Real meromorphic differentials: A language for describing meron configurations in planar magnetic nanoelements

We use the language of real meromorphic differentials from the theory of Klein surfaces to describe the metastable states of multiply connected planar ferromagnetic nanoelements that minimize the exchange energy and have no side magnetic charges. These solutions still have sufficient internal degrees of freedom, which can be used as Ritz parameters to minimize other contributions to the total energy or as slow dynamical variables in the adiabatic approximation. The nontrivial topology of the magnet itself leads to several effects first described for the annulus and observed in the experiment. We explain the connection between the numbers of topological singularities of various types in the magnet and the constraints on the positions of these singularities following from the Abel theorem. Using multivalued Prym differentials leads to new meron configurations that were not considered in the classic work by Gross.