Abstract:
Let (M,\omega) be a symplectic manifold, and suppose M is equipped with a Hamiltonian action of a compact Lie group G. If G=S^1, a theorem of T. Frankel shows that the moment map for this group action is a perfect Morse-Bott function on M, in both ordinary and equivariant cohomology. During the 1980's, this result of Frankel's was generalized in surprising ways, giving deep results in gauge theory and other areas. We review the work of Atiyah and Bott on two-dimensional gauge theory and the Kirwan surjectivity theorem, which shows, by a Morse-theoretic argument, that the equivariant cohomology H_G(M) surjects onto the cohomology of the Marsden-Weinstein (or GIT) quotient M//G. If time remains we will discuss recent generalizations of this result to actions of loop groups, and also the failure of surjectivity in the hyperkahler case of the Hitchin moduli space of pairs.
