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

TL;DR

Localization techniques for functional integrals in low-dimensional gauge theories and topological field theories are unified under three frameworks: the Mathai-Quillen formalism for Euler classes, equivariant localization (Duistermaat–Heckman), and the Weyl integral formula. The paper shows how infinite-dimensional path integrals can be treated as regularized finite-dimensional objects, yielding exact results in SUSY quantum mechanics, index theorems, and 2D Yang–Mills by reducing to moduli-space data, fixed points, or abelian sectors. It provides concrete constructions such as the Mathai-Quillen Euler form e_{s,∇}, the loop-space regularized Euler number χ_V, Niemi–Tirkkonen-like localization, and the Weyl diagonalization framework, connecting topological invariants with path integrals. Together these tools illuminate the deep connection between twisted supersymmetric theories and topological field theories and offer practical computational methods for moduli-space problems and gauge theory observables.

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.