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October 2011 Division algebras and supersymmetry II
John C. Baez, John Huerta
Adv. Theor. Math. Phys. 15(5): 1373-1410 (October 2011).

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 Poincaré Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincaré 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 Poincaré superalgebra in dimensions 3, 4, 6 and 10, and Lie 3-superalgebras extending the Poincaré 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.

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John C. Baez. John Huerta. "Division algebras and supersymmetry II." Adv. Theor. Math. Phys. 15 (5) 1373 - 1410, October 2011.

Information

Published: October 2011
First available in Project Euclid: 10 October 2012

zbMATH: 1260.81222
MathSciNet: MR2681758

Rights: Copyright © 2011 International Press of Boston

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Vol.15 • No. 5 • October 2011
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