How to distinguish a confining from a deconfining phase in gauge theories with fermions

We show that for SU(N) gauge theories with fermions, there is a gauge-invariant way of distinguishing a confining phase from a deconfining phase. We begin by considering the gauge theory with a constraint imposed which fixes the total fermion number. This leads us to a study of the gauge theory with a nonzero imaginary chemical potential. In this formulation of the theory, the $Z_N$ symmetry (which is known to be present in pure gauge theories) has important consequences. Deconfinement at high temperature and fermion number is related to a realization of this $Z_N$ symmetry which differs from that at low temperature. The implication for the critical behavior of gauge theories at a fixed chemical potential is discussed.