In this paper we define invariants of Hamiltonian group actions for central regular values of the moment map. The key hypotheses are that the moment map is proper and that the ambient manifold is symplectically aspherical. The invariants are based on the symplectic vortex equations. Applications include an existence theorem for relative periodic orbits, a computation for circle actions on a complex vector space, and a theorem about the relation between the invariants introduced here and the Seiberg-Witten invariants of a product of a Riemann surface with a two-sphere.where γ k,d denotes the spin c -structure determined by k and d. Moreover, if k > 2g − 2, then Φ M d,S ,µ d,S k,Σ (c m ) = Φ M k,Σ ,µ k,Σ d,S (c m ). Combining Theorems B and C one can recover the computation of the Seiberg-Witten invariants of product ruled surfaces by Li-Liu [21] and Ohta-Ono [28]. It is also interesting to examine the relation between our invariants and the Gromov-Witten invariants of the symplectic quotient M := M/ /G(τ ) := µ −1 (τ )/G whenever G acts freely on µ −1 (τ ). Such a relation was established in [15] under the hypothesis that the quotient is monotone. Under this condition (and hypotheses (H1 − 3)) it is shown in [15] that there exists a surjective ring homomorphism φ : H * G (M ) → QH * (M ) (with values in the quantum cohomology of the quotient) such that Φ M,µ−τ B,Σ

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).

The Fredholm property asserts that, for x ∈ M = S −1 (0), the vertical differential D x := DS(x) : T x B → E x is a Fredholm operator whose Fredholm index is independent of x. This implies that the differential of S, with respect to any trivialization of E, is Fredholm in a sufficiently small neighbourhood of M. The orientation hypothesis asserts that the determinant bundle is oriented over such a neighbourhood. We define the index of S by index(S) := index(D x) − dim G. This is the index of the elliptic complex 0 → g → T x B → E x → 0, where the map g → T x B is the infinitesimal action. G-moduli problems form a category as follows.

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