zero ’t Hooft coupling. The phase transition is signaled by a Hagedorn behavior [@Hagedorn:1965st] near the critical temperature, where the density of states grows exponentially with energy. This remarkable behavior can be ascribed to a zero mode of the gauge field (i.e. a mode with vanishing quadratic action) which cannot be integrated out in perturbation theory. Gauge invariance highly constrains the low energy effective action of this mode, resulting in non-trivial self interactions for it even at zero coupling.

To order $\lambda$ the phase transition is of first order and the critical temperature is identical with the Hagedorn temperature. However, in the next order in perturbation theory [@Aharony:2003sx], the transition might be of first order and slightly below the Hagedorn temperature, or second order and at the Hagedorn temperature. In order to determine the order of the phase transition one needs to perform a 3-loop computation in perturbation theory. For pure **YM** this has been done in [@Aharony:2005bq], where it was found that the transition is of first order. For **YM** theories with adjoint matter fields (including $\cal{N}=4$ **SYM**) the order of the phase transition at weak coupling is still an open question and examples which exhibit either behavior can 