TY - JOUR

T1 - Super Yang-Mills, division algebras and triality

AU - Anastasiou, C A

AU - Borsten, L.

AU - Duff, M. J.

AU - Hughes, L. J.

AU - Nagy, S.

PY - 2014/8

Y1 - 2014/8

N2 - 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 n N ) of division algebras, A n A n N =R,C,H,O, where the subscripts denote the dimension of the algebras. We present a master Lagrangian, defined over A n N -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 n 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.

AB - 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 n N ) of division algebras, A n A n N =R,C,H,O, where the subscripts denote the dimension of the algebras. We present a master Lagrangian, defined over A n N -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 n 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.

KW - Extended Supersymmetry

KW - Global Symmetries

KW - Space-Time Symmetries

KW - Supersymmetric gauge theory

UR - http://www.scopus.com/inward/record.url?scp=84906098534&partnerID=8YFLogxK

U2 - 10.1007/JHEP08(2014)080

DO - 10.1007/JHEP08(2014)080

M3 - Article

AN - SCOPUS:84906098534

SN - 1126-6708

VL - 2014

JO - Journal of High Energy Physics (JHEP)

JF - Journal of High Energy Physics (JHEP)

IS - 8

M1 - 80

ER -