Equivariant Cohomology, BRST Quantization, and Analytic Localization: A Unified Framework

TL;DR

This paper unifies equivariant cohomology models with BRST quantization, demonstrating their connection through gauge-fixing procedures and providing an analytic proof of the ABBV localization formula, supported by explicit examples.

Contribution

It establishes a unified framework linking equivariant cohomology, BRST quantization, and localization, with explicit isomorphisms and analytic proofs.

Findings

Explicit isomorphism between Cartan and Weil models via Kalkman transformation

Identification of BRST complex with the Cartan model in gauge theories

Analytic proof of the ABBV localization formula

Abstract

This paper provides a detailed exposition of the two main models for equivariant cohomology -- the Cartan and Weil models -- and their explicit isomorphism via the Kalkman (Mathai--Quillen) transformation. We then connect this framework to the BRST quantization of gauge theories, showing how the BRST complex can be identified with the Cartan model. Viewing both the Kalkman transformation and Witten's Morse-theoretic deformation as gauge-fixing procedures leads naturally to the \emph{equivariant Witten deformation}. This combined perspective yields a transparent analytic proof of the Atiyah--Bott--Berline--Vergne (ABBV) localization formula for integrals of equivariantly closed forms.The theory is richly illustrated with computations on $\mathbb{CP}^1$ and $\mathbb{CP}^n$, supplemented by explicit coordinate calculations.