The Moyal cohomology of the spinning particle

TL;DR

This paper investigates the cohomology of the spinning particle model, demonstrating that replacing the Poisson bracket with the Moyal bracket removes unwanted cohomology classes in the Batalin-Vilkovisky formalism.

Contribution

It introduces a Moyal deformation approach to eliminate negative degree cohomology in the spinning particle model within the BV formalism.

Findings

Replacing Poisson with Moyal bracket removes negative degree cohomology.

The cohomology of the algebra of functions is nontrivial in all negative degrees.

Moyal deformation resolves the cohomology conjecture violation for the spinning particle.

Abstract

Felder and Kazhdan conjecture that the local cohomology in the classical Batalin-Vilkovisky formalism vanishes in sufficiently negative degrees. This hypothesis is violated by the $N=1$ spinning particle. By Barnich-Grigoriev, this cohomology is isomorphic to the cohomology of the algebra of functions on the differential graded symplectic supermanifold of the associated Batalin-Fradkin-Vilkovisky model. This cohomology is nontrivial in all negative degrees. We show in this article that replacement in this symplectic supermanifold of the Poisson bracket by the Moyal bracket eliminates these spurious cohomology classes.