Equivariant Cohomology, BRST Quantization, and Analytic Localization: A Unified Framework
Lixin Xu
TL;DR
The paper addresses the unification of Cartan/Weil models of equivariant cohomology with BRST/BV quantization and analytic localization. It develops the Kalkman transformation and Witten deformation as gauge-fixing procedures, culminating in an equivariant Witten deformation whose large-$t$ limit localizes integrals to fixed points. Key contributions include an explicit isomorphism between Cartan and Weil formalisms, a BRST/BV interpretation of equivariant cohomology, and a rigorous ABBV localization proof with concrete estimates and CP$^1$ and CP$^n$ illustrations, expressed via the formula $\displaystyle \int_{M} [\omega]_{n} = \sum_{p\in M^{G}} \frac{i_{p}^{*}\omega}{e_{G}(N_{p})}$. This unified framework clarifies the connections between topology, group actions, and supersymmetric quantum mechanics, with broad relevance to localization in topological field theories and beyond.
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.