Topological Soliton with Nonzero Hopf Invariant in Yang-Mills-Higgs Model

TL;DR

This paper introduces a novel topological soliton with nonzero Hopf invariant in a 3+1D Yang-Mills-Higgs model, potentially affecting monopole statistics and contributing to the understanding of non-Abelian gauge theories.

Contribution

It presents the first explicit construction of a Hopf soliton with nonzero Hopf invariant in a 3+1D gauge theory with scalar fields, highlighting its possible physical implications.

Findings

Hopf soliton represents a spacetime event causing a $2\pi$ monopole rotation

Action of the Hopf soliton diverges logarithmically but may be relevant in finite systems

Potential for fractional statistics of monopoles via Chern-Simons term

Abstract

We propose a topological soliton or instanton solution with nonzero Hopf invariant to the 3+1D non-Abelian gauge theory coupled with scalar fields. This solution, which we call Hopf soliton, represents a spacetime event that makes a $2\pi$ rotation of the monopole. Although the action of this Hopf soliton is logarithmically divergent, it may still give relevant contributions in a finite-sized system. Since the Chern-Simons term for the unbroken $U(1)$ gauge field may appear in the low energy effective theory, the Hopf soliton may possibly generate fractional statistics for the monopoles.