Introduction to $Z_N$ Symmetry in $S{U}_N$ Gauge Theories at Finite Temperatures

There have been several talks at this workshop on the physical interpretation and consequences of the $Z_N$ symmetry which is present in the Euclidean Path Integral formulation of $S{U}_N$ Gauge Theories at finite temperature. The purpose of this paper is to present an introduction to this subject. After a brief review of Gluodynamics at finite temperature, the nature of the $Z_N$ symmetry in this system is described and its relationship to confinement is discussed. $Z_N$ domain walls and bubbles are then described as is their relationship to the confining–deconfining phase transition. The effect of Fermions is then considered and the presence of metastable extrema of the Effective Potential for the Polyakov–Wilson Line is described. It is then argued that these metastable extrema do not correspond to physically realizable metastable states.