Mathematical Research Letters

Volume 11 (2004)

Number 4

Kirwan map and moduli space of flat connections

Pages: 419 – 433

DOI: http://dx.doi.org/10.4310/MRL.2004.v11.n4.a2

Author

Sébastien Racanière (Imperial College London)

Abstract

If $K$ is a compact Lie group and $g\geq 2$ an integer, the space $K^{2g}$ is endowed with the structure of a Hamiltonian space with a Lie group valued moment map $\Phi$. Let $\beta$ be in the centre of $K$. The reduction $\Phi^{-1}(\beta)/K$ is homeomorphic to a moduli space of flat connections. When $K$ is simply connected, a direct consequence of a recent paper of Bott, Tolman and Weitsman is to give a set of generators for the $K$-equivariant cohomology of $\Phi^{-1}(\beta)$. Another method to construct classes in $H^*_K(\Phi^{-1}(\beta))$ is by using the so called universal bundle. When the group is $\Sun$ and $\beta$ is a generator of the centre, these last classes are known to also generate the equivariant cohomology of $\Phi^{-1}(\beta)$. The aim of this paper is to compare the classes constructed using the result of Bott, Tolman and Weitsman and the ones using the universal bundle. In particular, I prove that the set of cohomology classes coming from the universal bundle is indeed a set of multiplicative generators for the cohomology of the moduli space. With $K=\Sun$, this is a new proof of the well-known construction of generators for the cohomology of the moduli space of semi-stable vector bundles with fixed determinant.

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