AI 中文总结
本文在三维AdS引力共形边界条件下推导全息纠缠熵公式，证明外尔模无额外熵，验证Ryu-Takayanagi公式，计算特定几何的熵并从对偶边界理论得到一致结果，明确真空态非全局AdS的对偶态。
AI 中文摘要
我们在三维反德西特（AdS）引力中研究具有共形边界条件的全息纠缠熵，该条件固定了边界度量的共形类及其外曲率$K$，同时保留外尔模的动力学性质。通过推广Lewkowycz-Maldacena-Dong的复制构造，我们推导了对应的全息纠缠熵公式，证明了涨落的外尔模不会贡献额外熵。因此，整个边界的熵为贝肯斯坦-霍金熵，边界子区域的熵仍满足Ryu-Takayanagi公式，即极小曲面面积除以$4G_N$。我们对全局AdS、旋转及非旋转BTZ几何进行了显式计算：对于AdS$_3$中的一个区间，发现依赖于$K$的全息纠缠熵由$c_m=\frac{3 \ell}{2 G_N}$控制；对于高共形温度下的热态，熵由$c_{\rm eff}=\frac{3 \ell}{2 G_N} \frac{K \ell- \sqrt{K^2 \ell^2-4}}{2}$控制，该有效中心电荷与控制Cardy态密度的有效中心电荷一致，符合文献中的已有结果。最后，我们还直接从其对偶边界理论（耦合类时Liouville理论并被类边际$T \bar{T}$算子变形的全息共形场论（CFT））计算纠缠熵，得到$S_{EE} =\frac{c_{\rm eff}}{3} \ln\\!\left(\frac{2 R \sin\phi_0}{\epsilon}\right)$。无算子插入的真空态的这一结果，为$c_{\mathrm{eff}}$提供了独立的边界实现，同时明确该态并非全局AdS的对偶态。
英文摘要
We study holographic entanglement entropy in 3-dimensional AdS gravity with conformal boundary conditions which fix the conformal class of the boundary metric and its extrinsic curvature, $K$, while leaving the Weyl mode dynamical. Extending Lewkowycz-Maldacena-Dong's replica construction, we derive the corresponding holographic entanglement entropy formula. We show that the fluctuating Weyl mode does not contribute additional entropy. The entropy of the full boundary is therefore the Bekenstein-Hawking entropy, and the entropy of a boundary subregion continues to obey the Ryu-Takayanagi prescription, namely, the area of a minimal surface divided by $4G_N$. We carry out explicit calculations for global AdS, rotating and non-rotating BTZ geometries. For an interval in AdS$_3$ we find that the $K$-dependent holographic entanglement entropy is governed by $c_m=\frac{3 \ell}{2 G_N}.$ For a thermal state at high conformal temperatures we find that, the entropy is governed by $c_{\rm eff}=\frac{3 \ell}{2 G_N} \frac{K \ell- \sqrt{K^2 \ell^2-4}}{2}$, the same effective central charge that governs the Cardy-like density of states, in agreement with previous results in the literature. Finally, we also compute the entanglement entropy directly from the conjectured dual boundary theory - a holographic CFT coupled with time-like Liouville theory and deformed by a marginal $T \bar{T}$ like operator - and find $S_{EE} =\frac{c_{\rm eff}}{3} \ln\!\left(\frac{2 R \sinϕ_0}ε\right) .$ This result for the state with no operator insertions (vacuum state ), provides an independent boundary realization of $c_{\mathrm{eff}}$ while clarifying that this state is not the state dual to global AdS.
