The Spacetime of Double Field Theory: Review, Remarks, and Outlook

We review double field theory (DFT) with emphasis on the doubled spacetime and its generalized coordinate transformations, which unify diffeomorphisms and b-field gauge transformations. We illustrate how the composition of generalized coordinate transformations fails to associate. Moreover, in dimensional reduction, the O(d, d) T-duality transformations of fields can be obtained as generalized diffeomorphisms. Restricted to a half-dimensional subspace, DFT includes ‘generalized geometry’, but is more general in that local patches of the doubled space may be glued together with generalized coordinate transformations. Indeed, we show that for certain T-fold backgrounds with nongeometric fluxes, there are generalized coordinate transformations that induce, as gauge symmetries of DFT, the requisite O(d, d;Z) monodromy transformations. Finally we review recent results on the α′ extension of DFT which, reduced to the half-dimensional subspace, yields intriguing modifications of the basic structures of generalized geometry.