Abstract
Topological T-duality is a transformation taking a gerbe on a principal torus bundle to a gerbe on a principal dual-torus bundle. We give a new geometric construction of T-dualization, which allows the duality to be extended in the following two directions. First, bundles of groups other than tori, even bundles of some nonabelian groups, can be dualized. Second, bundles whose duals are families of noncommutative groups (in the sense of noncommutative geometry) can be treated, though in this case the base space of the bundles is best viewed as a topological stack. Some methods developed for the construction may be of independent interest. These are a Pontryagin type duality that interchanges commutative principal bundles with gerbes, a nonabelian Takai type duality for groupoids, and the computation of certain equivariant Brauer groups.
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Acknowledgment
I would like to thank Oren Ben-Bassat, Tony Pantev, Michael Pimsner, Jonathan Rosenberg, Jim Stasheff, and most of all Jonathan Block, for advice and helpful discussions. I am also grateful to the Institut Henri Poincaré, which provided a stimulating environment for some of this research.
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Communicated by A. Connes
The research reported here was supported in part by National Science Foundation grants DMS-0703718 and DMS-0611653.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Daenzer, C. A Groupoid Approach to Noncommutative T-Duality. Commun. Math. Phys. 288, 55–96 (2009). https://doi.org/10.1007/s00220-009-0767-7
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DOI: https://doi.org/10.1007/s00220-009-0767-7