Abstracts for Higher Differential Geometry Seminar

Rogers has defined a class of L$_\infty$-algebras that are naturally associated with manifolds equipped with closed higher-degree forms, and that reduce to Poisson bracket Lie algebras in the case of symplectic manifolds. Here we show that these L$_\infty$-algebras can be naturally identified with the L$_\infty$-algebras of infinitesimal autoequivalences of higher prequantum bundles. In particular, they are a Kostant-Souriau-type L$_\infty$ extension of (higher) Hamiltonian symplectomorphisms.

 We study principal bundles for strict Lie $n$-groups over simplicial manifolds. Given a Lie group $G$, one can construct a principal $G$-bundle on a manifold $M$ by taking a cover $U$ of $M$, specifying a transition cocycle, and then quotienting $U\times G$ by the equivalence relation generated by the cocycle. We demonstrate the existence of an analogous construction for arbitrary strict Lie $n$-groups. As an application, we show how our construction leads to a simple finite dimensional model of the Lie 2-group String($n$).