The Mile High Magic Pyramid

Abstract

Using a unified formulation of $\mathcal{N}=1,2,4,8$, super Yang-Mills theories in $D=3$ spacetime dimensions with fields valued respectively in $\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$, it was shown that tensoring left and right multiplets yields a Freudenthal magic square of $D=3$ supergravities. When tied in with the more familiar $\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$ description of super Yang-Mills in $D=3,4,6,10$ this results in a magic pyramid of supergravities: the known $4\times 4$ magic square at the base in $D=3$, a $3\times 3$ square in $D=4$, a $2\times 2$ square in $D=6$ and Type II supergravity at the apex in $D=10$.

Contribution to the proceedings of the 4th Mile High Conference on Nonassociative Mathematics University of Denver, Denver, Colorado, USA, July 29-August 5, 2017