The BRST Complex of Homological Poisson Reduction

Abstract

BRST complexes are differential graded Poisson algebras. They are associated to a coisotropic ideal $J$ of a Poisson algebra $P$ and provide a description of the Poisson algebra $(P/J)^J$ as their cohomology in degree zero. Using the notion of stable equivalence introduced by Felder and Kazhdan, we prove that any two BRST complexes associated to the same coisotropic ideal are quasi-isomorphic in the case $P = \mathbb{R}[V]$ where $V$ is a finite-dimensional symplectic vector space and the bracket on $P$ is induced by the symplectic structure on $V$. As a corollary, the cohomology of the BRST complexes is canonically associated to the coisotropic ideal $J$ in the symplectic case. We do not require any regularity assumptions on the constraints generating the ideal $J$. We finally quantize the BRST complex rigorously in the presence of infinitely many ghost variables and discuss uniqueness of the quantization procedure.