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

Abstract

We consider the action of a finite subgroup of the mapping class group $\mathrm{Mod}(S)$ of an oriented compact surface $S$ of genus $g⩾2$ 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 Teichmüller 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. 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 $Y=X/{\Gamma }^{+}$, 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 $G=\mathrm{SL}(2,\mathbb{R})$ and Boden, Andersen & Grove and Furuta & Steer for $G=\mathrm{SU}(n))$. 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.