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Analogous to how topological T-duality describes the aspects of T-duality that are not related to the pseudo-Riemannian metric and the background gauge field, differential T-duality still ignores the metric, but takes the background Kalb-Ramond field and its effect on the RR-fields into account: it is a refinement of topological T-duality from cohomology to differential cohomology.
This is also called geometric T-duality (KahleValentino), though this is a slight misnomer, since present technnology for the differential refinement does not take the metric into account.
We discuss here how the notion of differential T-duality pairs is a special case of the notion of twisted differential c-structures.
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for the moment see the remarks included in T-Duality and Differential K-Theory
differential T-duality
The first proposal for a description of differential T-duality is
A review is in section 7.4 of