Gauged Harmonic Maps, Born-Infeld Electromagnetism, and Magnetic Vortices

We study maps from a 2-surface into the standard 2-sphere coupled with Born-Infeld geometric electromagnetism through an Abelian gauge field. Such a formalism extends the classical harmonic map model, known as the σ-model, governing the spin vector orientation in a ferromagnet and allows us to obtain the coexistence of vortices and antivortices characterized by opposite, self-excited, magnetic flux lines. We show that the Born-Infeld free parameter may be used to achieve arbitrarily high local concentration of magnetic flux lines and that the total minimum energy is an additive function of these quantized flux lines realized as the numbers of vortices and antivortices. In the case where the underlying surface, or the domain, is compact, we obtain a necessary and sufficient condition for the existence of a unique solution representing a prescribed distribution of vortices and antivortices. In the case where the domain is the full plane, we prove the existence of a unique solution representing an arbitrary distribution of vortices and antivortices. Furthermore, we also consider the Einstein gravitation induced by these vortices, known as cosmic strings, and establish the existence of a solution representing a prescribed distribution of cosmic strings and cosmic antistrings under a necessary and sufficient condition that makes the underlying surface a complete surface with respect to the induced gravitational metric.