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Wild character varieties, meromorphic Hitchin systems and Dynkin diagrams

Philip Boalch

TL;DR

The work presents a unified framework tying integrable systems, nonabelian Hodge theory, and wild character varieties through meromorphic Hitchin systems and quasi-Hamiltonian/fission spaces. It defines nonabelian Hodge spaces and constructs rich families of examples, including H3 surfaces and Nakajima quiver varieties, with irregular data generating new symplectic and hyperkähler structures. By introducing irregular curves and multiplicative spaces, the paper extends the classical Dol/DR/B correspondence to wild settings and explores isomonodromy, dualities, and Dynkin-diagram-inspired classifications. The overall contribution is a broad program for cataloging nonabelian Hodge spaces and understanding their interrelations via Lax representations, fission, and modular interpretations across tame and wild regimes.

Abstract

The theory of Hitchin systems is something like a "global theory of Lie groups", where one works over a Riemann surface rather than just at a point. We'll describe how one can take this analogy a few steps further by attempting to make precise the class of rich geometric objects that appear in this story (including the non-compact case), and discuss their classification, outlining a theory of "Dynkin diagrams" as a step towards classifying some examples of such objects.

Wild character varieties, meromorphic Hitchin systems and Dynkin diagrams

TL;DR

The work presents a unified framework tying integrable systems, nonabelian Hodge theory, and wild character varieties through meromorphic Hitchin systems and quasi-Hamiltonian/fission spaces. It defines nonabelian Hodge spaces and constructs rich families of examples, including H3 surfaces and Nakajima quiver varieties, with irregular data generating new symplectic and hyperkähler structures. By introducing irregular curves and multiplicative spaces, the paper extends the classical Dol/DR/B correspondence to wild settings and explores isomonodromy, dualities, and Dynkin-diagram-inspired classifications. The overall contribution is a broad program for cataloging nonabelian Hodge spaces and understanding their interrelations via Lax representations, fission, and modular interpretations across tame and wild regimes.

Abstract

The theory of Hitchin systems is something like a "global theory of Lie groups", where one works over a Riemann surface rather than just at a point. We'll describe how one can take this analogy a few steps further by attempting to make precise the class of rich geometric objects that appear in this story (including the non-compact case), and discuss their classification, outlining a theory of "Dynkin diagrams" as a step towards classifying some examples of such objects.

Paper Structure

This paper contains 12 sections, 10 theorems, 32 equations, 2 figures.

Table of Contents

  1. The Lax Project
  2. Rational matrices
  3. Hitchin systems
  4. Nonabelian Hodge theory
  5. Isomonodromy and nonabelian Gauss--Manin connections
  6. Extending the nonabelian Hodge package
  7. Nonabelian Hodge spaces
  8. Non-perturbative symplectic manifolds
  9. New quasi-Hamiltonian spaces
  10. More examples
  11. H3 surfaces
  12. Quiver varieties

Key Result

Theorem 3

The fibre bundle ${\boldsymbol{\mathcal{M}}}\to \mathbb{B}$ has a natural complete flat (algebraic) Ehresmann connection on it, the nonabelian Gauss--Manin connection.

Figures (2)

  • Figure 1: Four readings of $\Gamma(221)=2\cdot\Gamma(21)=1\cdot\Gamma(22)$, from rsode.
  • Figure :

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Theorem 3: Sim94ab
  • Definition 4
  • Theorem 5: wnabhSab99
  • Remark 6
  • Definition 7
  • Remark 8
  • Theorem 9: Hitchin Hit95long
  • Lemma 10: quad ex.3, rsode Apx. C, hi-ya-nslcase
  • ...and 6 more