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

A recent Monte Carlo study of quenched QCD showed that the chiral condensate is non-vanishing above Tc 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 = 2π3 phase at a higher T than in the arg P = 0 phase. In the phase arg P = π of the SU(2) gauge theory the chiral condensate stays nonzero for all T.