Oblique Confinement at $\theta\neq 0$ in weakly coupled gauge theories with deformations

Abstract

The main focus of this work is to test the ideas related to the oblique confinement in a theoretically controllable manner using the "deformed QCD" as a toy model. We explicitly show that the oblique confinement in the weakly coupled gauge theories emerges as a result of condensation of $N$ types of monopoles shifted by the phase $\exp(i\frac{\theta+2\pi m}{N})$ in Bloch type construction. It should be contrasted with conventional and commonly accepted viewpoint that the confinement at $\theta\neq 0$ is due to the condensation of the electrically charged dyons which indeed normally emerge in the systems with $\theta\neq 0$ as a result of Witten's effect. We explain the basic reason why the "dyon" mechanism does not materialize-- it is because the Witten's effect holds for a static magnetic monopole treated as an external source. It should be contrasted with our case when $N$- types of monopoles are not static, but rather the dynamical degrees of freedom which fluctuate and themselves determine the ground state of the system.