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) subset of 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.