Monopoles on the Bryant-Salamon $G_2$ Manifolds
$G_2$-Monopoles are solutions to gauge theoretical equations on noncompact $7$-manifolds of $G_2$ holonomy. We shall study this equation on the $3$ Bryant-Salamon manifolds. We construct examples of $G_2$-monopoles on two of these manifolds, namely the total space of the bundle of anti-self-dual two forms over the $\mathbb{S}^4$ and $\mathbb{CP}^2$. These are the first nontrivial examples of $G_2$-monopoles. Associated with each monopole there is a parameter $m \in \mathbb{R}^+$, known as the mass of the monopole. We prove that under a symmetry assumption, for each given $m \in \mathbb{R}^+$ there is a unique monopole with mass $m$. We also find explicit irreducible $G_2$-instantons on $Λ^2_-(\mathbb{S}^4)$ and on $Λ^2_-(\mathbb{CP}^2)$. The third Bryant-Salamon $G_2$-metric lives on the spinor bundle over the $3$-sphere. In this case we produce a vanishing theorem for monopoles.