Shape Dependence of Holographic Renyi Entropy in Conformal Field Theories
Xi Dong
TL;DR
This work analyzes the shape dependence of Rényi entropy in four-dimensional holographic CFTs by focusing on the universal coefficients f_a(n), f_b(n), and f_c(n) that accompany geometric invariants of the entangling surface. By mapping to a deformed hyperboloid and relating f_b to the stress-tensor one-point function, the authors compute f_b holographically and compare it to f_c, finding a violation of the conjecture f_b(n)=f_c(n) in holographic theories. They provide both numerical and analytic results, including small-n and large-n limits, showing that f_b and f_c coincide only at n=1 but diverge otherwise. The results clarify the shape dependence of Rényi entropy in holographic CFTs and have implications for twist-operator analyses, suggesting possible approximate relations between f_b and f_c in certain regimes and guiding extensions to other dimensions.
Abstract
We develop a framework for studying the well-known universal term in the Renyi entropy for an arbitrary entangling region in four-dimensional conformal field theories that are holographically dual to gravitational theories. The shape dependence of the Renyi entropy $S_n$ is described by two coefficients: $f_b(n)$ for traceless extrinsic curvature deformations and $f_c(n)$ for Weyl tensor deformations. We provide the first calculation of the coefficient $f_b(n)$ in interacting theories by relating it to the stress tensor one-point function in a deformed hyperboloid background. The latter is then determined by a straightforward holographic calculation. Our results show that a previous conjecture $f_b(n) = f_c(n)$, motivated by surprising evidence from a variety of free field theories and studies of conical defects, fails holographically.