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.

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