Division Algebras and Supersymmetry IV

Abstract: Recent work applying higher gauge theory to the superstring has indicated the presence of 'higher symmetry', and the same methods work for the super-2-brane. In the previous paper in this series, we used a geometric technique to construct a 'Lie 2-supergroup' extending the Poincare supergroup in precisely those spacetime dimensions where the classical Green-Schwarz superstring makes sense: 3, 4, 6 and 10. In this paper, we use the same technique to construct a 'Lie 3-supergroup' extending the Poincare supergroup in precisely those spacetime dimensions where the classical Green-Schwarz super-2-brane makes sense: 4, 5, 7 and 11. Because the geometric tools are identical, our focus here is on the precise definition of a Lie 3-supergroup.