Gauged Hopfions

Abstract

We discuss the $U(1)$ gauged version of the $3+1$ dimensional Faddeev-Skyrme model supplemented by the Maxwell term. We show that there exist axially symmetric static solutions coupled to the noninteger toroidal flux of magnetic field, which revert to the usual Hopfions ${\mathcal{A}}_{m,n}$ of lower degrees $Q=mn$ in the limit of the gauge coupling constant vanishing. The masses of the static gauged Hopfions are found to be less than the corresponding masses of the usual ungauged solitons ${\mathcal{A}}_{1,1}$ and ${\mathcal{A}}_{2,1}$, respectively; they become lighter as the gauge coupling increases. The dependence of the solutions on the gauge coupling is investigated. We find that in the strong coupling regime the gauged Hopfion carries two magnetic fluxes, which are quantized in units of $2\ensuremath{\pi}$, carrying $n$ and $m$ quanta, respectively. The first flux encircles the position curve and the second one is directed along the symmetry axis. Effective quantization of the field in the gauge sector may allow us to reconsider the usual arguments concerning the lower topological bound in the Faddeev-Skyrme-Maxwell model.