$G_2$-instantons over asymptotically cylindrical manifolds

TL;DR

This paper constructs $G_2$-instantons on asymptotically cylindrical manifolds using gauge theory and stability conditions, advancing understanding of special holonomy spaces in higher-dimensional geometry.

Contribution

It introduces a concrete model for $G_2$-instantons on asymptotically cylindrical manifolds, linking gauge theory with the Hermitian Yang-Mills problem under stability assumptions.

Findings

Existence of $G_2$-instantons on specific asymptotically cylindrical manifolds.

Application of Simpson's methods to higher-dimensional gauge theory.

Establishment of stability conditions necessary for solutions.

Abstract

A concrete model for a 7-dimensional gauge theory under special holonomy is proposed, within the paradigm outlined by Donaldson and Thomas, over the asymptotically cylindrical G2-manifolds provided by Kovalev's noncompact version of the Calabi conjecture. One obtains a solution to the $G_2$-instanton equation from the associated Hermitian Yang-Mills problem, to which the methods of Simpson et al. are applied, subject to a crucial asymptotic stability assumption over the "boundary at infinity".