L -Algebra connections and applications to String- and Chern-Simons n-transport

Hisham Sati, Urs Schreiber, Jim Stasheff

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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.

Original languageEnglish (US)
Title of host publicationQuantum Field Theory
Subtitle of host publicationCompetitive Models
Pages303-424
Number of pages122
DOIs
StatePublished - 2009
Event3rd Workshop on Recent Developments in Quantum Field Theory - Leipzig, Germany
Duration: Jul 20 2007Jul 22 2007

Publication series

NameQuantum Field Theory: Competitive Models

Other

Other3rd Workshop on Recent Developments in Quantum Field Theory
CountryGermany
CityLeipzig
Period7/20/077/22/07

Keywords

  • 2-bundles
  • BF-theory
  • Cartan-Ehresman connection
  • Chern-Simons theory
  • Eilenberg-MacLane spaces
  • L -algebra
  • branes
  • differential greded algebras
  • strings

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics

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  • Cite this

    Sati, H., Schreiber, U., & Stasheff, J. (2009). L -Algebra connections and applications to String- and Chern-Simons n-transport. In Quantum Field Theory: Competitive Models (pp. 303-424). (Quantum Field Theory: Competitive Models). https://doi.org/10.1007/978-3-7643-8736-5-17