  We propose a QCD-like theory with the ZNc symmetry. The flavor-dependent
twisted boundary condition (TBC) is imposed on Nc degenerate flavor quarksin
the SU(Nc) gauge theory. The QCD-like theory is useful to understand the
mechanism of color confinement. Dynamics of the QCD-like theory is studied by
imposing the TBC on the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model.
The TBC model is applied to two- and three-color cases. The ZNc symmetry is
preserved below some temperature Tc, but spontaneously broken above Tc. The
color confinement below Tc preserves the flavor symmetry. Above Tc, the flavor
symmetry is broken, but the breaking is suppressed by the entanglement between
the Polyakov loop and the chiral condensate. Particularly at low temperature,
dynamics of the TBC model is similar to that of the PNJL model with the
standard fermion boundary condition, indicating that the ZNc symmetry is a good
approximate concept in the latter model even if the current quark mass is
small. The present prediction can be tested in future by lattice QCD, since the
QCD-like theory has no sign problem.
