Analytic topologically nontrivial solutions of the ($3+1$)-dimensional $U(1)$ gauged Skyrme model and extended duality

We construct the first analytic examples of topologically nontrivial solutions of the ($3+1$)-dimensional $U(1)$ gauged Skyrme model within a finite box in ($3+1$)-dimensional flat space-time. There are two types of gauged solitons. The first type corresponds to gauged Skyrmions living within a finite volume. The second corresponds to gauged time crystals (smooth solutions of the $U(1)$ gauged Skyrme model whose periodic time dependence is protected by a winding number). The notion of electromagnetic duality can be extended for these two types of configurations in the sense that the electric and one of the magnetic components can be interchanged. These analytic solutions show very explicitly the Callan-Witten mechanism (according to which magnetic monopoles may “swallow” part of the topological charge of the Skyrmion) since the electromagnetic field contributes directly to the conserved topological charge of the gauged Skyrmions. As it happens in superconductors, the magnetic field is suppressed in the core of the gauged Skyrmions. On the other hand, the electric field is strongly suppresed in the core of gauged time crystals.

Figure 1

The solutions for Eqs. (49) and (50) correspond to the values: $\lambda =0.04$, $l=0.47$, $K=1.0$, $p=1.0$, and $q=1.0$. Solving for $b\equiv -b_2=b_3$. The above plots clearly show the suppression of the magnetic field (which is nonvanishing only in the $\gamma$ and $\varphi$ directions) in the core of the Skyrmion.

Figure 2

The solutions for the Eqs. (64) and (65) correspond to the values: $\lambda =0.04$, $l=0.47$, $\omega =0.95$, $K=1.00$, $p=1.00$ and $q=1.00$. Solving for $b_{TC}\equiv b_1$. The above plots show clearly the suppression of the electric field in the core of the time crystal.