Advances in Theoretical and Mathematical Physics
Volume 15 (2011)
Division algebras and supersymmetry II
Pages: 1373 – 1410
Starting from the four normed division algebras – thereal numbers, complex numbers, quaternions and octonions– a systematic procedure gives a 3-cocycle on thePoincaré Lie superalgebra in dimensions 3, 4, 6 and 10.A related procedure gives a 4-cocycle on the Poincaré Liesuperalgebra 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 $n$-termchain complex equipped with a bracket satisfying the axiomsof a Lie superalgebra up to chain homotopy. We thus obtainLie 2-superalgebras extending the Poincaré superalgebrain dimensions 3, 4, 6 and 10, and Lie 3-superalgebrasextending the Poincaré superalgebra in dimensions 4, 5, 7and 11. As shown in Sati, Schreiber and Stasheff’s work onhigher gauge theory, Lie 2-superalgebra connectionsdescribe the parallel transport of strings, while Lie3-superalgebra connections describe the parallel transportof 2-branes. Moreover, in the octonionic case, theseconnections concisely summarize the fields appearing in10- and 11-dimensional supergravity.