of their "conformal dimensions". It is shown how the solutions of the oriented associativity equations can be reconstructed from the knowledge of the monodromy local moduli via a Riemann-Hilbert-Birkhoff boundary value problem. Furthermore, standing on results of B.Malgrange and C.Sabbah, it is proved that any formal homogeneous semisimple flat $F$-manifold, which is not strictly doubly resonant, is actually convergent. Our semisimplicity criterion for convergence is also reformulated in terms of solutions of Losev-Manin commutativity equations, growth estimates of correlators of $F$-cohomological field theories, and solutions of open Witten-Dijkgraaf-Verlinde-Verlinde equations.