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

## TL;DR

This paper investigates quiver gauge theories on specific seven-spheres and their cones, analyzing instanton moduli spaces using geometric structures like Sasaki-Einstein and hyper-Kähler manifolds, revealing new descriptions of these moduli spaces.

## Contribution

It provides a detailed description of equivariance conditions, quivers, and instanton moduli spaces on cones over round and squashed seven-spheres, extending known results to new geometric contexts.

## Key findings

- Moduli space of instantons on hyper-Kähler cone as intersection of Hermitian Yang-Mills moduli spaces
- Explicit description of equivariance conditions and quivers for specific Sasaki-Einstein structures
- Analysis of instantons on cones over orbifolds of seven-spheres

## Abstract

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 $S^7/\mathbb{Z}_k$.