  We study N =4 super Yang-Mills theories on a three sphere with two kinds of
chemical potentials. One is associated with the R-symmetry and the other with
the rotational symmetry of S^3 (SO(4) symmetry). These correspond to the
charged Kerr-AdS black holes via AdS/CFT. The exact partition functions at zero
coupling are computed and the thermodynamical properties are studied. We find a
nontrivial gap between the confinement/deconfinement transition line and the
boundary of the phase diagram when we include more than four chemical
potentials. In the dual gravity, we find such a gap in the phase diagram to
study the thermodynamics of the charged Kerr-AdS black hole. This shows that
the qualitative phase structures agree between the both sides. We also find
that the ratio of the thermodynamical quantities is almost well-known factor
3/4 even at the low temperature.

High Energy Physics - Theory;General Relativity and Quantum Cosmology