Finite group actions on Higgs bundle moduli spaces

Abstract

Let $\mathcal{M}(X,G)$ be the moduli space of $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a semisimple complex Lie group with centre $Z$. We describe the fixed points of the action of a finite group $\hat{I}“$ on $\mathcal{M}(X,G)$, induced by holomorphic actions of $\hat{I}“$ on $X$ and $G$, a character of $\hat{I}“$ and a homomorphism from $\hat{I}“$ to the group of $Z$-bundles over $X$. Two important ingredients in this study are provided by the theory of twisted $\hat{I}“$-equivariant bundles developed by Barajas--García-Prada--Gothen--Mundet i Riera, and the Prym--Narasimhan--Ramanan construction given by Barajas--García-Prada. Via the non-abelian Hodge correspondence, our results provide a description of the fixed-point subvarieties of certain finite group actions on the $G$-character variety of the fundamental group of $X$.

This new version of the paper merges results of the original 2020 version by the second and third authors, with results based on the 2023 PhD thesis of the first author (arXiv:2606.09710v1)