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 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.

💡 Research Summary

The paper “Division Algebras and Supersymmetry III” develops a concrete geometric method for integrating a Lie 2‑superalgebra that extends the Poincaré superalgebra into a full Lie 2‑supergroup. The construction is motivated by higher‑gauge theory applied to the superstring, where a “higher symmetry” appears precisely in the dimensions 3, 4, 6, 10 – the dimensions in which the classical Green–Schwarz superstring is consistent.

The authors begin by recalling that the normed division algebras ℝ, ℂ, ℍ, 𝕆 have dimensions 1, 2, 4, 8, and that supersymmetric Yang–Mills theory and the superstring live in spacetimes whose dimensions are two higher (n + 2). In the previous paper of the series they exhibited a Lie 2‑superalgebra, called superstring (n + 1, 1), built from a 3‑cocycle α on the Poincaré superalgebra \