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

TL;DR

The paper develops a method to construct $G_2$-instantons on resolved $G_2$-orbifolds, including explicit examples, by using a gluing technique involving orbifold instantons and Fueter sections, with relaxed assumptions in special cases.

Contribution

It introduces a new gluing construction for $G_2$-instantons on resolved orbifolds, expanding the class of known solutions and removing restrictive assumptions in certain cases.

Findings

Constructed many $G_2$-instantons on resolutions of $T^7/\Gamma$

Provided a new $G_2$-instanton example on $(T^3 \times K3)/\mathbb{Z}_2^2$

Achieved explicit examples with hundreds of instantons

Abstract

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