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Higgs bundles and surface group representations in the real symplectic group

In this paper, we study the moduli space of representations of a surface group (that is, the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n, ℝ). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor–Wood‐type inequality. Our main result is a count of the number of connected components of the moduli space of maximal representations, that is, representations with maximal Toledo invariant. Our approach uses the non‐abelian Hodge theory correspondence proved in a companion paper (O. García‐Prada, P. B. Gothen and I. Mundet i Riera, The Hitchin–Kobayashi correspondence, Higgs pairs and surface group representations, Preprint, 2012, arXiv:0909.4487 [math.AG].) to identify the space of representations with the moduli space of polystable Sp(2n, ℝ)‐Higgs bundles. A key step is provided by the discovery of new discrete invariants of maximal representations. These new invariants arise from an identification, in the maximal case, of the moduli space of Sp(2n, ℝ)‐Higgs bundles with a moduli space of twisted Higgs bundles for the group GL(n, ℝ).