Magnetic Z(N) symmetry in hot QCD and the spatial Wilson loop

TL;DR

The paper investigates how the deconfining transition in nonabelian gauge theories is governed by the realization of the magnetic $Z_N$ symmetry, using the 't Hooft loop $V(C)$ as a canonical order parameter and linking its behavior to the spatial Wilson loop. It shows that the magnetic $Z_N$ symmetry is spontaneously broken at zero temperature and restored above the deconfinement temperature, with distinct area- versus perimeter-law signatures for the spatial Wilson loop in 2+1 and 3+1 dimensions. A key physical picture is developed via vortex operators and domain-wall tensions, including a 2+1D toy model (U-loop) that illustrates how symmetry realization controls loop behavior. At finite temperature the relationship between symmetry realization and the Wilson loop becomes nuanced; the Wilson loop can retain area-law behavior in the high-temperature phase due to thermal excitations and vortex dynamics, consistent with lattice and strong-coupling insights. These results clarify the deep connection between global magnetic symmetry and observable loop observables and suggest lattice tests of magnetic flux free energies and vortex dynamics.

Abstract

We discuss the relation between the deconfining phase transition in gauge theories and the realization of the magnetic Z(N) symmetry. At low temperature the Z(N) symmetry is spontaneously broken while above the phase transition it is restored. This is intimately related to the change of behaviour of the spatial 't Hooft loop discussed in hep-ph/9909516. We also point out that the realization of the magnetic symmetry has bearing on the behaviour of the spatial Wilson loop. We give a physical argument to the effect that at zero temperature the spatial Wilson loop must have perimeter law behaviour in the symmetric phase but area law behaviour in the spontaneously broken phase. At high temperature the argument does not hold and the restoration of magnetic Z(N) is consistent with area law for the Wilson loop.