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

TL;DR

This paper reviews localization techniques for functional integrals, highlighting their mathematical foundations and applications in topological field theories and low-dimensional gauge theories, including supersymmetric quantum mechanics and 2D Yang-Mills.

Contribution

It provides a comprehensive overview of mathematical background, finite dimensional formulas, and applications of localization methods in various low-dimensional gauge and topological theories.

Findings

Clarifies the mathematical foundations of localization techniques.

Summarizes applications to supersymmetric quantum mechanics and 2D Yang-Mills.

Connects formal mathematical concepts with practical path integral computations.

Abstract

We review 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. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaat-Heckman theorem, and the Weyl integral formula respectively. In each case, we first introduce the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups), and describe the finite dimensional integration formulae. We then discuss some applications to path integrals and give an overview of the relevant literature. The applications we deal with include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two-dimensional Yang-Mills theory.