Generalized Classical BRST Cohomology and Reduction of Poisson Manifolds

TL;DR

This paper extends the classical BRST cohomology framework to handle the reduction of Poisson manifolds, including cases with second-class constraints, providing a new algebraic approach to Poisson reduction.

Contribution

It introduces a generalized BRST construction for Poisson manifold reduction, encompassing cases beyond symplectic reduction and including second-class constraints.

Findings

BRST cohomology vanishes in negative dimensions.

In zero dimension, BRST cohomology matches the algebra of smooth functions on the reduced Poisson manifold.

In general Poisson reduction, BRST cohomology differs from vertical differential forms cohomology.

Abstract

In this paper, we formulate a generalization of the classical BRST construction which applies to the case of the reduction of a poisson manifold by a submanifold. In the case of symplectic reduction, our procedure generalizes the usual classical BRST construction which only applies to symplectic reduction of a symplectic manifold by a coisotropic submanifold, \ie\ the case of reducible ``first class'' constraints. In particular, our procedure yields a method to deal with ``second-class'' constraints. We construct the BRST complex and compute its cohomology. BRST cohomology vanishes for negative dimension and is isomorphic as a poisson algebra to the algebra of smooth functions on the reduced poisson manifold in zero dimension. We then show that in the general case of reduction of poisson manifolds, BRST cohomology cannot be identified with the cohomology of vertical differential forms.