Polyakov loops and the Hosotani mechanism on the lattice

We explore the phase structure and symmetry breaking in four-dimensional SU(3) gauge theory with one spatial compact dimension on the lattice ($16^3 \times 4$ lattice) in the presence of fermions in the adjoint representation with periodic boundary conditions. We estimate numerically the density plots of the Polyakov loop eigenvalues phases, which reflect the location of minima of the effective potential in the Hosotani mechanism. We find strong indication that the four phases found on the lattice correspond to SU(3)-confined, SU(3)-deconfined, SU(2) x U(1), and U(1) x U(1) phases predicted by the one-loop perturbative calculation. The case with fermions in the fundamental representation with general boundary conditions, equivalent to the case of imaginary chemical potentials, is also found to support the $Z_3$ symmetry breaking in the effective potential analysis.