@inproceedings{f7ba9502f9b743798674a0849515144b,

title = "L ∞-Algebra connections and applications to String- and Chern-Simons n-transport",

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 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.",

keywords = "2-bundles, BF-theory, Cartan-Ehresman connection, Chern-Simons theory, Eilenberg-MacLane spaces, branes, differential greded algebras, strings",

author = "Hisham Sati and Urs Schreiber and Jim Stasheff",

note = "Copyright: Copyright 2021 Elsevier B.V., All rights reserved.; 3rd Workshop on Recent Developments in Quantum Field Theory ; Conference date: 20-07-2007 Through 22-07-2007",

year = "2009",

doi = "10.1007/978-3-7643-8736-5_17",

language = "English (US)",

isbn = "9783764387358",

series = "Quantum Field Theory: Competitive Models",

publisher = "Birkhauser Verlag AG",

pages = "303--424",

booktitle = "Quantum Field Theory",

}