Gauge theories with 16 supersymmetries on spheres

We give a unified approach to localization of maximally symmetric gauge theories on spheres, including S-6 and S-7. The approach follows Pestun's method of dimensionally reducing from 10 dimensional super Yang-Mills. The resulting theories have a reduced R-symmetry which includes an SU(1,1) subgroup, except in four dimensions where, because of conformal invariance, the full flat-space R-symmetry is maintained, and in seven dimensions where SU(1, 1) is the flat-space R-symmetry. For the case of S-6 and S-7 we discuss the localization of these theories and also present new results for the corresponding matrix models. The matrix models for S-6 and S-7 are qualitatively similar to the matrix models of a vector multiplet on S-4 and S-5 respectively. We also discuss the contributions of instantons in the six and seven dimensional cases