Global Seiberg–Witten Maps for U(n)-Bundles on Tori and T-duality

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.