Advances in Theoretical and Mathematical Physics

Volume 16 (2012)

Number 5

Division algebras and supersymmetry III

Pages: 1485 – 1589



John Huerta (CAMGSD, Instituto Superior Técnico, Technical University of Lisbon, Portugal)


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 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 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.

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