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We establish N=(1/8,1) supersymmetric Yang-Mills vector multiplet withgeneralized self-duality in Euclidian eight-dimensions with the original fullSO(8) Lorentz covariance reduced to SO(7). The key ingredient is the usage ofoctonion structure constants made compatible with SO(7) covariance andchirality in 8D. By a simple dimensional reduction together with extraconstraints, we derive N=1/8+7/8 supersymmetric self-dual vector multiplet in7D with the full SO(7) Lorentz covariance reduced to G_2. We find that extraconstraints needed on fields and supersymmetry parameter are not obtained froma simple dimensional reduction from 8D. We conjecture that other self-dualsupersymmetric theories in lower dimensions D =6 and 4 with respective reducedglobal Lorentz covariances such as SU(3) \subset SO(6) and SU(2) \subset SO(4)can be obtained in a similar fashion.Comment: 22 pages, no figures. Considerable revisions, in particular, a new  Appendix added for 8D self-dual Yang-Mills in Superspac

Octonions,G2Symmetry, Generalized Self-Duality and Supersymmetries in Dimensions D 8

Octonions,G₂Symmetry, Generalized Self-Duality and Supersymmetries in Dimensions D 8