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T-Duality for Torus Bundles with H-Fluxes via Noncommutative Topology

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Abstract

It is known that the T-dual of a circle bundle with H-flux (given by a Neveu-Schwarz 3-form) is the T-dual circle bundle with dual H-flux. However, it is also known that torus bundles with H-flux do not necessarily have a T-dual which is a torus bundle. A big puzzle has been to explain these mysterious “missing T-duals.” Here we show that this problem is resolved using noncommutative topology. It turns out that every principal T2-bundle with H-flux does indeed have a T-dual, but in the missing cases (which we characterize), the T-dual is non-classical and is a bundle of noncommutative tori. The duality comes with an isomorphism of twisted K-theories, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced by an isomorphism of twisted cyclic homology.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Adelaide, Adelaide, SA, 5005, Australia

    Varghese Mathai

  2. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Jonathan Rosenberg

Authors
  1. Varghese Mathai
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  2. Jonathan Rosenberg
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Correspondence to Varghese Mathai.

Additional information

Communicated by A. Connes

VM was supported by the Australian Research Council.

JR was partially supported by NSF Grant DMS-0103647, and thanks the Department of Pure Mathematics of the University of Adelaide for its hospitality in January 2004, which made this collaboration possible.

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Mathai, V., Rosenberg, J. T-Duality for Torus Bundles with H-Fluxes via Noncommutative Topology. Commun. Math. Phys. 253, 705–721 (2005). https://doi.org/10.1007/s00220-004-1159-7

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  • DOI: https://doi.org/10.1007/s00220-004-1159-7

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