Division Algebras and Supersymmetry I

TL;DR

This paper explores the deep connection between division algebras and supersymmetry, explaining why supersymmetric theories in specific spacetime dimensions are linked to the existence of normed division algebras.

Contribution

It provides a self-contained explanation of how division algebras underpin supersymmetry in certain spacetime dimensions.

Findings

Supersymmetry in 3, 4, 6, and 10 dimensions is related to division algebras.

The vanishing of a trilinear spinor expression is key to supersymmetry.

Division algebras exist only in dimensions 1, 2, 4, and 8.

Abstract

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.