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

TL;DR

This paper develops a theory of real Higgs bundles on Klein surfaces, establishing a correspondence with representations and analyzing their fixed point structures under involutions.

Contribution

It introduces real structures on Higgs pairs over Klein surfaces and proves a Hitchin--Kobayashi correspondence for them, linking moduli spaces with representation varieties.

Findings

Homeomorphism between real Higgs bundle moduli and orbifold fundamental group representations

Real Higgs bundles as fixed points of anti-holomorphic involutions

Extension of Higgs bundle theory to Klein surfaces with real structures

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 $G^{\mathbb{C}}$, constitute a particular class of examples of these pairs. The real structure in this case involves a conjugation of $G^{\mathbb{C}}$ 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 $G^{\mathbb{C}}$ and $X$.