R\'enyi entropy and conformal defects

TL;DR

This paper introduces a conformal defect framework to compute how Renyi entropies depend on the shape of the entangling surface in conformal field theories, linking defect operators to shape variations.

Contribution

It establishes a novel field theoretic approach using conformal defects and displacement operators to analyze shape dependence of Renyi entropies, including universal coefficients and second order variations.

Findings

Derived a constraint linking displacement operator two-point function and twist operator conformal weight.

Examined a conjecture on the universal coefficient for conical singularities in entangling surfaces.

Provided a formula for second order shape variations of Renyi entropy for spherical surfaces.

Abstract

We propose a field theoretic framework for calculating the dependence of R\'enyi 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 R\'enyi 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 R\'enyi entropy arising from small deformations of a spherical entangling surface, extending Mezei's results for the entanglement entropy.