Semidensities, Second-Class Constraints and Conversion in Anti-Poisson Geometry

TL;DR

This paper explores a geometric version of the BV operator on degenerate anti-Poisson manifolds, providing a local formula and applying it to Dirac antibrackets with second-class constraints, linking conversion to first-class constraints.

Contribution

It introduces a local formula for Khudaverdian's BV operator on degenerate anti-Poisson manifolds and connects the Dirac antibracket construction to constraint conversion.

Findings

Derived a local coordinate formula for elta_E operator.

Showed the Dirac BV operator results from converting second-class to first-class constraints.

Linked the geometric BV operator to the Dirac antibracket framework.

Abstract

We consider Khudaverdian's geometric version of a Batalin-Vilkovisky (BV) operator \Delta_E 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 \Delta_E 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 \Delta_{E_D}, exactly follows from conversion of the antisymplectic second-class constraints into first-class constraints on an extended manifold.