A magic square from Yang-Mills squared

TL;DR

This work uses division algebras $\mathds{R},\mathds{C},\mathds{H},\mathds{O}$ to give a unified description of $D=3$ Yang–Mills with $N=1,2,4,8$ and shows that tensoring left and right SYM multiplets yields a magic square of $D=3$ supergravities with corresponding U-duality groups $G$ and maximal compact subgroups $H$. The construction is grounded in a manifestly symmetric triality framework, yielding $G/H$ scalar manifolds with $\dim G/H = f/2$, where $f$ is the total number of degrees of freedom, and connects to the geometry of division-algebra projective planes. It further elaborates the octonionic realization of $N=8$ SYM, writes the $D=3$ SYM Lagrangian in $\mathds{A}$-valued language, and demonstrates how squaring SYM generates the full spectrum of $D=3$ supergravities, including a spectator field accounting for gauge indices. The results extend to a magic pyramid in higher dimensions, with the $D=10$ Type II theories at the apex, and illuminate the algebraic structure behind gravity as a square of Yang–Mills, tying U-duality to division-algebraic symmetries and projective-geometric interpretations.

Abstract

We give a unified description of D = 3 super-Yang-Mills theory with N = 1, 2, 4, and 8 supersymmeties in terms of the four division algebras: reals (R), complexes (C), quaternions (H) and octonions (O). Tensoring left and right super-Yang-Mills multiplets with N = 1, 2, 4, 8 we obtain a magic square RR, CR, CC, HR, HC, HH, OR, OC, OH, OO description of D = 3 supergravity with N = 2, 3, 4, 5, 6, 8, 9, 10, 12, 16.