Monopoles and the Polyakov loop

We propose a new order parameter of a ZN symmetry in SU(N) gauge theories in four dimensional Minkowski space-time, assuming spatial periodic boundary conditions. It is given by Tr(Pexp(i∫cAμdxμ)), where the spatial path c is taken, for example, along the x1 axis. The parameter vanishes when the ZN symmetry is preserved. We calculate the contribution of QCD monopoles to the order parameter and show that when the monopoles condense ⟨Φ⟩≠0, it vanishes, while it does not vanish when they do not condense. These calculations are performed using a monopole field Φ canonically quantized in a model of dual superconductor.