Division Algebras and Supersymmetry III

Recent work applying higher gauge theory to the superstring has indicated the presence of ‘higher symmetry’. Infinitesimally, this is realized by a ‘Lie 2-superalgebra’ extending the Poincaré superalgebra in precisely the dimensions where the classical superstring makes sense: 3, 4, 6 and 10. In the previous paper in this series, we constructed this Lie 2-superalgebra using the normed division algebras. In this paper, we use an elegant geometric technique to integrate this Lie 2-superalgebra to a ‘Lie 2-supergroup’ extending the Poincaré supergroup in the same dimensions. Briefly, a ‘Lie 2-superalgebra’ is a two-term chain complex with a bracket like a Lie superalgebra, but satisfying the Jacobi identity only up to chain homotopy. Simple examples of Lie 2-superalgebras arise from 3-cocycles on Lie superalgebras, and it is in this way that we constructed the Lie 2-superalgebra above. Because this 3-cocycle is supported on a nilpotent subalgebra, our geometric technique applies, and we obtain a Lie 2-supergroup integrating the Lie 2-superalgebra in the guise of a smooth 3-cocycle on the Poincaré supergroup.