Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Giordano Cotti

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.

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

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 -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 -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 -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 -cohomological field theories, and solutions of open Witten-Dijkgraaf-Verlinde-Verlinde equations.

Paper Structure

This paper contains 40 sections, 74 theorems, 265 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Flat $F$-manifolds
  3. Analytic flat $F$-manifolds
  4. Formal flat $F$-manifolds
  5. Local isomorphisms and pointed germs
  6. Semisimple flat $F$-manifolds
  7. Irreducible flat $F$-manifolds
  8. Extended deformed connections
  9. $\nabla^z$-flat coordinates and oriented associativity potentials
  10. Family of extended deformed connections
  11. $\widehat{\nabla}^\lambda$-flat covectors
  12. Matrices $\widetilde{\Psi},\widetilde{V}_i,\widetilde{V}^\lambda,\widetilde{\Gamma}$
  13. Darboux-Tsarev equations, and conformal dimensions
  14. Lamé coefficients, matrices $\Psi,V_i,V^\lambda,\Gamma$, and Darboux-Egoroff equations
  15. Monodromy moduli of admissible germs of semisimple flat $F$-manifolds
  16. ...and 25 more sections

Key Result

Theorem 1.2

Any homogeneous semisimple (analytic/formal) pointed germ of flat $F$-manifold, which is not strictly doubly resonant, is uniquely determined by its monodromy data. In particular, the vector potential $\bm F$ can be explicitly reconstructed from the monodromy data via a Riemann-Hilbert-Birkhoff boun

Figures (6)

  • Figure 1: Contour $\Gamma$, paths $\Gamma_{\pm\infty},\Gamma_1,\Gamma_2$, domains $\Pi_0,\Pi_L,\Pi_R$, and $\pm$ sides of $\Gamma$.
  • Figure 2: A stable 12-pointed chain of projective lines.
  • Figure 3: Isomorphism classes of one edges $n$-trees are parametrized by stable unordered 2-partition s $\sigma=\{S_1,S_2\}$ of $\{1,\dots,n\}$.
  • Figure 4:
  • Figure 5: Two stable $8$-trees $\tau_1$ and $\tau_2$.
  • ...and 1 more figures

Theorems & Definitions (130)

  • Remark 1.1
  • Theorem 1.2: Cf. Theorems \ref{['enF']}, \ref{['mthm1']}
  • Theorem 1.3: Cf. Theorem \ref{['thconv']}
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 120 more