2008/01/31 by Hisham Sati, Urs Schreiber, Jim Stasheff · 1 citation
Mathematics · Physics and Astronomy · #math.DG #hep-th #math.AT #msc:51P05 #msc:81T30 #msc:55R40 #msc:55U99
paper · pdf · doi:10.1007/978-3-7643-8736-5_17
published as Quantum Field Theory, (2009) 303-424 · 100 pages, references and clarifications added; correction to section 5.1 and further example to 9.3.1 added
arxiv created 2008/02/15 · arxiv updated 2015/05/12
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. Here the obstruction to the lift is a 3-bundle with connection (a 2-gerbe): the Chern-Simons 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 7-bundles classified by the second Pontrjagin class.