Semidensities, second-class constraints, and conversion in anti-Poisson geometry

Abstract

We consider Khudaverdian’s geometric version of a Batalin–Vilkovisky (BV) operator ΔE in the case of a degenerate anti-Poisson manifold. The characteristic feature of such an operator (aside from being a Grassmann-odd, nilpotent, second-order differential operator) is that it sends semidensities to semidensities. We find a local formula for the ΔE operator in arbitrary coordinates. As an important application of this setup, we consider the Dirac antibracket on an antisymplectic manifold with antisymplectic second-class constraints. We show that the entire Dirac construction, including the corresponding Dirac BV operator ΔED, exactly follows from conversion of the antisymplectic second-class constraints into first-class constraints on an extended manifold.