Finite group actions on Higgs bundle moduli spaces

Abstract

Let ${\cal M}(X,G)$ 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 $Γ$ on ${\cal M}(X,G)$, induced by holomorphic actions of $Γ$ on $X$ and $G$, a character of $Γ$ and a homomorphism from $Γ$ to the group of $Z$-bundles over $X$. Two important ingredients in this study are provided by the theory of twisted $Γ$-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 $X$.