Action of the mapping class group on character varieties and Higgs bundles

TL;DR

This paper explores how the mapping class group acts on character varieties and Higgs bundles, identifying fixed points via equivariant structures and extending known results to more general settings involving parabolic and pseudoreal Higgs bundles.

Contribution

It provides a new description of fixed points of the mapping class group action on character varieties using twisted equivariant Higgs bundles, generalizing previous work to include antiholomorphic automorphisms.

Findings

Fixed points characterized by twisted G-Higgs bundles with equivariant structures

Description of fixed points in terms of parabolic Higgs bundles on quotient surfaces

Extension of pseudoreal Higgs bundle theory to a broader setting

Abstract

We consider the action of a finite subgroup of the mapping class group $Mod(S)$ of an oriented compact surface $S$ of genus $g \geq 2$ on the moduli space $\mathcal{R}(S,G)$ of representations of $\pi_1(S)$ in a connected semisimple real Lie group $G$. Kerckhoff's solution of the Nielsen realization problem ensures the existence of an element $J$ in the Teichm\"uller space of $S$ for which $\Gamma$ can be realised as a subgroup of the group of automorphisms of $X=(S,J)$ which are holomorphic or antiholomorphic. We identify the fixed points of the action of $\Gamma$ on $\mathcal{R}(S,G)$ in terms of $G$-Higgs bundles on $X$ equipped with a certain twisted $\Gamma$-equivariant structure, where the twisting involves abelian and non-abelian group cohomology simultaneously. When the kernel of the isotropy representation of the maximal compact subgroup of $G$ is trivial, the fixed points can be described in terms of familiar objects on $Y=X/\Gamma^+$, where $\Gamma^+ \subset \Gamma$ is the maximal subgroup of $\Gamma$ consisting of holomorphic automorphisms of $X$. If $\Gamma=\Gamma^+$ one obtains actual $\Gamma$-equivariant $G$-Higgs bundles on $X$, which in turn correspond with parabolic Higgs bundles on $Y=X/\Gamma$ (this generalizes work of Nasatyr \& Steer for $G=SL(2,\mathbb{R})$ and Boden, Andersen & Grove and Furuta & Steer for $G=SU(n)$). If on the other hand $\Gamma$ has antiholomorphic automorphisms, the objects on $Y=X/\Gamma^+$ correspond with pseudoreal parabolic Higgs bundles. This is a generalization in the parabolic setup of the pseudoreal Higgs bundles studied by the first author in collaboration with Biswas & Hurtubise.