Symplectic Forms on Moduli Spaces of Flat Connections on 2-manifolds

Let $G$ be a compact connected semisimple Lie group. We extend the techniques of Weinstein [W] to give a construction in group cohomology of symplectic forms $\omega$ on \lq twisted' moduli spaces of representations of the fundamental group $\pi$ of a 2-manifold $\Sigma$ (the smooth analogues of ${\rm Hom} (\pi_1(\Sigma), G)/G$) and on relative character varieties of fundamental groups of 2-manifolds. We extend this construction to exhibit a symplectic form on the extended moduli space [J1] (a Hamiltonian $G$-space from which these moduli spaces may be obtained by symplectic reduction), and compute the moment map for the action of $G$ on the extended moduli space.Comment: 12 pages, LaTex version 2.09 This paper will appear in Proceedings of the Georgia International Topology Conference (Athens, GA, August 1993), ed. W. Kazez (International Press). The present version (April 1996) is essentially the version that will appear in print. Several sign errors have been corrected. Some errors have been corrected in the proof of nondegeneracy of the symplectic form (Section 5) on an open dense set in the extended moduli spac

Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface

We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of genus g greater than or equal to 2. We also use these formulas for intersection numbers to obtain a proof of the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).Comment: 77 pages, LaTeX version 2.09. This is the text of the revised version which will appear in Annals of Mathematics. An error in Section 2 of the previous version has been correcte