zbMATH — the first resource for mathematics

Homological reduction of constrained Poisson algebras. (English) Zbl 0874.58020
The “classical BRST construction”, at least as developed by Batalin-Fradkin-Vilkovisky and phrased in terms of constraints, is a homological construction for performing the reduction of the Poisson algebra \(C^\infty(W)\) of smooth functions on a Poisson manifold \(W\) by the ideal \(I\) of functions which vanish on a constraint locus. A set of generators \(\phi_\alpha\) for this ideal are referred to as constraints; Dirac calls the constraints first class if \(I\) is closed under the Poisson bracket and geometers refer to the constraint locus as coisotropic. (If the \(\mathbb{R}\)-linear span of the \(\phi_\alpha\) is closed under the bracket, physicists call the \(\phi_\alpha\) ‘close’ to a Lie algebra; in the more general ‘open’ first class case homological techniques are even more important.) The classical BRST-BFV construction has, in the nice cases, the same cohomology as this complex of longitudinal forms along the leaves of the Hamiltonian foliation.
The physicists’ model is itself crucially a Poisson algebra extension of a Poisson algebra \(P\) and its differential contains a piece which reinvented the Koszul complex for the ideal \(I\). The differential also contains a piece which looks like the Cartan-Chevalley-Eilenberg differential.
The present paper is concerned purely with the relevant homological (Poisson) algebraic structures. I adapt the notion of “model” from rational homotopy theory and use the techniques of homological perturbation theory to establish some of 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 work more invariantly in terms of the ideal generated by the constraints and take care to avoid assumptions of finite-dimensionality.

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI