Noncommutative associative superproduct for general supersymplectic forms

Abstract

We define a noncommutative and nonanticommutative associative product for general supersymplectic forms, allowing the explicit treatment of non(anti)commutative field theories from general nonconstant string backgrounds like a graviphoton field. We propose a generalization of deformation quantizationà la Fedosov to superspace, which considers noncommutativity in the tangent bundle instead of base space, by defining the Weyl super product of elements of Weyl super algebra bundles. Super Poincaré symmetry is not broken and chirality seems not to be compromised in our formulation. We show that, for a particular case, the projection of the Weyl super product to the base space gives rise the Moyal product for non(anti)commutative theories.

Supermanifolds, Symplectic Geometry and Curvature

“…While we will not deepen into the physical applications, neither of this odd symplectic curvature nor supersymplectic forms in general (for this, see [1,2,4,9]), we will offer a detailed review of the mathematics involved in this construction under quite general conditions, avoiding excessive technicalities with the aim of making this topic available to a wider audience.…”