tr}\left([\vev{A_y},~\vev{\Phi}]^2\right),$$ which is positive semidefinite. The diagonal form of $\vev{\Phi}$ makes the term vanish to minimize the effective potential. Then, the effective potential is divided into two parts written in terms of $\theta_i$ or $v$ alone. As a result, the minimization can be carried out separately with respect to the order parameters, so that the phase structure does not depend on the size of $S^1$. The residual gauge symmetry is determined by the generators of $SU(N)$ which commute with both $\vev{A_y}$ and $\vev{\Phi}$. Though we have already known that the gauge symmetry is (un)broken to $(SU(N))$ $U(1)^{N-1}$ through the Hosotani mechanism if the fermions belong to the (fundamental) adjoint representation under $SU(N)$, the actual residual gauge symmetry depends on the structure of the Higgs potential.

If we assume the same type of the Higgs potential as Eq.(\[potential\]), for example, there are generally flat directions parametrized like $\vbar_1^2 +\cdots +\vbar_{N-1}^2
+(\vbar_1 +\cdots + \vbar_{N-1})^2=\mbar^2/\lambda$ in the effective potential. The residual gauge symmetry is not uniquely determined in this case[^7]. For the fermions belonging to the fundamental representation under $SU(N)$, depending on the form of the Higgs VEV, the $SU(N)$ gauge symmetry is broken to its subgroup. On the other hand, for 