Abstract
We give a unified division algebraic description of (D = 3, \( \mathcal{N} \) = 1, 2, 4, 8), (D = 4, \( \mathcal{N} \) = 1, 2, 4), (D = 6, \( \mathcal{N} \) = 1, 2) and (D = 10, \( \mathcal{N} \) = 1) super Yang-Mills theories. A given (D = n + 2, \( \mathcal{N} \)) theory is completely specified by selecting a pair (\( {\mathbb{A}}_n \), \( {{\mathbb{A}}_n}_{\mathcal{N}} \)) of division algebras, \( {\mathbb{A}}_n \) ⊆ \( {{\mathbb{A}}_n}_{\mathcal{N}} \) = \( \mathbb{R},\mathbb{C},\mathrm{\mathbb{H}},\mathbb{O} \), where the subscripts denote the dimension of the algebras. We present a master Lagrangian, defined over \( {{\mathbb{A}}_n}_{\mathcal{N}} \) -valued fields, which encapsulates all cases. Each possibility is obtained from the unique (\( \mathbb{O},\mathbb{O} \)) (D = 10, \( \mathcal{N} \) = 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 \( {{\mathbb{A}}_n}_{\mathcal{N}} \) -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.
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Anastasiou, A., Borsten, L., Duff, M.J. et al. Super Yang-Mills, division algebras and triality. J. High Energ. Phys. 2014, 80 (2014). https://doi.org/10.1007/JHEP08(2014)080
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DOI: https://doi.org/10.1007/JHEP08(2014)080