  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 "doubly resonant" locus, namely the 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 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.
