Title: The Moyal bracket and quantization of the spinning particle in the Batalin-Vilkovisky formalism
The spinning particle is a one-dimensional toy model for the Neveu-Ramon-Schwartz superstring: it is obtained by coupling supersymmetric mechanics to a supergravity background on the world line. In earlier work, I discovered that the Batalin-Vilkovisky formalism for this model has a serious difficulty: the theory has very large local cohomology, unbounded below in degree, in violation of expectations. This local cohomology is isomorphic to the cohomology of the algebra of functions on a differential graded symplectic supermanifold: this is often referred to as the Batalin-Fradkin-Vilkovisky formalism of the theory, and bears the same relationship to the Batalin-Vilkovisky formalism as the Hamiltonian formalism bears to the Lagrangian formalism.
In this talk, I will calculate the cohomology of the BFV theory after quantization, which I perform by replacing the Poisson bracket by the Moyal bracket. This turns out to eliminate the parasitic cohomology classes. Conjecturally, a similar result is expected in the Batalin-Vilkovisky formalism for the NSR superstring.