A Kind of Magic

Abstract

We introduce the extended Freudenthal-Rosenfeld-Tits magic square based on six algebras: the reals $\mathbb{R}$, complexes $\mathbb{C}$, ternions $\mathbb{T}$, quaternions $\mathbb{H}$, sextonions $\mathbb{S}$ and octonions $\mathbb{O}$. The ternionic and sextonionic rows/columns of the magic square yield non-reductive Lie algebras, including $\mathfrak{e}_{7\scriptscriptstyle{\frac{1}{2}}}$. It is demonstrated that the algebras of the extended magic square appear quite naturally as the symmetries of supergravity Lagrangians. The sextonionic row (for appropriate choices of real forms) gives the non-compact global symmetries of the Lagrangian for the $D=3$ maximal $\mathcal{N}=16$, magic $\mathcal{N}=4$ and magic non-supersymmetric theories, obtained by dimensionally reducing the $D=4$ parent theories on a circle, with the graviphoton left undualised. In particular, the extremal intermediate non-reductive Lie algebra $\tilde{\mathfrak{e}}_{7(7)\scriptscriptstyle{\frac{1}{2}}}$ (which is not a subalgebra of $\mathfrak{e}_{8(8)}$) is the non-compact global symmetry algebra of $D=3$, $\mathcal{N}=16$ supergravity as obtained by dimensionally reducing $D=4$, $\mathcal{N}=8$ supergravity with $\mathfrak{e}_{7(7)}$ symmetry on a circle. The ternionic row (for appropriate choices of real forms) gives the non-compact global symmetries of the Lagrangian for the $D=4$ maximal $\mathcal{N}=8$, magic $\mathcal{N}=2$ and magic non-supersymmetric theories obtained by dimensionally reducing the parent $D=5$ theories on a circle. In particular, the Kantor-Koecher-Tits intermediate non-reductive Lie algebra $\mathfrak{e}_{6(6)\scriptscriptstyle{\frac{1}{4}}}$ is the non-compact global symmetry algebra of $D=4$, $\mathcal{N}=8$ supergravity as obtained by dimensionally reducing $D=5$, $\mathcal{N}=8$ supergravity with $\mathfrak{e}_{6(6)}$ symmetry on a circle.