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The Heavy Quark Free Energy in QCD and in Gauge Theories with Gravity Duals

We show that the Polyakov loop approaches its asymptotic value at high temperatures {\it from below} in any strongly-coupled gauge theory at large $N_c$ with a dual supergravity description. In these theories the Polyakov loop can be used to define a quantity that behaves as a free energy of a heavy quark in the medium. This should be compared to the recent lattice results in pure glue SU(3) theory at high temperatures where the Polyakov loop approaches its non-interacting limit {\it from above}. We show that the "heavy quark free energy" obtained from the renormalized loop computed on the lattice does not behave as a true thermodynamic free energy. We conjecture that this should always be the case in asymptotically free gauge theories.