The Mathai-Quillen Formalism and Topological Field Theory

TL;DR

This paper introduces the Mathai-Quillen formalism and its application to topological field theories, connecting finite and infinite dimensional vector bundles with cohomological and gauge theories.

Contribution

It provides an accessible explanation of the Mathai-Quillen formalism and its use in constructing topological gauge theories and interpreting supersymmetric quantum mechanics.

Findings

Regularized Euler numbers of infinite dimensional bundles are defined.

Supersymmetric quantum mechanics interpreted as Euler number of loop space.

Topological gauge theories constructed from infinite dimensional vector bundles.

Abstract

These lecture notes give an introductory account of an approach to cohomological field theory due to Atiyah and Jeffrey which is based on the construction of Gaussian shaped Thom forms by Mathai and Quillen. Topics covered are: an explanation of the Mathai-Quillen formalism for finite dimensional vector bundles; the definition of regularized Euler numbers of infinite dimensional vector bundles; interpretation of supersymmetric quantum mechanics as the regularized Euler number of loop space; the Atiyah-Jeffrey interpretation of Donaldson theory; the construction of topological gauge theories from infinite dimensional vector bundles over spaces of connections.