Duality between Different Geometries of a Resonant Level in a Luttinger Liquid

We prove an exact duality between the side-coupled and embedded geometries of a single level quantum dot attached to a quantum wire in a Luttinger liquid phase by a tunneling term and interactions. This is valid even in the presence of a finite bias voltage. Under this relation the Luttinger liquid parameter g goes into its inverse, and transmittance maps onto reflectance. We then demonstrate how this duality is revealed by the transport properties of the side-coupled case. Conductance is found to exhibit an antiresonance as a function of the level energy, whose width vanishes ( enhancing transport) as a power law for low temperature and bias voltage whenever g > 1, and diverges ( suppressing transport) for g < 1. On-resonance transmission is always destroyed, unless g is large enough.