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Twisted wild character varieties

Philip Boalch, Daisuke Yamakawa

TL;DR

The work develops a comprehensive framework for twisted wild character varieties by incorporating ramified irregular types and interior twists via local systems of groups. It extends the quasi-Hamiltonian/Poisson geometry of untwisted Stokes data to twisted Stokes local systems, establishing that the framed twisted Stokes representation spaces are twisted quasi-Hamiltonian with a corresponding twisted momentum map, yielding Poisson structures on the twisted character varieties. The paper also constructs twisted fission spaces and demonstrates how fusion, disc pullbacks, and reductions produce a broad class of twisted moduli spaces, including explicit examples tied to Painlevé hierarchies and torus-knot invariants. This work lays groundwork for twisted Dynkin diagrams, Langlands duality perspectives, and mirror-symmetric interpretations of twisted wild Hitchin-type moduli spaces.

Abstract

We will construct twisted versions of the wild character varieties.

Twisted wild character varieties

TL;DR

The work develops a comprehensive framework for twisted wild character varieties by incorporating ramified irregular types and interior twists via local systems of groups. It extends the quasi-Hamiltonian/Poisson geometry of untwisted Stokes data to twisted Stokes local systems, establishing that the framed twisted Stokes representation spaces are twisted quasi-Hamiltonian with a corresponding twisted momentum map, yielding Poisson structures on the twisted character varieties. The paper also constructs twisted fission spaces and demonstrates how fusion, disc pullbacks, and reductions produce a broad class of twisted moduli spaces, including explicit examples tied to Painlevé hierarchies and torus-knot invariants. This work lays groundwork for twisted Dynkin diagrams, Langlands duality perspectives, and mirror-symmetric interpretations of twisted wild Hitchin-type moduli spaces.

Abstract

We will construct twisted versions of the wild character varieties.

Paper Structure

This paper contains 20 sections, 14 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Torsors etc
  4. $\mathcal{G}$-local systems
  5. Graded local systems
  6. Covers
  7. Graded local systems
  8. Local system of exponents
  9. Points of maximal decay
  10. Irregular types and irregular classes
  11. Framings and standard fibres
  12. Gradings versus Reductions
  13. Singular directions and Stokes groups
  14. Twisted Stokes local systems
  15. Framed moduli spaces
  16. ...and 5 more sections

Key Result

Theorem 1

The moduli space $\mathop{\mathrm{THom}}\nolimits_\mathbb{S}(\Pi,G)$ of framed twisted Stokes $\mathcal{G}$-local systems is a smooth affine variety and is a twisted quasi-Hamiltonian ${\bf H}$-space.

Figures (2)

  • Figure 1: The Stokes diagram of the Airy equation, from stokes1857 p.116.
  • Figure :

Theorems & Definitions (28)

  • Theorem 1
  • Definition 2
  • Remark 3
  • Theorem 4
  • Definition 5
  • Theorem 6
  • Definition 7
  • Proposition 8
  • Definition 9
  • Definition 10
  • ...and 18 more