Connectedness of the moduli of Sp(2p,2q)-Higgs bundles

TL;DR

This paper proves the connectedness of the moduli space of Sp(2p,2q)-Higgs bundles over a compact Riemann surface, implying the connectedness of the corresponding representation space of the surface's fundamental group.

Contribution

It establishes the connectedness of the moduli space of Sp(2p,2q)-Higgs bundles using Morse-theoretic techniques, a novel application in this context.

Findings

Moduli space of Sp(2p,2q)-Higgs bundles is connected.

Connectedness of the representation space of the surface's fundamental group in Sp(2p,2q).

Application of Morse theory to Higgs bundle moduli spaces.

Abstract

Using the Morse-theoretic techniques introduced by Hitchin, we prove that the moduli space of $\Sp(2p,2q)$-Higgs bundles over a compact Riemann surface of genus $g\geq 2$ is connected. In particular, this implies that the moduli space of representations of the fundamental group of the surface in $\Sp(2p,2q)$ is connected.