Monopole and Polyakov loop

We propose a new order parameter of a $Z_N$ symmetry in SU(N) gauge theories in $4$ dimensional Minkowski space-time, assuming spatial periodic boundary conditions. It is given by $Tr(P\exp(i\int_c A_{\mu}dx^{\mu}))$ where the spatial path $c$ is taken, for example, along $x_1$ axis. The parameter vanishes when the $Z_N$ symmetry is preserved. We calculate the contribution of QCD monopoles to the order parameter and show that when the monopoles condense $\langle\Phi\rangle\neq 0$, it vanishes, while it does not vanish when they do not condense. These calculations are performed using a monopole field $\Phi$ canonically quantized in a model of dual superconductor.