  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 Poincare/Lorentz. This
construction initially doubles not only the (spacetime) coordinates for
translations but also those for Lorentz transformations (and their dual). As a
result, the Lorentz connection couples directly 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).
