Spin(7)-manifolds and symmetric Yang--Mills instantons

TL;DR

This paper links symmetric SU(2) Yang--Mills instantons to Spin(7)-holonomy metrics, extending Bryant and Salamon's method to explicitly construct such special holonomy manifolds using instantons.

Contribution

It introduces a novel construction method for Spin(7)-holonomy metrics from symmetric SU(2) instantons on spin-manifolds, generalizing previous approaches.

Findings

Constructed explicit Spin(7) metrics from instantons

Extended Bryant and Salamon's method for special holonomy

Provided a five-parameter family of deformed metrics

Abstract

In this Letter we establish a relationship between symmetric SU(2) Yang--Mills instantons and metrics with Spin(7)-holonomy. Our method is based on a slight extension of that of Bryant and Salamon developed to construct explicit manifolds with special holonomies in 1989. More precisely, we prove that making use of symmetric SU(2) Yang--Mills instantons on Riemannian spin-manifolds, we can construct metrics on the chiral spinor bundle whose holonomies are within Spin(7). Moreover if the resulting space is connected, simply connected and complete, the holonomy coincides with Spin(7). The basic example is the metric constructed on the chiral spinor bundle of the round four-sphere by using a generic SU(2)-instanton of unit action; hence it is a five-parameter deformation of the Bryant--Salamon example, also found by Gibbons, Page and Pope.