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We construct d<=7 dimensional maximally supersymmetric Yang-Mills theories ona class of curved backgrounds with off-shell supercharges. The off-shellsupersymmetry is mainly a generalization of on-shell supersymmetry constructedpreviously by Blau. We present several examples of backgrounds and discuss thenumber of the preserved supersymmetries on these backgrounds. We also constructanother maximally supersymmetric Yang-Mills theories on S^3 by dimensionalreducing along the R-direction of N=4 super Yang-Mills theory on RxS^3.

Maximal super Yang-Mills theories on curved background with off-shell supercharges