Vanishing of intersection numbers on the moduli space of Higgs bundles

Details Vanishing of intersection numbers on the moduli space of Higgs bundles Vanishing of intersection numbers on the moduli space of Higgs bundles

1998-05-15

arxiv.org/abs/math/9805071v3

Adv.Theor.Math.Phys.2:1011-1040,1998

Mathematics

Algebraic Geometry

30 pages (published version)

Scientific paper

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 of rank 2 stable Higgs bundles of fixed determinant of odd degree over a Riemann surface. We prove that all intersection numbers in the compactly supported cohomology vanish, i.e. "there are no topological L^2 harmonic forms on Hitchin's space". This result generalizes the well known vanishing of the Euler characteristic of the moduli space of rank 2 stable bundles of fixed determinant of odd degree over the given Riemann surface. Our proof shows that the vanishing of all intersection numbers in the compactly supported cohomology of Hitchin's space is given by relations analogous to Mumford's relations in the cohomology ring of the moduli space of stable bundles.

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