Basic Representations of Genus Zero Nonabelian Hodge Spaces

TL;DR

This paper introduces an enriched invariant called the enriched tree for genus zero nonabelian Hodge spaces, enabling the reconstruction of multiple deformation classes and linking to isomonodromy systems, with applications to Painlevé equations.

Contribution

It extends previous invariants by incorporating fission data, allowing the classification of deformation classes and connecting to isomonodromy systems in a broader singularity context.

Findings

Enriched invariant contains sufficient information for classifying deformations.

Reconstruction of multiple admissible deformation classes from the invariant.

Application to different Lax representations for Painlevé equations.

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 Painlev\'e equations.