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.
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Notes
There is a simple proof of (2.12), (2.13) [13]: multiplying the differential equations by \(\theta ^{\mu \nu }\) and analyzing them order by order yields
$$\begin{aligned} \theta ^{\mu \nu } { \partial \over \partial \theta ^{\mu \nu }} A_\rho ^{n+1}= & {} (n+1) A_\rho ^{n+1}= - \frac{\pi }{2} \theta ^{\mu \nu } \{ {{\hat{A}}}_{\mu }, \partial _{\nu } {{\hat{A}}}_\rho + {{\hat{F}}}_{\nu \rho } \}_\star ^n ,~~\\ \theta ^{\mu \nu } { \partial \over \partial \theta ^{\mu \nu }} \varepsilon ^{n+1}= & {} (n+1) \varepsilon ^{n+1}= - \frac{\pi }{2} \theta ^{\mu \nu } \{ {{\hat{A}}}_{\mu }, \partial _{\nu } {{\hat{\varepsilon }}}\}_\star ^n \end{aligned}$$since \(A_\rho ^{n+1}\) and \(\varepsilon ^{n+1}\) are homogeneous functions of \(\theta \) of order \(n+1\).
Since the bundle is a positive definite hermitian complex vector bundle and the algebra of continuous functions \(C(T)\) is a \(C^\star \)-algebra, we also have that \(\mathcal {E}_{n,m}\) is a Hilbert module over \(C(T)\).
With slight abuse of terminology, we call \(\mathbb {C}[[\theta ]]\)-modules (and \(\mathbb {C}[[\theta ]]\)-submodules) simply linear spaces (and subspaces).
There are different notions of Morita equivalence: The one just recalled for algebras (and more generally rings), a stronger notion for \(C^*\)-algebras, and, in the case of (multidimensional) tori, an even stronger one called complete Morita equivalence of smooth noncommutative tori [9] (called gauge Morita equivalence in [20]). These notions for two-dimensional tori are all equivalent (the bimodules \({\mathcal {E}}_{n,m}^\theta \) can be completed to full Hilbert modules providing \(C^*\)-algebra Morita equivalence, and they are Heisenberg modules with constant curvature connections that provide complete Morita equivalence).
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Acknowledgements
The authors are members of COST Action MP1405 QSpace - Quantum Structure of Spacetime and of INFN, CSN4, Iniziativa Specifica GSS, that have partially supported this project. This research has also a financial support of the Università del Piemonte Orientale. A.D. is grateful to Heriot-Watt University for hospitality. P.A. and A.D. acknowledge hospitality form LMU, Munich. P.A. is affiliated to INdAM, GNFM (Istituto Nazionale di Alta Matematica, Gruppo Nazionale di Fisica Matematica).
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Communicated by Boris Pioline.
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Aschieri, P., Deser, A. Global Seiberg–Witten Maps for U(n)-Bundles on Tori and T-duality. Ann. Henri Poincaré 20, 3197–3227 (2019). https://doi.org/10.1007/s00023-019-00823-1
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DOI: https://doi.org/10.1007/s00023-019-00823-1