A model for Hopfions on the space-time S^3 x R

TL;DR

This paper constructs exact static and oscillating soliton solutions, called hopfions, on the space-time S^3 x R with scalar fields valued in various target spaces, revealing a discrete spectrum of oscillation frequencies.

Contribution

It introduces a novel method to generate exact hopfion solutions with arbitrary oscillation frequencies on S^3 x R, expanding the understanding of topological solitons in this setting.

Findings

Existence of static hopfion solutions with non-trivial topological charge.

Oscillating hopfions with a discrete spectrum of frequencies.

Preservation of topological charge during oscillations.

Abstract

We construct static and time dependent exact soliton solutions for a theory of scalar fields taking values on a wide class of two dimensional target spaces, and defined on the four dimensional space-time S^3 x R. The construction is based on an ansatz built out of special coordinates on S^3. The requirement for finite energy introduces boundary conditions that determine an infinite discrete spectrum of frequencies for the oscillating solutions. For the case where the target space is the sphere S^2, we obtain static soliton solutions with non-trivial Hopf topological charges. In addition, such hopfions can oscillate in time, preserving their topological Hopf charge, with any of the frequencies belonging to that infinite discrete spectrum.