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We compute the effective action of the Polyakov loop in SU(2) and SU(3)Yang-Mills theory using a previously developed covariant variational approach.The formalism is extended to background gauge and it is shown how to relate thelow order Green's functions to the ones in Landau gauge studied earlier. Therenormalization procedure is discussed. The self-consistent effective action isderived and evaluated using the numerical solution of the gap equation. We finda clear signal for a deconfinement phase transition at finite temperatures,which is second order for SU(2) and first order for SU(3). The criticaltemperatures obtained are in reasonable agreement with high precision latticedata.

Covariant variational approach to Yang-Mills Theory: effective potential of the Polyakov loop