# Action of the Mapping Class Group on Character Varieties and Higgs Bundles

### Oscar Garcia-Prada

Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Nicolás Cabrera, 13-15, 28049 Madrid, Spain### Graeme Wilkin

Department of Mathematics, University of York YO10 5DD, United Kingdom

## Abstract

We consider the action of a finite subgroup of the mapping class group $Mod(S)$ of an oriented compact surface $S$ of genus $g⩾2$ on the moduli space $R(S,G)$ of representations of $π_{1}(S)$ 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 Teichmüller space of $S$ for which $Γ$ 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 $Γ$ on $R(S,G)$ in terms of $G$-Higgs bundles on $X$ equipped with a certain twisted $Γ$-equivariant structure, where the twisting involves abelian and non-abelian group cohomology simultaneously. These, in turn, correspond to certain representations of the orbifold fundamental group. 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 $Y=X/Γ_{+}$, where $Γ_{+}⊂Γ$ is the maximal subgroup of $Γ$ consisting of holomorphic automorphisms of $X$. If $Γ=Γ_{+}$ one obtains actual $Γ$-equivariant $G$-Higgs bundles on $X$, which in turn correspond with parabolic Higgs bundles on $Y=X/Γ$ (this generalizes work of Nasatyr & Steer for $G=SL(2,R)$ and Boden, Andersen & Grove and Furuta & Steer for $G=SU(n))$. If on the other hand $Γ$ has antiholomorphic automorphisms, the objects on $Y=X/Γ_{+}$ 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.

## Cite this article

Oscar Garcia-Prada, Graeme Wilkin, Action of the Mapping Class Group on Character Varieties and Higgs Bundles. Doc. Math. 25 (2020), pp. 841–868

DOI 10.4171/DM/764