Holographic Calculations of Renyi Entropy

TL;DR

This work extends the Casini–Huerta–Myers framework to compute Rényi entropy for general CFTs in $d$ dimensions with a spherical entangling surface by mapping to thermal entropy on $\mathbb{R}\times\mathbb{H}^{d-1}$ and then applying holography via bulk topological black holes. Across Einstein, Gauss–Bonnet, and quasi-topological gravity, the authors show Rényi entropy is a nonlinear function of central charges and additional CFT parameters, with universal features captured by horizon/Wald entropy and by the derivative $\partial_q h_q|_{q=1}$ tied to the stress-tensor two-point function $\widetilde{C}_T$. They connect the thermal calculus to the twist-operator replica-trick, deriving the scaling dimension $h_q$ holographically, reproducing the exact 2D results and generalizing to higher dimensions, where $S_q$ depends on richer CFT data beyond $(a,c)$. The analysis also clarifies regulator choices, reveals the universal divergence structure matches that of entanglement entropy, and highlights simplifications such as $\partial_q h_q|_{q=1}$ being governed by $\widetilde{C}_T$ across models. Overall, the results emphasize the complexity of higher-dimensional Rényi entropies and motivate further study of twist operators and spectral data in holographic CFTs.

Abstract

We extend the approach of Casini, Huerta and Myers to a new calculation of the Renyi entropy of a general CFT in d dimensions with a spherical entangling surface, in terms of certain thermal partition functions. We apply this approach to calculate the Renyi entropy in various holographic models. Our results indicate that in general, the Renyi entropy will be a complicated nonlinear function of the central charges and other parameters which characterize the CFT. We also exhibit the relation between this new thermal calculation and a conventional calculation of the Renyi entropy where a twist operator is inserted on the spherical entangling surface. The latter insight also allows us to calculate the scaling dimension of the twist operators in the holographic models.