Renyi entropy is a 1-parameter generalization of Entanglement entropy and is an ubiquitous quantity. Ryu and Takayanagi conjectured a general formula (RT formula) for computing the entanglement entropy of arbitrary regions of vacuum state in holographic CFTs. In an important paper validating the RT formula, Casini et al. computed the entanglement entropy of spherical regions in the CFT as the horizon area of a black hole with hyperbolic horizon. This black hole geometry, then presented a 1-parameter generalization, whose horizon area corresponds to the Renyi entropy of spherical regions in the CFT.
This black hole with hyperbolic horizon is dual to a defect-CFT with a co-dimension 2 twist defect. Two point function of the displacement operator (localized on the defect) is directly related to the change in Renyi entropy under small shape perturbations of the entangling surface. This two-point function is fixed by residual conformal symmetry up to a constant factor, and we compute this factor in holographic CFTs. This computation disproves an earlier conjectured formula in the literature.