Real Higgs pairs and Non-abelian Hodge correspondence on a Klein surface

2019/04/24 by Biswas, Indranil, Calvo, Luis Angel, Garcia-Prada, Oscar
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1904.10878

Abstract

We introduce real structures on L-twisted Higgs pairs over a compact Riemann surface equipped with an anti-holomorphic involution, and prove a Hitchin--Kobayashi correspondence for them. Real G-Higgs bundles, where G is a real form of a connected semisimple complex affine algebraic group G^ℂ, constitute a particular class of examples of these pairs. The real structure in this case involves a conjugation of G^ℂ commuting with the one defining the real form G. We establish a homeomorphism between the moduli space of real G-Higgs bundles and the moduli space of compatible representations of the orbifold fundamental group of X. Finally, we show how real G-Higgs bundles appear naturally as fixed points of certain anti-holomorphic involutions of the moduli space of G-Higgs bundles, that are constructed using the real structures on G^ℂ and X.

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