### 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 language | English (US) |
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Title of host publication | Quantum Field Theory |

Subtitle of host publication | Competitive Models |

Pages | 303-424 |

Number of pages | 122 |

DOIs | |

State | Published - 2009 |

Event | 3rd Workshop on Recent Developments in Quantum Field Theory - Leipzig, Germany Duration: Jul 20 2007 → Jul 22 2007 |

### Publication series

Name | Quantum Field Theory: Competitive Models |
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### Other

Other | 3rd Workshop on Recent Developments in Quantum Field Theory |
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Country | Germany |

City | Leipzig |

Period | 7/20/07 → 7/22/07 |

### 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

_{∞}-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