Phases of dual superconductivity and confinement in softly broken N=2 supersymmetric Yang-Mills theories

We study the electric flux tubes that undertake color confinement in N=2 supersymmetric Yang-Mills theories softly broken down to N=1 by perturbing with the first two Casimir operators. The relevant Abelian Higgs model is not the standard one due to the presence of an off-diagonal coupling among different magnetic U(1) factors. We perform a preliminary study of this model at a qualitative level. Bogomol'nyi-Prasad-Sommerfield (BPS) vortices are explicitly obtained for particular values of the soft breaking parameters. Generically, however, even in the ultrastrong scaling limit, vortices are not critical but exist in a ``hybrid'' type II phase. Also, ratios among string tensions are seen to follow no simple pattern. We examine the situation at the vacua half broken by the Higgs mechanism and find evidence for solutions with the behavior of superconducting strings. In some cases they are solutions to BPS equations.