The BRST Complex of Homological Poisson Reduction

TL;DR

This paper studies the BRST complex associated with coisotropic ideals in Poisson algebras, proving quasi-isomorphism invariance in symplectic cases and rigorously quantizing the complex without regularity assumptions.

Contribution

It establishes the invariance of BRST complexes under stable equivalence in symplectic cases and provides a rigorous quantization method without regularity constraints.

Findings

BRST complexes are quasi-isomorphic for the same coisotropic ideal in symplectic cases.

Cohomology of BRST complexes is canonically associated to the ideal J.

Quantization of the BRST complex is achieved rigorously with infinitely many ghost variables.

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.