  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 x 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 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.
