Non--Abelian Duality, Parafermions and Supersymmetry

TL;DR

This work analyzes how non-Abelian target-space dualities interact with supersymmetry in two complementary settings: left-invariant σ-models and WZW models. It shows that extended world-sheet supersymmetry is generally realized non-locally after non-Abelian duality, mediated by path-ordered Wilson lines and, in WZW cases, by classical non-Abelian parafermions, while a local $N=1$ SUSY is typically preserved. A key result is the canonical equivalence between WZW models and their non-Abelian duals with respect to a vector subgroup, demonstrated through matching current algebras expressed via parafermions. The findings illuminate how stringy (non-local) effects can restore or reinterpret supersymmetry after duality, with hyper-Kähler and axionic-instanton backgrounds explicitly illustrating the phenomenon and linking to broader duality frameworks such as Poisson–Lie duality and non-isometric backgrounds.

Abstract

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.