An octonionic formulation of the M-theory algebra

Abstract

We give an octonionic formulation of the $$ \mathcal{N}=1 $$ supersymmetry algebra in D = 11, including all brane charges. We write this interms of a novel outer product, which takes a pair of elements of the division algebra $$ \mathbb{A} $$ and returns a real linear operator on $$ \mathbb{A} $$ . More generally, with this product comes the power to rewrite any linear operation on $$ {\mathbb{R}}^n $$ (n =1, 2, 4, 8) in terms of multiplication in the n-dimensional division algebra $$ \mathbb{A} $$ . Finally, we consider the reinterpretation of the D =11 supersymmetry algebra as an octonionic algebra in D =4 and the truncation to division subalgebras.