Confinement/deconfinement transition of large N gauge theories with Nf fundamentals: nf/n finite

We consider large N zero-coupling d-dimensional U(N) gauge theories, with Nf matter fields in the fundamental representation on a compact spatial manifold Sd-1x time, with Nf/N finite. The Gauss' law constraint induces interactions among the fields, in spite of the zero-coupling. This class of theories undergo a 3rd order deconfinement phase transition at a temperature Tc proportional to the inverse length scale of the compact manifold. The low-temperature phase has a free-energy of O(Nf2), interpreted as that of a gas of (color singlet) mesons and glueballs. The high-temperature (deconfinement) phase has a free energy of order N2f(Nf/N,T), which is interpreted as that of a gas of gluons and of fundamental and anti-fundamental matter states. This suggests the existence of a dual string theory, and a transition to a black hole at high temperature