Phase Diagram of N=4 Super-Yang-Mills Theory with R-Symmetry Chemical   Potentials
  The phase diagram of large Nc, weakly-coupled N=4 supersymmetric Yang-Mills theory on a three-sphere with non-zero chemical potentials is examined. In the zero coupling limit, a transition line in the mu-T plane is found, separating a "confined" phase in which the Polyakov loop has vanishing expectation value from a "deconfined" phase in which this order parameter is non-zero. For non-zero but weak coupling, perturbative methods may be used to construct a dimensionally reduced effective theory valid for sufficiently high temperature. If the maximal chemical potential exceeds a critical value, then the free energy becomes unbounded below and no genuine equilibrium state exists. However, the deconfined plasma phase remains metastable, with a lifetime which grows exponentially with Nc (not Nc^2). This metastable phase persists with increasing chemical potential until a phase boundary, analogous to a spinodal decomposition line, is reached. Beyond this point, no long-lived locally stable quasi-equilibrium state exists.   The resulting picture for the phase diagram of the weakly coupled theory is compared with results believed to hold in the strongly coupled limit of the theory, based on the AdS/CFT correspondence and the study of charged black hole thermodynamics. The confinement/deconfinement phase transition at weak coupling is in qualitative agreement with the Hawking-Page phase transition in the gravity dual of the strongly coupled theory. The black hole thermodynamic instability line may be the counterpart of the spinodal decomposition phase boundary found at weak coupling, but no black hole tunneling instability, analogous to the instability of the weakly coupled plasma phase is currently known. 
Daisuke Yamada and Laurence G. Yaffe