Noncommutative supergeometry and duality

TL;DR

This paper introduces Q-algebras as a generalization of Q-manifolds, develops connection theory on modules over these algebras, and proves a duality theorem encompassing noncommutative torus gauge theories.

Contribution

It presents a new framework of Q-algebras, extending supergeometry, and establishes a duality theorem for gauge theories on these modules, including noncommutative tori.

Findings

Established a duality theorem for gauge theories on Q-algebra modules.

Generalized the concept of Q-manifolds to Q-algebras.

Connected the theory to noncommutative tori 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 contains as a simplest case SO(d,d,{\bf Z})-duality of gauge theories on noncommutative tori.