The geometry of induced currents in two-dimensional media

We present a framework that allows us to clearly identify the geometric features underlying the phenomenon of superconductivity in two-dimensional materials. In particular, we show that any such medium whose response to an externally applied electromagnetic field is a geodesically flowing induced current, must be a superconductor. In this manner, we conclude that the underlying geometry of this type of media is that of a Lorentzian contact manifold. Moreover, we show that the macroscopic hallmark of their superconducting state is a purely topological condition equivalent to the geodesic nature of the induced current: the non-vanishing of its helicity.