Chiral and deconfinement transition from Dyson-Schwinger equations

We determine the quark condensate and the dressed Polyakov loop from the finite temperature Landau gauge quark propagator evaluated with $U(1)$-valued boundary conditions in an approximation to quenched QCD. These gauge invariant quantities allow for an investigation of the chiral and deconfinement transitions. We compare results from Dyson-Schwinger equations on a lattice with infinite volume continuum results and study the temperature and quark mass dependence of both quantities. In particular we investigate the chiral condensate and the dressed Polyakov loop in the chiral limit. We also consider an alternative order parameter for the deconfinement transition, the dual scalar quark dressing, and compare it with the dressed Polyakov loop. As a result we find only slightly different transition temperatures for the chiral and the deconfinement transitions at finite quark masses; in the chiral limit both transitions coincide.