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
Action of the Mapping Class Group on Character Varieties and Higgs Bundles cover
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We consider the action of a finite subgroup of the mapping class group Mod(S)\mathrm{Mod}(S) of an oriented compact surface SS of genus g2g \geqslant 2 on the moduli space R(S,G)\mathcal{R}(S,G) of representations of π1(S)\pi_1(S) in a connected semisimple real Lie group GG. Kerckhoff's solution of the Nielsen realization problem ensures the existence of an element JJ in the Teichmüller space of SS for which Γ\Gamma can be realised as a subgroup of the group of automorphisms of X=(S,J)X=(S,J) which are holomorphic or antiholomorphic. We identify the fixed points of the action of Γ\Gamma on R(S,G)\mathcal{R}(S,G) in terms of GG-Higgs bundles on XX equipped with a certain twisted Γ\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 GG is trivial, the fixed points can be described in terms of familiar objects on Y=X/Γ+Y=X/\Gamma^+, where Γ+Γ\Gamma^+\subset \Gamma is the maximal subgroup of Γ\Gamma consisting of holomorphic automorphisms of XX. If Γ=Γ+\Gamma=\Gamma^+ one obtains actual Γ\Gamma-equivariant GG-Higgs bundles on XX, which in turn correspond with parabolic Higgs bundles on Y=X/ΓY=X/\Gamma (this generalizes work of Nasatyr & Steer for G=SL(2,R)G=\mathrm{SL}(2,\mathbb{R}) and Boden, Andersen & Grove and Furuta & Steer for G=SU(n))G=\mathrm{SU}(n)). If on the other hand Γ\Gamma has antiholomorphic automorphisms, the objects on Y=X/Γ+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.

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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