Higgs bundles and the real symplectic group

TL;DR

This paper reviews the development of non-abelian Hodge theory linking surface group representations to Higgs bundle moduli spaces, and discusses recent work on maximal Sp(2n,R)-Higgs bundles.

Contribution

It generalizes non-abelian Hodge theory to real Lie groups and surveys recent advances in the moduli space of maximal Sp(2n,R)-Higgs bundles.

Findings

Established correspondence between surface group representations and G-Higgs bundles.

Extended non-abelian Hodge theory to real Lie groups.

Presented new results on the structure of maximal Sp(2n,R)-Higgs bundle moduli spaces.

Abstract

We give an overview of the work of Corlette, Donaldson, Hitchin and Simpson leading to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface and the moduli space of Higgs bundles. We then explain how this can be generalized to a correspondence between character varieties for representations of surface groups in real Lie groups G and the moduli space of G-Higgs bundles. Finally we survey recent joint work with Bradlow, Garc\'ia-Prada and Mundet i Riera on the moduli space of maximal Sp(2n,R)-Higgs bundles.