L-Algebra Connections and Applications to String- and Chern-Simons n-Transport

  • Hisham Sati
  • Urs Schreiber
  • Jim Stasheff

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

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

Mathematics Subject Classification (2000)

Primary 83E30 Secondary 55P20 81T30 55R45 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Hisham Sati
    • 1
  • Urs Schreiber
    • 2
  • Jim Stasheff
    • 3
  1. 1.Department of MathematicsYale UniversityNew Haven
  2. 2.Fachbereich MathematikUniversität HamburgHamburg
  3. 3.Department of MathematicsUniversity of PennsylvaniaPhiladelphia

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