Noncommutative supergeometry, duality and deformations

Abstract

Following Lett. Math. Phys. 50 (1999) 309, 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, 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.