Gauge theories on spheres with 16 supercharges and non-constant couplings

We construct a class of theories with 16 supersymmetries on spheres of dimension nine and less. The gauge coupling and mass terms for the scalar fields depend on the polar angle away from the north pole. Assuming finite coupling on the north pole, this leads to zero coupling at the south pole for d  >  4 and infinite coupling at the south pole for d  <  4. The underlying supersymmetry algebra of these theories is shown to be isomorphic to the Poincaré superalgebra in d-dimensions. We also give a localization procedure which leads to nontrivial results for d  =  2. (paper)