Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Basic representations of genus zero nonabelian Hodge spaces

Jean Douçot

TL;DR

This work develops a comprehensive combinatorial framework to identify conjectural isomorphisms between genus-zero nonabelian Hodge spaces by using enriched trees, fission forests, and core diagrams to encode irregular data and their symplectic SL2(C) actions. The enriched tree invariant refines the diagram data and enables explicit reconstruction of k+1 nearby representations within an SL2(C) orbit, clarifying how isomonodromic systems may be isomorphic across different Lax representations. The theory is applied to Painlevé moduli spaces, where all basic representations and their readings recover known alternative Lax representations and reveal new ones in an abstract setting. The results connect Fourier transforms, Möbius/SL2(C) actions, and admissible deformations to a unified description of wild character varieties, offering a path toward explicit isomorphisms between nonabelian Hodge spaces and their isomonodromic dynamics.

Abstract

In some previous work, we defined an invariant of genus zero nonabelian Hodge spaces taking the form of a diagram. Here, enriching the diagram by fission data to obtain a refined invariant, the enriched tree, including a partition of the core diagram into $k$ subsets, we show that this invariant contains sufficient information to reconstruct $k$+1 different classes of admissible deformations of wild Riemann surfaces, that are all representations of one single nonabelian Hodge space, so that the isomonodromy systems defined by these representations are expected to be isomorphic. This partially generalises to the case of arbitrary singularity data the picture of the simply-laced case featuring a diagram with a complete $k$-partite core. We illustrate this framework by discussing different Lax representations for Painlevé equations.

Basic representations of genus zero nonabelian Hodge spaces

TL;DR

This work develops a comprehensive combinatorial framework to identify conjectural isomorphisms between genus-zero nonabelian Hodge spaces by using enriched trees, fission forests, and core diagrams to encode irregular data and their symplectic SL2(C) actions. The enriched tree invariant refines the diagram data and enables explicit reconstruction of k+1 nearby representations within an SL2(C) orbit, clarifying how isomonodromic systems may be isomorphic across different Lax representations. The theory is applied to Painlevé moduli spaces, where all basic representations and their readings recover known alternative Lax representations and reveal new ones in an abstract setting. The results connect Fourier transforms, Möbius/SL2(C) actions, and admissible deformations to a unified description of wild character varieties, offering a path toward explicit isomorphisms between nonabelian Hodge spaces and their isomonodromic dynamics.

Abstract

In some previous work, we defined an invariant of genus zero nonabelian Hodge spaces taking the form of a diagram. Here, enriching the diagram by fission data to obtain a refined invariant, the enriched tree, including a partition of the core diagram into subsets, we show that this invariant contains sufficient information to reconstruct +1 different classes of admissible deformations of wild Riemann surfaces, that are all representations of one single nonabelian Hodge space, so that the isomonodromy systems defined by these representations are expected to be isomorphic. This partially generalises to the case of arbitrary singularity data the picture of the simply-laced case featuring a diagram with a complete -partite core. We illustrate this framework by discussing different Lax representations for Painlevé equations.
Paper Structure (32 sections, 26 theorems, 61 equations, 1 figure)

This paper contains 32 sections, 26 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General motivation: classification of (wild) nonabelian Hodge spaces
  3. Diagrams and isomorphisms between moduli spaces of connections
  4. Basic operations on connections in genus zero and weak representations
  5. Main results: from enriched trees to classes of nearby representations
  6. Outlook
  7. Contents
  8. Symplectic action on fission data of irregular connections on $\mathbb P^1$
  9. Formal data of irregular connections
  10. The exponential local system
  11. Local and global formal data
  12. Modified formal data
  13. Admissible deformations, fission data, and forests
  14. Local version
  15. Global version
  16. ...and 17 more sections

Key Result

Theorem 1.4

Any algebraic connection $(E,\nabla)$ on a Zariski open subset of $\mathbb P^1$ canonically determines an enriched tree $\mathscr T(E,\nabla)$, which depends only on its wild Riemann surface with boundary data. If $(E,\nabla)$ is irreducible, and is not a rank one connection with a pole of order les

Figures (1)

  • Figure 1: Example of different readings of a (core) diagram in the simply-laced case. On the left, a complete 3-partite graph, with the different colours of the vertices corresponding to the partition. The three other figures on the right correspond to the nongeneric readings, each reading consisting in singling out one of the susbsets of the partition.

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Prop. \ref{['prop: definition_of_enriched_trees']}, Def. \ref{['def:enriched_tree']}, Cor. \ref{['cor:symplectic_invariance_enriched_tree']}
  • Theorem 1.5: Cor. \ref{['cor:we_get_classes_of_nearby_reps']}
  • Theorem 1.6: Prop. \ref{['prop:rank_of_representations']}, \ref{['prop:number_of_singularities']}
  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 45 more