Vanishing of intersection numbers on the moduli space of Higgs bundles

In this paper we consider the topological side of a problem which is the analogue of Sen’s S-duality testing conjecture for Hitchin’s moduli space M of rank 2 stable Higgs bundles of fixed determinant of odd degree over a Riemann surface Σ. We prove that all intersection numbers in the compactly supported cohomology ofM vanish, i.e. “there are no topological L harmonic forms on M”. This result generalizes the well known vanishing of the Euler characteristic of the moduli space of rank 2 stable bundles N of fixed determinant of odd degree over Σ. Our proof shows that the vanishing of all intersection numbers of H∗ cpt(M) is given by relations analogous to the Mumford relations in the cohomology ring of N .