Localization and diagonalization: A review of functional integral techniques for low-dimensional gauge theories and topological field theories

Abstract

Localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories are reviewed. These are the functional integral counterparts of the Mathai–Quillen formalism, the Duistermaat–Heckman theorem, and the Weyl integral formula, respectively. In each case, the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups) is introduced, and the finite dimensional integration formulae described. Then some applications to path integrals are discussed and an overview of the relevant literature is given. The applications include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two-dimensional Yang–Mills theory.