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 - Dec 1 2009
Event3rd Workshop on Recent Developments in Quantum Field Theory - Leipzig, Germany
Duration: Jul 20 2007Jul 22 2007

Other

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

Fingerprint

bundles
algebra
strings
functionals

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

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) 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.

Quantum Field Theory: Competitive Models. 2009. p. 303-424.

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

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, 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
Sati H, Schreiber U, Stasheff J. L -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
Sati, Hisham ; Schreiber, Urs ; Stasheff, Jim. / L -Algebra connections and applications to String- and Chern-Simons n-transport. Quantum Field Theory: Competitive Models. 2009. pp. 303-424
@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, branes, Cartan-Ehresman connection, Chern-Simons theory, differential greded algebras, Eilenberg-MacLane spaces, L -algebra, strings",
author = "Hisham Sati and Urs Schreiber and Jim Stasheff",
year = "2009",
month = "12",
day = "1",
doi = "10.1007/978-3-7643-8736-5-17",
language = "English (US)",
isbn = "9783764387358",
pages = "303--424",
booktitle = "Quantum Field Theory",

}

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 -