Connectedness of the moduli of \(\mathrm{Sp}(2p,2q)\)-Higgs bundles (Q2922458)

The aim of this paper is to use the Morse-theoretic techniques introduced by \textit{N. J. Hitchin} [Proc. Lond. Math. Soc. (3) 55, 59--126 (1987; Zbl 0634.53045)], to prove that the moduli space \(\mathcal{M}_{\mathrm{Sp}(2p,2q)}\) of \(\mathrm{Sp}(2p,2q)\)-Higgs bundles over a compact Riemann surface of genus \(g>2\) is connected (here \(\mathrm{Sp}(2p,2q)\) is the real form of \(\mathrm{Sp}(2p + 2q,\mathbb{C})\)). In particular, this implies that the moduli space of representations of the fundamental group of the surface in \(\mathrm{Sp}(2p, 2q)\) is connected. The main results of this paper are consistent with the recent results independently obtained by \textit{L. Schaposnik} in her PhD. Thesis [Spectral data for \(G\)-Higgs bundles, University of Oxford, 2012; preprint, \url{arXiv:1301.1981}], using other methods, namely through the study of the Hitchin map.