Universality in the Shape Dependence of Holographic R\'enyi Entropy for General Higher Derivative Gravity

TL;DR

This paper derives universal relations for shape-dependent coefficients of Renyi entropy in four-dimensional CFTs using higher derivative gravity, revealing connections between holographic and free field theories and proposing a universal law across dimensions.

Contribution

It introduces universal relations for shape coefficients of Renyi entropy in 4D CFTs from higher derivative gravity and explores their applicability to various free and interacting theories.

Findings

Shape coefficients satisfy known differential relations.

Relations hold for free fermions and vectors, but not scalars.

A combined relation is universal for all free CFTs and conjectured to be universal for all CFTs.

Abstract

We consider higher derivative gravity and obtain universal relations for the shape coefficients $(f_a, f_b, f_c)$ of the shape dependent universal part of the R\'enyi entropy for four dimensional CFTs in terms of the parameters $(c, t_2, t_4)$ of two-point and three-point functions of stress tensors. As a consistency check, these shape coefficients $f_a$ and $f_c$ satisfy the differential relation as derived previously for the R\'enyi entropy. Interestingly, these holographic relations also apply to weakly coupled conformal field theories such as theories of free fermions and vectors but are violated by theories of free scalars. The mismatch of $f_a$ for scalars has been observed in the literature and is due to certain delicate boundary contributions to the modular Hamiltonian. Interestingly, we find a combination of our holographic relations which are satisfied by all free CFTs including scalars. We conjecture that this combined relation is universal for general CFTs in four dimensional spacetime. Finally, we find there are similar universal laws for holographic R\'enyi entropy in general dimensions.