Analytic (3+1)-dimensional gauged Skyrmions, Heun, and Whittaker-Hill equations and resurgence

We show that one can reduce the coupled system of seven field equations of the (3 + 1)-dimensional gauged Skyrme model to the Heun equation (which, for suitable choices of the parameters, can be further reduced to the Whittaker-Hill equation) in two nontrivial topological sectors. Hence, one can get a complete analytic description of gauged solitons in (3 + 1) dimensions living within a finite volume in terms of classic results in the theory of differential equations and Kummer's confluent functions. We consider here two types of gauged solitons: gauged Skyrmions and gauged time crystals (namely, gauged solitons periodic in time, whose time period is protected by a winding number). The dependence of the energy of the gauged Skyrmions on the baryon charge can be determined explicitly. The theory of Kummer's confluent functions leads to a quantization condition for the period of the time crystals. Likewise, the theory of Sturm-Liouville operators gives rise to a quantization condition for the volume occupied by the gauged Skyrmions. The present analysis also discloses that resurgent techniques are very well suited to deal with the gauged Skyrme model as well. In particular, we discuss a very nice relation between the electromagnetic perturbations of the gauged Skyrmions and the Mathieu equation which allows to use many of the modern resurgence techniques to determine the behavior of the spectrum of these perturbations.