T-duality for torus bundles via noncommutative topology

TL;DR

This paper extends T-duality to torus bundles with H-flux by employing noncommutative topology, revealing that missing T-duals are noncommutative torus bundles with preserved twisted K-theory isomorphisms.

Contribution

It demonstrates that all principal 2-torus-bundles with H-flux have T-duals within noncommutative topology, resolving the puzzle of missing classical T-duals.

Findings

Missing T-duals are noncommutative torus bundles.

Twisted K-theory isomorphism is preserved in noncommutative T-duality.

Twisted cyclic homology replaces twisted cohomology in the noncommutative setting.

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 2-torus-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.