T-Duality via Gerby Geometry and Reductions

TL;DR

This paper explores topological T-duality for torus bundles with S^1-gerbes, showing how gerbe geometry determines T-duals and illustrating the theory with examples including Lie groups and flag manifolds.

Contribution

It introduces a geometric approach to T-duality via gerbe reductions, connecting gerbe data to the construction of T-dual pairs and providing explicit examples.

Findings

Gerbe geometry determines T-dual pairs via band reductions.

The canonical lifting gerbe on compact Lie groups relates to Langlands dual groups.

Explicit examples illustrate the theory's application to Lie groups and flag manifolds.

Abstract

We consider topological T-duality of torus bundles equipped with S^{1}-gerbes. We show how a geometry on the gerbe determines a reduction of its band to the subsheaf of S^{1}-valued functions which are constant along the torus fibres. We observe that such a reduction is exactly the additional datum needed for the construction of a T-dual pair. We illustrate the theory by working out the example of the canonical lifting gerbe on a compact Lie group which is a torus bundles over the associated flag manifold. It was a recent observation of Daenzer and van Erp (arXiv1211.0763) that for certain compact Lie groups and a particular choice of the gerbe, the T-dual torus bundle is given by the Langlands dual group.