Noncommutative supergeometry, duality and deformations

Abstract

We introduce a notion of $Q$-algebra that can be considered as a generalization of the notion of $Q$-manifold (a supermanifold equipped with an odd vector field obeying $\{Q,Q\}=0$). We develop the theory of connections on modules over $Q$-algebras and prove a general duality theorem for gauge theories on such modules. This theorem containing as a simplest case $SO(d,d,\mathbf{Z})$-duality of gauge theories on noncommutative tori can be applied also in more complicated situations. We show that $Q$-algebras appear naturally in Fedosov construction of formal deformation of commutative algebras of functions and that similar $Q$-algebras can be constructed also in the case when the deformation parameter is not formal.

Extended version of hep-th/9912212