Natural curvature for manifest T-duality

We reformulate the manifestly T-dual description of the massless sector of the closed bosonic string, directly from the geometry associated with the (left and right) affine Lie algebra of the coset space Poincaré/Lorentz. This construction initially dou-bles not only the (spacetime) coordinates for translations but also those for Lorentz transformations (and their “dual”). As a result, the Lorentz connection couples di-rectly to the string (as does the vielbein), rather than being introduced ad hoc to the covariant derivative as previously. This not only reproduces the old definition of T-dual torsion, but automatically gives a general, covariant definition of T-dual curvature (but still with some undetermined connections)