\(G_2\)-instantons on Resolutions of \(G_2\)-orbifolds

We explain a construction of \(G_2\) $G_2$ -instantons on manifolds obtained by resolving \(G_2\) $G_2$ -orbifolds. This includes the case of \(G_2\) $G_2$ -instantons on resolutions of \(T^7/\Gamma \) $T^7/\mathrm{\Gamma }$ as a special case. The ingredients needed are a \(G_2\) $G_2$ -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 \(T^7/\Gamma \) $T^7/\mathrm{\Gamma }$ , improved control over the torsion-free \(G_2\) $G_2$ -structure allows to remove this assumption. As an application, we construct a large number of \(G_2\) $G_2$ -instantons on the simplest example of a resolution of \(T^7/\Gamma \) $T^7/\mathrm{\Gamma }$ . We also construct one new example of a \(G_2\) $G_2$ -instanton on the resolution of \((T^3 \times \text {K3})/\mathbb {Z}^2_2\) $(T^3\times \text{K3})/{\mathbb{Z}}_2^2$ .