Noncommutative supergeometry, duality and deformations

TL;DR

This paper introduces $Q$-algebras as a generalization of $Q$-manifolds, develops their connection theory, proves a duality theorem for gauge theories on these modules, and explores their natural appearance in deformation quantization.

Contribution

It generalizes the concept of $Q$-manifolds to $Q$-algebras, establishes a duality theorem for gauge theories on them, and links these structures to deformation quantization.

Findings

Duality theorem for gauge theories on $Q$-algebras.

$Q$-algebras naturally arise in Fedosov deformation quantization.

Application to noncommutative tori and gauge theory dualities.

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 $SO(d,d,{\bf Z})$-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.