Division Algebras and Supersymmetry II
John C. Baez, John Huerta
TL;DR
This work builds a division-algebra–driven bridge between supersymmetry and higher gauge theory by constructing cocycles from the $3$-$\psi$'s rule and the $4$-$\Psi$'s rule in dimensions $3,4,6,10$ and $4,5,7,11$. These cocycles yield nontrivial Lie $2$- and Lie $3$-superalgebras extending the Poincaré superalgebras, namely ${\rm superstring}(n+1,1)$ and ${\rm 2\text{-}brane}(n+2,1)$, which in turn underpin higher gauge connections describing string and membrane parallel transport. A uniform, algebraic treatment uses $L_\infty$-superalgebras to realize these extensions and to relate them to WZW terms and background fields ($A$, $B$, $C$) in string theory and $11$-dimensional supergravity, with the octonionic case summarizing the field content of $10$- and $11$-dimensional theories. The results illuminate a tight network of cocycles, higher-algebra structures, and physical backgrounds, providing a principled route to higher gauge-theoretic descriptions of extended objects across relevant dimensions.
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.