Monopoles, instantons and the Helmholtz equation

TL;DR

This paper explores the relationship between circle-invariant Yang-Mills instantons on certain 4-manifolds and hyperbolic monopoles, revealing how solutions to the Helmholtz equation generate a broad class of monopoles with geometric insights.

Contribution

It demonstrates how specific conformal choices lead to hyperbolic monopoles from instantons and connects these results to the JNR construction with a geometric perspective.

Findings

Helmholtz equation solutions generate hyperbolic monopoles.

Conformal metric choices produce smooth and singular monopoles.

Relation established between instantons, monopoles, and the JNR construction.

Abstract

In this work we study the dimensional reduction of smooth circle invariant Yang-Mills instantons defined on 4-manifolds which are non-trivial circle fibrations over hyperbolic 3-space. A suitable choice of the 4-manifold metric within a specific conformal class gives rise to singular and smooth hyperbolic monopoles. A large class of monopoles is obtained if the conformal factor satisfies the Helmholtz equation on hyperbolic 3-space. We describe simple configurations and relate our results to the JNR construction, for which we provide a geometric interpretation.