String principal bundles and Courant algebroids

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. In this talk, we'll explain the stack of transitive Courant algebroids built from local data. Then, in the case of the string principal bundle, the higher/noncommutative Atiyah algebroid turns out to be a transitive Courant algebroid. This 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. (Joint work in progress with Yunhe Sheng and Xiaomeng Xu).