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

Hausel T. 1998. Vanishing of intersection numbers on the moduli space of Higgs bundles. Advances in Theoretical and Mathematical Physics. 2(5), 1011–1040.

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Abstract

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 ∑. We prove that all intersection numbers in the compactly supported cohomology of M vanish, i.e. "there are no topological L2 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 ∑. 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.

Publishing Year

Date Published

1998-09-01

Journal Title

Advances in Theoretical and Mathematical Physics

Acknowledgement

First of all I would like to thank my supervisor Nigel Hitchin for suggesting Problem 1, and for his help and
encouragement. I am grateful to Michael Thaddeus for his inspiring paper [Thai], enlightening communications and his constant interest in my work. I am also indebted to Manfred Lehn for the idea of the proof of Theorem 6.2. I have found
conversations with Michael Atiyah, Frances Kirwan and Graeme Segal very stimulating. I thank the Mathematical Institute and St. Catherine's College, Oxford for their hospitality during the preparation of this work. Finally I thank Trinity College, Cambridge for financial support.

Volume

2

Issue

5

Page

1011 - 1040

ISSN

IST-REx-ID

### Cite this

Hausel T. Vanishing of intersection numbers on the moduli space of Higgs bundles.

*Advances in Theoretical and Mathematical Physics*. 1998;2(5):1011-1040. doi:10.4310/ATMP.1998.v2.n5.a3Hausel, T. (1998). Vanishing of intersection numbers on the moduli space of Higgs bundles.

*Advances in Theoretical and Mathematical Physics*. International Press. https://doi.org/10.4310/ATMP.1998.v2.n5.a3Hausel, Tamás. “Vanishing of Intersection Numbers on the Moduli Space of Higgs Bundles.”

*Advances in Theoretical and Mathematical Physics*. International Press, 1998. https://doi.org/10.4310/ATMP.1998.v2.n5.a3.T. Hausel, “Vanishing of intersection numbers on the moduli space of Higgs bundles,”

*Advances in Theoretical and Mathematical Physics*, vol. 2, no. 5. International Press, pp. 1011–1040, 1998.Hausel, Tamás. “Vanishing of Intersection Numbers on the Moduli Space of Higgs Bundles.”

*Advances in Theoretical and Mathematical Physics*, vol. 2, no. 5, International Press, 1998, pp. 1011–40, doi:10.4310/ATMP.1998.v2.n5.a3.**All files available under the following license(s):**

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arXiv math/9805071