$$G_2$$-instantons on Resolutions of $$G_2$$-orbifolds

Abstract

AbstractWe 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 / Γ 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 / Γ , 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 / Γ . 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 × K3 ) / Z 2 2 .