Division Algebras and Supersymmetry IV

TL;DR

This paper extends the geometric construction of supergroups to Lie 3-supergroups in specific spacetime dimensions relevant to super-2-branes, advancing the mathematical framework of higher gauge theories in supersymmetry.

Contribution

It introduces a method to construct Lie 3-supergroups extending the Poincare supergroup in dimensions relevant to super-2-branes, building on previous work with Lie 2-supergroups.

Findings

Constructed Lie 3-supergroups in dimensions 4, 5, 7, 11

Extended the geometric techniques for higher gauge theories

Clarified the definition of Lie 3-supergroups

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.