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

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