Noncommutative line bundle and Morita equivalence. (English) Zbl 1036.53070

In this letter, the authors discuss the global properties of abelian noncommutative gauge theories based on \(*\) products which are deformation quantizations of arbitrary Poisson structures over some manifolds M. Starting from classical (complex) line bundles over a manifold \(M\), the authors give an explicit form of finite noncommutative gauge transformations and develop the concept of a noncommutative line bundle in the sense of deformation quantization. They study the essential properties of a noncommutative line bundle in the framework of deformation quantization. In the course of their investigations a new star product \( *'\) is defined by means of the local covariantizing maps, the consistency condition for finite noncommutative gauge transformations and its explicit solution in the abelian case are obtained. Moreover, it is shown that the local existence of invertible covariantizing maps which are closely related to the Seiberg-Witten map, naturally leads to the notion of noncommutative line bundle with noncommutative transition functions.
Further, the authors introduce the space of sections of such a line bundle and describe the notion of connection and its curvature \( K\) on such a space. Thereafter, it is precisely shown that the space of sections is, in fact, a projective module. Meanwhile, it is also pointed out that the curvature \( K \) of the space of sections of the noncommutative line bundle measures the difference between two star products \( * \) and \( * '\) . Finally it is shown that the star product \( *' \) is Morita equivalent to \( * \).


53D55 Deformation quantization, star products
16D90 Module categories in associative algebras
55R65 Generalizations of fiber spaces and bundles in algebraic topology
58J42 Noncommutative global analysis, noncommutative residues
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