Equivariant and supersymmetric localization in QFT

TL;DR

This paper reviews equivariant localization techniques in quantum field theories with supersymmetry, highlighting their mathematical foundations and applications in computing partition functions and expectation values across various dimensions.

Contribution

It provides a comprehensive overview of equivariant localization formalism and its extension to infinite-dimensional QFT path integrals, with examples from supersymmetric theories.

Findings

Localization computes partition functions in supersymmetric theories.

Application of localization to index theorems and Wilson loops.

Formalism connects physical theories with cohomological models.

Abstract

Equivariant localization theory is a powerful tool that has been extensively used in the past thirty years to elegantly obtain exact integration formulas, in both mathematics and physics. These integration formulas are proved within the mathematical formalism of equivariant cohomology, a variant of standard cohomology theory that incorporates the presence of a symmetry group acting on the space at hand. A suitable infinite-dimensional generalization of this formalism is applicable to a certain class of Quantum Field Theories (QFT) endowed with supersymmetry. In this thesis we review the formalism of equivariant localization and some of its applications in Quantum Mechanics (QM) and QFT. We start from the mathematical description of equivariant cohomology and related localization theorems of finite-dimensional integrals in the case of an Abelian group action, and then we discuss their formal application to infinite-dimensional path integrals in QFT. We summarize some examples from the literature of computations of partition functions and expectation values of supersymmetric operators in various dimensions. For 1-dimensional QFT, that is QM, we review the application of the localization principle to the derivation of the Atiyah-Singer index theorem applied to the Dirac operator on a twisted spinor bundle. In 3 and 4 dimensions, we examine the computation of expectation values of certain Wilson loops in supersymmetric gauge theories and their relation to 0-dimensional theories described by "matrix models". Finally, we review the formalism of non-Abelian localization applied to 2-dimensional Yang-Mills theory and its application in the mapping between the standard "physical" theory and a related "cohomological" formulation.