Lie n-algebras, supersymmetry and division algebras
From CRCG-Wiki
Starting from the four normed division algebras---the real numbers, complex numbers, quaternions and octonions---a systematic procedure gives a 3-cocycle on the Poincar\'e Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincar\'e 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 Poincar\'e superalgebra in dimensions 3, 4, 6, and 10, and Lie 3-superalgebras extending the Poincar\'e 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.
