  String or M-theory in the background of Kerr-AdS black holes is thought to be
dual to the large n limit of certain conformal field theories on a rotating
sphere at finite temperature. The five dimensional black hole is associated to
N=4 supersymmetric Yang-Mills theory on a rotating three-sphere and the four
dimensional one to the superconformal field theory of coinciding M2 branes on a
rotating two-sphere. The thermodynamic potentials can be expanded in inverse
powers of the radius of the sphere. We compute the leading and subleading terms
of this expansion in the field theory at one loop and compare them to the
corresponding supergravity expressions. The ratios between these terms at weak
and strong coupling turns out not to depend on the rotation parameters in the
case of N=4 SYM. For the field theory living on one M2 brane we find a
subleading logarithmic term. No such term arises from the supergravity
calculation.
