1.“Higher category for L_infty groupoids”

Lie ∞-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie ∞-groupoids called "Lie ∞-groups" by integrating L∞-algebras.In order to study the compatibility between this integration procedure and the homotopy theory of L∞-algebras, we present a homotopy theory for Lie ∞-groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie ∞-groupoids do not form a category of fibrant objects (CFO), since the category of manifolds lacks pullbacks. Instead, we show that Lie ∞-groupoids form an "incomplete category of fibrant objects" in which the weak equivalences correspond to "stalkwise" weak equivalences of simplicial sheaves. This homotopical structure enjoys many of the same properties as a CFO, such as having, in the presence of functorial path objects, a convenient realization of its simplicial localization. We further prove that the acyclic fibrations are precisely the hypercovers, which implies that many of Behrend and Getzler's results also hold in this more general context. As an application, we show that homotopy equivalent L∞-algebras integrate to "Morita equivalent" Lie ∞-groups.

2.“String principal bundles and Courant algebroids”

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 their corresponding 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).