  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.
