The authors study how entanglement entropy behaves in three-dimensional anti-de Sitter (AdS3) gravity when one imposes conformal boundary conditions. These boundary rules fix the shape of the boundary metric up to an overall scale (the conformal class) and fix the trace of the boundary’s extrinsic curvature K, but they allow the overall scale (the Weyl mode) to fluctuate. Using a holographic setup, the paper shows that these extra boundary fluctuations do not produce new entropy. The entropy of the full boundary is the usual Bekenstein–Hawking black hole entropy (area of the horizon divided by 4G_N) and the entropy of a boundary subregion is still given by the Ryu–Takayanagi prescription (area of a minimal bulk surface divided by 4G_N).

To reach these conclusions the authors extend the Lewkowycz–Maldacena–Dong replica trick, a method that computes entanglement by making multiple copies of the spacetime and studying singular surfaces that join them. They adapt this construction to the conformal boundary conditions (CBC) and to known arguments by Dong and by Casini–Huerta–Myers (CHM). They then carry out concrete bulk calculations for global AdS and for rotating and non-rotating BTZ black hole geometries. These bulk computations reproduce the expected Bekenstein–Hawking entropy for the whole boundary and the Ryu–Takayanagi result for subregions.

The paper gives some explicit, K-dependent formulas. For an interval in AdS3 the entanglement entropy is controlled by the matter central charge c_m = 3ℓ/(2G_N), where ℓ is the AdS radius and G_N is Newton’s constant. For a high-temperature thermal state the entropy is governed by an effective central charge c_eff = (3ℓ/2G_N)·(Kℓ − sqrt(K^2ℓ^2 − 4))/2. The authors also compute the same subregion entropy directly in the conjectured dual boundary theory — a holographic conformal field theory (CFT) coupled to a time-like Liouville field and deformed by a marginal T-bar-T-like operator — and find S_EE = (c_eff/3) ln(2R sin φ0 / ε), where R and φ0 set the interval on the cylinder and ε is a short-distance cutoff. This provides an independent boundary-side realization of c_eff. The paper stresses that the boundary state used in that CFT calculation (the “vacuum” with no operator insertions) is not the same state that is dual to global AdS in the bulk.