## Advances in Theoretical and Mathematical Physics

### Division algebras and supersymmetry II

#### 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.

#### Article information

Source
Adv. Theor. Math. Phys., Volume 15, Number 5 (2011), 1373-1410.

Dates
First available in Project Euclid: 10 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.atmp/1349879112

Mathematical Reviews number (MathSciNet)
MR2681758

Zentralblatt MATH identifier
1260.81222

#### Citation

Baez, John C.; Huerta, John. Division algebras and supersymmetry II. Adv. Theor. Math. Phys. 15 (2011), no. 5, 1373--1410. https://projecteuclid.org/euclid.atmp/1349879112