A Kind of Magic

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.Comment: 38 pages. Reference added and minor corrections mad

Explicit Orbit Classification of Reducible Jordan Algebras and Freudenthal Triple Systems

We determine explicit orbit representatives of reducible Jordan algebras and of their corresponding Freudenthal triple systems. This work has direct application to the classification of extremal black hole solutions of N = 2, 4 locally supersymmetric theories of gravity coupled to an arbitrary number of Abelian vector multiplets in D = 4, 5 space-time dimensions.Comment: 18 pages. Updated to match published versio

The pure BRST Einstein-Hilbert Lagrangian from the double-copy to cubic order

We construct the pure gravity Becchi-Rouet-Stora-Tyutin (BRST) Einstein-Hilbert Lagrangian, to cubic order, using the BRST convolution product of two Yang-Mills theories, in conjunction with the Bern-Carrasco-Johansson (BCJ) double-copy.Comment: 20 page

Global symmetries of Yang-Mills squared in various dimensions

Tensoring two on-shell super Yang-Mills multiplets in dimensions $D\leq 10$ yields an on-shell supergravity multiplet, possibly with additional matter multiplets. Associating a (direct sum of) division algebra(s) $\mathbb{D}$ with each dimension $3\leq D\leq 10$ we obtain formulae for the algebras $\mathfrak{g}$ and $\mathfrak{h}$ of the U-duality group $G$ and its maximal compact subgroup $H$, respectively, in terms of the internal global symmetry algebras of each super Yang-Mills theory. We extend our analysis to include supergravities coupled to an arbitrary number of matter multiplets by allowing for non-supersymmetric multiplets in the tensor product.Comment: 25 pages, 2 figures, references added, minor typos corrected, further comments on sec. 2.4 included, updated to match version to appear in JHE

Super Yang-Mills, division algebras and triality

We give a unified division algebraic description of (D=3, N=1,2,4,8), (D=4, N=1,2,4), (D=6, N=1,2) and (D=10, N=1) super Yang-Mills theories. A given (D=n+2, N) theory is completely specified by selecting a pair (A_n, A_{nN}) of division algebras, A_n, A_{nN} = R, C, H, O, where the subscripts denote the dimension of the algebras. We present a master Lagrangian, defined over A_{nN}-valued fields, which encapsulates all cases. Each possibility is obtained from the unique (O, O) (D=10, N=1) theory by a combination of Cayley-Dickson halving, which amounts to dimensional reduction, and removing points, lines and quadrangles of the Fano plane, which amounts to consistent truncation. The so-called triality algebras associated with the division algebras allow for a novel formula for the overall (spacetime plus internal) symmetries of the on-shell degrees of freedom of the theories. We use imaginary A_{nN}-valued auxiliary fields to close the non-maximal supersymmetry algebra off-shell. The failure to close for maximally supersymmetric theories is attributed directly to the non-associativity of the octonions.Comment: 24 pages, 2 figures. Updated to match published version. References adde