Connectedness of Higgs bundle moduli for complex reductive Lie groups

TL;DR

This paper investigates the connectedness of the moduli space of G-Higgs bundles for complex reductive Lie groups, establishing that the number of components is determined by topological invariants, providing an alternative proof for flat G-connection moduli.

Contribution

It offers an intrinsic approach to determine the connectedness of G-Higgs bundle moduli spaces for complex reductive Lie groups, linking components to topological invariants.

Findings

Number of connected components indexed by topological invariants

Provides an alternative proof for connected semisimple groups

Extends understanding to non-connected reductive groups

Abstract

We carry an intrinsic approach to the study of the connectedness of the moduli space $\mathcal{M}_G$ of $G$-Higgs bundles, over a compact Riemann surface, when $G$ is a complex reductive (not necessarily connected) Lie group. We prove that the number of connected components of $\mathcal{M}_G$ is indexed by the corresponding topological invariants. In particular, this gives an alternative proof of the counting by J. Li of the number of connected components of the moduli space of flat $G$-connections in the case in which $G$ is connected and semisimple.