Phase of the Wilson line

This paper discusses some aspects of the global Z(N) symmetry of finite-temperature, SU(N), pure Yang-Mills lattice gauge theory. It contributes to the recent discussion of the physics of the phase of the Wilson line expectation value. In the high T phase, left-angle L right-angle takes one of N distinct values proportional to the Nth roots of unity in Z(N), and the Z(N) symmetry is broken. Only one of these is consistent with the usual interpretation left-angle L right-angle=e-F/T with F the excess free energy due to a source of color-electric flux at the position of the line. This relation should be generalized to left-angle L right-angle=ze-F/T with z element-of Z(N) so that it is consistent with the negative or complex values. In the Hamiltonian description, the physical variables are the group elements on the links of the spatial lattice. In a Lagrangian formulation, there are also group elements on links in the inverse-temperature direction from which the Wilson line is constructed. These are unphysical, auxiliary variables introduced to enforce the Gauss law constraints. The following results are obtained. The generalized relation left-angle L right-angle=ze-F/T, which has appeared in earlier papers, is derived. The value of z element-of Z(N) is determined by the external field that is needed for taking the infinite-volume limit. There is a single physical, high-temperature phase, which is the same for all z. The global Z(N) symmetry is not physical; it acts as the identity on all physical states. In the Hamiltonian formulation, the high-temperature phase is not distinguished by physical broken symmetry. Rather the high-temperature phase has a percolating flux network that is not present in the low-temp phase