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We study quiver gauge theories on the round and squashed seven-spheres, andorbifolds thereof. They arise by imposing $G$-equivariance on the homogeneousspace $G/H=\mathrm{SU}(4)/\mathrm{SU}(3)$ endowed with its Sasaki-Einsteinstructure, and $G/H=\mathrm{Sp}(2)/\mathrm{Sp}(1)$ as a 3-Sasakian manifold. Inboth cases we describe the equivariance conditions and the resulting quivers.We further study the moduli spaces of instantons on the metric cones over thesespaces by using the known description for Hermitian Yang-Mills instantons onCalabi-Yau cones. It is shown that the moduli space of instantons on thehyper-Kahler cone can be described as the intersection of three HermitianYang-Mills moduli spaces. We also study moduli spaces of translationallyinvariant instantons on the metric cone $\mathbb{R}^8/\mathbb{Z}_k$ over$S^7/\mathbb{Z}_k$.

Sasakian quiver gauge theories and instantons on cones over round and squashed seven-spheres