Title: String principal bundles and Courant algebroids,
Abstract: Just like Atiyah Lie algebroids encode the infinitesimal symmetries of
principal bundles, exact Courant algebroids are believed to encode the
infinitesimal symmetries of $S^1$-gerbes. At the same time, transitive Courant
algebroids may be viewed as the higher analogue of Atiyah Lie algebroids, and
the non-commutative analogue of exact Courant algebroids. In this article, we
explore what the "principal bundle" behind transitive Courant algebroids are,
and they turn out to be principal 2-bundles of string groups. First, we
construct the stack of principal 2-bundles of string groups with connection
data. We prove a lifting theorem for the stack of string principal bundles with
connections and show the multiplicity of the lifts once they exist. This is a
differential geometrical refinement of what is known for string structures by
Redden, Waldorf and Stolz-Teichner. We also extend the result of Bressler and
Chen-Stiénon-Xu on extension obstruction involving transitive Courant
algebroids to the case of transitive Courant algebroids with connections, as a
lifting theorem with the description of multiplicity once liftings exist. At
the end, we build a morphism between these two stacks. The morphism turns out
to be neither injective nor surjective in general, which shows that the process
of associating the "higher Atiyah algebroid" loses some information and at the
same time, only some special transitive Courant algebroids come from string
bundles.