Gravitational dual of the Renyi displacement operator

We give a recipe for computing correlation functions of the displacement operator localized on a spherical or planar higher dimensional twist defect using AdS/CFT. Such twist operators are typically used to construct the nth Renyi entropies of spatial entanglement in conformal field theories and are holographically dual to black holes with hyperbolic horizons. The displacement operator then tells us how the Renyi entropies change under small shape deformations of the entangling surface. We explicitly construct the bulk to boundary propagator for the displacement operator insertion as a linearized metric fluctuation of the hyperbolic black hole and use this to extract the coefficient of the displacement operator two point function C-D in any dimension. The n -> 1 limit of the twist displacement operator gives the same bulk response as the insertion of a null energy operator in vacuum, which is consistent with recent results on the shape dependence of entanglement entropy and modular energy.