$$G_2$$ -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 \(T^7/\Gamma \) 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 \(T^7/\Gamma \), 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 \(T^7/\Gamma \). We also construct one new example of a \(G_2\)-instanton on the resolution of \((T^3 \times \text {K3})/\mathbb {Z}^2_2\).