# RT surface The famous HR-HRT formula relates an [[0301 Entanglement entropy|entanglement entropy]] of the [[0001 AdS-CFT|holographic]] CFT to a bulk quantity -- the area of an extremal surface: $S=\mathrm{ext}_\,A[\gamma],$where $\gamma$ is a codimension-two surface. This is the leading order result in $G_N$ and $\alpha^\prime$: at higher orders in $\alpha^\prime$, the area is replaced with the [[0145 Generalised area|generalised area]]; at higher orders in $G_N$, the RT/HRT surface is replaced by the [[0212 Quantum extremal surface|quantum extremal surface]]. ## Refs - originals - RT: [[2006#Ryu, Takayanagi (Mar)]] (short) and [[2006#Ryu, Takayanagi (May)]] (long) - HRT: [[2007#Hubeny, Rangamani, Takayanagi]] - generalisations - FLM: [[2013#Faulkner, Lewkowycz, Maldacena]] (1-loop corrections) - [[0212 Quantum extremal surface|QES]]: [[2014#Engelhardt, Wall]] (all-loop corrections) - [[0145 Generalised area|generalised area]]: [[2013#Dong]] (higher derivative corrections) - proofs - [[2013#Lewkowycz, Maldacena]] proved RT - [[2016#Dong, Lewkowycz, Rangamani]] proved HRT - [[2017#Dong, Lewkowycz]] proved [[0212 Quantum extremal surface|QES]] ## Properties - [[0252 Causality of HEE|causality of RT-HRT]] - invariance under RG flow: [[2023#Dong, Remmen, Wang, Weng, Wu]] ## Homology constraints - early discussion - [[Fursaev2006]] and [[HeadrickTakayanagi2007]] - clean rephrasing - [[Headrick2013]] - [[2014#Headrick, Hubeny, Lawrence, Rangamani]] - RT v.s. HRT on static surfaces - always use HRT it seems (RT is ambiguous sometimes) - [[2013#Hubeny, Maxfield, Rangamani, Tonni]] ## Generalisations - [[Tsujimura2020]]: proposal for more general case, and include a sum over many cosmic branes - two intervals - [[Tsujimura2021]][](https://arxiv.org/pdf/2106.00015.pdf): sum of signed RTs - bad: unjustified to throw away homology constraint - better understanding: [[2020#Headrick]] - for brane world - [[2024#Miao, Xie]] ## Annealed vs quenched - [[2024#Hernandez-Cuenca (Jul)]] - [[2025#Antonini, Iliesiu, Rath, Tran]] ## Understanding RT/HRT - via matrix quantum mechanics - [[FrenkelHartnoll2021]][](https://arxiv.org/pdf/2111.05967.pdf) - [[2024#Fliss, Frenkel, Hartnoll, Soni]]: entanglement entropy in matrix quantum mechanics is given as a minimisation, similar to RT ## Related - [[0145 Generalised area]] - [[0219 Entanglement wedge reconstruction]] - [[0302 Gravity from entanglement]] - [[0565 Entanglement entropy in string theory]]