Finite group actions on Higgs bundle moduli spaces and twisted equivariant structures

TL;DR

This paper studies how finite group actions on a Riemann surface induce actions on Higgs bundle moduli spaces, revealing fixed points as moduli spaces with twisted equivariant structures and connecting them to twisted non-abelian Hodge theory.

Contribution

It introduces a framework for understanding fixed points of finite group actions on Higgs moduli spaces via twisted equivariant structures and non-abelian cohomology, extending the non-abelian Hodge correspondence.

Findings

Fixed points correspond to moduli of Higgs bundles with twisted equivariant structures.

Fixed points are parameterized by non-abelian first cohomology of the group.

A twisted non-abelian Hodge correspondence describes fixed points in representation spaces.

Abstract

We consider the moduli space ${\cal M}(G)$ of $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a semisimple complex Lie group, and study the action of a finite group $\Gamma$ on ${\cal M}(G)$ induced by a holomorphic action of $\Gamma$ on $X$ and $G$, and a character of $\Gamma$. The fixed-point subvariety for this action is given by a union of moduli spaces of $G$-Higgs bundles equipped with a certain twisted $\Gamma$-equivariant structure involving a $2$-cocycle of $\Gamma$ with values in the centre of $G$. This union is paremeterized by the non-abelian first cohomology set of $\Gamma$ in the adjoint group of $G$. We also describe the fixed points in the moduli space of representations of the fundamental group of $X$ in $G$, via a twisted equivariant version of the non-abelian Hodge correspondence, which involves the $\Gamma$-equivariant fundamental group of $X$.