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.
Categories: math.DG math.AT