Self-duality in quantum impurity problems

We establish the existence of an exact non-perturbative self-duality in a variety of quantum impurity problems, including the Luttinger liquid or quantum wire with impurity. The former is realized in the fractional quantum Hall effect, where the duality interchanges electrons with Laughlin quasiparticles. We discuss the mathematical structure underlying this property, which bears an intriguing resemblance with the work of Seiberg and Witten on supersymmetric non-abelian gauge theory.