Yang-Mills instantons on cones and sine-cones over nearly K\"ahler manifolds

TL;DR

This paper develops a unified eight-dimensional framework for instanton equations on seven-dimensional manifolds related to nearly K"ahler structures, revealing new connections between G_2 and Spin(7) instantons and generalizing known octonionic instantons.

Contribution

It introduces a unified approach to instantons on cones and sine-cones over nearly K"ahler manifolds, linking G_2 and Spin(7) structures and extending classical instanton solutions.

Findings

Existence of G_2-instantons on various seven-dimensional manifolds.

Construction of Spin(7)-instantons from G_2-instantons.

Octonionic instantons as special cases within this framework.

Abstract

We present a unified eight-dimensional approach to instanton equations on several seven-dimensional manifolds associated to a six-dimensional homogeneous nearly K\"ahler manifold. The cone over the sine-cone on a nearly K\"ahler manifold has holonomy group Spin(7) and can be foliated by submanifolds with either holonomy group G_2, a nearly parallel G_2-structure or a cocalibrated G_2-structure. We show that there is a G_2-instanton on each of these seven-dimensional manifolds which gives rise to a Spin(7)-instanton in eight dimensions. The well-known octonionic instantons on R^7 and R^8 are contained in our construction as the special cases of an instanton on the cone and on the cone over the sine-cone, both over the six-sphere, respectively.