Rényi entropy and conformal defects

TL;DR

The paper develops a defect-CFT framework in which twist operators are treated as conformal defects to study how Rényi entropies depend on entangling-surface geometry. It introduces the displacement operator $D^a$ that implements local deformations and shows the second-order shape variation of $S_n$ is controlled by the two-point function $igra D^a D^bigra_n$, tying geometric responses to defect data $h_n$ and $C_D$. A central result is a simple constraint $C_D(n)= d \, ext{Gamma}((d+1)/2) (2/ ext sqrt{π})^{d-1} h_n$, which unifies several conjectures about cone/cusp contributions and, in four dimensions, yields explicit relations $f_c(n)=rac{3 ext{π}}{2}rac{h_n}{n-1}$ and $f_b(n)=rac{ ext{π}^2}{16}rac{C_D(n)}{n-1}$. The authors apply these ideas to deformations of spheres and conical singularities, verify consistency with Mezei’s EE results in the $n o1$ limit, and discuss the viability of the constraint in holographic theories, SUSY Wilson lines, and free field cases. Overall, the work provides a unifying defect-CFT perspective on Rényi entropy shape dependence and identifies precise regimes where universal geometric coefficients are dictated by defect data, while highlighting limits and open questions for broader universality.

Abstract

We propose a field theoretic framework for calculating the dependence of Rényi entropies on the shape of the entangling surface in a conformal field theory. Our approach rests on regarding the corresponding twist operator as a conformal defect and in particular, we define the displacement operator which implements small local deformations of the entangling surface. We identify a simple constraint between the coefficient defining the two-point function of the displacement operator and the conformal weight of the twist operator, which consolidates a number of distinct conjectures on the shape dependence of the Rényi entropy. As an example, using this approach, we examine a conjecture regarding the universal coefficient associated with a conical singularity in the entangling surface for CFTs in any number of spacetime dimensions. We also provide a general formula for the second order variation of the Rényi entropy arising from small deformations of a spherical entangling surface, extending Mezei's results for the entanglement entropy.