L-∞ algebra connections

An article of ours

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)U(H)PU(H)U(1) \to U(H) \to PU(H) to higher categorical central extensions, like the String-extension BU(1)StringSpin\mathbf{B}U(1) \to String \to Spin. 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 G=Spin(n)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)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.

Revised on February 23, 2015 17:55:36 by Urs Schreiber (