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Symmetry breaking caused by large β-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 S 2 Γ S 1 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 β-charge is necessary. With such large β-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 β-charge is not so large