An article of ours
-algebra connections and applications to String- and Chern-Simons -transport
in Quantum Field Theory, Birkhäuser (2009), 303-424, DOI: 10.1007/978-3-7643-8736-5_17
Abstract We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L-∞ algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their higher parallel transport.
It is known that over a D-brane the Kalb-Ramond background gauge field of the string restricts to a 2-bundle with connection (a gerbe) which can be seen as the obstruction to lifting the PU(H)-bundle on the D-brane to a U(H)-bundle. We discuss how this phenomenon generalizes from the ordinary central extension to higher categorical central extensions, like the String-extension . Here the obstruction to the lift is a [[nLab:principal infinity-bundle|3-bundle] with connection (a bundle 2-gerbe): the Chern-Simons circle 3-bundle classified by the first Pontrjagin class. For this obstructs the existence of a String-structure. We discuss how to describe this obstruction problem in terms of Lie n-algebras and their corresponding categorified Cartan-Ehresmann connections. Generalizations even beyond String-extensions are then straightforward. For the next step is “Fivebrane structures” whose existence is obstructed by certain generalized Chern-Simons circle 7-bundles classified by the second Pontrjagin class.