# Topological Quantum Field Theory Seminar

### Witten-Style Nonabelian Localization For a Noncompact Manifold

Witten in [1] offered a clever scheme to express certain integrals over a Hamiltonian (i.e., symplectic, with group action and a moment map) manifold as a sum of local contributions from the critical points of the square of the moment map. In particular this allows one to read off the ring structure of the cohomology of the symplectic reduction (when it is nice enough) from integrating equivariant cohomology classes in the original space. His elegant argument ignores most analytic subtleties and thus is purely heuristic, but Jeffrey and Kirwan in [2] were able to reproduce his key results in the compact case, by relating the question to one accessible by older abelian localization techniques. I will argue that the noncompact case is particularly important by relating to some outstanding cases, and that the abelian localization argument is unlikely to extend here. I will prove Witten's results rigorously using his version of nonabelian localization, and suggest ways to extend these results further.
1. E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), no. 4, 303-368.
hep-th/9204083
2. L. C. Jeffrey, F. Kirwan, Localization for nonabelian group action, Topology 34 (1995) no. 2, 291-327.
alg-geom/9307001