Bouwknegt-Evslin-Mathai and Bunke-Schick proved that the twisted K-theory for the T-dual pairs are isomorphic. On the level of differential geometric objects, Cavalcanti-Gualtieri proved that the exact Courant algebroid associated to the T-dual $S^1$-gerbes are the same.
Now we extend this story Spin(n)-equivariantly. Leaving alone what the cohomological invariants should be, Baraglia and Hekmati showed that, on the level of differential geometric objects, the transitive Courant algebroids associated to both sides are isomorphic.
As we know, to a usual principal bundle, one can associate an Atiyah algebroid. For an S^1 gerbe, the higher version of an Atiyah algebroid is an exact Courant algebroid whose Severa class is the Dixmier-Douady class of the gerbe. Then, in the case of the string principal bundle, the higher/noncommutative Atiyah algebroid turns out to be a transitive Courant algebroid.
In this talk, we make the connection between transitive Courant algebroids and string principal bundles explicit and functorial by constructing a morphism between theircorresponding stacks.
This also explains why the obstruction to lifting a principal G-bundle to a principal String(G)-bundle (controlled by one half the Pontryagin class) coincides with the one for a twisted Courant algebroid to be Courant. (Based on joint work arXiv:1701.00959, with Yunhe Sheng and Xiaomeng Xu).