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Diagrams and irregular connections on the Riemann sphere

Jean Douçot

TL;DR

The paper develops a general diagrammatic framework encoding irregular, ramified meromorphic connections on the Riemann sphere via a core diagram plus legs labeled by conjugacy data. It proves that these diagrams are invariant under the SL2(C) action on Weyl-algebra modules, including Fourier-Laplace transforms, enabling a canonical diagram for connections with multiple irregular singularities. The authors derive explicit combinatorial rules for the core diagram, Legendre transforms of exponential factors, and a dimension formula for wild character varieties, then apply the construction to read multiple Lax representations of Painlevé-type equations from a single diagram. They illustrate the framework with numerous examples, including H3 surfaces and higher Painlevé systems, revealing a unifying diagrammatic perspective on isomonodromic systems and their symmetries.

Abstract

We define a diagram associated to any algebraic connection on a vector bundle on a Zariski open subset of the Riemann sphere, extending the definition of Boalch-Yamakawa to the general case featuring several irregular singularities, possibly ramified. We prove that the diagram is invariant under the symplectic automorphisms of the Weyl algebra, encompassing the Fourier-Laplace transform. As an application, we establish several new cases of the observation that different Lax representations of a given Painlevé-type equation may be read off directly from the diagram, corresponding to connections with different formal data, usually on different rank bundles.

Diagrams and irregular connections on the Riemann sphere

TL;DR

The paper develops a general diagrammatic framework encoding irregular, ramified meromorphic connections on the Riemann sphere via a core diagram plus legs labeled by conjugacy data. It proves that these diagrams are invariant under the SL2(C) action on Weyl-algebra modules, including Fourier-Laplace transforms, enabling a canonical diagram for connections with multiple irregular singularities. The authors derive explicit combinatorial rules for the core diagram, Legendre transforms of exponential factors, and a dimension formula for wild character varieties, then apply the construction to read multiple Lax representations of Painlevé-type equations from a single diagram. They illustrate the framework with numerous examples, including H3 surfaces and higher Painlevé systems, revealing a unifying diagrammatic perspective on isomonodromic systems and their symmetries.

Abstract

We define a diagram associated to any algebraic connection on a vector bundle on a Zariski open subset of the Riemann sphere, extending the definition of Boalch-Yamakawa to the general case featuring several irregular singularities, possibly ramified. We prove that the diagram is invariant under the symplectic automorphisms of the Weyl algebra, encompassing the Fourier-Laplace transform. As an application, we establish several new cases of the observation that different Lax representations of a given Painlevé-type equation may be read off directly from the diagram, corresponding to connections with different formal data, usually on different rank bundles.

Paper Structure

This paper contains 27 sections, 27 theorems, 125 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The construction
  3. Organisation of the paper
  4. Formal data of irregular connections
  5. The exponential local system
  6. Graded local systems
  7. Modified formal data
  8. Action of $\mathop{\mathrm{SL}}\nolimits_2(\mathbb{C})$
  9. The stationary phase formula
  10. Symplectic transformations
  11. Invariance of the diagram for one irregular singularity
  12. Diagram for one irregular singularity
  13. An explicit formula for the number of edges
  14. Form of the Legendre transform
  15. Invariance of the diagram under Fourier-Laplace transform
  16. ...and 12 more sections

Key Result

Theorem 1.2

If the connection $(E,\nabla)$ is irreducible, the dimension of its wild character variety $\mathcal{M}_{\mathrm{B}}(\bm\Sigma,\bm\mathcal{C})$ is given by the quiver variety dimension formula: where $\mathbf d$ is the dimension vector coming from any choice of minimal marking of $(E,\nabla)$, and $(\,\cdot\,,\,\cdot\,)$ is the bilinear form determined by the (symmetrized) Cartan matrix $C=2-B$ o

Figures (4)

  • Figure 1: Diagrams for the Painlevé equations VI, V, IV, III, II, I.
  • Figure 2: Diagrams associated to all known H3 surfaces. The names of the surfaces are as in boalch2018wild. All unspecified multiplicities are equal to 1.
  • Figure 3: Diagrams for higher Painlevé systems $hP_{VI}^{(n)}$, $hP_{V}^{(n)}$, $hP_{IV}^{(n)}$, and $hP_{III}^{(n)}$.
  • Figure 4: Diagrams for the Lax representations of 4-dimensional Painlevé-type equations of kawakami2018degeneration featuring several irregular singularities. Notice that the last line is none other than a higher Painlevé III system.

Theorems & Definitions (67)

  • Definition 1.1: boalch2020diagrams
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: deligne1978lettermalgrange1982classification
  • Definition 2.4
  • Theorem 3.1
  • Remark 3.1
  • ...and 57 more