# Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi ## Causality, Analyticity and an IR Obstruction to UV Completion \[Links: [arXiv](https://arxiv.org/abs/hep-th/0602178), [PDF](https://arxiv.org/pdf/hep-th/0602178.pdf)\] \[Abstract: We argue that certain apparently consistent low-energy effective field theories described by local, Lorentz-invariant Lagrangians, secretly exhibit macroscopic non-locality and cannot be embedded in any UV theory whose S-matrix satisfies canonical [[0120 Analyticity constraints|analyticity constraints]]. The obstruction involves the signs of a set of leading irrelevant operators, which must be strictly positive to ensure UV analyticity. An IR manifestation of this restriction is that the "wrong" signs lead to superluminal fluctuations around non-trivial backgrounds, making it impossible to define local, causal evolution, and implying a surprising IR breakdown of the effective theory. Such effective theories can not arise in quantum field theories or weakly coupled string theories, whose S-matrices satisfy the usual analyticity properties. This conclusion applies to the DGP brane-world model modifying gravity in the IR, giving a simple explanation for the difficulty of embedding this model into controlled stringy backgrounds, and to models of electroweak symmetry breaking that predict negative anomalous quartic couplings for the W and Z. Conversely, any experimental support for the DGP model, or measured negative signs for anomalous quartic gauge boson couplings at future accelerators, would constitute direct evidence for the existence of superluminality and macroscopic non-locality unlike anything previously seen in physics, and almost incidentally falsify both local quantum field theory and perturbative string theory.\] # Blau, Thompson ## Chern-Simons Theory on $S^1$-Bundles: Abelianisation and q-deformed Yang-Mills Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/0601068), [PDF](https://arxiv.org/pdf/hep-th/0601068.pdf)\] \[Abstract: We study [[0089 Chern-Simons theory|Chern-Simons theory]] on 3-manifolds $M$ that are circle-bundles over 2-dimensional surfaces $\Sigma$ and show that the method of Abelianisation, previously employed for trivial bundles $\Sigma \times S^1$, can be adapted to this case. This reduces the non-Abelian theory on $M$ to a 2-dimensional Abelian theory on $\Sigma$ which we identify with $q$-deformed Yang-Mills theory, as anticipated by Vafa et al. We compare and contrast our results with those obtained by [[2005#Beasley, Witten|Beasley and Witten]] using the method of non-Abelian localisation, and determine the surgery and framing prescription implicit in this path integral evaluation. We also comment on the extension of these methods to [[0557 BF theory|BF theory]] and other generalisations.\] # Brower, Polchinski, Strassler, Tan ## The Pomeron and Gauge/String Duality \[Links: [arXiv](https://arxiv.org/abs/hep-th/0603115), [PDF](https://arxiv.org/pdf/hep-th/0603115.pdf)\] \[Abstract: The traditional description of high-energy small-angle scattering in QCD has two components -- a soft Pomeron Regge pole for the tensor glueball, and a hard BFKL Pomeron in leading order at weak coupling. On the basis of [[0001 AdS-CFT|gauge/string duality]], we present a coherent treatment of the Pomeron. In large-$N$ QCD-like theories, we use curved-space string-theory to describe simultaneously both the BFKL regime and the classic Regge regime. The problem reduces to finding the spectrum of a single $j$-plane Schrodinger operator. For ultraviolet-conformal theories, the spectrum exhibits a set of Regge trajectories at positive $t$, and a leading $j$-plane cut for negative $t$, the cross-over point being model-dependent. For theories with logarithmically-running couplings, one instead finds a discrete spectrum of poles at all $t$, where the Regge trajectories at positive $t$ continuously become a set of slowly-varying and closely-spaced poles at negative $t$. Our results agree with expectations for the BFKL Pomeron at negative $t$, and with the expected glueball spectrum at positive $t$, but provide a framework in which they are unified. Effects beyond the single Pomeron exchange are briefly discussed.\] ## Comments - techniques in this paper are used in [[2014#Shenker, Stanford]] # Copsey, Horowitz ## Gravity dual of gauge theory on $S^2 \times S^1 \times R$ \[Links: [arXiv](https://arxiv.org/abs/hep-th/0602003), [PDF](https://arxiv.org/pdf/hep-th/0602003.pdf)\] \[Abstract: We (numerically) construct new static, [[0231 Bulk solutions for CFTs on non-trivial geometries|asymptotically AdS solutions]] where the conformal infinity is the product of time and $S^2 \times S^1$. There always exist a family of solutions in which the $S^1$ is not contractible and, for small $S^1$, there are two additional families of solutions in which the $S^1$ smoothly pinches off. This shows that (when fermions are antiperiodic around the $S^1$) there is a quantum phase transition in the gauge theory as one decreases the radius of the $S^1$ relative to the $S^2$. We also compare the masses of our solutions and argue that the one with lowest mass should minimize the energy among all solutions with conformal boundary $S^2 \times S^1 \times R$. This provides a new [[0116 Positive energy theorem|positive energy conjecture]] for asymptotically locally AdS metrics. A simple analytic continuation produces AdS black holes with topology $S^2 \times S^1$.\] ## Generalisations - $d s^{2}=-\alpha(r) d t^{2}+\frac{d r^{2}}{\alpha(r) \beta(r)}+e^{\gamma(r)} d x_{p}^{2}+r^{2} d \Omega_{d-p-1}^{2}$ for general $d$ and $p$ in [[2018#Harlow, Ooguri (Long)]] Appendix I (the letter I not 1) - there, it is called vacuum solution for $\alpha>0$ everywhere and wormhole solution for $\alpha(r_0)=0$ for some $r_0>0$ - the focus was on generalisation of the black string, not of the bubble - $S^{d-3}\times R$ -> $H^{d-3}\times R$: [[MannRaduStelea2006]] - with a electric charge and rotation - [[BrihayeRaduStelea2007]] - non-Abelian charge - [[BrihayeRadu2007]] - connecting supersymmetric to uncharged magnetic strings - [[BernamontiCaldarelliKlemmOleaSiegZorzan2007]] ## Static solutions - $d s^{2}=-e^{\gamma(r)} d t^{2}+\alpha(r) d \chi^{2}+\frac{d r^{2}}{\alpha(r) \beta(r)}+r^{2} d \Omega$ - $\alpha(r)$ determines everything ($\beta$ and $\gamma^\prime$ determined by it algebraically, and integration constant of $\gamma$ absorbed in redefinition of time) - **static vacuum solutions** - a solution with $\alpha(r)$ positive everywhere - $\alpha^\prime(0)=0$ to be smooth - unique with topology $R^4\times S^1$ - $\chi$ can be compactified with have any periodicity - **static bubble solutions** - ![[CopseyHorowitz2006_s-r0.png|200]] - two solutions for $s$ small enough, where $s$ is the size of $S^1$ - **black string solutions** - analytically continue the bubble solutions ## Initial data for non-static solutions with static boundary - choose $\alpha(r)$ freely and use initial data constraints to determine $\beta(r)$ - in all examples, they always have higher energy than the static solution, which is believed to be the ground state ## Initial data for non-static solutions with expanding boundary - generalises Witten's bubble of nothing - n.b. can only compare energies of two solutions if the conformal metrics are the same (at least at $t=0$, and 1st and 2nd derivatives in $t$ should also agree) # Cornalba, Costa, Penedones, Schiappa (a) ## Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions \[Links: [arXiv](https://arxiv.org/abs/hep-th/0611122), [PDF](https://arxiv.org/pdf/hep-th/0611122.pdf)\] \[Abstract: We initiate a program to generalize the standard eikonal approximation to compute amplitudes in Anti-de Sitter spacetimes. Inspired by the shock wave derivation of the [[0436 Eikonal approximation|eikonal amplitude]] in flat space, we study the two-point function $E \sim\langle O_1 O_1 \rangle_{shock}$ in the presence of a shock wave in Anti-de Sitter, where $O_1$ is a scalar primary operator in the dual conformal field theory. At tree level in the gravitational coupling, we relate the shock two-point function E to the discontinuity across a kinematical branch cut of the conformal field theory four-point function $A \sim \langle O_1 O_2 O_1 O_2 \rangle$, where $O_2$ creates the shock geometry in Anti-de Sitter. Finally, we extend the above results by computing $E$ in the presence of shock waves along the horizon of Schwarzschild [[0086 Banados-Teitelboim-Zanelli black hole|BTZ]] black holes. This work gives new tools for the study of Planckian physics in Anti-de Sitter spacetimes.\] ## Refs - accompanying paper [[2006#Cornalba, Costa, Penedones, Schiappa (b)]] - later on resumming the gravitational loop expansion [[2007#Cornalba, Costa, Penedones]] - [[0129 Dual of shockwaves]] ## Summary - study two-point functions in ==AdS== with shock wave - at ==tree level== in the gravitational coupling, relates the two-point function to the discontinuity across a kinematical branch cut of the CFT 4-point function $\mathcal{A}\sim\langle\mathcal{O}_1\mathcal{O}_2\mathcal{O}_1\mathcal{O}_2\rangle$, where $\mathcal{O}_2$ creates the shock wave - extend the above to a shock wave along the horizon of ==Schwarzschild BTZ== black hole ## Eikonal limit - small scattering angle - $t$-channel exchange dominates over full tree level amplitude - alternative way of computing it - using [[0117 Shockwave]] - established in flat space -> extend to AdS in this paper ## Shock wave v.s. scattering amplitude - in flat space, one can approximately reconstruct the full amplitude from the tree level phase shift (I guess related to the shockwave phase shift) - in AdS, need extra information - because tree-level eikonal two-point function $\mathcal{E}_1$ is not related to amplitude $\mathcal{A}_1$ but to the *discontinuity* $\mathcal{M}_1$ of $\mathcal{A}_1$ across a branch cute ## 4-pt. function ![[FitzpatrickHuangLi2019_4pt.png]] - two heavy operators create the shock wave - two light operators are connected by a geodesic penetrating the shock wave ## Notations - 4-pt. function and scattering amplitude - $\left\langle\mathcal{O}_{1}\left(\mathbf{p}_{1}\right) \mathcal{O}_{2}\left(\mathbf{p}_{2}\right) \mathcal{O}_{1}\left(\mathbf{p}_{3}\right) \mathcal{O}_{2}\left(\mathbf{p}_{4}\right)\right\rangle_{\mathrm{CFT}_{d}} \equiv \frac{1}{\mathbf{p}_{13}^{\Delta_{1}} \mathbf{p}_{24}^{\Delta_{2}}} \mathcal{A}(z, \bar{z})$ - $\mathcal{A}=\mathcal{A}_{0}+\mathcal{A}_{1}+\cdots$ - 2-pt. function when there is shock - $\mathcal{E} \sim\left\langle\mathcal{O}_{1} \mathcal{O}_{1}\right\rangle_{\text {shock }}$ - discontinuity - $\mathcal{M}_{1}(z, \bar{z})=\operatorname{Disc}_{z} \mathcal{A}_{1}(z, \bar{z}) \equiv \frac{1}{2 \pi i}\left(\mathcal{A}_{1}^{\circlearrowright}(z, \bar{z})-\mathcal{A}_{1}(z, \bar{z})\right)$ - $\mathcal{A}_{1}^{\circlearrowright}$ is the analytic continuation of $\mathcal{A}_1$ with some contour definition - result: - $\mathcal{M}_{1} \simeq-8 G \mathcal{N}_{\Delta_{1}} \mathcal{N}_{\Delta_{2}} \Gamma\left(2 \Delta_{1}-1+j\right) \Gamma\left(2 \Delta_{2}-1+j\right)$\times\left(-q^{2}\right)^{\Delta_{1}}\left(-p^{2}\right)^{\Delta_{2}} \int_{H_{d-1}} \widetilde{d x} \widetilde{d y} \frac{\Pi(x, y)}{(2 q \cdot x)^{2 \Delta_{1}-1+j}(2 p \cdot y)^{2 \Delta_{2}-1+j}}$ - propagators - massless Minkowskian scalar propagator on AdS${}_{d+1}$: - $\square_{\mathrm{AdS}_{d+1}} \boldsymbol{\Pi}(\mathbf{x}, \mathbf{y})=i \delta(\mathbf{x}, \mathbf{y})$ - massive Euclidean scalar propagator on $H_{d-1}$ of mass-squared $d-1$ - $\left[\square_{H_{d-1}}-(d-1)\right] \Pi(x, y)=-\delta(x, y)$ ## Higher spin - interaction # Cornalba, Costa, Penedones, Schiappa (b) ## Eikonal Approximation in AdS/CFT: Conformal Partial Waves and Finite N Four-Point Functions \[Links: [arXiv](https://arxiv.org/abs/hep-th/0611123), [PDF](https://arxiv.org/pdf/hep-th/0611123.pdf)\] \[Abstract: \] # Do, Norbury ## Weil-Petersson volumes and cone surfaces \[Links: [arXiv](https://arxiv.org/abs/math/0603406), [PDF](https://arxiv.org/pdf/math/0603406)\] \[Abstract: The moduli spaces of hyperbolic surfaces of genus $g$ with $n$ geodesic boundary components are naturally symplectic manifolds. [[0627 Mirzakhani recursion|Mirzakhani]] proved that their volumes are polynomials in the lengths of the boundaries by computing the volumes recursively. In this paper we give new recursion relations between the volume polynomials.\] # Duhr, Hoche, Maltoni ## Color-dressed recursive relations for multi-parton amplitudes \[Links: [arXiv](https://arxiv.org/abs/hep-ph/0607057), [PDF](https://arxiv.org/pdf/hep-ph/0607057.pdf)\] \[Abstract: Remarkable progress inspired by [[0330 Twistor theory|twistors]] has lead to very simple analytic expressions and to new recursive relations for multi-parton color-ordered amplitudes. We show how such relations can be extended to include color and present the corresponding color-dressed formulation for the [[0353 Berends-Giele recursion relations|Berends-Giele]], [[0058 BCFW|BCF]] and a new kind of [[0352 CSW relations|CSW]] recursive relations. A detailed comparison of the numerical efficiency of the different approaches to the calculation of multi-parton cross sections is performed.\] ## Summary - *derives* colour-dressed [[0353 Berends-Giele recursion relations|Berends-Giele recursion relations]] (as an appetiser), [[0058 BCFW|BCFW]] and [[0352 CSW relations|CSW]] - with *purposes*: - there is an interesting similarity between a colour-decomposition based on the adjoint representation and [[0058 BCFW|CSW]] so it would be interesting to find a formulation that embodies both - improve numerical efficiency ## Colour-dressed [[0058 BCFW|BCFW]] - main result (4.12):$\mathcal{A}_{n}(1,2, \ldots, n)=\sum_{\pi \in O P(n-2,2)} \mathcal{A}_{k+1}\left(\hat{1}, \pi_{1},-\hat{P}_{1, \pi_{1}}^{-h, x}\right) \frac{1}{P_{1, \pi_{1}}^{2}} \mathcal{A}_{n-k+1}\left(\hat{P}_{1, \pi_{1}}^{h, x}, \pi_{2}, \hat{n}\right)$ - $OP$: ordered partitions, i.e., $(\pi_1,\pi_2)\ne(\pi_2,\pi_1)$ - $x$: colour of intermediate gluons - proof (fairly easy) - substitute [[0058 BCFW|BCFW]] into the adjoint-rep.-based [[0354 Colour decomposition|colour decomposition]] - then write $\sum_{k=2}^{n-2} \sum_{\sigma \in S_{n-2}} \rightarrow \sum_{\pi \in O P(n-2,2)} \sum_{\sigma \in S_{k-1}} \sum_{\sigma^{\prime} \in S_{n-k-1}}$ - next, identify the colour-dressed subamplitudes - only the sum over partitions remain # Fjelstad, Fuchs, Runkel, Schweigert ## Uniqueness of open/closed rational CFT with given algebra of open states \[Links: [arXiv](https://arxiv.org/abs/hep-th/0612306), [PDF](https://arxiv.org/pdf/hep-th/0612306)\] \[Abstract: We study the [[0602 Moore-Seiberg construction|sewing constraints]] for rational [[0003 2D CFT|two-dimensional conformal field theory]] on oriented surfaces with possibly non-empty [[0548 Boundary CFT|boundary]]. The boundary condition is taken to be the same on all segments of the boundary. The following uniqueness result is established: For a solution to the sewing constraints with nondegenerate closed state vacuum and nondegenerate two-point correlators of boundary fields on the disk and of bulk fields on the sphere, up to equivalence all correlators are uniquely determined by the one-, two,- and three-point correlators on the disk. Thus for any such theory every consistent collection of correlators can be obtained by the TFT approach of [hep-th/0204148](https://arxiv.org/abs/hep-th/0204148), [hep-th/0503194](https://arxiv.org/abs/hep-th/0503194). As morphisms of the category of world sheets we include not only homeomorphisms, but also sewings; interpreting the correlators as a natural transformation then encodes covariance both under homeomorphisms and under sewings of world sheets.\] # Hirata, Takayanagi ## AdS/CFT and Strong Subadditivity of Entanglement Entropy \[Links: [arXiv](https://arxiv.org/abs/hep-th/0608213), [PDF](https://arxiv.org/pdf/hep-th/0608213.pdf)\] \[Abstract: Recently, a [[0007 RT surface|holographic computation of the entanglement entropy]] in conformal field theories has been proposed via the AdS/CFT correspondence. One of the most important properties of the entanglement entropy is known as the [[0218 Strong subadditivity|strong subadditivity]]. This requires that the entanglement entropy should be a concave function with respect to geometric parameters. It is a non-trivial check on the proposal to see if this property is indeed satisfied by the entropy computed holographically. In this paper we examine several examples which are defined by annuli or [[0362 Entanglement surface with cusps|cusps]], and confirm the strong subadditivity via direct calculations. Furthermore, we conjecture that Wilson loop correlators in strongly coupled gauge theories satisfy the same relation. We also discuss the relation between the holographic entanglement entropy and the [[0171 Covariant entropy bound|Bousso bound]].\] # Hollands, Marolf ## Asymptotic generators of fermionic charges and boundary conditions preserving supersymmetry \[Links: [arXiv](https://arxiv.org/abs/gr-qc/0611044), [PDF](https://arxiv.org/pdf/gr-qc/0611044.pdf)\] \[Abstract: We use a [[0019 Covariant phase space|covariant phase space]] formalism to give a general prescription for defining Hamiltonian generators of bosonic and fermionic symmetries in diffeomorphism invariant theories, such as [[0332 Supergravity|supergravities]]. A simple and general criterion is derived for a choice of boundary condition to lead to conserved generators of the symmetries on the phase space. In particular, this provides a criterion for the preservation of supersymmetries. For bosonic symmetries corresponding to diffeomorphisms, our prescription coincides with the method of Wald et al. We then illustrate these methods in the case of certain supergravity theories in $d=4$. In minimal AdS supergravity, the boundary conditions such that the supercharges exist as Hamiltonian generators of supersymmetry transformations are unique within the usual framework in which the boundary metric is fixed. In extended ${\mathcal N}=4$ AdS supergravity, or more generally in the presence of chiral matter superfields, we find that there exist many boundary conditions preserving ${\mathcal N}=1$ supersymmetry for which corresponding generators exist. These choices are shown to correspond to a choice of certain arbitrary boundary ''superpotentials,'' for suitably defined ''boundary superfields.'' We also derive corresponding formulae for the conserved bosonic charges, such as energy, in those theories, and we argue that energy is always positive, for any supersymmetry-preserving boundary conditions. We finally comment on the relevance and interpretation of our results within the [[0001 AdS-CFT|AdS-CFT]] correspondence.\] # Hubeny, Liu, Rangamani ## Bulk-cone singularities & signatures of horizon formation in AdS/CFT \[Links: [arXiv](https://arxiv.org/abs/hep-th/0610041), [PDF](https://arxiv.org/pdf/hep-th/0610041.pdf)\] \[Abstract: \] ## Refs - related to [[0091 Boundary causality]] - [[0026 Bulk reconstruction]] using lightcone cuts ## Summary - **bulk cone singularities**: *additional* singularities in the boundary correlators when two points are separated by a null *geodesic* through the *bulk* - when null separated through boundary: usual CFT singularities - these lie *inside* the boundary light cone - a sharp feature in boundary observables corresponding to black hole event horizon formation ## Divergence of propagator - argue that thee boundary correlator $G(x,x^\prime)=\langle\mathcal{O}(x)\mathcal{O}(x^\prime)\rangle$ is singular <=> there exist null geodesics connecting the two points - a subtlety of when null geodesics not contributing to the propagator - if the P.I. contour cannot be deformed to the saddle point corresponding to null geodesic - an example of an almost null geodesic in [[FidkowskiHubenyKlebanShenker2003]] # Luo ## On Teichmuller Space of Surface with Boundary \[Links: [arXiv](https://arxiv.org/abs/math/0601364), [PDF](https://arxiv.org/pdf/math/0601364)\] \[Abstract: We characterization hyperbolic metrics on compact surfaces with boundary using a variational principle. As a consequence, a new parametrization of the [[0626 Teichmuller TQFT|Teichmuller space]] of compact surface with boundary is produced. In the new parametrization, the Teichmuller space becomes an open convex polytope. It is conjectured that the Weil-Petersson symplectic form can be expressed explicitly in terms of the new coordinate.\] # Maeda ## Final fate of spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity \[Links: [arXiv](https://arxiv.org/abs/gr-qc/0602109), [PDF](https://arxiv.org/pdf/gr-qc/0602109.pdf)\] \[Abstract: We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in [[0425 Gauss-Bonnet gravity|Einstein-Gauss-Bonnet gravity]]. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fluid and a cosmological constant. This is a generalization of the [[0595 Quasi-local energy#Misner-Sharp energy|Misner-Sharp]] formalism of the four-dimensional spherically symmetric spacetime with a perfect fluid in general relativity. The whole picture and the final fate of the gravitational collapse of a dust cloud differ greatly between the cases with $n=5$ and $n \ge 6$. There are two families of solutions, which we call plus-branch and the minus-branch solutions. Bounce inevitably occurs in the plus-branch solution for $n \ge 6$, and consequently singularities cannot be formed. Since there is no trapped surface in the plus-branch solution, the singularity formed in the case of $n=5$ must be naked. In the minus-branch solution, naked singularities are massless for $n \ge 6$, while massive naked singularities are possible for $n=5$. In the homogeneous collapse represented by the flat Friedmann-Robertson-Walker solution, the singularity formed is spacelike for $n \ge 6$, while it is ingoing-null for $n=5$. In the inhomogeneous collapse with smooth initial data, the strong cosmic censorship hypothesis holds for $n \ge 10$ and for $n=9$ depending on the parameters in the initial data, while a naked singularity is always formed for $5 \le n \le 8$. These naked singularities can be globally naked when the initial surface radius of the dust cloud is fine-tuned, and then the [[0221 Weak cosmic censorship|weak cosmic censorship]] hypothesis is violated.\] # Moore, Segal ## D-branes and K-theory in 2D topological field theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/0609042), [PDF](https://arxiv.org/pdf/hep-th/0609042)\] \[Abstract: This expository paper describes [[0602 Moore-Seiberg construction|sewing conditions]] in two-dimensional open/closed topological field theory. We include a description of the $G$-equivariant case, where $G$ is a finite group. We determine the category of boundary conditions in the case that the closed string algebra is semisimple. In this case we find that sewing constraints -- the most primitive form of worldsheet locality -- already imply that D-branes are ($G$-twisted) vector bundles on spacetime. We comment on extensions to cochain-valued theories and various applications. Finally, we give uniform proofs of all relevant sewing theorems using Morse theory.\] # Mukhopadhyay, Padmanabhan ## Holography of gravitational action functionals \[Links: [arXiv](https://arxiv.org/abs/hep-th/0608120), [PDF](https://arxiv.org/pdf/hep-th/0608120.pdf)\] \[Abstract: \] ## Summary - Einstein gravity has a holographic feature: - separation into bulk and surface term - one can construct the other - ${Q_{a}}^{bcd}{R^a}_{bcd}$ where $Q_{abcd}$ has the same symmetry as $R_{abcd}$ and satisfy $\nabla_c Q^{abcd}=0$ share a *holographic* feature - might use to constrain semi-classical corrections to Einstein gravity # Ryu, Takayanagi (Mar) ## Holographic Derivation of Entanglement Entropy from AdS/CFT \[Links: [arXiv](https://arxiv.org/abs/hep-th/0603001), [PDF](https://arxiv.org/pdf/hep-th/0603001.pdf)\] \[Abstract: A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from AdS/CFT correspondence. We argue that the entanglement entropy in $d+1$ dimensional conformal field theories can be obtained from the area of $d$ dimensional minimal surfaces in AdS$_{d+2}$, analogous to the Bekenstein-Hawking formula for [[0004 Black hole entropy|black hole entropy]]. We show that our proposal perfectly reproduces the correct [[0301 Entanglement entropy|entanglement entropy]] in 2D CFT when applied to AdS$_3$. We also compare the entropy computed in AdS$_5 \times S^5$ with that of the free $\mathcal{N}=4$ super Yang-Mills.\] # Ryu, Takayanagi (May) ## Aspects of Holographic Entanglement Entropy \[Links: [arXiv](https://arxiv.org/abs/hep-th/0605073), [PDF](https://arxiv.org/pdf/hep-th/0605073.pdf)\] \[Abstract: This is an extended version of our short report [hep-th/0603001](https://arxiv.org/abs/hep-th/0603001), where a holographic interpretation of entanglement entropy in conformal field theories is proposed from AdS/CFT correspondence. In addition to a concise review of relevant recent progresses of entanglement entropy and details omitted in the earlier letter, this paper includes the following several new results : We give a more direct derivation of our claim which relates the entanglement entropy with the minimal area surfaces in the AdS$_3$/CFT$_2$ case as well as some further discussions on higher dimensional cases. Also the relation between the entanglement entropy and central charges in 4D conformal field theories is examined. We check that the logarithmic part of the 4D entanglement entropy computed in the CFT side agrees with the AdS$_5$ result at least under a specific condition. Finally we estimate the entanglement entropy of massive theories in generic dimensions by making use of our proposal.\]