Existence of a confinement phase in quantum electrodynamics

We show that four-dimensional U(1) gauge theory in the continuum formulation has a confining phase (exhibiting the area law of the Wilson loop) in the strong coupling region above a critical coupling $g_c.$ This result is obtained by taking into account topological nontrivial sectors in U(1) gauge theory. The derivation is based on the reformulation of gauge theory as a deformation of topological quantum field theory and a subsequent dimensional reduction of the $D$-dimensional topological quantum field theory to the $(D$-2)-dimensional nonlinear $\sigma$ model. The topological quantum field theory part of four-dimensional U(1) gauge theory is exactly equivalent to the two-dimensional O(2) nonlinear $\sigma$ model. The confining (r. Coulomb) phase of U(1) gauge theory corresponds to the high- (r. low-) temperature phase of the O(2) nonlinear $\sigma$ model and the critical point $g_c$ is determined by the Berezinskii-Kosterlitz-Thouless phase transition temperature. The quark (charge) confinement in the strong coupling phase is caused by vortex condensation. Thus the continuum gauge theory has direct correspondence to the compact formulation of lattice gauge theory.