Kirwan map and moduli space of flat connections

Abstract

Abstract. If K is a compact Lie group and g ≥ 2 an integer, the space K 2g is endowed with the structure of a Hamiltonian space with a Lie group valued moment map Φ. Let β be in the centre of K. The reduction Φ −1 (β)/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 Φ −1 (β).Another method to construct classes in H * K (Φ −1 (β)) is by using the so called universal bundle. When the group is SU(n) and β is a generator of the centre, these last classes are known to also generate the equivariant cohomology of Φ −1 (β). 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 = SU(n), 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.

Mixed Hodge polynomials of character varieties

“…We note that [43,4,54,59] work with the compact groups SU(n), however the arguments are correct with complex groups too. Another way to see that Jeffrey's formulas (4.1.5) , (4.1.6) and (4.1.7) for the universal classes are valid for G := SL(n, C) is to note that Lemma 4.1.12 below implies that the natural inclusion map of the twisted SU(n)-character variety into the twisted SL(n, C)-character variety M ′ n induces an isomorphism on (µ 2g n -invariant) cohomology below degree 2(g − 1)(n − 1) + 2.…”

“…However Jeffrey's formula for β n is trivially correct for the complex character varieties as we will see below. Another difference in our application of [43,4,54,59] is that we work on the level of cohomology instead of differential forms or cochains, but our cohomological interpretation of [43,4,54,59] is straightforward using the last paragraph in Construction 4.1.2.…”

“…First we know from (2.2.6) that the homogenous weight of ǫ j is 1. To determine the weight of the rest of the universal classes we will use Jeffrey's [43] group cohomology description of them as interpreted in [4,54,59].…”

Character varieties

“…They appear as moduli spaces of hyperbolic, projective, and other geometric structures on surfaces, c.f. [BIW,Go1,Go3,Go9,GM1,GM2,GW,JM,KM,Li,Sa,Wa], as well as and the moduli spaces of flat connections, holomorphic bundles, and of Higgs bundles, [AB, AMW, CHM, Da, DDW, DWWW, DWW, GGM, FL1, FL2, GGM, Hi1, Hi2, HLR, HT, Je2, Je3, JK, Ki, NS, Me, MW,Ol,Rac,Sim1,Sim2,Th1,Th2,Th3,Wi2,Wi3,Za]. These connections inspired an investigation of topology (and, more specifically, cohomology) of character varieties, in many of the papers cited here.…”

Witten's formulas for intersection pairings on moduli spaces of flat G-bundles

“…, n. Beauville [7] gave an alternative construction of the Atiyah-Bott classes, and Biswas-Raghavendra [10] obtained generators for moduli spaces of parabolic bundles. See also Racanière [35].…”