Gravitational dual of the Rényi twist displacement operator

Abstract

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 $n$'th Renyi entropies of spatial entanglement in CFTs 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 $n \rightarrow 1$ 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.

Figures (2)

• Figure 1: Plots showing $C_D$ as a function of $n$ in holographic theories with an Einstein gravitational dual. In the left plot we have normalized to the coefficient of the stress tensor two point function $C_T$ and in the right plot we have shown the relative error of the holographic answer from the conjecture of Bianchi:2015liz which used results from free theories, and other conjectures, to guess that $C_D^{conj} = S_d h_\Sigma$ in the conventions used here.
• Figure 2: Witten diagrams Witten:1998qj for the displacement operator. The ball represents the Euclidean hyperbolic black hole and for clarity we have shown 1 co-dimension to the defect. The loop diagram on the left would compute the perturbative $1/N$ correction to $C_D$. The four point function of displacement operators is the next correction to the expansion of the shape dependence of the Renyi entropies about the spherical or planar cut: $S_n( \Sigma) \sim S_n(\Sigma_0 ) +\sum_N (N!)^{-1} \int \delta w^{\alpha_1} \ldots \int \delta w^{\alpha_N} \left< D_{\alpha_1} \ldots D_{\alpha_N} \right>$