A Skyrme-like model with an exact BPS bound

TL;DR

This paper introduces a new Skyrme-like model with an exact BPS bound, featuring a first order Bogomolny equation that simplifies finding solutions and is connected to force free equations in physics.

Contribution

The paper presents a novel Skyrme-like model with an exact BPS bound and solutions satisfying a Bogomolny equation related to force free equations, with explicit solutions on S^3.

Findings

Existence of a first order Bogomolny equation for the model.

Construction of explicit finite energy solutions on S^3.

Connection between the Bogomolny equation and force free equations.

Abstract

We propose a new Skyrme-like model with fields taking values on the sphere S^3 or, equivalently, on the group SU(2). The action of the model contains a quadratic kinetic term plus a quartic term which is the same as that of the Skyrme-Faddeev model. The novelty of the model is that it possess a first order Bogomolny type equation whose solutions automatically satisfy the second order Euler-Lagrange equations. It also possesses a lower bound on the static energy which is saturated by the Bogomolny solutions. Such Bogomolny equation is equivalent to the so-called force free equation used in plasma and solar Physics, and which possesses large classes of solutions. An old result due to Chandrasekhar prevents the existence of finite energy solutions for the force free equation on the entire tridimensional space R^3. We construct new exact finite energy solutions to the Bogomolny equations for the case where the space is the three-sphere S^3, using toroidal like coordinates.