Global Seiberg-Witten maps for U(n)-bundles on tori and T-duality

Abstract

Seiberg-Witten maps are a well-established method to locally construct noncommutative gauge theories starting from commutative gauge theories. We revisit and classify the ambiguities and the freedom in the definition. Geometrically, Seiberg-Witten maps provide a quantization of bundles with connections. We study the case of U(n)-vector bundles on two-dimensional tori, prove the existence of globally defined Seiberg-Witten maps (induced from the plane to the torus) and show their compatibility with Morita equivalence.

28 pages. Revised version: sharpened in Sec. 4.3 the study of the Seiberg-Witten maps for sections in the adjoint, related to their ordering ambiguities; added sum of connections for tensor product bundles in Sec. 5; improved in Sec. 5.1 the compatibility between Seiberg-Witten map and T-duality transformations