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.

中文翻译：

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半单平面 F 流形的 Riemann-Hilbert-Birkhoff 反问题和定向结合势的收敛

在本文中，我们解决了准齐次形式幂级数的分类问题，提供了定向结合性方程的解。这种分类是通过在同质半简单平面的形式胚芽空间上引入单向局部模量系统来执行的$F$-歧管。这种局部模量系统在严格双共振轨迹的补充上得到了很好的定义，即平面胚的形式胚轨迹。$F$-流形表现出原点处的规范坐标的合并以及它们的共形维度的共振。展示了如何通过 Riemann-Hilbert-Birkhoff 边值问题从单向局部模量的知识重构定向结合性方程的解。此外，根据B. Malgrange和C. Sabbah的结果，证明了任何形式齐次半单平面$F$-流形，不是严格的双谐振，实际上是收敛的。我们的收敛半简单准则也根据 Losev-Manin 交换性方程的解、相关器的增长估计重新表述$F$-上同调场论，以及开放的 Witten–Dijkgraaf–Verlinde–Verlinde 方程的解。