A universal feature of CFT Renyi entropy

We show that for a d-dimensional CFT in flat space, the Renyi entropy S-q across a spherical entangling surface has the following property: in an expansion around q = 1, the first correction to the entanglement entropy is proportional to C-T, the coefficient of the stress tensor vacuum two-point function, with a fixed d-dependent coefficient. This is equivalent to a similar statement about the free energy of CFTs living on S-1 x Hd-1 with inverse temperature beta = 2 pi q. In addition to furnishing a direct argument applicable to all CFTs, we exhibit this result using a handful of gravity and field theory computations. Knowledge of C-T thus doubles as knowledge of Renyi entropies in the neighborhood of q = 1, which we use to establish new results in 3d vector models at large N.