A Note on Dual Superconductivity and Connnement

Electric self-dual vortices arising as BPS states in the strong coupling limit of N=2 supersymmetric Yang-Mills theory, softly broken to N = 1, are reported. 1. About twenty years ago, 't Hooft and Mandelstam 1], 2] proposed a qualitative description of the quark connnement phenomenon based on an analogy with superconductivity. According to this interpretation, the QCD vacuum behaves as a \dual" superconductor, a material where magnetic charge condensation and the electric Meissner eeect take place. At that time, it was not known how local quantum eld operators that create or annihilate magnetically charged quanta could arise in a non-abelian gauge theory and the 't Hooft-Mandelstam picture remained as a \natural" but descriptive scheme of quark connnement. Pursuing another brilliant idea, the Olive-Montonen electromagnetic duality conjecture 3], Seiberg-Witten 4] showed that in the family of quantum eld theories describing N = 2 supersymmetric Yang-Mills systems at a speciic quantum vacuum, the 't Hooft insight does indeed hold. In certain patch of the quantum moduli space of vacua the local elds carry and/or are coupled to magnetic charge. The conventional local elds of N = 2 SUSY YM theory appropriately describe the system in another patch and are related with those previously mentioned by a non-local transformation suggested by electromagnetic duality. The Olive-Montonen conjecture is subsequently proven in a far from trivial manner as a duality of the Wilsonian eeective theories at weak and strong coupling, and there is a third patch to completely cover the quantum moduli space of vacua where the local elds bring both electric and magnetic charge. Moreover, by softly breaking N = 2 to N = 1 supersymmetry, Seiberg and Witten were also able to show that magnetic charge condensation and the electric Higgs mechanism are 1