Higher-dimensional magnetic Skyrmions

We propose a generalization of the theory of magnetic Skyrmions in chiral magnets in two dimensions to a higher-dimensional theory with magnetic Skyrmions in three dimensions and an $S^3$ target space, requiring a 4-dimensional magnetization vector. A physical realization of our theory could be made using a synthetic dimension, recently promoted and realized in condensed matter physics. In the simplest incarnation of the theory, we find a Skyrmion and a sphaleron - the latter being an unstable soliton. Including also the Skyrme term in the theory enriches the spectrum to a small metastable Skyrmion, an unstable sphaleron and a large stable Skyrmion.