# Almheiri, Marolf, Polchinski, Stanford, Sully ## An Apologia for Firewalls \[Links: [arXiv](https://arxiv.org/abs/1304.6483), [PDF](https://arxiv.org/pdf/1304.6483.pdf)\] \[Abstract: We address proposed alternatives to the black hole [[0195 Firewall|firewall]]. We show that embedding the interior Hilbert space of an old black hole into the Hilbert space of the early radiation is inconsistent, as is embedding the semi-classical interior of an AdS black hole into any dual CFT Hilbert space. We develop the use of large AdS black holes as a system to sharpen the firewall argument. We also reiterate arguments that unitary non-local theories can avoid firewalls only if the non-localities are suitably dramatic.\] # Andersen, Kashaev ## A new formulation of the Teichmüller TQFT \[Links: [arXiv](https://arxiv.org/abs/1305.4291), [PDF](https://arxiv.org/pdf/1305.4291.pdf)\] \[Abstract: By using the Weil-Gel'fand-Zak transform of Faddeev's quantum dilogarithm, we propose a new state-integral model for the Teichmüller TQFT, where the circle valued state variables live on the edges of oriented leveled shaped triangulations.\] # Arnold, Szepietowski ## Spin 1/2 quasinormal mode frequencies in Schwarzschild-AdS spacetime \[Links: [arXiv](https://arxiv.org/abs/1308.0341), [PDF](https://arxiv.org/pdf/1308.0341.pdf)\] \[Abstract: We find the asymptotic formula for [[0325 Quasi-normal modes|quasinormal mode]] frequencies $\omega_n$ of the Dirac equation in a Schwarzschild-AdS$_D$ background in space-time dimension $D > 3$, in the large black-hole limit appropriate to many applications of the [[0001 AdS-CFT|AdS/CFT]] correspondence. By asymptotic, we mean large overtone number $n$ with everything else held fixed, and we find the $O(n^0)$ correction to the known leading $O(n)$ behavior of $\omega_n$. The result has the schematic form $\omega_n \simeq n \Delta\omega + A \ln n + B$, where $\Delta\omega$ and $A$ are constants and $B$ depends logarithmically on the $(D-2)$-dimensional spatial momentum $k$ parallel to the horizon. We show that the asymptotic result agrees well with exact quasinormal mode frequencies computed numerically.\] ## Refs - spin-3/2 analogue [[2013#Arnold, Szepietowski, Vaman]] # Arnold, Szepietowski, Vaman ## Gravitino and other spin-3/2 quasinormal modes in Schwarzschild-AdS spacetime \[Links: [arXiv](https://arxiv.org/abs/1311.6409), [PDF](https://arxiv.org/pdf/1311.6409.pdf)\] \[Abstract: We investigate quasinormal mode frequencies $\omega_n$ of gravitinos and generic massive spin-3/2 fields in a Schwarzschild-AdS$_D$ background in spacetime dimension $D>3$, in the black brane (large black hole) limit appropriate to many applications of the AdS/CFT correspondence. First, we find asymptotic formulas for $\omega_n$ in the limit of large overtone number $n$. Asymptotically, $\omega_n \simeq n \Delta\omega + O(\ln n) + O(n^0)$, where $\Delta\omega$ is a known constant, and here we compute the $O(\ln n)$ and $O(n^0)$ corrections to the leading $O(n)$ behavior. Then we compare to numerical calculations of exact quasinormal mode frequencies. Along the way, we also improve the reach and accuracy of an earlier, similar analysis of spin-1/2 fields.\] ## Refs - spin-1/2 analogue [[2013#Arnold, Szepietowski]] # Boulanger, Ponomarev, Skvortsov, Taronna ## On the uniqueness of higher-spin symmetries in AdS and CFT \[Links: [arXiv](https://arxiv.org/abs/1305.5180), [PDF](https://arxiv.org/pdf/1305.5180.pdf)\] \[Abstract: We study the uniqueness of higher-spin algebras which are at the core of higher-spin theories in AdS and of CFTs with exact [[0621 Higher-spin conserved currents in CFT|higher-spin symmetry]], i.e. conserved tensors of rank greater than two. The Jacobi identity for the gauge algebra is the simplest consistency test that appears at the quartic order for a gauge theory. Similarly, the algebra of charges in a CFT must also obey the Jacobi identity. These algebras are essentially the same. Solving the Jacobi identity under some simplifying assumptions spelled out, we obtain that the Eastwood-Vasiliev algebra is the unique solution for $d=4$ and $d>6$. In $5d$ there is a one-parameter family of algebras that was known before. In particular, we show that the introduction of a single higher-spin gauge field/current automatically requires the infinite tower of higher-spin gauge fields/currents. The result implies that from all the admissible non-Abelian cubic vertices in AdS($d$), that have been recently classified for totally symmetric higher-spin gauge fields, only one vertex can pass the Jacobi consistency test. This cubic vertex is associated with a gauge deformation that is the germ of the Eastwood-Vasiliev's higher-spin algebra.\] # Bzowski, McFadden, Skenderis ## Implications of conformal invariance in momentum space \[Links: [arXiv](https://arxiv.org/abs/1304.7760), [PDF](https://arxiv.org/pdf/1304.7760)\] \[Abstract: We present a comprehensive analysis of the implications of conformal invariance for [[0633 CFT correlators|3-point functions]] of the stress-energy tensor, conserved currents and scalar operators in general dimension and in momentum space. Our starting point is a novel and very effective decomposition of tensor correlators which reduces their computation to that of a number of scalar form factors. For example, the most general 3-point function of a conserved and traceless stress-energy tensor is determined by only five form factors. Dilatations and special conformal Ward identities then impose additional conditions on these form factors. The special conformal Ward identities become a set of first and second order differential equations, whose general solution is given in terms of integrals involving a product of three Bessel functions ('triple-K integrals'). All in all, the correlators are completely determined up to a number of constants, in agreement with well-known position space results. We develop systematic methods for explicitly evaluating the triple-K integrals. In odd dimensions they are given in terms of elementary functions while in even dimensions the results involve dilogarithms. In some cases, the triple-K integrals diverge and subtractions are necessary and we show how such subtractions are related to conformal anomalies. This paper contains two parts that can be read independently of each other. In the first part, we explain the method that leads to the solution for the correlators in terms of triple-K integrals and how to evaluate these integrals, while the second part contains a self-contained presentation of all results. Readers interested only in results may directly consult the second part of the paper.\] # Casini, Huerta, Rosabal ## Remarks on entanglement entropy for gauge fields \[Links: [arXiv](https://arxiv.org/abs/1312.1183), [PDF](https://arxiv.org/pdf/1312.1183.pdf)\] \[Abstract: \] ## Refs - used in [[2018#Dong, Harlow, Marolf]] to argue that the area operator is in the center of the algebra - deals with [[0004 Black hole entropy]] when there is gauge field. i.e. talks about the [[0046 Non-factorisation of Hilbert space in QG]] ## Summary - ambiguities arise on the correspondence between algebras and regions - possible to choose (in many ways) local algebras with *trivial* center -> genuine entanglement entropy can be defined for given region - choices correspond to maximal trees of links on the boundary (partial gauge fixing) - -> gauge fixing dependence of entanglement entropy - continuum limit: relative entropy and mutual information are finite and gauge independent ## Puzzles for entanglement entropy for gauge fields - [[Kabat1995]] negative contact term for black holes - [4-7] # Cordova, Jafferis ## Complex Chern-Simons from M5-branes on the Squashed Three-Sphere \[Links: [arXiv](https://arxiv.org/abs/1305.2891), [PDF](https://arxiv.org/pdf/1305.2891)\] \[Abstract: We derive an equivalence between the $(2,0)$ superconformal M5-brane field theory dimensionally reduced on a squashed three-sphere, and [[0089 Chern-Simons theory|Chern-Simons theory]] with complex gauge group. In the reduction, the massless fermions obtain an action which is second order in derivatives and are reinterpreted as ghosts for gauge fixing the emergent non-compact gauge symmetry. A squashing parameter in the geometry controls the imaginary part of the complex Chern-Simons level.\] # Cvetic, Gibbons ## Exact quasi-normal modes for the near horizon Kerr metric \[Links: [arXiv](https://arxiv.org/abs/1312.2250), [PDF](https://arxiv.org/pdf/1312.2250.pdf)\] \[Abstract: We study the [[0325 Quasi-normal modes|quasi-normal modes]] of a massless scalar field in a general sub-extreme Kerr back- ground by exploiting the hidden $SL(2, R) \times SL(2, R) \times SO(3)$ symmetry of the subtracted geometry approximation. This faithfully models the near horizon geometry but locates the black hole in a confining asymptotically conical box analogous to the anti-de-Sitter backgrounds used in string theory. There are just two series of modes, given in terms of hypergeometric functions and spherical harmonics, reminiscent of the left-moving and right-moving degrees in string theory: one is over- damped, the other is underdamped and exhibits rotational splitting. The remarkably simple exact formulae for the complex frequencies would in principle allow the determination of the mass and angular momentum from observations of a black hole. No black hole bomb is possible because the Killing field which co-rotates with the horizon is everywhere timelike outside the black hole.\] # Dong ## Holographic Entanglement Entropy for General Higher Derivative Gravity \[Links: [arXiv](https://arxiv.org/abs/1310.5713), [PDF](https://arxiv.org/pdf/1310.5713.pdf)\] \[Abstract: We propose a general formula for calculating the [[0145 Generalised area|entanglement entropy]] in theories dual to [[0006 Higher-derivative gravity|higher derivative gravity]] where the Lagrangian is a contraction of Riemann tensors. Our formula consists of [[0559 Wald entropy|Wald]]'s formula for the [[0004 Black hole entropy|black hole entropy]], as well as corrections involving the extrinsic curvature. We derive these corrections by noting that they arise from naively higher order contributions to the action which are enhanced due to would-be logarithmic divergences. Our formula reproduces the Jacobson-Myers entropy in the context of [[0341 Lovelock gravity|Lovelock gravity]], and agrees with existing results for general four-derivative gravity. We emphasise that the formula should be evaluated on a particular bulk surface whose location can in principle be determined by solving the equations of motion with conical boundary conditions. This may be difficult in practice, and an alternative method is desirable. A natural prescription is simply minimising our formula, analogous to the [[0007 RT surface|Ryu-Takayanagi prescription]] for Einstein gravity. We show that this is correct in several examples including Lovelock and general four-derivative gravity.\] ## Refs - [[0145 Generalised area]] ## Summary - computes [[0145 Generalised area|HEE]] for $f$(Riemann) gravity ([[0006 Higher-derivative gravity|HDG]]) using the method of [[2013#Lewkowycz, Maldacena]] - the formula is given by $S_{E E}=2 \pi \int d^d y \sqrt{g}\left\{\frac{\partial L}{\partial R_{z \bar{z} z \bar{z}}}+\sum_\alpha\left(\frac{\partial^2 L}{\partial R_{z i z j} \partial R_{\bar{z} k \bar{z} l}}\right)_\alpha \frac{8 K_{z i j} K_{\bar{z} k l}}{q_\alpha+1}\right\}$ ## Caveats - below eq.2.8, $\tau$ has range 2$\pi$, so $z=\rho e^{i \tau}$ is a coordinate in the ==quotient space== - the original space has coordinates $\tilde\rho$ and $\tilde \tau$ and the expansion requires regularity in the original space; but later one goes to the quotient space coordinates $\rho$ and $\tau$ and define $z$ and $\bar z$ using these (explained in Appendix B) - extract linear term in $\epsilon$ first ==before taking the regulator away== ## Regularised cone method - first go to quotient space - first way of thinking about it - $S_{EE}$ is obtained from the linear order in $(n-1)$ of the difference between the singular solution at $n$ and the $n=1$ solution (smooth) - $S_{E E}=\left.\partial_{\epsilon} S[\hat{B}_{n}]\right|_{\epsilon=0}$ - we can just replace the latter by a regularised cone at $n$ so that both terms are calculated at $n$, one singular, one regularised - this is the expression eq. 3.18 used to obtain the anomaly term - second way of thinking about it - ![[Dong2013_fig2.png|350]] - varying w.r.t. $(n-1)$ of the 1st term is zero - need to take the size of cap to zero - difficult in general (e.g. does not integrate easily for anomaly term) - but in obtaining the Wald term, it happens to be a delta function # Faulkner ## The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT \[Links: [arXiv](https://arxiv.org/abs/1303.7221), [PDF](https://arxiv.org/pdf/1303.7221.pdf)\] \[Abstract: We study [[0293 Renyi entropy|entanglement Renyi entropies]] (EREs) of 1+1 dimensional CFTs with classical gravity duals. Using the replica trick the EREs can be related to a partition function of n copies of the CFT glued together in a particular way along the intervals. In the case of two intervals this procedure defines a genus $n-1$ surface and our goal is to find smooth three dimensional gravitational solutions with this surface living at the boundary. We find two families of handlebody solutions labelled by the replica index $n$. These particular bulk solutions are distinguished by the fact that they do not spontaneously break the replica symmetries of the boundary surface. We show that the regularized classical action of these solutions is given in terms of a simple numerical prescription. If we assume that they give the dominant contribution to the gravity partition function we can relate this classical action to the EREs at leading order in $G_N$. We argue that the prescription can be formulated for non-integer $n$. Upon taking the limit $n \to 1$ the classical action reproduces the predictions of the [[0007 RT surface|Ryu-Takayanagi formula]] for the [[0301 Entanglement entropy|entanglement entropy]].\] ## Refs - [[0293 Renyi entropy]] - companioned by [[2013#Hartman]] # Faulkner, Guica, Hartman, Myers, van Raamsdonk ## Gravitation from Entanglement in Holographic CFTs \[Links: [arXiv](https://arxiv.org/abs/1312.7856), [PDF](https://arxiv.org/pdf/1312.7856.pdf)\] \[Abstract: Entanglement entropy obeys a 'first law', an exact quantum generalization of the ordinary first law of thermodynamics. In any CFT with a semiclassical holographic dual, this first law has an interpretation in the dual gravitational theory as a constraint on the spacetimes dual to CFT states. For small perturbations around the CFT vacuum state, we show that the set of such constraints for all ball-shaped spatial regions in the CFT is exactly equivalent to the requirement that the dual geometry satisfy the gravitational equations of motion, linearized about pure AdS. For theories with entanglement entropy computed by the Ryu-Takayanagi formula $S=A/(4G_N)$, we obtain the linearized Einstein equations. For theories in which the vacuum entanglement entropy for a ball is computed by more general Wald functionals, we obtain the linearized equations for the associated higher-curvature theories. Using the first law, we also derive the holographic dictionary for the stress tensor, given the holographic formula for entanglement entropy. This method provides a simple alternative to holographic renormalization for computing the stress tensor expectation value in arbitrary higher derivative gravitational theories.\] ## Refs - [[0302 Gravity from entanglement]] ## Summary - obtains ==linearised EOM around pure AdS== for ==Einstein and [[0006 Higher-derivative gravity|HDG]]== for ==spherical boundary regions== - although for higher derivative gravity, only **Wald** term is used: it is okay because $K=0$ for RT surfaces for spherical boundary regions in pure AdS # Faulkner, Lewkowycz, Maldacena ## Quantum corrections to holographic entanglement entropy \[Links: [arXiv](https://arxiv.org/abs/1307.2892), [PDF](https://arxiv.org/pdf/1307.2892.pdf)\] \[Abstract: We consider [[0301 Entanglement entropy|entanglement entropy]] in quantum field theories with a gravity dual. In the gravity description, the leading order contribution comes from the area of a minimal surface, as proposed by [[0007 RT surface|Ryu-Takayanagi]]. Here we describe the one loop correction to this formula. The minimal surface divides the bulk into two regions. The bulk loop correction is essentially given by the bulk entanglement entropy between these two bulk regions. We perform some simple checks of this proposal.\] ## Refs - [[0212 Quantum extremal surface]] - [[0007 RT surface]] # Fursaev, Patrushev, Solodhukin ## Distributional Geometry of Squashed Cones \[Links: [arXiv](https://arxiv.org/abs/1306.4000), [PDF](https://arxiv.org/pdf/1306.4000.pdf)\] \[Abstract: \] ## Summary - precursor to [[2013#Dong]] # Grumiller, Riedler, Rosseel, Zojer ## Holographic applications of logarithmic conformal field theories \[Links: [arXiv](https://arxiv.org/abs/1302.0280), [PDF](https://arxiv.org/pdf/1302.0280.pdf)\] \[Abstract: We review the relations between Jordan cells in various branches of physics, ranging from quantum mechanics to massive gravity theories. Our main focus is on holographic correspondences between critically tuned gravity theories in Anti-de Sitter space and [[0563 Log CFT|logarithmic conformal field theories]] in various dimensions. We summarize the developments in the past five years, include some novel generalizations and provide an outlook on possible future developments.\] # Harlow, Hayden ## Quantum Computation vs. Firewalls \[Links: [arXiv](https://arxiv.org/abs/1301.4504), [PDF](https://arxiv.org/pdf/1301.4504.pdf)\] \[Abstract: In this paper we discuss quantum computational restrictions on the types of thought experiments recently used by [[2012#Almheiri, Marolf, Polchinski, Sully|Almheiri, Marolf, Polchinski, and Sully]] to argue against the smoothness of black hole horizons. We argue that the quantum computations required to do these experiments take a time which is exponential in the entropy of the black hole under study, and we show that for a wide variety of black holes this prevents the experiments from being done. We interpret our results as motivating a broader type of non-locality than is usually considered in the context of black hole thought experiments, and claim that once this type of non-locality is allowed there may be no need for firewalls. Our results do not threaten the unitarity of of black hole evaporation or the ability of advanced civilizations to test it.\] # Hartman ## Entanglement Entropy at Large Central Charge \[Links: [arXiv](https://arxiv.org/abs/1303.6955), [PDF](https://arxiv.org/pdf/1303.6955.pdf)\] \[Abstract: Two-dimensional conformal field theories with a large [[0033 Central charge|central charge]] and a small number of low-dimension operators are studied using the conformal block expansion. A universal formula is derived for the [[0293 Renyi entropy|Renyi entropies]] of $N$ disjoint intervals in the ground state, valid to all orders in a series expansion. This is possible because the full perturbative answer in this regime comes from the exchange of the stress tensor and other descendants of the vacuum state. Therefore, the Renyi entropy is related to the [[0032 Virasoro algebra|Virasoro]] vacuum [[0031 Conformal block|block]] at large central charge. The entanglement entropy, computed from the Renyi entropy by an analytic continuation, decouples into a sum of single-interval entanglements. This field theory result agrees with the [[0007 RT surface|Ryu-Takayanagi formula]] for the [[0145 Generalised area|holographic entanglement entropy]] of a 2d CFT, applied to any number of intervals, and thus can be interpreted as a microscopic calculation of the area of minimal surfaces in 3d gravity.\] ## Refs - [[0293 Renyi entropy]] - companioned by [[2013#Faulkner]] # Hartman, Maldacena ## Time Evolution of Entanglement Entropy from Black Hole Interiors \[Links: [arXiv](https://arxiv.org/abs/1303.1080), [PDF](https://arxiv.org/pdf/1303.1080.pdf)\] \[Abstract: We compute the time-dependent [[0301 Entanglement entropy|entanglement entropy]] of a CFT which starts in relatively simple initial states. The initial states are the thermofield double for thermal states, dual to eternal black holes, and a particular pure state, dual to a black hole formed by gravitational collapse. The [[0301 Entanglement entropy|entanglement entropy]] grows linearly in time. This linear growth is directly related to the growth of the black hole interior measured along "nice" spatial slices. These nice slices probe the spacelike direction in the interior, at a fixed special value of the interior time. In the case of a two-dimensional CFT, we match the bulk and boundary computations of the entanglement entropy. We briefly discuss the long time behavior of various correlators, computed via classical geodesics or surfaces, and point out that their exponential decay comes about for similar reasons. We also present the time evolution of the wavefunction in the tensor network description.\] ## Examples 1. two-sided eternal BH 2. a particular BH state formed from collapse ## Late-time analysis # Hubeny, Maxfield, Rangamani, Tonni ## Holographic entanglement plateaux \[Links: [arXiv](https://arxiv.org/abs/1306.4004), [PDF](https://arxiv.org/pdf/1306.4004.pdf)\] \[Abstract: \] ## Refs - [[0007 RT surface]] - [[0298 MOTS]] ## Summary - *argues* to modify the homology surface to be everywhere spacelike ## Multiple candidates - for the same boundary region, there are infinitely many minimal surfaces, obtained by folding around the horizon many times ## Ambiguity in RT ### Situation I - in a *globally* static spacetime, RT = HRT - but there are static but non-globally static ones e.g. ![[HubenyMaxfieldRangamaniTonni2013fig10.png|300]] - resolution: require the homology surface to be spacelike ### Situation II - bag of gold - RT: the horizon - HRT: the empty set - also ![[HubenyMaxfieldRangamaniTonni2013fig12.png|500]] - RT is red because it only knows about the exterior static patch but HRT is an inner one with smaller area - resolution: not provided but HRT should be right # Huse, Nandkishore, Oganesyan, Pal, Sondhi ## Localization protected quantum order \[Links: [arXiv](https://arxiv.org/abs/1304.1158), [PDF](https://arxiv.org/pdf/1304.1158.pdf)\] \[Abstract: Closed quantum systems with quenched randomness exhibit many-body localized regimes wherein they do not equilibrate even though prepared with macroscopic amounts of energy above their ground states. We show that such localized systems can order in that individual many-body eigenstates can break symmetries or display [[0158 Topological order|topological order]] in the infinite volume limit. Indeed, isolated localized quantum systems can order even at energy densities where the corresponding thermally equilibrated system is disordered, i.e.: localization protects order. In addition, localized systems can move between ordered and disordered localized phases via non-thermodynamic transitions in the properties of the many-body eigenstates. We give evidence that such transitions may proceed via localized critical points. We note that localization provides protection against decoherence that may allow experimental manipulation of macroscopic quantum states. We also identify a 'spectral transition' involving a sharp change in the spectral statistics of the many-body Hamiltonian.\] # Iorgov, Lisovyy, Tykhyy ## Painlevé VI connection problem and monodromy of $c=1$ conformal blocks \[Links: [arXiv](https://arxiv.org/abs/1308.4092), [PDF](https://arxiv.org/pdf/1308.4092.pdf)\] \[Abstract: Generic $c=1$ four-point conformal blocks on the Riemann sphere can be seen as the coefficients of Fourier expansion of the tau function of Painlevé VI equation with respect to one of its integration constants. Based on this relation, we show that $c=1$ fusion matrix essentially coincides with the connection coefficient relating tau function asymptotics at different critical points. Explicit formulas for both quantities are obtained by solving certain functional relations which follow from the tau function expansions. The final result does not involve integration and is given by a ratio of two products of Barnes G-functions with arguments expressed in terms of conformal dimensions/monodromy data. It turns out to be closely related to the volume of hyperbolic tetrahedron.\] # Jafferis, Lupsasca, Lysov, Ng, Strominger ## Quasinormal Quantization in deSitter Spacetime \[Links: [arXiv](https://arxiv.org/abs/1305.5523), [PDF](https://arxiv.org/pdf/1305.5523.pdf)\] \[Abstract: A scalar field in four-dimensional deSitter spacetime (dS$_4$) has [[0325 Quasi-normal modes|quasinormal modes]] which are singular on the past horizon of the south pole and decay exponentially towards the future. These are found to lie in two complex highest-weight representations of the dS$_4$ isometry group $SO(4,1)$. The Klein-Gordon norm cannot be used for quantization of these modes because it diverges. However a modified 'R-norm', which involves reflection across the equator of a spatial $S^3$ slice, is nonsingular. The quasinormal modes are shown to provide a complete orthogonal basis with respect to the R-norm. Adopting the associated R-adjoint effectively transforms $SO(4,1)$ to the symmetry group $SO(3,2)$ of a $2+1$-dimensional CFT. It is further shown that the conventional Euclidean vacuum may be defined as the state annihilated by half of the quasinormal modes, and the Euclidean Green function obtained from a simple mode sum. Quasinormal quantization contrasts with some conventional approaches in that it maintains manifest dS-invariance throughout. The results are expected to generalize to other dimensions and spins.\] ## Summary - surprisingly, quasinormal modes form a complete basis in dS ## Related - [[2023#Cotler, Strominger]] # Kim, Huse ## Ballistic spreading of entanglement in a diffusive nonintegrable system \[Links: [arXiv](https://arxiv.org/abs/1306.4306), [PDF](https://arxiv.org/pdf/1306.4306.pdf)\] \[Abstract: We study the time evolution of the [[0301 Entanglement entropy|entanglement entropy]] of a one-dimensional nonintegrable spin chain, starting from random nonentangled initial pure states. We use exact diagonalization of a nonintegrable quantum Ising chain with transverse and longitudinal fields to obtain the exact quantum dynamics. We show that the entanglement entropy increases linearly with time before finite-size saturation begins, demonstrating a ballistic spreading of the entanglement, while the energy transport in the same system is diffusive. Thus we explicitly demonstrate that the spreading of entanglement is much faster than the energy diffusion in this nonintegrable system.\] ## Refs - [[0522 Entanglement dynamics]] # Kong, Li, Runkel ## Cardy algebras and sewing constraints, II \[Links: [arXiv](https://arxiv.org/abs/1310.1875), [PDF](https://arxiv.org/pdf/1310.1875)\] \[Abstract: This is the part II of a two-part work started in [arXiv:0807.3356](https://arxiv.org/abs/0807.3356) [math.QA]. In part I, Cardy algebras were studied, a notion which arises from the classification of genus-0,1 open-closed rational conformal field theories. In this part, we prove that a Cardy algebra also satisfies the higher genus factorisation and modular-invariance properties formulated in [arXiv:hep-th/0612306](https://arxiv.org/abs/hep-th/0612306) in terms of the notion of a solution to the [[0602 Moore-Seiberg construction|sewing constraints]]. We present the proof by showing that the latter notion, which is defined as a monoidal natural transformation, can be expressed in terms of generators and relations, which correspond exactly to the defining data and axioms of a Cardy algebra.\] # Lee, Neves ## The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass \[Links: [arXiv](https://arxiv.org/abs/1310.3002), [PDF](https://arxiv.org/pdf/1310.3002.pdf)\] \[Abstract: In the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature at least -6 and negative mass when the genus of the conformal boundary at infinity is positive. Using inverse mean curvature flow, we prove a [[0476 Penrose inequality|Penrose inequality]] for these negative mass metrics. The motivation comes from a previous result of P. Chruściel and W. Simon, which states that the Penrose inequality we prove implies a static uniqueness theorem for negative mass Kottler metrics.\] # Lewkowycz, Maldacena ## Generalized gravitational entropy \[Links: [arXiv](https://arxiv.org/abs/1304.4926), [PDF](https://arxiv.org/pdf/1304.4926.pdf)\] \[Abstract: We consider classical Euclidean gravity solutions with a boundary. The boundary contains a non-contractible circle. These solutions can be interpreted as computing the trace of a density matrix in the full quantum gravity theory, in the classical approximation. When the circle is contractible in the bulk, we argue that the entropy of this density matrix is given by the area of a minimal surface. This is a generalization of the usual black hole entropy formula to euclidean solutions without a Killing vector. A particular example of this set up appears in the computation of the entanglement entropy of a subregion of a field theory with a gravity dual. In this context, the [[0007 RT surface|minimal area prescription]] was proposed by Ryu and Takayanagi. Our arguments explain their conjecture.\] ## Refs - [[0007 RT surface]] - [[0145 Generalised area]] ## Regularised cone - done in covering space - 4 geometries in Fig.3 all have period of $2\pi n$ in $\tau$ - the correct $n=1$ solution is actually the 4th one, but it is singular because of change in the period - c.f. [[2013#Dong]] which was done in quotient space # Liu, Suh (May) ## Entanglement Tsunami: Universal Scaling in Holographic Thermalization \[Links: [arXiv](https://arxiv.org/abs/1305.7244), [PDF](https://arxiv.org/pdf/1305.7244.pdf)\] \[Abstract: We consider the [[0522 Entanglement dynamics|time evolution of entanglement entropy]] after a global quench in a strongly coupled [[0001 AdS-CFT|holographic system]], whose subsequent equilibration is described in the gravity dual by the gravitational collapse of a thin shell of matter resulting in a black hole. In the limit of large regions of entanglement, the evolution of entanglement entropy is controlled by the geometry around and inside the event horizon of the black hole, resulting in regimes of pre-local- equilibration quadratic growth (in time), post-local-equilibration linear growth, a late-time regime in which the evolution does not carry any memory of the size and shape of the entangled region, and a saturation regime with critical behavior resembling those in continuous phase transitions. Collectively, these regimes suggest a picture of entanglement growth in which an "entanglement tsunami" carries entanglement inward from the boundary. We also make a conjecture on the maximal rate of entanglement growth in relativistic systems.\] ## Summary - a bound on entanglement growth # Liu, Suh (Nov) ## Entanglement growth during thermalization in holographic systems \[Links: [arXiv](https://arxiv.org/abs/1311.1200), [PDF](https://arxiv.org/pdf/1311.1200.pdf)\] \[Abstract: We derive in detail several universal features in the time evolution of [[0301 Entanglement entropy|entanglement entropy]] and other nonlocal observables in [[0558 Quantum quench|quenched]] holographic systems. The quenches are such that a spatially uniform density of energy is injected at an instant in time, exciting a strongly coupled CFT which eventually equilibrates. Such quench processes are described on the gravity side by the gravitational collapse of a thin shell that results in a black hole. Various nonlocal observables have a unified description in terms of the area of extremal surfaces of different dimensions. In the large distance limit, the evolution of an extremal surface, and thus the corresponding boundary observable, is controlled by the geometry around and inside the event horizon of the black hole, allowing us to identify regimes of pre-local- equilibration quadratic growth, post-local-equilibration linear growth, a memory loss regime, and a saturation regime with behavior resembling those in phase transitions. We also discuss possible bounds on the maximal [[0327 Entanglement velocity|rate of entanglement growth]] in relativistic systems.\] # Maldacena, Susskind ## Cool horizons for entangled black holes \[Links: [arXiv](https://arxiv.org/abs/1306.0533), [PDF](https://arxiv.org/pdf/1306.0533.pdf)\] \[Abstract: \] ## An assumption that turns out wrong - since EPR cannot be used to transfer information, for [[0220 ER=EPR]], ER needs to be un-traversable - but there are now [[0083 Traversable wormhole]]s after this paper, so this assumption is invalidated - **resolution**: on the EPR side one can add couplings which would correspond to traversable wormholes ## A misnomer - "maximally entangled" here refers to the thermofield double state, although the density matrix is not the identity matrix ## Summary - proposes [[0220 ER=EPR]] - shows that this is true not just for the maximally entangled pair - formulates versions of [[0195 Firewall]] and revolves them ## Statement of [[0220 ER=EPR]] - ER is created by EPR like correlations between microstates of the BHs - bases on previous observations [[Israel1976]] and [[2001#Maldacena]] - special case - the ER bridge is a special kind of EPR correlation in which the EPR correlated quantum systems have a weakly coupled Einstein gravity description - on the EPR side it is special because it is one of many possible entanglement profiles ## 2. Getting different entangled states 1. do time evolution, i.e. move the WdW patch 2. insert operators in Euclidean section - they correspond to creating particles. see [[2001#Maldacena]] 3. left and right horizons not touching: add matter to each side or use time-dependent Hamiltonian in Euclidean evolution ## 3. What supports ER=EPR - no superluminal signals - no creation by LOCC - restoring the thermofield double - messages from Alice to Bob - clouds - Hawking radiation ## 4. Implications for [[0195 Firewall]] ## 5. Comments for [[0195 Firewall]] and construction of interior # Marolf, Polchinski ## Gauge/Gravity Duality and the Black Hole Interior \[Links: [arXiv](https://arxiv.org/abs/), [PDF](https://arxiv.org/pdf/.pdf)\] \[Abstract: We present a further argument that typical black holes with field theory duals have [[0195 Firewall|firewalls]] at the horizon. This argument makes no reference to entanglement between the black hole and any distant system, and so is not evaded by identifying degrees of freedom inside the black hole with those outside. We also address the [[0220 ER=EPR|ER=EPR]] conjecture of Maldacena and Susskind, arguing that the correlations in generic highly entangled states cannot be geometrized as a smooth wormhole.\] # McGough, Verlinde ## Bekenstein-Hawking entropy as topological entanglement entropy \[Links: [arXiv](https://arxiv.org/abs/1308.2342), [PDF](https://arxiv.org/pdf/1308.2342.pdf)\] \[Abstract: \] <!-- Wayne Weng recommended this when talking about using entanglement entropy to detect topological order (as in CMT) [[0158 Topological order]] --> # Miao ## A note on holographic Weyl anomaly and entanglement entropy \[Links: [arXiv](https://arxiv.org/abs/1309.0211), [PDF](https://arxiv.org/pdf/1309.0211.pdf)\] \[Abstract: \] ## Summary - *obtains* the [[0209 Holographic renormalisation|holographic anomaly]] for general [[0006 Higher-derivative gravity]] in AdS5 and AdS7 - *proposes* a formula for [[0145 Generalised area|HEE]] for [[0006 Higher-derivative gravity]] in AdS5 - which *agrees* with [[2013#Dong]] ## Method for anomaly - use a referenced curvature $\bar{R}_{\mu \nu \rho \sigma}=-\left(\hat{G}_{\mu \rho} \hat{G}_{\nu \sigma}-\hat{G}_{\mu \sigma} \hat{G}_{\nu \rho}\right)$ - expand $f$ in $S=\frac{1}{2 \kappa_{d+1}^{2}} \int d^{d+1} x \sqrt{-\hat{G}} f\left(\hat{R}_{\mu \nu \rho \sigma}\right)+S_{B}$ around this referenced curvature - rewrite it as $f_{n}=\sum_{i=1}^{m_{n}} c_{i}^{n} \tilde{K}_{i}^{n}$ where $\tilde{K}_{i}^{n}=\left.K_{i}^{n}\right|_{[\hat{R} \rightarrow(\hat{R}-\bar{R})]}$ and $K$ are scalars constructed from curvature tensors - notice $\hat{R}-\bar{R}=o(\rho), \quad \hat{R}_{\mu \nu}-\bar{R}_{\mu \nu}=o(1), \quad \hat{R}_{\mu \nu \rho \sigma}-\bar{R}_{\mu \nu \rho \sigma}=o\left(\frac{1}{\rho}\right)$ etc - determine $c^n_i$ ## Method for HEE - [[2011#Hung, Myers, Smolkin]] - [[2013#Fursaev, Patrushev, Solodhukin]] # Murata, Reall, Tanahashi ## What happens at the horizon(s) of an extreme black hole? \[Links: [arXiv](https://arxiv.org/abs/1307.6800), [PDF](https://arxiv.org/pdf/1307.6800.pdf)\] \[Abstract: A massless scalar field exhibits an [[0340 Aretakis instability|instability]] at the event horizon of an extreme black hole. We study numerically the nonlinear evolution of this instability for spherically symmetric perturbations of an extreme Reissner-Nordstrom (RN) black hole. We find that generically the endpoint of the instability is a non-extreme RN solution. However, there exist fine-tuned initial perturbations for which the instability never decays. In this case, the perturbed spacetime describes a time-dependent extreme black hole. Such solutions settle down to extreme RN outside, but not on, the event horizon. The event horizon remains smooth but certain observers who cross it at late time experience large gradients there. Our results indicate that these dynamical extreme black holes admit a $C^1$ extension across an inner (Cauchy) horizon.\] # Neiman ## The imaginary part of the gravity action and black hole entropy \[Links: [arXiv](https://arxiv.org/abs/1301.7041), [PDF](https://arxiv.org/pdf/1301.7041.pdf)\] \[Abstract: \] ## Refs - [[0004 Black hole entropy]] - earlier work [[Neimann2012]] gives only the null boundary case ## Summary - proposes that the [[0357 Imaginary action]] is related to the [[0004 Black hole entropy]] ## The general result - $\operatorname{Im} S=\frac{1}{4} \sum_{\text {flips }} \sigma_{\text {flip }}$ ## Relation to entropy - argues that the imaginary action is more general than the [[0004 Black hole entropy]] ## Generalisation to Lovelock - procedures 1. use [[Myers1987]] to write down the boundary action 2. use a [[0102 Hayward term]]-like procedure to obtain the corner terms 3. write down the imaginary part of the action just like for GR but with area replaced by [[1993#Jacobson, Myers]] entropy - comment - will not work for f(Riem) etc # Nozaki, Numasawa, Takayagani ## Holographic Local Quenches and Entanglement Density \[Links: [arXiv](https://arxiv.org/abs/1302.5703), [PDF](https://arxiv.org/pdf/1302.5703.pdf)\] \[Abstract: We propose a free falling particle in an AdS space as a holographic model of local [[0558 Quantum quench|quench]]. Local quenches are triggered by local excitations in a given quantum system. We calculate the time-evolution of [[0007 RT surface|holographic entanglement entropy]]. We confirm a logarithmic time-evolution, which is known to be typical in two dimensional local quenches. To study the structure of quantum entanglement in general quantum systems, we introduce a new quantity which we call entanglement density and apply this analysis to quantum quenches. We show that this quantity is directly related to the energy density in a small size limit. Moreover, we find a simple relationship between the amount of quantum information possessed by a massive object and its total energy based on the AdS/CFT.\] # Sarkar, Wall ## GSL at linear order for actions that are functions of Lovelock densities \[Links: [arXiv](https://arxiv.org/abs/1306.1623), [PDF](https://arxiv.org/pdf/1306.1623.pdf)\] \[Abstract: \] ## Summary - show that f(Locklock) has an increasing entropy for linear perturbation to Killing horizon - for classical sources obtaining [[0480 Null energy condition|NEC]] - both in terms of classical [[0005 Black hole second law]] and [[0082 Generalised second law]] ## Caveat: - Gravitons neglected (because they require second order perturbation of the metric) although they are of order $\hbar$. - Location of graviton horizon may differ from metric horizon. ## Idea - need $\Theta=\frac{1}{4 \hbar G}\left(\frac{d \rho}{d \lambda}+\theta \rho\right)$ to be positive - want to show that $\frac{d \Theta}{d \lambda} \approx-\frac{1}{4} E_{a b} k^{a} k^{b}=-2 \pi T_{a b} k^{a} k^{b}$ - so it's negative if [[0480 Null energy condition|NEC]] - => $\Theta$ always positive if [[0480 Null energy condition|NEC]] because $\Theta(\lambda=+\infty)=0$ ## Calculation - for Lovelock, already done in [[KolekarPadmanabhanSarkar2012]] - for f(Lovelock), the extra term from variation is not a total derivative - so have more terms to deal with # Serbyn, Papic, Abanin ## Universal slow growth of entanglement in interacting strongly disordered systems \[Links: [arXiv](https://arxiv.org/abs/1304.4605), [PDF](https://arxiv.org/pdf/1304.4605.pdf)\] \[Abstract: Recent numerical work by Bardarson et. al. ([[2012#Bardarson, Pollman. Moore]]) revealed a slow, logarithmic in time, growth of [[0301 Entanglement entropy|entanglement entropy]] for initial product states in a putative many-body localized phase. We show that this surprising phenomenon results from the dephasing due to exponentially small interaction-induced corrections to the eigenenergies of different states. For weak interactions, we find that the entanglement entropy grows as $\xi \ln (Vt/\hbar)$, where $V$ is the interaction strength, and $\xi$ is the single-particle localization length. The saturated value of the entanglement entropy at long times is determined by the participation ratios of the initial state over the eigenstates of the subsystem. The proposed mechanism is illustrated with numerical simulations of small systems. Our work shows that the logarithmic [[0522 Entanglement dynamics|entanglement growth]] is a universal phenomenon characteristic of the many-body localized phase in any number of spatial dimensions, and reveals a broad hierarchy of dephasing time scales present in such a phase.\] ## Refs - [[0522 Entanglement dynamics]] - [[0541 Thermalisation]] # Shenker, Stanford (Jun) ## Black holes and the butterfly effect \[Links: [arXiv](https://arxiv.org/abs/1306.0622), [PDF](https://arxiv.org/pdf/1306.0622.pdf)\] \[Abstract: We use [[0001 AdS-CFT|holography]] to study sensitive dependence on initial conditions in strongly coupled field theories. Specifically, we mildly perturb a thermofield double state by adding a small number of quanta on one side. If these quanta are released a scrambling time in the past, they destroy the local two-sided correlations present in the unperturbed state. The corresponding bulk geometry is a two-sided AdS black hole, and the key effect is the blueshift of the early infalling quanta relative to the $t = 0$ slice, creating a [[0117 Shockwave|shock wave]]. We comment on string- and Planck-scale corrections to this setup, and discuss points that may be relevant to [[0195 Firewall|the firewall controversy]].\] ## Summary - original: holographic description of the perturbation on the CFT as a shock wave - comment on string and Planck scale corrections ## Reasoning and intuition 1. typical entangled states -> atypical under evolution (chaos destroys entanglement) 2. mutual information between left and right subregions decreases over time (see [[2013#Hartman, Maldacena]]) 3. at past time $-t_w$ the states are "aimed" precisely to give the perfect entanglement at $t=0$ -> so this aiming can fail dramatically with a small quanta at large $t_w$ ## Creation of shockwave - a particle thrown into the black hole at very early time gains lots of energy in the local frame of $t=0$ - if thrown early enough, it is a shock wave # Shenker, Stanford (Dec) ## Multiple shocks \[Links: [arXiv](https://arxiv.org/abs/1312.3296), [PDF](https://arxiv.org/pdf/1312.3296.pdf)\] \[Abstract: Using [[0001 AdS-CFT|gauge/gravity duality]], we explore a class of states of two CFTs with a large degree of entanglement, but with very weak local two-sided correlation. These states are constructed by perturbing the thermofield double state with thermal-scale operators that are local at different times. Acting on the dual black hole geometry, these perturbations create an intersecting network of shock waves, supporting a very long wormhole. [[0008 Quantum chaos|Chaotic]] CFT dynamics and the associated fast scrambling time play an essential role in determining the qualitative features of the resulting geometries.\] # Strominger (Aug) ## Asymptotic Symmetries of Yang-Mills Theory \[Links: [arXiv](https://arxiv.org/abs/1308.0589), [PDF](https://arxiv.org/pdf/1308.0589.pdf)\] \[Abstract: [[0060 Asymptotic symmetry|Asymptotic symmetries]] at future null infinity ($\mathcal{I}^+$) of Minkowski space for electrodynamics with massless charged fields, as well as non-Abelian gauge theories with gauge group $G$, are considered at the semiclassical level. The possibility of charge/color flux through $\mathcal{I}^+$ suggests the symmetry group is infinite-dimensional. It is conjectured that the symmetries include a $G$ [[0069 Kac-Moody algebra|Kac-Moody]] symmetry whose generators are "large" gauge transformations which approach locally holomorphic functions on the conformal two-sphere at $\mathcal{I}^+$ and are invariant under null translations. The Kac-Moody currents are constructed from the gauge field at the future boundary of $\mathcal{I}^+$. The current [[0106 Ward identity|Ward identities]] include Weinberg's [[0009 Soft theorems|soft photon theorem]] and its colored extension.\] # Strominger (Dec) ## On BMS Invariance of Gravitational Scattering \[Links: [arXiv](https://arxiv.org/abs/1312.2229), [PDF](https://arxiv.org/pdf/1312.2229.pdf)\] \[Abstract: $\text{BMS}^+$ transformations act nontrivially on outgoing gravitational scattering data while preserving intrinsic structure at future null infinity ($\mathcal{I}^+$). $\text{BMS}^-$ transformations similarly act on ingoing data at past null infinity ($\mathcal{I}^-$). In this paper we apply - within a suitable finite neighborhood of the Minkowski vacuum - results of Christodoulou and Klainerman to link $\mathcal{I}^{+}$ to $\mathcal{I}^{-}$ and thereby identify "diagonal" elements $\text{BMS}^0$ of $\text{BMS}^+\times\text{BMS}^-$. We argue that $\text{BMS}^0$ is a nontrivial infinite-dimensional symmetry of both classical gravitational scattering and the quantum gravity $S$-matrix. It implies the conservation of net accumulated energy flux at every angle on the conformal $S^2$ at $\mathcal{I}$. The associated Ward identity is shown to relate $S$-matrix elements with and without soft gravitons. Finally, $\text{BMS}^0$ is recast as a $U(1)$ [[0069 Kac-Moody algebra|Kac-Moody symmetry]] and an expression for the Kac-Moody current is given in terms of a certain soft graviton operator on the boundary of null infinity.\] ## Refs - [[0010 Celestial holography]] - [[0009 Soft theorems]]