Homological Reduction of Constrained Poisson Algebras

Abstract

The ``classical BRST construction'' as developed by Batalin-Fradkin-Vilkovisky is a homological construction for the reduction of the Poisson algebra $P = C^\infty (W)$ of smooth functions on a Poisson manifold $W$ by the ideal $I$ of functions which vanish on a constraint locus. This ideal is called first class if $I$ is closed under the Poisson bracket; geometers refer to the constraint locus as coisotropic. The physicists' model is crucially a differential Poisson algebra extension of a Poisson algebra $P$; its differential contains a piece which reinvented the Koszul complex for the ideal $I$ and a piece which looks like the Cartan-Chevalley-Eilenberg differential. The present paper is concerned purely with the homological (Poisson) algebraic structures, using the notion of ``model'' from rational homotopy theory and the techniques of homological perturbation theory to establish some of the basic results explaining the mathematical existence of the classical BRST-BFV construction. Although the usual treatment of BFV is basis dependent (individual constraints) and nominally finite dimensional, I take care to avoid assumptions of finite dimensionality and work more invariantly in terms of the ideal. In particular, the techniques are applied to the `irregular' case (the ideal is not generated by a regular sequence of constraints), although the geometric interpretation is less complete.