Phase structure of a compact $U\left(1\right)\mathrm{}$ gauge theory from the viewpoint of a sine-Gordon model

We discuss the phase structure of the four-dimensional compact $U\left(1\right)\mathrm{}$ gauge theory at finite temperature using a deformation of the topological model. Its phase structure can be determined by the behavior of the Coulomb gas (CG) system on the cylinder. We utilize the relation between the CG system and the sine-Gordon (SG) model, and investigate the phase structure of the gauge theory in terms of the SG model. Especially, the critical-line equation of the gauge theory in the strong-coupling and high-temperature region is obtained by calculating the one-loop effective potential of the SG model.