Division Algebras and Supersymmetry III

TL;DR

This paper constructs a Lie 2-supergroup extending the Poincare supergroup in specific dimensions using geometric techniques, building on previous work that used division algebras to define related algebraic structures.

Contribution

It introduces an elegant geometric method to integrate Lie 2-superalgebras into Lie 2-supergroups, extending the Poincare supergroup in dimensions relevant to superstring theory.

Findings

Constructed Lie 2-supergroups in dimensions 3, 4, 6, 10

Extended previous algebraic models using geometric integration

Applied to higher gauge theory and superstring symmetry

Abstract

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 Poincare superalgebra in precisely the dimensions where the classical supersymmetric string 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 Poincare 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 Poincare supergroup.