CMSA Quantum Field Theory and Physical Mathematics Seminar: The Moyal bracket and the BV cohomology of the spinning particle

The spinning particle is the one-dimensional reduction of the Neveu-Schwartz-Ramond superstring. It consists of a supersymmetric particle moving in a one-dimensional supergravity background, and its quantization is the Hilbert superspace of harmonic spinors. (These models are classified by N, the number of copies of fermionic fields. In this talk, N=1. The extension to N=2 is work in progress with Ivo.) It is actually an AKSZ model (so a generalization of one-dimensional Chern-Simons), and so associated to a differential graded symplectic supermanifold, by which we mean a pair (ω,Q), where ω is a(n exact) symplectic form and Q is an odd function of degree 1. The cohomology of the ring of functions of this supermanifold with differential the Poisson bracket with Q determines the classical BV cohomology of the spinning particle, so is important for understanding perturbative BV quantization of this model. I calculated this cohomology in earlier work for N=1, and showed that it is somewhat bizarre, with two series of cohomology classes in arbitrary negative degrees, each a copy of the functions on the target manifold.

In the study of quantum BFV, we should instead consider the Moyal bracket on the target, and lift Q to an element Q satisfying [Q,Q]=0. The cohomology of the differential [Q,-] is the Moyal cohomology of the differential graded symplectic supermanifold. (This lift corresponds to the choice of a Spinc structure on the target manifold.) In this talk, I prove that the Moyal cohomology, unlike the Poisson cohomology, is well-behaved: in the spectral sequence from Poisson to Moyal cohomology, the extra cohomology classes of negative degree cancel each other pairwise at the E1 page.