**Title:** Flat connections on Riemann surfaces

**Speaker:** Professor Lisa Jeffrey

**Speaker Info:** McGill University

**Brief Description:**

**Special Note**:

**Abstract:**

Let G be a compact connected Lie group and S a compact Riemann surface. The space M(S) of conjugacy classes of representations of the fundamental group of S into G is identified (via a fundamental 1965 theorem of Narasimhan and Seshadri) with a moduli space of semistable holomorphic bundles on S In gauge theory it has a third description as the space of gauge equivalence classes of flat connections on a principal G-bundle over S Although M(S) is not smooth, if n and d are any pair of coprime positive integers there is a smooth compact Kaehler manifold M(n,d) which is a natural analogue of the singular space M(S) associated to G = SU(n) .The generators of the cohomology ring of M(n,d) were identified in a seminal 1982 paper of Atiyah and Bott, in which the Betti numbers of M(n,d) (in other words the dimensions of the cohomology groups as vector spaces) were determined. Only recently however have the intersection numbers between products of powers of these generators (which encode the structure of the cohomology ring) been understood. In 1992 Witten found formulas for these intersection numbers (using methods from quantum field theory): in joint work with F. Kirwan (available electronically as alg-geom/9608029) we give a mathematical proof of these formulas using methods from symplectic geometry.

This lecture will describe the context in which these moduli spaces arise and outline the origins and the consequences of the formulas determining their cohomology.

Copyright © 1997-2024 Department of Mathematics, Northwestern University.