  The 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 $Q_{b} \in \pi_2(S^2)$ (a winding number), are dual to vortices in a BPS
vortex model with a topological charge $Q_{v}\in \pi_1(S^1)$ (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.
