Localization, Equivariant Cohomology, and Integration Formulas

Abstract

In this chapter we review the derivation of the DuistermaatHeckman integration formula and its path integral generalizations, and explain the underlying formalism of equivariant cohomology. We evaluate the quantum mechanical partition function for a general integrable model by localizing onto an ordinary integral of an equivariant characteristic class. We also describe the Mathai-Quillen formalism and its equivariant loop space extensions. We show how certain standard relations in classical Morse theory can be derived from this formalism, and generalize these relations to the infinite-dimensional and equivariant context. We also explain how Poincare supersymmetric quantum field theories can be formulated using equivariant cohomology in the loop space.