  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.
