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Degenerate Riemann-Hilbert-Birkhoff problems, semisimplicity, and convergence of WDVV-potentials

Giordano Cotti

Abstract

In the first part of this paper, we give a new analytical proof of a theorem of C. Sabbah on integrable deformations of meromorphic connections on $\mathbb P^1$ with coalescing irregular singularities of Poincaré rank 1, and generalizing a previous result of B. Malgrange. In the second part of this paper, as an application, we prove that any semisimple formal Frobenius manifold (over $\mathbb C$), with unit and Euler field, is the completion of an analytic pointed germ of a Dubrovin-Frobenius manifold. In other words, any formal power series, which provides a quasi-homogenous solution of WDVV equations and defines a semisimple Frobenius algebra at the origin, is actually convergent under no further tameness assumptions.

Degenerate Riemann-Hilbert-Birkhoff problems, semisimplicity, and convergence of WDVV-potentials

Abstract

In the first part of this paper, we give a new analytical proof of a theorem of C. Sabbah on integrable deformations of meromorphic connections on with coalescing irregular singularities of Poincaré rank 1, and generalizing a previous result of B. Malgrange. In the second part of this paper, as an application, we prove that any semisimple formal Frobenius manifold (over ), with unit and Euler field, is the completion of an analytic pointed germ of a Dubrovin-Frobenius manifold. In other words, any formal power series, which provides a quasi-homogenous solution of WDVV equations and defines a semisimple Frobenius algebra at the origin, is actually convergent under no further tameness assumptions.

Paper Structure

This paper contains 27 sections, 41 theorems, 132 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Degenerations of Riemann-Hilbert-Birkhoff inverse problems
  3. Riemann-Hilbert-Birkhoff inverse problems
  4. Universal integrable deformations of Birkhoff normal forms: Malgrange Theorems
  5. Integrable deformations of degenerate Birkhoff normal forms: Sabbah Theorem
  6. An analytical proof of Sabbah Theorem
  7. Admissible data and Riemann-Hilbert-Birkhoff boundary value problem
  8. Factorization of solutions
  9. Solvability as an open property
  10. Proof of Sabbah Theorem
  11. Formal Frobenius and Dubrovin-Frobenius manifold s
  12. Formal Frobenius manifolds
  13. Dubrovin-Frobenius manifold s
  14. From Dubrovin-Frobenius to formal Frobenius structures, and vice-versa
  15. Semisimplicity of Frobenius structures.
  16. ...and 12 more sections

Key Result

Theorem 2.3

Assume that the matrix $\Lambda_o$ is regular. The connection $\nabla^o$ with matrix con1 has a germ of universal deformation.

Figures (2)

  • 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: Contours $\Gamma'$ and $\Gamma"$, their orientations and $\pm$ sides.

Theorems & Definitions (62)

  • Definition 2.2
  • Theorem 2.3: Mal83aMal86
  • Theorem 2.4: JMU81Mal83b
  • Theorem 2.5: Sab18
  • Definition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Remark 3.5
  • Definition 3.6
  • Remark 3.7
  • ...and 52 more