Shape Dependence of Holographic R\'enyi Entropy in General Dimensions

TL;DR

This paper develops a holographic approach to analyze how Reny entropies in conformal field theories respond to small shape deformations of the entangling surface, providing explicit results across multiple dimensions and challenging existing conjectures.

Contribution

It introduces a new holographic method linking stress tensor responses to shape deformations and extends analysis beyond leading order in bulk theories, discrediting previous conjectures.

Findings

Explicit numerical results for dimensions 3 to 6

Analytical limits for Reny index approaching 1 and 0

Disproof of existing conjectures relating shape dependence and twist operator weight

Abstract

We present a holographic method for computing the response of R\'enyi entropies in conformal field theories to small shape deformations around a flat (or spherical) entangling surface. Our strategy employs the stress tensor one-point function in a deformed hyperboloid background and relates it to the coefficient in the two-point function of the displacement operator. We obtain explicit numerical results for $d=3,\dots,6$ spacetime dimensions, and also evaluate analytically the limits where the R\'enyi index approaches 1 and 0 in general dimensions. We use our results to extend the work of 1602.08493 and disprove a set of conjectures in the literature regarding the relations between the R\'enyi shape dependence and the conformal weight of the twist operator. We also extend our analysis beyond leading order in derivatives in the bulk theory by studying Gauss-Bonnet gravity.