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N=4 SYM on S^3 with Near Critical Chemical Potentials

We study the N = 4 theory at weak coupling, on a three sphere in the grand canonical ensemble with R symmetry chemical potentials. We focus attention on near critical values for the chemical potentials, above which the classical theory has no ground state. By computing a one loop effective potential for the light degrees of freedom in this regime, we show the existence of flat directions of complex dimension N, 2N and 3N for one, two and three critical chemical potentials respectively; these correspond to one half, one quarter and one-eighth BPS states becoming light respectively at the critical values. At small finite temperature we show that the chemical potentials can be continued beyond their classical limiting values to yield a deconfined metastable phase with lifetime diverging in the large N limit. Our low temperaure analysis complements the high temperature metastability found by Yamada and Yaffe. The resulting phase diagram at weak coupling bears a striking resemblance to the strong coupling phase diagram for charged AdS black holes. Our analysis also reveals subtle qualitative differences between the two regimes.