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In this paper, we address the problem of classification of quasi-homogeneousformal power series providing solutions of the oriented associativityequations. Such a classification is performed by introducing a system ofmonodromy local moduli on the space of formal germs of homogeneous semisimpleflat $F$-manifolds. This system of local moduli is well-defined on thecomplement of the "doubly resonant" locus, namely the locus of formal germs offlat $F$-manifolds manifesting both coalescences of canonical coordinates atthe origin, and resonances of their "conformal dimensions". It is shown how thesolutions of the oriented associativity equations can be reconstructed from theknowledge of the monodromy local moduli via a Riemann-Hilbert-Birkhoff boundaryvalue problem. Furthermore, standing on results of B. Malgrange and C. Sabbah,it is proved that any formal homogeneous semisimple flat $F$-manifold, which isnot doubly resonant, is actually convergent. Our semisimplicity criterion forconvergence is also reformulated in terms of solutions of Losev-Manincommutativity equations, growth estimates of correlators of $F$-cohomologicalfield theories, and solutions of open Witten-Dijkgraaf-Verlinde-Verlindeequations.

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