  Using the overlap Dirac operator I show that, contrary to some expectations,
even well above the critical temperature there is not necessarily a gap in the
Dirac spectrum in pure SU(2) gauge theory. This happens when the Polyakov loop
and the fermion boundary condition combine to give close to periodic boundary
condition for the fermions in the time direction. In this Polyakov loop sector
there is a non-vanishing density of Dirac eigenvalues around zero which implies
that chiral symmetry is spontaneously broken. I demonstrate this both directly
and also by finding good agreement with the random matrix theory prediction for
the distribution of the lowest Dirac eigenvalue. I show that the chiral
condensate increases with the temperature therefore it is very unlikely to be
explained by topological fluctuations that become rapidly smaller above T_c.
Finally I show that it is only a small fraction of the lowest Dirac eigenvalues
that decide which Polyakov loop sector is favored by the fermion determinant if
dynamical fermions are turned on. This provides a qualitative understanding of
how the loss of confinement above T_c implies the restoration of chiral
symmetry.
