For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, show that certain integrals of equivariant cohomology classes localize as a sum of contributions from these compact critical sets, and bound the contribution from each critical set. In the case (1) that the contribution from higher critical sets grows slowly enough that the overall integral converges rapidly and (2) that 0 is a regular value of the moment map, we recover Witten's result [E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303–368; http://xxx.lanl.gov/abs/hep-th/9204083] identifying the polynomial part of these integrals as the ordinary integral of the image of the class under the Kirwan map to the symplectic quotient.
Differential Geometry and its Applications
Sawin, Stephen F., "Witten's nonabelian localization for noncompact Hamiltonian spaces" (2007). Mathematics Faculty Publications. 17.
Sawin, Stephen F. 2007. Witten's nonabelian localization for noncompact Hamiltonian spaces. Differential Geometry and its Applications 25 (2), 191-206.