Speaker: Ezra Getzler

Title:BV cohomology and the Moyal bracket

Abstract: The N=1 spinning particle is the one-dimensional worldsheet
reduction of the field theory underlying the N=1 RNS superstring: it
consists of a supersymmetric particle moving in a general Riemannian
manifold M (possibly with a background magnetic field) coupled to
supergravity on the worldline. It is actually an example of an AKSZ
field theory, whose associated target supermanifold is the phase space
of the Dirac operator, with coordinates Here are the momenta with
respect to a moving frame on M, are the fermionic coordinates, and are
the fermionic and bosonic ghosts for local supersymmetry, and and are
the corresponding momenta. This phase space carries a cohomological
vector field Q, the Hamiltonian vector field for the Hamiltonian

(This phase space is also called BFV theory.) In earlier work, we
calculated the cohomology of the associated complex of functions of the
coordinates : this was used to calculate the BV cohomology of the N=1
spinning particle. In past work, we showed that surprisingly, there is a
large amount of cohomology in negative degrees, but this cohomology does
not appear to be associated with higher ghosts as is more usual in the
BV formalism. The culprit seems to be the ghost sector of the local
supersymmetry; perhaps this is an echo of picture-changing in the RNS
superstring.

In this talk, we replace the Poisson bracket on this phase space by the
Moyal bracket of first-quantization. It turns out that this cohomology
is much better behaved: the cohomology in negative degrees disappears.
This seems to point to the need for first quantization of the BFV theory
to completely describe local supersymmetry in the BV formalism.