The Mathai-Quillen Formalism and Topological Field Theory

TL;DR

The notes address how cohomological topological field theories can be understood as regularized Euler numbers realized through the Mathai-Quillen Thom form. Starting from the finite‑dimensional case, they extend to infinite dimensions, defining $χ_s(E)$ and illustrating this with loop space, SUSY quantum mechanics, and gauge theories. The Atiyah–Jeffrey interpretation is developed to show Donaldson theory arises as the regularized Euler number of a bundle over ${\cal A}/{\cal G}$, with observables tied to BRST cohomology and localization arguments. The framework also yields 3D/topological gauge theories and connects flat connection moduli to Casson-type invariants, providing a unifying, differential‑geometric route to a broad class of cohomological theories.

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.