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Division Algebras, Supersymmetry and Higher Gauge Theory

Abstract

Starting from the four normed division algebras---the real numbers, complex

numbers, quaternions and octonions, with dimensions k=1, 2, 4 and 8,

respectively---a systematic procedure gives a 3-cocycle on the Poincar'e Lie

superalgebra in dimensions k+2=3, 4, 6 and 10. A related procedure gives a

4-cocycle on the Poincare Lie superalgebra in dimensions k+3=4, 5, 7 and

11. The existence of these cocycles follow from certain spinor identities that

hold only in these dimensions, and which are closely related to the existence

of superstring and super-Yang--Mills theory in dimensions k+2, and

super-2-brane theory in dimensions k+3.

In general, an (n+1)-cocycle on a Lie superalgebra yields a `Lie

n-superalgebra': that is, roughly speaking, an n-term chain complex

equipped with a bracket satisfying the axioms of a Lie superalgebra up to chain

homotopy. We thus obtain Lie 2-superalgebras extending the Poincare

superalgebra in dimensions 3, 4, 6, and 10, and Lie 3-superalgebras extending

the Poincar'e superalgebra in dimensions 4, 5, 7 and 11. As shown in Sati,

Schreiber and Stasheff's work on generalized connections valued in Lie

n-superalgebras, Lie 2-superalgebra connections describe the parallel

transport of strings, while Lie 3-superalgebra connections describe the

parallel transport of 2-branes. Moreover, in the octonionic case, these

connections concisely summarize the fields appearing in 10- and 11-dimensional

supergravity.

Generically, integrating a Lie n-superalgebra to a Lie n-supergroup yields a

`Lie n-supergroup' that is hugely infinite-dimensional. However,

when the Lie n-superalgebra is obtained from an (n+1)-cocycle on a

nilpotent Lie superalgebra, there is a geometric procedure to integrate the

cocycle to one on the corresponding nilpotent Lie supergroup.

In general, a smooth (n+1)-cocycle on a supergroup yields a `Lie

n-supergroup': that is, a weak n-group internal to supermanifolds. Using

our geometric procedure to integrate the 3-cocycle in dimensions 3, 4, 6 and

10, we obtain a Lie 2-supergroup extending the Poincare supergroup in those

dimensions, and similarly integrating the 4-cocycle in dimensions 4, 5, 7 and

11, we obtain a Lie 3-supergroup extending the Poincar'e supergroup in those

dimensions.

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