Division algebras and supersymmetry IV

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 Poincaré supergroup in precisely those spacetime dimensions where the classical Green–Schwarz superstring makes sense: $3$, $4$, $6$ and $1$$0$. In this paper, we use the same technique to construct a ‘Lie $3$-supergroup’ extending the Poincaré 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.

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This work was partially supported by FQXi grant RFP2- 08-04, by the Geometry and Mathematical Physics Project (EXCL/MATGEO/ 0222/2012), and by a postdoctoral grant from the Portuguese science foundation, SFRH/BPD/92915/2013.

Published 7 July 2017