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We give a recipe for computing correlation functions of the displacementoperator localized on a spherical or planar higher dimensional twist defectusing AdS/CFT. Such twist operators are typically used to construct the $n$'thRenyi entropies of spatial entanglement in CFTs and are holographically dual toblack holes with hyperbolic horizons. The displacement operator then tells ushow the Renyi entropies change under small shape deformations of the entanglingsurface. We explicitly construct the bulk to boundary propagator for thedisplacement operator insertion as a linearized metric fluctuation of thehyperbolic black hole and use this to extract the coefficient of thedisplacement operator two point function $C_D$ in any dimension. The $n\rightarrow 1$ limit of the twist displacement operator gives the same bulkresponse as the insertion of a null energy operator in vacuum, which isconsistent with recent results on the shape dependence of entanglement entropyand modular energy.

Gravitational dual of the R\'{e}nyi twist displacement operator