Dual variables for lattice gauge theories and the phase structure ofZ(N)systems

The 't Hooft disorder parameters are constructed within the framework of $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ lattice gauge theories in three or four dimensions. It is found that these operators arise naturally from a duality transformation which is similar to the standard transformation for $Z\left(N\right)\mathrm{}$ gauge theories. To illustrate the behavior of dual variables in a simpler context, we study the Villain form of the $Z\left(N\right)\mathrm{}$ gauge system in three and four dimensions. The techniques include duality, strong-coupling expansions, and the electrodynamic representation. In four dimensions it is found that for $N>\mathrm{}{N}_c\approx 4$, the system possesses at least three phases: a strong-coupling phase with electric confinement, a weak-coupling phase with magnetic confinement, and an intermediate phase which resembles QED, with a massless photon and no confinement. We also study an $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$-Higgs system, which interpolates between the $Z\left(N\right)\mathrm{}$ and $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ systems.