Non-Abelian duality, parafermions, and supersymmetry.

Non–Abelian duality in relation to supersymmetry is examined. When the action of the isometry group on the complex structures is non–trivial, extended supersymmetry is realized non–locally after duality, using path ordered Wilson lines. Prototype examples considered in detail are, hyper–Kahler metrics with SO(3) isometry and supersymmetric WZW models. For the latter, the natural objects in the non–local realizations of supersymmetry arising after duality are the classical non–Abelian parafermions. The canonical equivalence of WZW models and their non–Abelian duals with respect to a vector subgroup is also established.