# Generalised area In [[0001 AdS-CFT|holography]], the [[0301 Entanglement entropy|entanglement entropies]] of the boundary system can be computed in the bulk as geometric quantities (plus corrections coming from *bulk* quantum fields). This led to tremendous progress in our understanding of the holographic dictionary, including the emergence of spacetime, the [[0131 Information paradox|information paradox]] and etc. It also led to ideas such as [[0219 Entanglement wedge reconstruction|entanglement wedge reconstruction]], [[0146 Quantum error correction|quantum error correction]] (in realising holography), etc. In the original [[0007 RT surface|HR-HRT]] proposal, the [[0301 Entanglement entropy|entanglement entropy]] of a region on the boundary CFT is related to the *area* of an extremal surface in the bulk. This is accurate for Einstein gravity minimally coupled to scalar fields at leading order in Newton's constant, $G_N$. For other theories, the functional is more non-trivial. Writing $S=\operatorname{ext}A_\text{gen},$the expectation is that $A_\text{gen}$, the *generalised area*, is the integral of a *local* geometric quantity on a codimension-two surface. This page is devoted to the functional form of the generalised area, including its dependence on the Lagrangian and its derivation(s). ## Related - [[0004 Black hole entropy]]: more general than HEE because they do not necessarily compute anything on the boundary - [[0007 RT surface]] - [[0212 Quantum extremal surface]] ## Derivations - Dong method (specifically the regularised cone version of [[2013#Lewkowycz, Maldacena|LM]] method) - should be the correct one - but suffers from splitting problem - Cosmic brane method - trivial for Einstein gravity because the brane action is known; but trivially useless for general theories because working out the brane action amounts to deriving the entropy function - relate to [new paper on cosmic brane in Lovelock-CS theory](https://arxiv.org/pdf/2006.02803.pdf) #todo - [[2019#Dong, Marolf]] method - should solve the splitting problem - relies on a subtraction prescription - GHY term method - used in [[2016#Dong, Lewkowycz, Rangamani]] - need to show that they are independent of BC/boundary term #todo - Hayward term/TakayanagiTamaoka method - correct for Einstein and Lovelock but seemingly wrong result for f(Riem) - [[2023#Kastikainen, Svesko (Dec, b)]]: get correct results for f(Riem) using special periodic boundary conditions - from boundary term for near extremal BH - [[ChowdhurySahaGangopadhyay2021]][](https://arxiv.org/pdf/2103.01483.pdf) - using [[0483 Boundary-condition-preserving kink transformation|BC-preserving area flow]] - DongMarolfRath to appear - checked for higher derivative matter action - not sure whether it works for [[0006 Higher-derivative gravity|HDG]], and even if so, whether it has the splitting problem ## Finite Renyi number $n$ - Einstein: cosmic brane - [[2019#Dong, Marolf]]: general prescription for higher derivatives ## Ambiguities - [[Camps2016]] - [[2014#Miao, Guo]] - [[CampsKelly2014]] - [[2015#Miao]] - [[2020#Caceres, Vasquez, Lopez]] - [[AnastasiouArayaArgandonaOlea2022]][](https://arxiv.org/pdf/2208.00093.pdf): different splitting leads to different stress tensor correlation in CFT*n* ## Lorentzian replica trick (For HRT) - so far only in Einstein case: [[2016#Dong, Lewkowycz, Rangamani]] ## Properties - field redefinition invariance - [[2016#Mozaffar, Mollabashi, Sheikh-Jabbari, Vahidinia]] - (local) total derivative action has no entropy - [[2015#Dong, Miao]] ## Renormalisation issues - see also [[0372 Holographic first law of entanglement entropy]] - [[2016#Taylor, Woodhead]] - via [[0395 Volume renormalisation|volume renormalisation]] - Einstein - [[2018#Anastasiou, Araya, Arias, Olea]] - for Lovelock - [[AnastasiouArayaMannOlea2021]][](https://arxiv.org/pdf/2103.14640.pdf) - 4-derivative gravity - [[AnastasiouArayaMorenoOleaRivera-Betancour2021]][](https://arxiv.org/pdf/2102.11242.pdf) ![[A0011 Digital light bulb.jpeg]]