  It is known that the deconfining transition of QCD is accompanied by the
appearance of localized eigenmodes at the low end of the Dirac spectrum. In the
quenched case localization appears exactly at the critical temperature of
deconfinement. In the present work, using quenched simulations exactly at the
critical temperature, we show that the localization properties of low Dirac
modes change abruptly between the confined and deconfined phase. This means
that in the real Polyakov loop sector, the mobility edge has a discontinuity at
the critical temperature. In contrast, in the complex sector, there is no such
discontinuity at $T_c$, even the lowest Dirac modes remain delocalized at the
critical temperature in the deconfined phase.
