The Moyal cohomology of 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.