G2-instantons over Kovalev manifolds II

TL;DR

This paper constructs the first known instantons on compact manifolds with exactly G_2 holonomy by gluing HYM connections over asymptotically stable bundles on Kovalev's Calabi-Yau 3-folds, under specific nondegeneracy conditions.

Contribution

It provides a novel method to produce G_2-instantons on compact 7-manifolds using a gluing technique with asymptotically stable bundles, advancing the understanding of gauge theory in G_2 geometry.

Findings

First nontrivial G_2-instantons on compact manifolds

Gluing of HYM connections over Kovalev's Calabi-Yau 3-folds

Construction under nondegeneracy conditions on bundles

Abstract

This is the first nontrivial construction to date of instantons over a compact manifold with holonomy exactly $G_2$. The HYM connections on asymptotically stable bundles over Kovalev's noncompact Calabi-Yau 3-folds, obtained in the first article, are glued compatibly with a twisted connected sum, to produce a $G_2-$instanton over the resulting compact 7-manifold. This is accomplished under a nondegeneracy acyclic assumption on the bundle `at infinity', which occurs e.g. over certain projective varieties $X_{22}$ in $\mathbb{C}P^{13}$ equipped with an asymptotically rigid bundle.