Intersection Theory on Moduli Spaces of Holomorphic Bundles of Arbitrary Rank on a Riemann Surface

Abstract

Let n and d be coprime positive integers, and define M(n,d) to be the moduli space of (semi)stable holomorphic vector bundles of rank n, degree d and fixed determinant on a compact Riemann surface E. This moduli space is a compact Kifhler manifold which has been studied from many different points of view for more than three decades (see for instance Narasimhan and Seshadri [41]). The subject of this article is the characterization of the intersection pairings in the cohomology ring1 H* (M (n, d)). A set of generators of this ring was described by Atiyah and Bott in their seminal 1982 paper [2] on the YangMills equations on Riemann surfaces (where in addition inductive formulas for the Betti numbers of M (n, d) obtained earlier using number-theoretic methods [13], [25] were rederived). By Poincare duality, knowledge of the intersection pairings between products of these generators (or equivalently knowledge of the evaluation on the fundamental class of products of the generators) completely determines the structure of the cohomology ring. In 1991 Donaldson [15] and Thaddeus [47] gave formulas for the intersection pairings between products of these generators in H* (M (2, 1)) (in terms of Bernoulli numbers). Then using physical methods, Witten [50] found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in H* (M (n, d)) for general rank n. These generalized his (rigorously proved) formulas [49] for the symplectic volume of M (n, d): For instance, the symplectic volume of M (2, 1) is given by