Abstract
We present a holographic method for computing the response of R\'enyi entropies in conformal field theories to small shape deformations around a flat (or spherical) entangling surface. Our strategy employs the stress tensor one-point function in a deformed hyperboloid background and relates it to the coefficient in the two-point function of the displacement operator. We obtain explicit numerical results for $d=3,\dots,6$ spacetime dimensions, and also evaluate analytically the limits where the R\'enyi index approaches 1 and 0 in general dimensions. We use our results to extend the work of 1602.08493 and disprove a set of conjectures in the literature regarding the relations between the R\'enyi shape dependence and the conformal weight of the twist operator. We also extend our analysis beyond leading order in derivatives in the bulk theory by studying Gauss-Bonnet gravity.
Highlights
The Renyi entropies Sn form a one-parameter family labeled by the index n, which is often taken to be an integer [9, 10]
We present a holographic method for computing the response of Renyi entropies in conformal field theories to small shape deformations around a flat entangling surface
We use our results to extend the work of 1602.08493 and disprove a set of conjectures in the literature regarding the relation between the Renyi shape dependence and the conformal weight of the twist operator
Summary
The main object of interest here will be the twist operators which appear in evaluating the Renyi entropies as in eq (1.4). For the remainder of our discussion, we will consider the case where the underlying field theory is a CFT, which allows us to take advantage of the description of the twist operators as conformal defects [33]. In the above expression and throughout the following, expectation values labeled by n are implicitly taken in the presence of the twist operator. CD is the parameter which we wish to determine here Extracting it from a direct computation of δSn in eq (2.6) would involve second order perturbation theory around a flat entangling surface. The coefficient hn which appears in these expressions is the so-called conformal weight of the twist operator. It is defined by the expectation value of the stress tensor with a planar twist operator.
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