The helicity modulus in gauge field theories

We consider the 4d compact U(1) theory with Wilson action and characterize its phase diagram using the notion of electromagnetic flux, instead of the more usual magnetic monopole. Taking inspiration from the flux picture, we consider the helicity modulus (h.m.) for this theory, and show that it is an order parameter for the confinement deconfinement phase transition. We extend the definition of the h.m. to an Abelian projected Yang-Mills theory, and discuss its behavior in SU(2).