A ring of instantons inducing a monopole loop

TL;DR

This paper constructs a self-dual Yang-Mills solution with a ring-shaped instanton superposition, revealing a gauge-invariant monopole loop and analyzing gauge detection methods, contributing to understanding monopole-instanton correlations.

Contribution

It presents an explicit analytic example of a monopole loop induced by a ring of instantons, with detailed gauge detection analysis and implications for monopole-topology relations.

Findings

Monopole detected in Maximal Abelian and Laplacian Abelian gauges

Configuration has infinite action but well-defined magnetic charge

Polyakov gauge does not detect the monopole

Abstract

We consider the superposition of infinitely many instantons on a circle in R^4. The construction yields a self-dual solution of the Yang-Mills equations with action density concentrated on the ring. We show that this configuration is reducible in which case magnetic charge can be defined in a gauge invariant way. Indeed, we find a unit charge monopole (worldline) on the ring. This is an analytic example of the correlation between monopoles and action/topological density, however with infinite action. We show that both the Maximal Abelian Gauge and the Laplacian Abelian Gauge detect the monopole, while the Polyakov gauge does not. We discuss the implications of this configuration.