Self-dual Hopfions

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.

💡 Research Summary

The paper investigates a reduced version of the extended Skyrme‑Faddeev model in four‑dimensional spacetime, where the target space is the two‑sphere S² and the usual quadratic (sigma‑model) kinetic term is omitted. The resulting Lagrangian contains only quartic derivative contributions, namely the Skyrme‑type term and a higher‑order “extended” term, with coupling constants α and β. By removing the quadratic piece the theory retains full conformal invariance in the spatial coordinates, a property that plays a central role in the construction of exact solutions.

The authors introduce an ansatz that exploits both the conformal symmetry of the base space and the internal U(1) symmetry of the target sphere. In practice one writes the unit vector field n(x)∈S² in terms of a complex stereographic coordinate z and parametrises the spacetime coordinates by complex variables (ξ,η) that are related to the usual spherical coordinates (r,θ,φ). The ansatz takes the form

 z = f(r) exp