Interaction of superconducting vortices: lectures presented at the Canadian Mathematical Society Summer Research Institute on gauge theories

Results recently obtained on multi-vortex configurations are described. After a brief review of the model, a numerical analysis, performed by variational methods, of the interaction between two vortices is illustrated. The study, done in collaboration with Laurence Jacobs, shows that two vortices attract or repel each other according to whether a dimensionless coupling constant lambda, characterizing the relative strength of the matter self-coupling versus the gauge coupling, takes a value smaller or greater than one. This agrees with results previously obtained for asymptotic separations of the vortices. For lambda = 1, in particular, the vortices appear in equilibrium at any separation, hinting to the existence of a much wider class of solutions to the field equations. In the second lecture, the case lambda = 1 is considered in detail, illustrating analytical results which demonstrate that for this special value of the coupling constant solutions with any number of vortices at arbitrary positions do indeed exist