found by studying classical type IIB supergravity. This study showed that the theory has a low temperature confining phase, in which the free energy scales as $N^0$, and a high temperature deconfining phase, in which the free energy scales as $N^2$. The two phases were found to be separated by a first order phase transition.

It is then interesting to ask whether this behavior persists even at small ’t Hooft coupling. In that limit the conformal $\cal{N}=4$ theory can be studied by using perturbation theory. However the study needs not be restricted to weakly coupled conformal theories, and can be extended to asymptotically free **YM** theories. As opposed to conformal field theories, asymptotically free gauge theories generate an infrared scale $\Lambda_{YM}$ which leads to the breakdown of perturbation theory in flat space at low temperatures/energy scales. But on $S^3$ with radius $R_{S^3}$ that bad infrared behavior is cut off by the finite size of space, i.e. if $R_{S^3}\Lambda_{YM}\ll 1$ one can use perturbation theory to compute the partition function at all temperatures.

This has been done in [@Sundborg:1999ue],[@Aharony:2003sx], where it was shown that for a wide class of models the first order deconfinement transition persists even in the limit of 