Holographic entanglement entropy with conformal boundary conditions

Abstract

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 $S_{EE} =\frac{c_{\rm eff}}{3} \ln\!\left(\frac{2 R \sin\phi_0}{\epsilon}\right) .$ 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.