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

Abstract

In a 1992 paper [41], Witten gave a formula for the intersection pairings of the moduli space of flat G-bundles over an oriented surface, possibly with markings. In this paper, we give a general proof of Witten's formula, for arbitrary compact, simple groups, and any markings for which the moduli space has at most orbifold singularities.

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].…”

The index formula for the moduli ofG-bundles on a curve

“…In Section 5, we show how Witten's integration formulae over M arise from our index formula in the large level limit; we only give full details for SL.2/. (The formulae were proven for SL.r/ by JeffreyKirwan [JK98] and, independently of our work but simultaneously, by Meinrenken [Mei05] for compact, 1-connected G.) Section 6 enhances our index formulae by incorporating Kähler differentials, needed in our next application in Section 7 to a conjecture of Newstead and Ramanan. The original version, proved by Gieseker [Gie84], asserted the vanishing of the top 2g 1 Chern classes of the moduli space of stable, odd degree vector bundles of rank 2 on †.…”

Geometry of the intersection ring and vanishing relations in the cohomology of the moduli space of parabolic bundles on a curve

“…For some other references on this subject, see [1,2,3,4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,28,29,30,31,33,35,36,37].…”