Division Algebras and Supersymmetry I

Details Division Algebras and Supersymmetry I Division Algebras and Supersymmetry I

2009-09-02

arxiv.org/abs/0909.0551v2

in Superstrings, Geometry, Topology, and C*-algebras, eds. R. Doran, G. Friedman and J. Rosenberg, Proc. Symp. Pure Math. 81,

Physics

High Energy Physics

High Energy Physics - Theory

16 pages, various errors corrected, new title

Scientific paper

Supersymmetry is deeply related to division algebras. Nonabelian Yang--Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green--Schwarz superstring. In both cases, supersymmetry relies on the vanishing of a certain trilinear expression involving a spinor field. The reason for this, in turn, is the existence of normed division algebras in dimensions 1, 2, 4 and 8: the real numbers, complex numbers, quaternions and octonions. Here we provide a self-contained account of how this works.

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