Hot gauge theories andZNphases

In this paper several aspects of $Z_N$ symmetry in gauge theories at high temperatures are discussed. The metastable $Z_N$ bubbles in $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ gauge theories with fermions may have, generically, unacceptable thermodynamic behavior. Their free energy $F\propto T^4$, with a positive proportionality constant. This leads not only to negative pressure but also to negative specific heat and, more seriously, to negative entropy. We argue that although such domains are important in the Euclidean theory, they cannot be interpreted as physical domains in Minkowski space. A related problem is connected with the analysis of the high-temperature limit of the confining phase. Using two-dimensional QCD with adjoint fermions as a toy model we shall demonstrate that in the light fermion limit there is no breaking of the $Z_N$ symmetry in the high-temperature limit and thus there are no $Z_N$ bubbles.