Gravity duals of boundary cones

TL;DR

The work develops regular gravity duals to boundary conical singularities arising from the replica trick, using a double expansion in the cone strength and distance to the singular surface to extract vacuum polarization and Rényi entropy divergences. It constructs explicit bulk geometries for squashed cones in AdS5, computing the boundary stress-energy response and verifying the expected logarithmic structure, including a notable f_b(n) ≠ f_c(n) outcome for holographic theories. It further analyzes the splitting problem in higher-derivative gravity, showing a non-minimal resolution in GR that influences holographic entropy formulas and their higher-derivative extensions. The results illuminate how bulk regularity conditions regulate boundary singularities and offer a framework for extending holographic entanglement concepts to more general geometries and gravity theories.

Abstract

The replica trick defines Renyi entropies as partition functions on conically singular geometries. We discuss their gravity duals: regular bulk solutions to the Einstein equations inducing conically singular metrics at the boundary. When the conical singularity is supported on a flat or spherical surface, these solutions are rewritings of the hyperbolic black hole. For more general shapes, these solutions are new. We construct them perturbatively in a double expansion in the distance and strength of the conical singularity, and extract the vacuum polarisation due to the cone. Recent results about the structure of logarithmic divergences of Renyi entropies are reproduced ---in particular, $f_b\neq f_c$. We discuss in detail the dynamical resolution of the singularity in the bulk. This resolution is in agreement with a previous proposal, and indicates a non-minimal settling to the `splitting problem': an apparent ambiguity in the holographic entropy formula of certain theories with higher derivatives.