What makes magnetic skyrmions different from magnetic bubbles?

•There are topological constraints on positions of topological solitons•The constraints are due to the magnetization continuity and boundary conditions.•The constraints are global, integral across all the magnetization vector field.•Magnetic domains are free from the constraints and can be moved individually.•Magnetic domains (bubbles) are not skyrmions. A large enough piece of ferromagnet is usually not magnetized uniformly, but develops a magnetization texture. In thin films these textures can be doubly-periodic. Such are the well known magnetic bubble domains and the recently observed “skyrmion” magnetization textures in MnSi. In this paper we develop a theory of periodic magnetization textures, based on complex calculus to answer the question – is there a difference between those two textures even if they seem to carry the same topological winding number (or topological charge)? We find that such difference exists, facilitated by a different role played by the magnetization vector’s in-plane phase. We separate classical-like and quantum-like features of magnetization textures and highlight the role of magnetic anisotropy in favouring either of these cases.