### 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) |
---|---|

Title of host publication | Quantum Field Theory |

Subtitle of host publication | Competitive Models |

Pages | 303-424 |

Number of pages | 122 |

DOIs | |

State | Published - Dec 1 2009 |

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

### Other

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

Country | Germany |

City | Leipzig |

Period | 7/20/07 → 7/22/07 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

_{∞}-Algebra connections and applications to String- and Chern-Simons n-transport. In

*Quantum Field Theory: Competitive Models*(pp. 303-424) https://doi.org/10.1007/978-3-7643-8736-5-17

**L _{∞}-Algebra connections and applications to String- and Chern-Simons n-transport.** / Sati, Hisham; Schreiber, Urs; Stasheff, Jim.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

_{∞}-Algebra connections and applications to String- and Chern-Simons n-transport. in

*Quantum Field Theory: Competitive Models.*pp. 303-424, 3rd Workshop on Recent Developments in Quantum Field Theory, Leipzig, Germany, 7/20/07. https://doi.org/10.1007/978-3-7643-8736-5-17

_{∞}-Algebra connections and applications to String- and Chern-Simons n-transport. In Quantum Field Theory: Competitive Models. 2009. p. 303-424 https://doi.org/10.1007/978-3-7643-8736-5-17

}

TY - GEN

T1 - L ∞-Algebra connections and applications to String- and Chern-Simons n-transport

AU - Sati, Hisham

AU - Schreiber, Urs

AU - Stasheff, Jim

PY - 2009/12/1

Y1 - 2009/12/1

N2 - 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.

AB - 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.

KW - 2-bundles

KW - BF-theory

KW - branes

KW - Cartan-Ehresman connection

KW - Chern-Simons theory

KW - differential greded algebras

KW - Eilenberg-MacLane spaces

KW - L -algebra

KW - strings

UR - http://www.scopus.com/inward/record.url?scp=79952801542&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79952801542&partnerID=8YFLogxK

U2 - 10.1007/978-3-7643-8736-5-17

DO - 10.1007/978-3-7643-8736-5-17

M3 - Conference contribution

AN - SCOPUS:79952801542

SN - 9783764387358

SP - 303

EP - 424

BT - Quantum Field Theory

ER -