Self-dual Hopfions

TL;DR

This paper constructs exact self-dual soliton solutions with Hopf charge in a reduced Skyrme-Faddeev model, revealing infinite conserved currents and linking the profile equation to the sine-Gordon Bogomolny equation.

Contribution

It introduces a class of self-dual Hopfion solutions in a simplified Skyrme-Faddeev model using an ansatz based on symmetries, with solutions satisfying first order equations.

Findings

Solutions are exact, static and time-dependent with non-trivial Hopf charge.

Solutions satisfy first order differential equations implying second order equations of motion.

The profile function obeys the Bogomolny equation of the sine-Gordon model.

Abstract

We construct static and time-dependent exact soliton solutions with non-trivial Hopf topological charge for a field theory in 3+1 dimensions with the target space being the two dimensional sphere S**2. The model considered is a reduction of the so-called extended Skyrme-Faddeev theory by the removal of the quadratic term in derivatives of the fields. The solutions are constructed using an ansatz based on the conformal and target space symmetries. The solutions are said self-dual because they solve first order differential equations which together with some conditions on the coupling constants, imply the second order equations of motion. The solutions belong to a sub-sector of the theory with an infinite number of local conserved currents. The equation for the profile function of the ansatz corresponds to the Bogomolny equation for the sine-Gordon model.