# Schreiber L-infinity algebra connections

Related projects

An article we once wrote:

• $L_\infty$-algebra connections and applications to String- and Chern-Simons $n$-transport

in:

Quantum Field Theory,

Birkhäuser (2009), 303-424

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.