Symmetry breaking caused by large ℛ-charge

We discuss the gauge symmetry breaking via the Hosotani mechanism by using exact results on supersymmetric gauge theories based on the localization method. We use the theories on S2 × S1 Euclidean space, and study how the effective potential for the Wilson line phase varies by running an imaginary chemical potential. In order to break the symmetry, we find that large ℛ-charge is necessary. With such large ℛ-charge, we study the phase structure of the theory. In addition, we observed that a finite size effect on our curved space when we take ℛ-charge is not so large.