Noncommutative line bundle and Morita equivalence

TL;DR

This paper explores the structure of abelian noncommutative gauge theories, introducing noncommutative line bundles and demonstrating Morita equivalence between different star products, thus advancing the understanding of noncommutative geometry.

Contribution

It introduces the concept of noncommutative line bundles with transition functions and proves Morita equivalence between star products via covariantizing maps.

Findings

Noncommutative line bundles have noncommutative transition functions.

The space of sections forms a projective module.

Star products related by covariantizing maps are Morita equivalent.

Abstract

Global properties of abelian noncommutative gauge theories based on $\star$-products which are deformation quantizations of arbitrary Poisson structures are studied. The consistency condition for finite noncommutative gauge transformations and its explicit solution in the abelian case are given. It is shown that the local existence of invertible covariantizing maps (which are closely related to the Seiberg-Witten map) leads naturally to the notion of a noncommutative line bundle with noncommutative transition functions. We introduce the space of sections of such a line bundle and explicitly show that it is a projective module. The local covariantizing maps define a new star product $\star'$ which is shown to be Morita equivalent to $\star$.