Division Algebras and Supersymmetry I

TL;DR

The work addresses why supersymmetry in Yang–Mills theory and Green–Schwarz strings is tied to spacetime dimensions $3,4,6,10$ by developing a purely equational framework based on the four normed division algebras. It constructs vectors, spinors, and intertwiners without gamma matrices, using $V=\mathfrak{h}_2({\mathbb K})$ and $S_\pm={\mathbb K}^2$, and proves the crucial $3$-$\psi$'s rule via alternativity. The central contribution is a self-contained proof that the trilinear obstruction $\operatorname{tri}\psi$ vanishes in $3$, $4$, $6$, and $10$ dimensions, enabling supersymmetry for pure super-Yang–Mills and informing the Green–Schwarz superstring within a division-algebra framework. This work clarifies the algebraic origin of the dimensionality of supersymmetry and provides a robust, gamma-matrix–free approach to intertwiners, spinors, and Clifford actions rooted in normed division algebras.

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.

Theorems & Definitions (29)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof