T-duality via gerby geometry and reductions

Abstract

We consider topological T-duality of torus bundles equipped with [Formula: see text]-gerbes. We show how a geometry on the gerbe determines a reduction of its band to the subsheaf of S1-valued functions which are constant along the torus fibers. 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 bundle over the associated flag manifold. It was a recent observation of Daenzer and van Erp [16] 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.