Topological duality between vortices and planar Skyrmions in BPS theories with area-preserving diffeomorphism symmetries

Abstract

The Bogomol'nyi-Prasad-Sommerfield (BPS) baby Skyrme models are submodels of baby Skyrme models, where the nonlinear sigma model term is suppressed. They have Skyrmion solutions saturating a BPS bound, and the corresponding static energy functional is invariant under area-preserving diffeomorphisms (APDs). Here we show that the solitons in the BPS baby Skyrme model, which carry a nontrivial topological charge ${Q}_{b}\ensuremath{\in}{\ensuremath{\pi}}_{2}({S}^{2})$ (a winding number), are dual to vortices in a BPS vortex model with a topological charge ${Q}_{v}\ensuremath{\in}{\ensuremath{\pi}}_{1}({S}^{1})$ (a vortex number), in the sense that there is a map between the BPS solutions of the two models. The corresponding energy densities of the BPS solutions of the two models are identical. A further consequence of the duality is that the dual BPS vortex models inherit the BPS property and the infinitely many symmetries (APDs) of the BPS baby Skyrme models. Finally, we demonstrate that the same topological duality continues to hold for the $U(1)$ gauged versions of the models.