139 research outputs found

    A Kind of Magic

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    We introduce the extended Freudenthal-Rosenfeld-Tits magic square based on six algebras: the reals R\mathbb{R}, complexes C\mathbb{C}, ternions T\mathbb{T}, quaternions H\mathbb{H}, sextonions S\mathbb{S} and octonions O\mathbb{O}. The ternionic and sextonionic rows/columns of the magic square yield non-reductive Lie algebras, including e712\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=3D=3 maximal N=16\mathcal{N}=16, magic N=4\mathcal{N}=4 and magic non-supersymmetric theories, obtained by dimensionally reducing the D=4D=4 parent theories on a circle, with the graviphoton left undualised. In particular, the extremal intermediate non-reductive Lie algebra e~7(7)12\tilde{\mathfrak{e}}_{7(7)\scriptscriptstyle{\frac{1}{2}}} (which is not a subalgebra of e8(8)\mathfrak{e}_{8(8)}) is the non-compact global symmetry algebra of D=3D=3, N=16\mathcal{N}=16 supergravity as obtained by dimensionally reducing D=4D=4, N=8\mathcal{N}=8 supergravity with e7(7)\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=4D=4 maximal N=8\mathcal{N}=8, magic N=2\mathcal{N}=2 and magic non-supersymmetric theories obtained by dimensionally reducing the parent D=5D=5 theories on a circle. In particular, the Kantor-Koecher-Tits intermediate non-reductive Lie algebra e6(6)14\mathfrak{e}_{6(6)\scriptscriptstyle{\frac{1}{4}}} is the non-compact global symmetry algebra of D=4D=4, N=8\mathcal{N}=8 supergravity as obtained by dimensionally reducing D=5D=5, N=8\mathcal{N}=8 supergravity with e6(6)\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

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    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

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    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

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    Tensoring two on-shell super Yang-Mills multiplets in dimensions D10D\leq 10 yields an on-shell supergravity multiplet, possibly with additional matter multiplets. Associating a (direct sum of) division algebra(s) D\mathbb{D} with each dimension 3D103\leq D\leq 10 we obtain formulae for the algebras g\mathfrak{g} and h\mathfrak{h} of the U-duality group GG and its maximal compact subgroup HH, 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

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    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