Equivariant Localization and Stationary Phase

Abstract

Equivariant cohomology in general and the equivariant localization theorems in particular have taken on a role of increasing significance in theoretical physics of late (see e. g. [3], [4] and [10]). These lectures are an attempt to provide a self-contained and elementary introduction to the Cartan model of equivariant cohomology, a complete proof of the simplest of the localization theorems, and, as an application, a proof of the famous Duistermaat–Heckman theorem on exact stationary phase approximations. 1. Stationary Phase Approximation We consider a compact, oriented, smooth manifold M of dimension n = 2k and denote by ν a volume form on M . Suppose H : M → R is a Morse function on M , i. e., a smooth function whose critical points p (dH(p) = 0) are all nondegenerate (this means that the Hessian Hp : Tp(M)×Tp(M)→ R, defined by Hp(Vp,Wp) = Vp(W (H)), where Vp,Wp ∈ Tp(M) and W is a vector field on M with W (p) =Wp, is a nondegenerate bilinear form). Finally, let T denote some real parameter. We consider the integral ∫