The phase diagram of large N c , 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 µ-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 N c (not N 2 c ). 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.

Properties of Schrödinger black holes are derived from those of AdS black holes expressed in light-cone coordinates with a particular normalization. Unlike the usual construction from an AdS black hole using a null Melvin twist, an AdS black hole in light-cone is simple and has a well-defined Brown-York procedure with the standard counterterms. Our procedure is demonstrated by the computation of the DC conductivity and the derivation of the R-charged black hole thermodynamic properties.

Free scalar field theory in the sector with a large number of particles can be interpreted as bosonic string theory on anti-de Sitter space of a vanishing radius. Different ways of writing the field theory Hamiltonian translate to different ways of reparametrizing the world-sheet coordinate. Adding a mass term in the field theory corresponds to cutting off the warped AdS direction, with the cutoff inversely proportional to the mass. The string theory has neither a tachyon nor a critical dimension.

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We argue that the "vortex-finding" property of maximal center gauge, i.e. the ability of this gauge to locate center vortices inserted by hand on any given lattice, is the key to its success in extracting the vortex content of thermalized lattice configurations. We explain how this property comes about, and why it is expected not only in maximal center gauge, but also in an infinite class of gauge conditions based on adjoint-representation link variables. In principle, the vortex-finding property can be foiled by Gribov copies. This fact is relevant to a gauge-fixing procedure devised by Kovács and Tomboulis, where we show that the loss of center dominance, found in their procedure, is explained by a corresponding loss of the vortex-finding property. The association of center dominance with the vortex-finding property is demonstrated numerically in a number of other gauges.