Phases ofZN lattice gauge theory

We use the Lagrangian formulation of theZN gauge theory in four dimensions and demonstrate its equivalence with the partition function for a Coulomb gas of electric and magnetic loops. This model is shown to be self-dual when expressed in the Villai form. It is found, as a lower bound, that forN 4 three phases appear. The explicit calculation of the Wilson loop identifies them as electric confining, nonconfining and magnetic confining.