L-infinity algebra connections and applications to String- and Chern-Simons n-transport

Understand

We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L-infinity 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 parallel transport and quantization.
• It is known that over a D-brane the Kalb-Ramond background 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 U(1) -> U(H) -> PU(H) to higher categorical central extensions, like the String-extension BU(1) -> String(G) -> G.