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

TL;DR

This paper studies the construction of noncommutative gauge theories via Seiberg-Witten maps on U(n)-bundles over tori, establishing their global existence and relation to T-duality and Morita equivalence.

Contribution

It proves the existence of globally defined Seiberg-Witten maps for U(n)-bundles on tori and analyzes their compatibility with Morita equivalence.

Findings

Existence of global Seiberg-Witten maps on tori.

Compatibility with Morita equivalence.

Classification of ambiguities in Seiberg-Witten maps.

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.