$G_2$-instantons on resolutions of $G_2$-orbifolds

We explain a construction of $G_2$-instantons on manifolds obtained by resolving $G_2$-orbifolds. This includes the case of $G_2$-instantons on resolutions of $T^7\mathrm{/}\mathrm{\Gamma }$ as a special case. The ingredients needed are a $G_2$-instanton on the orbifold and a Fueter section over the singular set of the orbifold which are used in a gluing construction. In the general case, we make the very restrictive assumption that the Fueter section is pointwise rigid. In the special case of resolutions of $T^7\mathrm{/}\mathrm{\Gamma }$, improved control over the torsion-free $G_2$-structure allows to remove this assumption. As an application, we construct a large number of $G_2$-instantons on the simplest example of a resolution of $T^7\mathrm{/}\mathrm{\Gamma }$ with hundreds of distinct ones among them. We also construct one new example of a $G_2$-instanton on the resolution of $(T^3\times \text{K3})\mathrm{/}{\mathbb{Z}}_2^2$.