# AdS/BCFT The so-called AdS/BCFT is an extension of [[0001 AdS-CFT|AdS/CFT]]. Here, the boundary theory is a $d$-dimensional [[0548 Boundary CFT|BCFT]], i.e., a CFT living on a manifold $X$ with a boundary $\partial X$, and the bulk theory is a gravitational theory living on a $(d+1)$-dimensional manifold $M$, whose boundary is the union of $X$ and $Q$, where $Q$ can be thought of as some end-of-the-world (ETW) brane that cuts off the spacetime. We will focus on the 3d bulk case. The bulk (Euclidean) action is therefore$I_E=-\frac{1}{16 \pi G_N} \int_M \sqrt{g}(R-2 \Lambda)-\frac{1}{8 \pi G_N} \int_Q \sqrt{h}(K-T),$where $K$ is the trace of the extrinsic curvature (defined with the normal pointing outwards) and $T$ is a constant interpreted as the tension of the ETW brane. Whereas the standard boundary condition at the asymptotic boundary is the usual Dirichlet boundary condition, Neumann boundary conditions are imposed at $Q$. This allows the "brane" to fluctuate, and therefore its position is fixed dynamically. The Neumann boundary condition sets$K_{i j}-(K-T) h_{i j}=0.$ One important data characterising a BCFT is the so-called $g$-function. It is simply the partition function of the disk. This can be computed holographically as follows. The bulk spacetime dual to an empty disk is obtained by removing part of the global AdS spacetime and adding an ETW brane. The relation between the tension and $g$ is given by$\log g = \frac{1}{4G_N}{\rm arctanh} (L_{\rm AdS}T) = \frac{c}{6}{\rm arctanh} (L_{\rm AdS}T).$ ## Refs - originals - [[2011#Takayanagi]] - [[2011#Fujita, Takayanagi, Tonni]] - entanglement inequalities - [[ChouLinWangYang2020]][](https://arxiv.org/pdf/2011.02790.pdf) - spectrum of end-of-world-branes in BCFT - [[MiyajiTakayanagiUgajin2021]][](https://arxiv.org/pdf/2103.06893.pdf) - BCFT at large charge - [[CuomoMezeiRaviv-Moshe2021]][](https://arxiv.org/pdf/2108.06579.pdf) - general comments - [[2021#Reeves, Rozali, Simidzija, Sully, Waddell, Wakeham]]: almost never holographic - bootstrap bounds on the brane tensor - [[2021#Collier, Mazac, Wang]] - [[0089 Chern-Simons theory|Chern-Simons]] formulation of the bulk dual - [[2020#Takayanagi, Uetoko]] - Holographic FG - [[2012#Nozaki, Takayanagi, Ugajin]] - branes with kinks - [[2022#Miyaji, Murdia]] - alternative proposals - [[2017#Miao, Chu, Guo (Short)]] and [[2017#Chu, Miao, Guo (Long)]]: proposes to use traceless condition rather than Neumann condition at the brane; reproduces non-trivial boundary anomaly (3d and 4d) ## Examples - in 3d (BTZd): [[AliSuneeta2021]][](https://arxiv.org/pdf/2112.07188.pdf) - top-down constructions from [[0332 Supergravity|SUGRA]]: - [[2011#Chiodaroli, D'Hoker, Gutperle]] - [[2012#Chiodaroli, D'Hoker, Gutperle]] - two boundary: [[GengLustMishraWakeham2021]][](https://arxiv.org/pdf/2104.07039.pdf) - top-down vs bottom up: [[2025#Harvey, Jensen, Uzu]] ## Correlators - [[KastikainenShashi2021]][](https://arxiv.org/pdf/2109.00079.pdf) using geodesics ## Boundary conditions - [[ChuMiao2021]][](https://arxiv.org/pdf/2110.03159.pdf) ## Entanglement entropy - [[SuzukiTakayanagi2022]][](https://arxiv.org/abs/2202.08462) - [[SullyVanRaamsdonkWakeham2022]][](https://arxiv.org/pdf/2004.13088.pdf) - [[2021#Geng, Karch, Perez-Pardavila, Raju, Randall, Riojas, Shashi]]