Chiral symmetry at finite T, the phase of the Polyakov loop and the spectrum of the Dirac operator

A recent Monte Carlo study of {\em quenched} QCD showed that the chiral condensate is non-vanishing above $T_c$ in the phase where the average of the Polyakov loop $P$ is complex. We show how this is related to the dependence of the spectrum of the Dirac operator on the boundary conditions in Euclidean time. We use a random matrix model to calculate the density of small eigenvalues and the chiral condensate as a function of $\arg P$. The chiral symmetry is restored in the $\arg P=2\pi \mathrm{/}3$ phase at a higher $T$ than in the $\arg P=0$ phase. In the phase $\arg P=\pi$ of the $SU(2)$ gauge theory the chiral condensate stays nonzero for all~$T$.