BCFT and Islands in Two Dimensions

TL;DR

This work establishes and tests the Island/BCFT correspondence in two dimensions, showing that entanglement entropy computed via the Island prescription in a CFT coupled to 2d gravity matches the holographic BCFT result. It identifies the 2d induced gravity on the AdS$_2$ end-of-the-world brane as the gravitational dual to BCFT boundary dynamics, deriving it from both holographic and field-theoretic viewpoints and connecting it to Liouville/dilaton gravity. The authors demonstrate that nontrivial BCFT one-point functions can be reproduced by turning on bulk scalar backgrounds (analytically for Δ=2 via a Janus solution and numerically for Δ≠2), providing a concrete mechanism to encode boundary data in the gravity side. They also relate energy flux boundary conditions and replica-wormhole arguments to the quantum extremal surface framework, reinforcing the equivalence of the Island and BCFT pictures and offering a pathway to generalize these insights to higher dimensions and richer holographic setups.

Abstract

By combining the AdS/BCFT correspondence and the brane world holography, we expect an equivalence relation between a boundary conformal field theory (BCFT) and a gravitational system coupled to a CFT. However, it still remains unclear how the boundary condition of the BCFT is translated in the gravitational system. We examine this duality relation in a two-dimensional setup by looking at the computation of entanglement entropy and energy flux conservation. We also identify the two-dimensional gravity which is dual to the boundary dynamics of a BCFT. Moreover, we show that by considering a gravity solution with scalar fields turned on, we can reproduce one point functions correctly in the AdS/BCFT.

Figures (5)

• Figure 1: A sketch of AdS/BCFT setup.
• Figure 2: A sketch of setup of calculating entanglement entropy when we couple a CFT on a half line to a gravity on AdS$_2$ (left) and an equivalent setup in BCFT (right).
• Figure 3: The tension parameter $T$ (left) and the boundary condition parameter $a$ (right) as functions of $\rho$ for fixed values of $\gamma$. The blue, red, yellow and green curves correspond to $\gamma=0$, $\gamma=0.9/\sqrt{2}$, $\gamma=0.99/\sqrt{2}$ and $\gamma=1/\sqrt{2}$, respectively. We set $\phi_0=0$.
• Figure 4: Numerical plots of singular solutions of $g(u)$, $\chi(u)$, $f(u)$ and $\phi(u)$ with $\Delta=\Delta_+$ for various dimensions in the range of $1.6 \le \Delta_+ \le 2.4$. Here we have Log plot for $f(u)$ and choose the parameter $\alpha = 2.0$.
• Figure 5: Numerical plots of non-singular solutions of $g(u)$, $\chi(u)$, $f(u)$ and $\phi(u)$ with $\Delta=\Delta_+$ for various dimensions in the range of $1.2 \le \Delta_+ \le 2.0$. At least, adjusting the values of $\alpha$, we can find a non-singular solution for a wide range of non-irrelevant dimension $\Delta$. Here we took $\alpha=2.101$ (for $\Delta=1.2$), $\alpha=2.083$ (for $\Delta=1.4$), $\alpha=1.962$ (for $\Delta=1.6$), $\alpha=1.663$ (for $\Delta=1.8$) and $\alpha=1.414$ (for $\Delta=2.0$).