The classical master equation

TL;DR

This paper formalizes the construction of solutions to the classical master equation in algebraic geometry, establishing their uniqueness up to stable equivalence and exploring the associated BRST cohomology's geometric and algebraic properties.

Contribution

It introduces the notion of stable equivalence of solutions, proves existence and uniqueness up to this equivalence, and provides a geometric interpretation of BRST cohomology sheaves.

Findings

Existence and uniqueness of solutions up to stable equivalence.

BRST cohomology is independent of choices and uniquely determined.

Geometric interpretation of BRST cohomology sheaves as Lie-Rinehart algebra cohomology.

Abstract

We formalize the construction by Batalin and Vilkovisky of a solution of the classical master equation associated with a regular function on a nonsingular affine variety (the classical action). We introduce the notion of stable equivalence of solutions and prove that a solution exists and is unique up to stable equivalence. A consequence is that the associated BRST cohomology, with its structure of Poisson_0-algebra, is independent of choices and is uniquely determined up to unique isomorphism by the classical action. We give a geometric interpretation of the BRST cohomology sheaf in degree 0 and 1 as the cohomology of a Lie-Rinehart algebra associated with the critical locus of the classical action. Finally we consider the case of a quasi-projective varieties and show that the BRST sheaves defined on an open affine cover can be glued to a sheaf of differential Poisson_0-algebras.