An article of ours
Hisham Sati, Urs Schreiber, Jim Stasheff,
$L_\infty$-algebra connections and applications to String- and Chern-Simons $n$-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 $U(1) \to U(H) \to PU(H)$ to higher categorical central extensions, like the String-extension $\mathbf{B}U(1) \to String \to Spin$. Here the obstruction to the lift is a 3-bundle with connection (a bundle 2-gerbe): the Chern-Simons circle 3-bundle classified by the first Pontrjagin class. For $G = Spin(n)$ 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 $G = Spin(n)$ the next step is “Fivebrane structures” whose existence is obstructed by certain generalized Chern-Simons circle 7-bundles classified by the second Pontrjagin class.
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