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

2004; Springer Science+Business Media; Volume: 253; Issue: 3 Linguagem: Inglês

10.1007/s00220-004-1159-7

1432-0916

Varghese Mathai, Jonathan Rosenberg,

Noncommutative and Quantum Gravity Theories

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.