Ramond-Ramond cohomology and O(D, D) T-duality

In the name of supersymmetric double field theory, superstring effective actions can be reformulated into simple forms. They feature a pair of vielbeins corresponding to the same spacetime metric, and hence enjoy double local Lorentz symmetries. In a manifestly covariant manner - with regard to O(D, D) T-duality, diffeomorphism, B-field gauge symmetry and the pair of local Lorentz symmetries we incorporate R-R potentials into double field theory. We take them as a single object which is in a bi-fundamental spinorial representation of the double Lorentz groups. We identify cohomological structure relevant to he field strength. A priori, the R-R sector as well as all the fermions are O(D, D), singlet. Yet, gauge fixing the two vielbeins equal to each other modifies the O(D, D), transformation rule to call for a compensating local Lorentz rotation, such that the R-R potential may turn into an O(D, D) spinor and T-duality can flip the chirality exchanging type IIA and IIB supergravities.