Universality in the geometric dependence of Renyi entropy

We derive several new results for Renyi entropy, S-n, across generic entangling surfaces. We establish a perturbative expansion of the Renyi entropy, valid in generic quantum field theories, in deformations of a given density matrix. When applied to even-dimensional conformal field theories, these results lead to new constraints on the n-dependence, independent of any perturbative expansion. In 4d CFTs, we show that the n-dependence of the universal part of the ground state Renyi entropy for entangling surfaces with vanishing extrinsic curvature contribution is in fact fully determined by the Renyi entropy across a sphere in flat space. Using holography, we thus provide the first computations of Renyi entropy across non-spherical entangling surfaces in strongly coupled 4d CFTs. Furthermore, we address the possibility that in a wide class of 4d CFTs, the flat space spherical Renyi entropy also fixes the n-dependence of the extrinsic curvature contribution, and hence that of arbitrary entangling surfaces. Our results have intriguing implications for the structure of generic modular Hamiltonians.