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Division Algebras and Supersymmetry II | | John C. Baez
; John Huerta
; | | Date: |
17 Mar 2010 | | Abstract: | Starting from the four normed division algebras - the real numbers, complex
numbers, quaternions and octonions - a systematic procedure gives a 3-cocycle
on the Poincare Lie superalgebra in dimensions 3, 4, 6 and 10. A related
procedure gives a 4-cocycle on the Poincare Lie superalgebra in dimensions 4,
5, 7 and 11. In general, an (n+1)-cocycle on a Lie superalgebra yields a "Lie
n-superalgebra": that is, roughly speaking, an n-term chain complex equipped
with a bracket satisfying the axioms of a Lie superalgebra up to chain
homotopy. We thus obtain Lie 2-superalgebras extending the Poincare
superalgebra in dimensions 3, 4, 6, and 10, and Lie 3-superalgebras extending
the Poincare superalgebra in dimensions 4, 5, 7 and 11. As shown in Sati,
Schreiber and Stasheff’s work on higher gauge theory, Lie 2-superalgebra
connections describe the parallel transport of strings, while Lie
3-superalgebra connections describe the parallel transport of 2-branes.
Moreover, in the octonionic case, these connections concisely summarize the
fields appearing in 10- and 11-dimensional supergravity. | | Source: | arXiv, 1003.3436 | | Services: | Forum | Review | PDF | Favorites | |
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