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

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.