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 |
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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
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 proceeding › Conference contribution
}
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 -