Natural curvature for manifest T-duality

Understand

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).