# Afkhami-Jeddi, Hartman, Kundu, Tajdini ## Einstein gravity 3-pt. functions from CFT \[Links: [arXiv](https://arxiv.org/abs/1610.09378), [PDF](https://arxiv.org/pdf/1610.09378.pdf)\] \[Abstract: We study stress tensor correlation functions in four-dimensional conformal field theories with large $N$ and a sparse spectrum. Theories in this class are expected to have local holographic duals, so effective field theory in anti-de Sitter suggests that the stress tensor sector should exhibit universal, gravity-like behavior. At the linearized level, the hallmark of locality in the emergent geometry is that stress tensor three-point functions $\langle TTT\rangle$, normally specified by three constants, should approach a universal structure controlled by a single parameter as the gap to higher spin operators is increased. We demonstrate this phenomenon by a direct CFT calculation. Stress tensor exchange, by itself, violates [[0132 Causality constraints in CFT|causality]] and [[0035 Unitarity of CFT|unitarity]] unless the three-point functions are carefully tuned, and the unique consistent choice exactly matches the prediction of [[0554 Einstein gravity|Einstein gravity]]. Under some assumptions about the other potential contributions, we conclude that this structure is universal, and in particular, that the [[0306 Weyl anomaly|anomaly]] coefficients satisfy $a\approx c$ as conjectured by [[2014#Camanho, Edelstein, Maldacena, Zhiboedov|Camanho et al]]. The argument is based on [[0118 Causality constraints for gravity|causality]] of a four-point function, with kinematics designed to probe bulk locality, and invokes the [[0474 Chaos bound|chaos bound]] of [[2015#Maldacena, Shenker, Stanford|Maldacena, Shenker, and Stanford]].\] ## Summary - studies stress tensor correlation functions in 4d [[0122 Holographic CFT|holographic CFTs]] - local holographic duals - => should exhibit universal, gravity-like behaviour - locality of emergent geometry => $\langle TTT\rangle$ has university structure with a single parameter - stress tensor exchange can violate causality and unitarity unless tuned - -> tuning gives Einstein gravity (joke: "tunes out to be") - [[0118 Causality constraints for gravity]] ## Refs - later paper [[2017#Afkhami-Jeddi, Hartman, Kundu, Tajdini]] # Akers, Koeller, Leichenauer, Levine ## Geometric Constraints from Subregion Duality Beyond the Classical Regime \[Links: [arXiv](https://arxiv.org/abs/1610.08968), [PDF](https://arxiv.org/pdf/1610.08968.pdf)\] \[Abstract: Subregion duality in AdS/CFT implies certain constraints on the geometry: entanglement wedges must contain causal wedges, and nested boundary regions must have nested entanglement wedges. We elucidate the logical connections between these statements and the [[0243 Quantum focusing conjecture|Quantum Focussing Conjecture]], [[0405 Quantum null energy condition|Quantum Null Energy Condition]], [[0091 Boundary causality|Boundary Causality Condition]], and [[0417 Averaged null energy condition|Averaged Null Energy Condition]]. Our analysis does not rely on the classical limit of bulk physics, but instead works to all orders in $G\hbar \sim 1/N$. This constitutes a nontrivial check on the consistency of subregion duality, [[0219 Entanglement wedge reconstruction|entanglement wedge reconstruction]], and [[0007 RT surface|holographic entanglement entropy]] beyond the classical regime.\] ## Restrictions - the boundary theory is on ==Minkowski== when talking about third column (see [[2017#Akers, Chandrasekaran, Leichenauer, Levine, Shahbazi-Moghaddam]] for curved CFT) ## Refs - a main paper on [[0142 Entanglement wedge nesting|EWN]] - extensions - [[2017#Akers, Chandrasekaran, Leichenauer, Levine, Shahbazi-Moghaddam]] ## Summary ![[AkersKoellerLeichenauerLevine2016_relation9.png|450]] - everything is to all orders in $(G\hbar)^{1/2}$ - the classical result is simpler and has been studied before # Appels, Gregory, Kubiznak ## Thermodynamics of Accelerating Black Holes \[Links: [arXiv](https://arxiv.org/abs/1604.08812), [PDF](https://arxiv.org/pdf/1604.08812.pdf)\] \[Abstract: We address a long-standing problem of describing the thermodynamics of a charged [[0336 C-metric|accelerating black hole]]. We derive a standard first law of [[0127 Black hole thermodynamics|black hole thermodynamics]], with the usual identification of entropy proportional to the area of the event horizon -- even though the event horizon contains a conical singularity. This result not only extends the applicability of black hole thermodynamics to realms previously not anticipated, it also opens a possibility for studying novel properties of an important class of exact radiative solutions of Einstein equations describing accelerated objects. We discuss the thermodynamic volume, stability and phase structure of these black holes.\] # Bagchi, Gary, Zodinmawia ## The BMS Bootstrap \[Links: [arXiv](https://arxiv.org/abs/1612.01730), [PDF](https://arxiv.org/pdf/1612.01730.pdf)\] \[Abstract: We initiate a study of the bootstrap programme for field theories with [[0064 BMS group|BMS]] symmetry. Specifically, we look at two-dimensional field theories with BMS3 symmetry and, using highest weight representations, we construct the [[0123 BMS bootstrap|BMS bootstrap]] equation by formulating the notion of crossing symmetry in the four-point functions of these field theories. In the limit of large central charges, we find analytic expressions for the BMS blocks that are the basic ingredients for the solution of the bootstrap equation. This constitutes, to the best of our knowledge, the first example of the formulation and significant steps towards the solution of a bootstrap equation in a theory which is not a relativistic conformal field theory.\] ## Summary 1. shows that many features of CFT also apply to BMSFT. 2. construct the [[0123 BMS bootstrap]] equation. 3. derives [[0098 BMS blocks]] for large central charge. ## Comments - First paper on [[0123 BMS bootstrap]]. - ref.15 explains why boundary of 3d gravity is 2d field theory on the null boundary # Bhattacharyya, Haehl, Kundu, Loganayagam, Rangamani ## Towards a second law for Lovelock theories \[Links: [arXiv](https://arxiv.org/abs/1612.04024), [PDF](https://arxiv.org/pdf/1612.04024.pdf)\] \[Abstract: In classical general relativity described by Einstein-Hilbert gravity, black holes behave as thermodynamic objects. In particular, the laws of black hole mechanics can be interpreted as laws of thermodynamics. The first law of black hole mechanics extends to [[0006 Higher-derivative gravity|higher derivative theories]] via the Noether charge construction of Wald. One also expects the statement of the [[0005 Black hole second law|second law]], which in Einstein-Hilbert theory owes to Hawking's area theorem, to extend to higher derivative theories. To argue for this however one needs a notion of entropy for dynamical black holes, which the Noether charge construction does not provide. We propose such an entropy function for the family of Lovelock theories, treating the higher derivative terms as perturbations to the Einstein-Hilbert theory. Working around a dynamical black hole solution, and making no assumptions about the amplitude of departure from equilibrium, we construct a candidate entropy functional valid to all orders in the low energy effective field theory. This entropy functional satisfies a second law, modulo a certain subtle boundary term, which deserves further investigation in non-spherically symmetric situations.\] ## Coordinate choice - detailed derivation in app.A # Blake (Mar) ## Universal Charge Diffusion and the Butterfly Effect \[Links: [arXiv](https://arxiv.org/abs/1603.08510), [PDF](https://arxiv.org/pdf/1603.08510.pdf)\] \[Abstract: \] ## Refs - OG for [[0434 Diffusivity]] and its relation to butterfly velocity # Bueno, Cano ## Einsteinian cubic gravity \[Links: [arXiv](https://arxiv.org/abs/1607.06463), [PDF](https://arxiv.org/pdf/1607.06463.pdf)\] \[Abstract: We drastically simplify the problem of linearizing a general higher-order theory of gravity. We reduce it to the evaluation of its Lagrangian on a particular Riemann tensor depending on two parameters, and the computation of two derivatives with respect to one of those parameters. We use our method to construct a $D$-dimensional cubic theory of gravity which satisfies the following properties: 1) it shares the spectrum of Einstein gravity, i.e., it only propagates a transverse and massless graviton on a maximally symmetric background; 2) the relative coefficients of the different curvature invariants involved are the same in all dimensions; 3) it is neither trivial nor topological in four dimensions. Up to cubic order in curvature, the only previously known theories satisfying the first two requirements are the Lovelock ones: Einstein gravity, Gauss-Bonnet and cubic-Lovelock. Of course, the last two theories fail to satisfy requirement 3 as they are, respectively, topological and trivial in four dimensions. We show that, up to cubic order, there exists only one additional theory satisfying requirements 1 and 2. Interestingly, this theory is, along with Einstein gravity, the only theory which also satisfies 3.\] # Caron-Huot, Dixon, McLeod, Hippel ## Bootstrapping a five-loop amplitude using Steinmann relations \[Links: [arXiv](https://arxiv.org/abs/1609.00669), [PDF](https://arxiv.org/pdf/1609.00669.pdf)\] \[Abstract: \] ### Summary - use [[0135 Steinmann relations]] to constrain planar $\mathcal{N}=4$ SYM amplitudes - ==5-loop 6-particle== amplitude - 4 and 5 particles: trivial by conformal symmetry ($\mathcal{N}=4$ SYM is conformal by itself, without talking about holography) ### Steinmann relations - amplitude can have no double discontinuities in overlapping channels - overlapping channels <-> cut lines that intersect (Cutkosky rules) # Chang, Lin ## Semiclassical OPE coefficients from 3D gravity \[Links: [arXiv](https://arxiv.org/abs/1604.01774), [PDF](https://arxiv.org/pdf/1604.01774)\] \[Abstract: We present a closed form expression for the semiclassical [[0030 Operator product expansion|OPE]] coefficients that are universal for all 2D CFTs with a "weak" light spectrum, by taking the semiclassical limit of the [[0573 Crossing kernel|fusion kernel]]. We match this with a properly regularized and normalized bulk action evaluated on a geometry with three conical defects, analytically continued in the deficit angles beyond the range for which a metric with positive signature exists. The analytically continued geometry has a codimension-one coordinate singularity surrounding the heaviest conical defect. This singularity becomes a horizon after Wick rotating to Lorentzian signature, suggesting a connection between universality and the existence of a horizon.\] # Cotler, Gur-Ari, Hanada, Polchinski, Saad, Shenker, Stanford, Streicher, Tezuka ## Black Holes and Random Matrices \[Links: [arXiv](https://arxiv.org/abs/1611.04650), [PDF](https://arxiv.org/pdf/1611.04650.pdf)\] \[Abstract: We argue that the late time behavior of horizon fluctuations in large anti-de Sitter (AdS) black holes is governed by the [[0579 Random matrix theory|random matrix]] dynamics characteristic of [[0008 Quantum chaos|quantum chaotic systems]]. Our main tool is the [[0201 Sachdev-Ye-Kitaev model|Sachdev-Ye-Kitaev (SYK) model]], which we use as a simple model of a black hole. We use an analytically continued partition function $|Z(\beta +it)|^2$ as well as correlation functions as diagnostics. Using numerical techniques we establish random matrix behavior at late times. We determine the early time behavior exactly in a double scaling limit, giving us a plausible estimate for the crossover time to random matrix behavior. We use these ideas to formulate a conjecture about general large AdS black holes, like those dual to 4D super-Yang-Mills theory, giving a provisional estimate of the crossover time. We make some preliminary comments about challenges to understanding the late time dynamics from a bulk point of view.\] # Couch, Fischler, Nguyen ## Noether charge, black hole volume, and complexity \[Links: [arXiv](https://arxiv.org/abs/1610.02038), [PDF](https://arxiv.org/pdf/1610.02038.pdf)\] \[Abstract: In this paper, we study the physical significance of the thermodynamic volumes of AdS black holes using the [[0019 Covariant phase space|Noether charge formalism]] of Iyer and Wald. After applying this formalism to study the extended thermodynamics of a few examples, we discuss how the extended thermodynamics interacts with the recent [[0204 Quantum complexity|complexity]] = action proposal of [[2015#Brown, Roberts, Susskind, Swingle, Zhao (Sep)|Brown et al.]] (CA-duality). We, in particular, discover that their proposal for the late time rate of change of complexity has a nice decomposition in terms of thermodynamic quantities reminiscent of the Smarr relation. This decomposition strongly suggests a geometric, and via CA-duality holographic, interpretation for the thermodynamic volume of an AdS black hole. We go on to discuss the role of thermodynamics in complexity = action for a number of black hole solutions, and then point out the possibility of an alternate proposal, which we dub "complexity = volume 2.0". In this alternate proposal, the complexity would be thought of as the spacetime volume of the Wheeler-DeWitt patch. Finally, we provide evidence that, in certain cases, our proposal for complexity is consistent with the Lloyd bound whereas CA-duality is not.\] ## Refs - this is the proposal of CV2.0 # Czech, Lamprou, McCandlish, Mosk, Sully ## A stereoscopic look into the bulk \[Links: [arXiv](https://arxiv.org/abs/1604.03110), [PDF](https://arxiv.org/pdf/1604.03110.pdf)\] \[Abstract: \] ## Refs - independent work: [[DeBoerHaehlHellerMyers2016]] ## Summary - OPE blocks are bulk operators smeared along geodesics - generalises [[0007 RT surface]] - OPE block kinematics can be equivalently written as KG equations in the space of CFT bi-locals - gives a simple derivation of the [[0185 Dual of conformal blocks]] ## Scale - scale must be important: otherwise cannot probe the bulk - -> local operators are not good, but **bi-local** could work, e.g. pairs of insertions (the distance gives a scale) - they are not very non-local, but they can be organised into OPE blocks, which are non-local ## Conformal blocks ###### 3d - geodesic line connecting two geodesic lines ![[CzechLamprouMcCandlishMoskSully2016_3dblock.png|400]] - left: kinematic space propagator - right: geodesic Witten diagram ###### higher dimensions ![[CzechLamprouMcCandlishMoskSully2016_hdblock.png|400]] # Dong ## The Gravity Dual of Renyi Entropy \[Links: [arXiv](https://arxiv.org/abs/1601.06788), [PDF](https://arxiv.org/pdf/1601.06788.pdf)\] \[Abstract: A remarkable yet mysterious property of black holes is that their entropy is proportional to the horizon area. This area law inspired the holographic principle, which was later realized concretely in [[0001 AdS-CFT|gauge/gravity duality]]. In this context, [[0301 Entanglement entropy|entanglement entropy]] is given by the area of a minimal surface in a dual spacetime. However, discussions of area laws have been constrained to entanglement entropy, whereas a full understanding of a quantum state requires [[0293 Renyi entropy|Renyi entropies]]. Here we show that all Renyi entropies satisfy a similar area law in holographic theories and are given by the areas of dual cosmic branes. This geometric prescription is a one-parameter generalization of the [[0007 RT surface|minimal surface prescription]] for entanglement entropy. Applying this we provide the first holographic calculation of mutual Renyi information between two disks of arbitrary dimension. Our results provide a framework for efficiently studying Renyi entropies and understanding entanglement structures in strongly coupled systems and quantum gravity.\] ## Refs - [[0293 Renyi entropy]] ## Example - two intervals, to linear order in $\delta n$ # Dong, Harlow, Wall ## Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality \[Links: [arXiv](https://arxiv.org/abs/1601.05416), [PDF](https://arxiv.org/pdf/1601.05416.pdf)\] \[Abstract: In this Letter we prove a simple theorem in quantum information theory, which implies that bulk operators in the Anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence can be reconstructed as CFT operators in a spatial subregion $A$, provided that they lie in its [[0219 Entanglement wedge reconstruction|entanglement wedge]]. This is an improvement on existing reconstruction methods, which have at most succeeded in the smaller causal wedge. The proof is a combination of the recent work of [[0048 JLMS|Jafferis, Lewkowycz, Maldacena, and Suh]] on the quantum relative entropy of a CFT subregion with earlier ideas interpreting the correspondence as a [[0146 Quantum error correction|quantum error correcting code]].\] ## Refs - later work [[2016#Harlow]] on RT surface from [[0146 Quantum error correction]] ## Summary - proves [[0219 Entanglement wedge reconstruction|EWR]] - gives a cleaner version/proof of [[0048 JLMS|JLMS]] ## Code subspace - not unique - a simple way to define - choose a state that we know has a bulk geometric interpretation - then consider subspace of all states where the backreaction is perturbatively small ## The theorem - assume the bulk operator acts only on $\mathcal{H}_a$, where $\mathcal{H}_{\text {code }}=\mathcal{H}_{a} \otimes \mathcal{H}_{\bar{a}}$ is a subspace of $\mathcal{H}=\mathcal{H}_{A} \otimes \mathcal{H}_{\bar{A}}$ (here $\mathcal{H}$ is the full Hilbert space in the bulk, which is also the boundary Hilbert space) - then there is some the (boundary) operator $O_A$ acting only on $\mathcal{H}_A$ that reproduces the (bulk) action of $O$ on the code subspace, i.e., $O_{A}|\phi\rangle=O|\phi\rangle$ <!-- ## Comments - Marija wonders how this works in finite N with the prescription of [[Hernandez-CuencaHorowitzTrevinoDW2021]] - maybe QEC is only a feature of perturbative gravity? --> # Dong, Lewkowycz, Rangamani ## Deriving covariant holographic entanglement \[Links: [arXiv](https://arxiv.org/abs/1607.07506), [PDF](https://arxiv.org/pdf/1607.07506.pdf)\] \[Abstract: We provide a gravitational argument in favour of the covariant [[0007 RT surface|holographic entanglement entropy]] proposal. In general time-dependent states, the proposal asserts that the [[0301 Entanglement entropy|entanglement entropy]] of a region in the boundary field theory is given by a quarter of the area of a bulk extremal surface in Planck units. The main element of our discussion is an implementation of an appropriate [[0042 Schwinger-Keldysh techniques|Schwinger-Keldysh contour]] to obtain the reduced density matrix (and its powers) of a given region, as is relevant for the replica construction. We map this contour into the bulk gravitational theory, and argue that the saddle point solutions of these replica geometries lead to a consistent prescription for computing the field theory Renyi entropies. In the limiting case where the replica index is taken to unity, a local analysis suffices to show that these saddles lead to the extremal surfaces of interest. We also comment on various properties of holographic entanglement that follow from this construction.\] ## Summary - derives [[0007 RT surface|HRT]] ## Key techniques - replica trick but in Lorentzian signature - used the bulk dual of [[0042 Schwinger-Keldysh techniques|SK techniques]] developed in [[2008#Skenderis, van Rees (May)]] and [[2008#Skenderis, van Rees (Dec)]] # Donnay, Giribet, Gonzales, Pino ## Extended Symmetries at the Black Hole Horizon \[Links: [arXiv](https://arxiv.org/abs/1607.05703), [PDF](https://arxiv.org/pdf/1607.05703.pdf)\] \[Abstract: We prove that non-extremal black holes in four-dimensional general relativity exhibit an infinite-dimensional [[0561 Near-horizon symmetry|symmetry in their near horizon region]]. By prescribing a physically sensible set of boundary conditions at the horizon, we derive the algebra of asymptotic Killing vectors, which is shown to be infinite-dimensional and includes, in particular, two sets of supertranslations and two mutually commuting copies of the [[0032 Virasoro algebra|Virasoro algebra]]. We define the surface charges associated to the asymptotic diffeomorphisms that preserve the boundary conditions and discuss the subtleties of this definition, such as the integrability conditions and the correct definition of the Dirac brackets. When evaluated on the stationary solutions, the only non-vanishing charges are the zero-modes. One of them reproduces the [[0004 Black hole entropy|Bekenstein-Hawking entropy]] of Kerr black holes. We also study the extremal limit, recovering the NHEK geometry. In this singular case, where the algebra of charges and the integrability conditions get modified, we find that the computation of the zero-modes correctly reproduces the black hole entropy. Furthermore, we analyze the case of three spacetime dimensions, in which the integrability conditions notably simplify and the field equations can be solved analytically to produce a family of exact solutions that realize the boundary conditions explicitly. We examine other features, such as the form of the algebra in the extremal limit and the relation to other works in the literature.\] # Donnelly, Freidel ## Local subsystems in gauge theory and gravity \[Links: [arXiv](https://arxiv.org/abs/1601.04744), [PDF](https://arxiv.org/pdf/1601.04744.pdf)\] \[Abstract: \] ## Summary - In the context of [[0019 Covariant phase space]] - Extend the extended *Hilbert* space in gauge theory to its classical analog [[0044 Extended phase space]] - Add boundary dof - [[0071 Yang-Mills|YM]]: choice of gauge - Einstein: location of the codim-2 surface and choice of *conformal* normal frame - Extend to gravity theories. - Obtains ==area== as a charge. ## Full derivation - symplectic form - $L[g]=\frac{1}{2} R \epsilon$ - $\epsilon:=\frac{s_{g}}{D !} \epsilon_{a_{1} \ldots a_{D}} \mathrm{d} x^{a_{1}} \wedge \cdots \wedge \mathrm{d} x^{a_{D}}=\sqrt{g}\, \mathrm{d}^{D} x$ - $s_g=-1$ for Lorentzian ## GR - the bulk symplectic potential under diffeomorphism: $\Theta_{\left(Y^{-1} \circ X\right)(\sigma)}\left[Y^{*} g, \delta Y^{*} g\right]=\Theta_{\Sigma}[g, \delta g]+\int_{S} \pi_{g}\left[\delta_{Y}\right]$ - the boundary symplectic potential under diffeomorphism: $\Theta_{Y^{-1}(S)}\left[Y^{*} g, Y^{-1} \circ X, \delta\left(Y^{-1} \circ X\right)\right]=\int_{S} \pi_{g}\left[\delta_{X}-\delta_{Y}\right]$ - => the extra pieces cancel # Du, Teng, Wu ## Direct Evaluation of n-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY Formalism \[Links: [arXiv](https://arxiv.org/abs/1608.00883), [PDF](https://arxiv.org/pdf/1608.00883.pdf)\] \[Abstract: In this paper we extend our techniques, developed in a previous paper (Du, etc, JHEP 05(2016)086) for direct evaluation of arbitrary $n$-point tree-level [[0061 Maximally helicity violating amplitudes|MHV]] amplitudes in 4d Yang-Mills and gravity theory using the [[0543 Cachazo-He-Yuan formalism|Cachazo-He-Yuan (CHY) formalism]], to the 4d Einstein-Yang-Mills (EYM) theory. Any single-trace color-ordered $n$-point tree-level MHV amplitude in EYM theory, obtained by a direct evaluation of the CHY formula, is of an elegant factorized form of a [[0072 Parke-Taylor n-gluon tree amplitude|Parke-Taylor]] factor and a Hodges determinant, much simpler and more compact than the existing formulas in the literature. We prove that our new expression is equivalent to the conjectured Selivanov-Bern-De Freitas-Wong (SBDW) formula, with the help of a new theorem showing that the SBDW generating function has a graph theory interpretation. Together with Ref. (Du, etc, JHEP 05(2016)086), we provide strong analytic evidence for hidden simplicity in quantum field theory.\] # Elvang, Jones, Naculich ## Soft Photon and Graviton Theorems in Effective Field theory \[Links: [arXiv](https://arxiv.org/abs/1611.07534), [PDF](https://arxiv.org/pdf/1611.07534.pdf)\] \[Abstract: Extensions of the photon and graviton [[0009 Soft theorems|soft theorems]] are derived in 4d local effective field theories with massless particles of arbitrary spin. We prove that effective operators can result in new terms in the soft theorems at subleading order for photons and subsubleading order for gravitons. The new soft terms are unique and we provide a complete classification of all local operators responsible for such modifications. We show that no local operators can modify the subleading soft graviton theorem. The soft limits are taken in a manifestly on-locus manner using a complex double deformation of the external momenta. In addition to the new soft theorems, the resulting master formula yields consistency conditions such as the conservation of electric charge, the Einstein equivalence principle, supergravity Ward identities, and the Weinberg-Witten theorem.\] ## Comments - corrections to subleading soft photon and subsubleading soft graviton theorems studied in this paper match the corresponding CCFT expressions in [[2021#Jiang (Aug)]] ## Summary - *extends* photon and graviton [[0009 Soft theorems|soft theorems]] to ==EFT with massless particles of arbitrary spin== - soft photon theorem: corrected at subleading order - soft graviton theorem: corrected at subsubleading order - only operators with 3-point interactions can affect the single-particle soft theorems - because having too many derivatives makes its interaction too soft to matter - e.g. $F^3$ does not affect the leading and subleading soft theorems (similarly for $R^3$) ## Soft photon theorem in EFT - $A_{n+1}^{\mathrm{ph}}=\left(\frac{S^{(0)}}{\epsilon^2}+\frac{S^{(1)}}{\epsilon}\right) A_n+\frac{\tilde{S}^{(1)}}{\epsilon} \tilde{A}_n+O(\epsilon)$ ## Soft graviton theorem in EFT - $A_{n+1}^{\mathrm{grav}}=\left(\frac{\mathcal{S}^{(0)}}{\epsilon^3}+\frac{\mathcal{S}^{(1)}}{\epsilon^2}+\frac{\mathcal{S}^{(2)}}{\epsilon}\right) A_n+\frac{\tilde{\mathcal{S}}^{(2)}}{\epsilon} \tilde{A}_n+O(\epsilon)$ - where $\tilde{\mathcal{S}}^{(2)} \tilde{A}_n=\sum_k g_k \frac{[s k]^3}{\langle s k\rangle} \tilde{A}_n^{(k)}$ - no corrections at leading and subleading orders - the subsubleading correction is not universal: it depends on the momenta of the legs in the remaining diagram - operators that give corrections: - $\phi R_{\mu \nu \rho \sigma} R^{\mu \nu \rho \sigma}, \quad R^{\mu \nu \rho \sigma} \bar{\psi}_{\rho} \gamma_{\mu \nu} \partial_{\sigma} \chi, \quad R_{\mu \nu \rho \sigma} F^{\mu \nu} F^{\rho \sigma}$ # Engelhardt, Fischetti ## The gravity dual of boundary causality \[Links: [arXiv](https://arxiv.org/abs/1604.03944), [PDF](https://arxiv.org/pdf/1604.03944.pdf)\] \[Abstract: In [[0001 AdS-CFT|gauge/gravity duality]], points which are not causally related on the boundary cannot be causally related through the bulk; this is the statement of [[0091 Boundary causality|boundary causality]]. By the [[0477 Gao-Wald theorem|Gao-Wald theorem]], the [[0417 Averaged null energy condition|averaged null energy condition]] in the bulk is sufficient to ensure this property. Here we proceed in the converse direction: we derive a necessary as well as sufficient condition for the preservation of boundary causality under perturbative (quantum or stringy) corrections to the bulk. The condition that we find is a (background-dependent) constraint on the amount by which light cones can "open" over all null bulk geodesics. We show that this constraint is weaker than the averaged null energy condition.\] ## Summary - derive a *necessary and sufficient* condition for [[0091 Boundary causality|BCC]] under *perturbative* corrections (quantum or stringy) to the bulk - result: a perturbation to AdS must cause light cones to close when averaged over along any complete null geodesic ## [[0477 Gao-Wald theorem|Gao-Wald]] - two points are causally related through the bulk only if they are causally related on the boundary - requires ANCC, i.e. ANCC => BCC ## Result - **averaged light cone tilt** along $\gamma$: $I(\gamma) \equiv \int_{\gamma} \delta g_{a b} k^{a} k^{b} d \lambda$ - $I(\gamma)\ge0$ for all $\gamma$ ## Relation to other conditions ![[EngelhardtFischetti2016_relations.png]] # Engelhardt, Horowitz (May) ## Towards a Reconstruction of General Bulk Metrics \[Links: [arXiv](https://arxiv.org/abs/1605.01070), [PDF](https://arxiv.org/pdf/1605.01070)\] \[Abstract: We prove that the metric of a general holographic spacetime can be [[0026 Bulk reconstruction|reconstructed]] (up to an overall conformal factor) from distinguished spatial slices - "[[0027 Bulk reconstruction using lightcone cuts|light-cone cuts]]" - of the conformal boundary. Our prescription is covariant and applies to bulk points in causal contact with the boundary. Furthermore, we describe a procedure for determining the light-cone cuts corresponding to bulk points in the causal wedge of the boundary in terms of the divergences of correlators in the dual field theory. Possible extensions for determining the conformal factor and including the cuts of points outside of the causal wedge are discussed. We also comment on implications for [[0219 Entanglement wedge reconstruction|subregion-subregion duality]].\] # Engelhardt, Horowitz (Dec) ## Recovering the spacetime metric from a holographic dual \[Links: [arXiv](https://arxiv.org/abs/1612.00391), [PDF](https://arxiv.org/pdf/1612.00391)\] \[Abstract: We review our recent proposal to use certain spatial cross-sections of the boundary at infinity -- [[0027 Bulk reconstruction using lightcone cuts|light-cone cuts]] -- to recover the conformal metric in the bulk. We discuss some extensions of this work, including how to obtain the conformal metric near the horizon of a collapsing black hole. We also show how to obtain the conformal factor under certain conditions.\] # Engelsoy, Mertens, Verlinde ## An Investigation of AdS$_2$ Backreaction and Holography \[Links: [arXiv](https://arxiv.org/abs/1606.03438), [PDF](https://arxiv.org/pdf/1606.03438)\] \[Abstract: We investigate a [[0050 JT gravity|dilaton gravity model]] in AdS$_2$ proposed by Almheiri and Polchinski and develop a 1d effective description in terms of a dynamical boundary time with a Schwarzian derivative action. We show that the effective model is equivalent to a 1d version of [[0562 Liouville theory|Liouville theory]], and investigate its dynamics and symmetries via a standard canonical framework. We include the coupling to arbitrary conformal matter and analyze the effective action in the presence of possible sources. We compute commutators of local operators at large time separation, and match the result with the time shift due to a gravitational [[0117 Shockwave|shockwave]] interaction. We study a black hole evaporation process and comment on the role of entropy in this model.\] # Faulkner, Leigh, Parrikar, Wang ## Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition \[Links: [arXiv](https://arxiv.org/abs/1605.08072), [PDF](https://arxiv.org/pdf/1605.08072.pdf)\] \[Abstract: We study [[0416 Modular Hamiltonian|modular Hamiltonians]] corresponding to the vacuum state for deformed half-spaces in relativistic quantum field theories on $\mathbb{R}^{1,d-1}$. We show that in addition to the usual boost generator, there is a contribution to the modular Hamiltonian at first order in the shape deformation, proportional to the integral of the null components of the stress tensor along the Rindler horizon. We use this fact along with monotonicity of [[0199 Relative entropy|relative entropy]] to prove the [[0417 Averaged null energy condition|averaged null energy condition]] in Minkowski space-time. This subsequently gives a new proof of the [[0493 Conformal collider bounds|Hofman-Maldacena bounds]] on the parameters appearing in CFT three-point functions. Our main technical advance involves adapting newly developed perturbative methods for calculating [[0301 Entanglement entropy|entanglement entropy]] to the problem at hand. These methods were recently used to prove certain results on the shape dependence of entanglement in CFTs and here we generalize these results to excited states and real time dynamics. We also discuss the [[0001 AdS-CFT|AdS/CFT]] counterpart of this result, making connection with the recently proposed gravitational dual for modular Hamiltonians in holographic theories.\] # Fischetti, Hickling, Wiseman ## Bounds on the local energy density of holographic CFTs from bulk geometry \[Links: [arXiv](https://arxiv.org/abs/1605.00007), [PDF](https://arxiv.org/pdf/1605.00007.pdf)\] \[Abstract: The stress tensor is a basic local operator in any field theory; in the context of [[0001 AdS-CFT|AdS/CFT]], it is the operator which is dual to the bulk geometry itself. Here we exploit this feature by using the bulk geometry to place constraints on the local energy density in static states of holographic (2+1)-dimensional CFTs living on a closed (but otherwise generally curved) spatial geometry. We allow for the presence of a marginal scalar deformation, dual to a massless scalar field in the bulk. For certain vacuum states in which the bulk geometry is well-behaved at zero temperature, we find that the bulk equations of motion imply that the local energy density integrated over specific boundary domains is negative. In the absence of scalar deformations, we use the inverse mean curvature flow to show that if the CFT spatial geometry has spherical topology but non-constant curvature, the local energy density must be positive somewhere. This result extends to other topologies, but only for certain types of vacuum; in particular, for a generic toroidal boundary, the vacuum's bulk dual must be the zero-temperature limit of a toroidal black hole.\] ## Summary - *shows* that certain integrals of the energy density are bounded below by geometric data at *bulk horizons* ## Setting - deform the CFT by a static but otherwise general ==scalar== operator - dual to massless bulk scalar - boundary is 3d: 2d arbitrary spatial metric cross time (static) # Fu, Gaiotto, Maldacena, Sachdev ## Supersymmetric SYK models \[Links: [arXiv](https://arxiv.org/abs/1610.08917), [PDF](https://arxiv.org/pdf/1610.08917.pdf)\] \[Abstract: We discuss a [[0359 Supersymmetry|supersymmetric]] generalization of the [[0201 Sachdev-Ye-Kitaev model|Sachdev-Ye-Kitaev model]]. These are quantum mechanical models involving $N$ Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The ${\cal N}=1$ model with a single supercharge has unbroken supersymmetry at large $N$, but non-perturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large $N$ equation, and also by performing numerical computations for small values of $N$. We also compute the large $N$ spectrum of "singlet" operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an ${\cal N}=2$ version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large $N$ computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.\] # Gao, Jafferis, Wall ## Traversable Wormholes via a Double Trace Deformation \[Links: [arXiv](https://arxiv.org/abs/1608.05687), [PDF](https://arxiv.org/pdf/1608.05687.pdf)\] \[Abstract: After turning on an interaction that couples the two boundaries of an eternal [[0086 Banados-Teitelboim-Zanelli black hole|BTZ]] black hole, we find a quantum matter stress tensor with negative average null energy, whose gravitational backreaction renders the Einstein-Rosen bridge traversable. Such a traversable wormhole has an interesting interpretation in the context of [[0220 ER=EPR|ER=EPR]], which we suggest might be related to quantum teleportation. However, it cannot be used to violate [[0091 Boundary causality|causality]]. We also discuss the implications for the energy and holographic entropy in the dual CFT description.\] ## Refs - [[0083 Traversable wormhole]] - [[0451 GJW wormhole]] # Gralla, Zimmerman, Zimmerman ## Transient Instability of Rapidly Rotating Black Holes \[Links: [arXiv](https://arxiv.org/abs/1608.04739), [PDF](https://arxiv.org/pdf/1608.04739.pdf)\] \[Abstract: We analytically study the linear response of a near-extremal Kerr black hole to external scalar, electromagnetic, and gravitational field perturbations. We show that the energy density, electromagnetic field strength, and tidal force experienced by infalling observers exhibit transient growth near the horizon. The growth lasts arbitrarily long in the extremal limit, reproducing the horizon instability of extremal Kerr. We explain these results in terms of [[0561 Near-horizon symmetry|near-horizon geometry]] and discuss potential astrophysical implications.\] ## Summary - near-extremal version of [[0340 Aretakis instability|Aretakis instability]] - analogous story for RN: [[2016#Zimmerman]] # Grozdanov, Kaplis, Starinets ## From strong to weak coupling in holographic models of thermalization \[Links: [arXiv](https://arxiv.org/abs/1605.02173), [PDF](https://arxiv.org/pdf/1605.02173.pdf)\] \[Abstract: \] ## Summary - [[0425 Gauss-Bonnet gravity|GB gravity]] correction to [[0325 Quasi-normal modes|QNM]] including - [[0430 Holographic shear viscosity]] - [[0445 Hod's universal relaxation bound]] - QNM spectrum (qualitative difference) # Grumiller, Riegler ## Most general AdS$_3$ boundary conditions \[Links: [arXiv](https://arxiv.org/abs/1608.01308), [PDF](https://arxiv.org/pdf/1608.01308)\] \[Abstract: We consider the most general asymptotically anti-de Sitter boundary conditions in [[0002 3D gravity|three-dimensional Einstein gravity]] with negative cosmological constant. The metric contains in total twelve independent functions, six of which are interpreted as chemical potentials (or non-normalizable fluctuations) and the other half as canonical boundary charges (or normalizable fluctuations). Their presence modifies the usual [[0011 Fefferman-Graham expansion|Fefferman-Graham expansion]]. The [[0085 Asymptotic symmetry of AdS3|asymptotic symmetry]] algebra consists of two $sl(2)_k$ current algebras, the levels of which are given by $k=l/(4G_N)$, where $l$ is the AdS radius and $G_N$ the three-dimensional Newton constant.\] # Gu, Qi, Stanford ## Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models \[Links: [arXiv](https://arxiv.org/abs/1609.07832), [PDF](https://arxiv.org/pdf/1609.07832.pdf)\] \[Abstract: The [[0201 Sachdev-Ye-Kitaev model|Sachdev-Ye-Kitaev model]] is a (0+1)-dimensional model describing Majorana fermions or complex fermions with random interactions. This model has various interesting properties such as approximate local criticality (power law correlation in time), zero temperature entropy, and [[0008 Quantum chaos|quantum chaos]]. In this article, we propose a higher dimensional generalization of the Sachdev-Ye-Kitaev model, which is a lattice model with $N$ Majorana fermions at each site and random interactions between them. Our model can be defined on arbitrary lattices in arbitrary spatial dimensions. In the large $N$ limit, the higher dimensional model preserves many properties of the Sachdev-Ye-Kitaev model such as local criticality in two-point functions, zero temperature entropy and chaos measured by the [[0482 Out-of-time-order correlator|out-of-time-ordered correlation functions]]. In addition, we obtain new properties unique to higher dimensions such as diffusive energy transport and a "[[0167 Butterfly velocity|butterfly velocity]]" describing the propagation of chaos in space. We mainly present results for a (1+1)-dimensional example, and discuss the general case near the end.\] # Gubser, Knaute, Parikh, Samberg, Witaszczyk ## $p$-adic AdS/CFT \[Links: [arXiv](https://arxiv.org/abs/1605.01061), [PDF](https://arxiv.org/pdf/1605.01061.pdf)\] \[Abstract: We construct a [[0084 p-adic holography|p-adic]] analog to AdS/CFT, where an unramified extension of the $p$-adic numbers replaces Euclidean space as the boundary and a version of the Bruhat-Tits tree replaces the bulk. Correlation functions are computed in the simple case of a single massive scalar in the bulk, with results that are strikingly similar to ordinary holographic correlation functions when expressed in terms of local zeta functions. We give some brief discussion of the geometry of $p$-adic chordal distance and of Wilson loops. Our presentation includes an introduction to $p$-adic numbers.\] # Harlow ## The Ryu-Takayanagi Formula from Quantum Error Correction \[Links: [arXiv](https://arxiv.org/abs/1607.03901), [PDF](https://arxiv.org/pdf/1607.03901.pdf)\] \[Abstract: I argue that a version of the quantum-corrected [[0007 RT surface|Ryu-Takayanagi formula]] holds in any [[0146 Quantum error correction|quantum error-correcting code]]. I present this result as a series of theorems of increasing generality, with the final statement expressed in the language of operator-algebra quantum error correction. In [[0001 AdS-CFT|AdS/CFT]] this gives a "purely boundary" interpretation of the formula. I also extend a recent theorem, which established [[0219 Entanglement wedge reconstruction|entanglement-wedge reconstruction]] in AdS/CFT, when interpreted as a subsystem code, to the more general, and I argue more physical, case of subalgebra codes. For completeness, I include a self-contained presentation of the theory of [[0415 Von Neumann algebra|von Neumann algebras]] on finite-dimensional Hilbert spaces, as well as the algebraic definition of entropy. The results confirm a close relationship between bulk gauge transformations, [[0556 Edge mode|edge-modes]]/[[0459 Soft hair|soft-hair]] on black holes, and the Ryu-Takayanagi formula. They also suggest a new perspective on the homology constraint, which basically is to get rid of it in a way that preserves the validity of the formula, but which removes any tension with the linearity of quantum mechanics. Moreover they suggest a boundary interpretation of the "[[0211 Bit thread|bit threads]]" recently introduced by Freedman and Headrick.\] ## Summary - *argues* that a version of quantum-corrected [[0007 RT surface|RT]] holds in any [[0146 Quantum error correction|QEC]] code ## Refs - generalised by [[2021#Akers, Penington]] # Hartman, Jain, Kundu ## A New Spin on Causality Constraints \[Links: [arXiv](https://arxiv.org/abs/1601.07904), [PDF](https://arxiv.org/pdf/1601.07904.pdf)\] \[Abstract: Causality in a [[0117 Shockwave|shockwave]] state is related to the analytic properties of a four-point correlation function. Extending recent results for scalar probes, we show that this constrains the couplings of the stress tensor to light spinning operators in conformal field theory, and interpret these constraints in terms of the interaction with null energy. For spin-1 and spin-2 conserved currents in four dimensions, the resulting inequalities are a subset of the [[0493 Conformal collider bounds|Hofman-Maldacena]] conditions for positive energy deposition. It is well known that energy conditions in holographic theories are related to [[0118 Causality constraints for gravity|causality]] on the gravity side; our results make a connection on the CFT side, and extend it to non-holographic theories.\] # Hartman, Kundu, Tajdini ## Averaged Null Energy Condition from Causality \[Links: [arXiv](https://arxiv.org/abs/1610.05308), [PDF](https://arxiv.org/pdf/1610.05308.pdf)\] \[Abstract: Unitary, Lorentz-invariant quantum field theories in flat spacetime obey microcausality: commutators vanish at spacelike separation. For ==interacting theories in more than two dimensions==, we show that this implies that the [[0417 Averaged null energy condition|averaged null energy]], $\int du\, T_{uu}$, must be positive. This non-local operator appears in the [[0030 Operator product expansion|operator product expansion]] of local operators in the lightcone limit, and therefore contributes to $n$-point functions. We derive a sum rule that isolates this contribution and is manifestly positive. The argument also applies to certain higher spin operators other than the stress tensor, generating an infinite family of new constraints of the form $\int du X_{uuu\cdots u} \geq 0$. These lead to new inequalities for the coupling constants of spinning operators in conformal field theory, which include as special cases (but are generally stronger than) the existing constraints from the lightcone bootstrap, deep inelastic scattering, [[0493 Conformal collider bounds|conformal collider]] methods, and [[0199 Relative entropy|relative entropy]]. We also comment on the relation to the recent derivation of the averaged null energy condition from relative entropy, and suggest a more general connection between causality and information-theoretic inequalities in QFT.\] # He, Zhang ## New Formulas for Amplitudes from Higher-Dimensional Operators \[Links: [arXiv](https://arxiv.org/abs/1608.08448), [PDF](https://arxiv.org/pdf/1608.08448.pdf)\] \[Abstract: In this paper we study tree-level amplitudes from higher-dimensional operators, including $F^3$ operator of gauge theory, and $R^2$, $R^3$ operators of gravity, in the [[0543 Cachazo-He-Yuan formalism|Cachazo-He-Yuan formulation]]. As a generalization of the reduced Pfaffian in Yang-Mills theory, we find a new, gauge-invariant object that leads to gluon amplitudes with a single insertion of $F^3$, and gravity amplitudes by [[0398 KLT relations|Kawai-Lewellen-Tye relations]]. When reduced to four dimensions for given helicities, the new object vanishes for any solution of scattering equations on which the reduced Pfaffian is non-vanishing. This intriguing behavior in four dimensions explains the vanishing of graviton helicity amplitudes produced by the [[0425 Gauss-Bonnet gravity|Gauss-Bonnet]] $R^2$ term, and provides a scattering-equation origin of the decomposition into [[0234 Self-dual gravity|self-dual]] and anti-self-dual parts for $F^3$ and $R^3$ amplitudes.\] # Heydeman, Marcolli, Saberi, Stoica ## Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the AdS$_3$/CFT$_2$ correspondence \[Links: [arXiv](https://arxiv.org/abs/1605.07639), [PDF](https://arxiv.org/pdf/1605.07639.pdf)\] \[Abstract: One of the many remarkable properties of conformal field theory in two dimensions is its connection to algebraic geometry. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. This opens the door to interesting generalizations, obtained by taking another choice of field: for instance, the [[0084 p-adic holography|p-adics]]. We generalize the [[0001 AdS-CFT|AdS/CFT]] correspondence according to this principle; the result is a formulation of holography in which the bulk geometry is discrete---the Bruhat--Tits tree for $\mathrm{PGL}(2,\mathbb{Q}_p)$---but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. We suggest that this forms the natural geometric setting for [[0054 Tensor network|tensor networks]] that have been proposed as models of [[0026 Bulk reconstruction|bulk reconstruction]] via [[0146 Quantum error correction|quantum error correcting codes]]; in certain cases, geodesics in the Bruhat--Tits tree reproduce those constructed using quantum error correction. Other aspects of holography also hold: Standard holographic results for massive free scalar fields in a fixed background carry over to the tree, whose vertical direction can be interpreted as a renormalization-group scale for modes in the boundary CFT. Higher-genus [[0002 3D gravity|bulk geometries]] (the [[0086 Banados-Teitelboim-Zanelli black hole|BTZ]] black hole and its generalizations) can be understood straightforwardly in our setting, and the [[0007 RT surface|Ryu-Takayanagi formula]] for the [[0301 Entanglement entropy|entanglement entropy]] appears naturally.\] # Hickling (Thesis) ## Bulk space-time geometries in AdS/CFT \[Links: [Inspire](https://inspirehep.net/literature/1756146)\] \[Abstract: \] ## Comments - bdry $R_t \times S^2\times S^2$ and $R_t \times S^1\times S^3$ for [[0231 Bulk solutions for CFTs on non-trivial geometries]] - see sec. 7.4 - BUT only for BH solutions, not bubbles - I suspect that you can get bubble by analytic continuation in the first case but not the second (need to swap the role of time circle with a spatial circle) # Hofman, Li, Meltzer, Poland, Rejon-Barrera ## A Proof of the Conformal Collider Bounds \[Links: [arXiv](https://arxiv.org/abs/1603.03771), [PDF](https://arxiv.org/pdf/1603.03771.pdf)\] \[Abstract: In this paper, we prove that the "[[0493 Conformal collider bounds|conformal collider bounds]]" originally proposed by Hofman and Maldacena hold for any unitary parity-preserving conformal field theory (CFT) with a unique stress tensor in spacetime dimensions larger than 2. In particular this implies that the ratio of [[0033 Central charge|central charges]] for a unitary 4d CFT lies in the interval $\frac{31}{18} \geq \frac{a}{c} \geq \frac{1}{3}$. For superconformal theories this is further reduced to $\frac{3}{2} \geq \frac{a}{c} \geq \frac{1}{2}$. The proof relies only on CFT first principles - in particular, bootstrap methods - and thus constitutes the first complete field theory proof of these bounds. We further elaborate on similar bounds for non-conserved currents and relate them to results obtained recently from deep inelastic scattering.\] # Jafferis, Lewkowycz, Maldacena, Suh ## Relative entropy equals bulk relative entropy \[Links: [arXiv](https://arxiv.org/abs/1512.06431), [PDF](https://arxiv.org/pdf/1512.06431.pdf)\] \[Abstract: \] ## Refs - [[0145 Generalised area|HEE]] - original paper of [[0048 JLMS]] ## Summary - to leading order in $G_\text{N}$, the [[0199 Relative entropy]] of nearby states is given by relative entropy in the bulk - the boundary modular flow is dual to bulk modular flow in the entanglement wedge - implications for [[0219 Entanglement wedge reconstruction]] # Jensen ## Chaos in AdS$_2$ holography \[Links: [arXiv](https://arxiv.org/abs/1605.06098), [PDF](https://arxiv.org/pdf/1605.06098)\] \[Abstract: We revisit AdS$_2$ holography with the [[0201 Sachdev-Ye-Kitaev model|SYK]] models in mind. Our main result is to rewrite a generic theory of gravity near an AdS$_2$ throat as a novel hydrodynamics coupled to the correlation functions of a conformal quantum mechanics. This gives a prescription for the computation of $n$-point functions in the dual quantum mechanics. We thereby find that the dual is maximally chaotic.\] # Jubb, Samuel, Sorkin, Surya ## Boundary and corner terms in the action for GR \[Links: [arXiv](https://arxiv.org/abs/1612.00149), [PDF](https://arxiv.org/pdf/1612.00149.pdf)\] \[Abstract: \] # Kapec, Mitra, Raclariu, Strominger ## A 2D Stress Tensor for 4D Gravity \[Links: [arXiv](https://arxiv.org/abs/1609.00282), [PDF](https://arxiv.org/pdf/1609.00282.pdf)\] \[Abstract: \] ## Summary - establishes that [[0009 Soft theorems|subleading soft graviton]] theorem gives a [[0032 Virasoro algebra]] on [[0022 Celestial sphere]] a la [[2015#He, Mitra, Strominger]] - using [[0060 Asymptotic symmetry]] methods ## Comments - the OPE between 2 gravitons does not give you the Virasoro commutator but a the OPE between stress tensor and graviton does, which is what this paper talks about <!-- comment due to #alfredoguevara ---> # Kaufman, Tai, Lukin, Rispoli, Schittko, Preiss, Greiner ## Quantum thermalization through entanglement in an isolated many-body system \[Links: [arXiv](https://arxiv.org/abs/1603.04409), [PDF](https://arxiv.org/pdf/1603.04409.pdf), [Science](https://www.science.org/doi/10.1126/science.aaf6725)\] \[Abstract: The concept of entropy is fundamental to thermalization, yet appears at odds with basic principles in quantum mechanics. Statistical mechanics relies on the maximization of entropy for a system at thermal equilibrium. However, an isolated many-body system initialized in a pure state will remain pure during Schrödinger evolution, and in this sense has static, zero entropy. The underlying role of quantum mechanics in many-body physics is then seemingly antithetical to the success of statistical mechanics in a large variety of systems. Here we experimentally study the emergence of statistical mechanics in a quantum state, and observe the fundamental role of quantum entanglement in facilitating this emergence. We perform microscopy on an evolving quantum system, and we see thermalization occur on a local scale, while we measure that the full quantum state remains pure. We directly measure [[0301 Entanglement entropy|entanglement entropy]] and observe how it assumes the role of the thermal entropy in thermalization. Although the full state remains measurably pure, entanglement creates local entropy that validates the use of statistical physics for local observables. In combination with number-resolved, single-site imaging, we demonstrate how our measurements of a pure quantum state agree with the [[0040 Eigenstate thermalisation hypothesis|Eigenstate Thermalization Hypothesis]] and thermal ensembles in the presence of a near-volume law in the entanglement entropy.\] ## The lesson - quantum entanglement drives thermalisation - even though the entire system remain pure, small sets of atoms look thermal ## The system - strings of six rubidium atoms # Krasnov (Review) ## Self-Dual Gravity \[Links: [arXiv](https://arxiv.org/abs/1610.01457), [PDF](https://arxiv.org/pdf/1610.01457.pdf)\] \[Abstract: Self-dual gravity is a diffeomorphism invariant theory in four dimensions that describes two propagating polarisations of the graviton and has a negative mass dimension coupling constant. Nevertheless, this theory is not only renormalisable but quantum finite, as we explain. We also collect various facts about self-dual gravity that are scattered across the literature.\] ## Summary - reviews mainly self-dual gravity and why it is renormalisable and quantum finite - discusses [[0136 Self-dual Yang-Mills|SDYM]] too ## Features of self-dual gravity - only known example of a consistent quantum *pure* gravity - needs either Riemannian or split signature # Kraus, Maloney ## A Cardy Formula for Three-Point Coefficients: How the Black Hole Got its Spots \[Links: [arXiv](https://arxiv.org/abs/1608.03284), [PDF](https://arxiv.org/pdf/1608.03284.pdf)\] \[Abstract: [[0612 Modular invariance|Modular covariance]] of torus one-point functions constrains the three point function coefficients of a two dimensional CFT. This leads to an asymptotic formula for the average value of light-heavy-heavy three point coefficients, generalizing Cardy's formula for the high energy density of states. The derivation uses certain asymptotic properties of one-point conformal blocks on the torus. Our asymptotic formula matches a dual AdS$_3$ computation of one point functions in a black hole background. This is evidence that the BTZ black hole geometry emerges upon course-graining over a suitable family of heavy microstates.\] # Kubiznak, Mann, Teo ## Black hole chemistry: thermodynamics with Lambda \[Links: [arXiv](https://arxiv.org/abs/1608.06147), [PDF](https://arxiv.org/pdf/1608.06147.pdf)\] \[Abstract: We review recent developments on the thermodynamics of black holes in extended phase space, where the cosmological constant is interpreted as thermodynamic pressure and treated as a thermodynamic variable in its own right. In this approach, the mass of the black hole is no longer regarded as internal energy, rather it is identified with the chemical enthalpy. This leads to an extended dictionary for black hole thermodynamic quantities, in particular a notion of thermodynamic volume emerges for a given black hole spacetime. This volume is conjectured to satisfy the reverse isoperimetric inequality - an inequality imposing a bound on the amount of entropy black hole can carry for a fixed thermodynamic volume. New thermodynamic phase transitions naturally emerge from these identifications. Namely, we show that black holes can be understood from the viewpoint of chemistry, in terms of concepts such as Van der Waals fluids, reentrant phase transitions, and triple points. We also review the recent attempts at extending the AdS/CFT dictionary in this setting, discuss the connections with horizon thermodynamics, applications to Lifshitz spacetimes, and outline possible future directions in this field.\] ## Refs - [[0525 Black hole chemistry]] # Lewkowycz, Turiaci, Verlinde ## A CFT Perspective on Gravitational Dressing and Bulk Locality \[Links: [arXiv](https://arxiv.org/abs/1608.08977), [PDF](https://arxiv.org/pdf/1608.08977)\] \[Abstract: We revisit the construction of local bulk operators in AdS/CFT with special focus on [[0180 Dressing|gravitational dressing]] and its consequences for bulk locality. Specializing to 2+1-dimensions, we investigate these issues via the proposed identification between bulk operators and [[0620 Non-orientable CFT|cross-cap boundary states]]. We obtain explicit expressions for correlation functions of bulk fields with boundary stress tensor insertions, and find that they are free of non-local branch cuts but do have non-local poles. We recover the [[0016 HKLL|HKLL]] recipe for restoring bulk locality for interacting fields as the outcome of a natural CFT crossing condition. We show that, in a suitable gauge, the cross-cap states solve the bulk wave equation for general background geometries, and satisfy a conformal Ward identity analogous to a [[0009 Soft theorems|soft graviton theorem]], [[0032 Virasoro algebra|Virasoro symmetry]], the large $N$ conformal bootstrap and the uniformization theorem all play a key role in our derivations.\] # Maldacena, Stanford ## Comments on the Sachdev-Ye-Kitaev model \[Links: [arXiv](https://arxiv.org/abs/1604.07818), [PDF](https://arxiv.org/pdf/1604.07818.pdf)\] \[Abstract: We study a quantum mechanical model proposed by [[0201 Sachdev-Ye-Kitaev model|Sachdev, Ye and Kitaev]]. The model consists of $N$ Majorana fermions with random interactions of a few fermions at a time. It it tractable in the large $N$ limit, where the classical variable is a bilocal fermion bilinear. The model becomes strongly interacting at low energies where it develops an emergent conformal symmetry. We study two and four point functions of the fundamental fermions. This provides the spectrum of physical excitations for the bilocal field. The emergent conformal symmetry is a reparametrization symmetry, which is spontaneously broken to $SL(2,R)$, leading to zero modes. These zero modes are lifted by a small residual explicit breaking, which produces an enhanced contribution to the four point function. This contribution displays a maximal [[0466 Lyapunov exponent|Lyapunov exponent]] in the chaos region ([[0482 Out-of-time-order correlator|out of time ordered correlator]]). We expect these features to be universal properties of large $N$ quantum mechanics systems with emergent reparametrization symmetry. This article is largely based on talks given by Kitaev, which motivated us to work out the details of the ideas described there.\] # Maldacena, Stanford, Yang ## Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space \[Links: [arXiv](https://arxiv.org/abs/1606.01857), [PDF](https://arxiv.org/pdf/1606.01857.pdf)\] \[Abstract: We study a two dimensional dilaton gravity system, recently examined by Almheiri and Polchinski, which describes near extremal black holes, or more generally, nearly AdS$_2$ spacetimes. The [[0060 Asymptotic symmetry|asymptotic symmetries]] of AdS$_2$ are all the time reparametrizations of the boundary. These symmetries are spontaneously broken by the AdS$_2$ geometry and they are explicitly broken by the small deformation away from AdS$_2$. This pattern of spontaneous plus explicit symmetry breaking governs the gravitational backreaction of the system. It determines several gravitational properties such as the linear in temperature dependence of the near extremal entropy as well as the gravitational corrections to correlation functions. These corrections include the ones determining the growth of [[0482 Out-of-time-order correlator|out of time order correlators]] that is indicative of [[0008 Quantum chaos|chaos]]. These gravitational aspects can be described in terms of a Schwarzian derivative effective action for a reparametrization.\] # Maloney, Ross ## Holography on Non-Orientable Surfaces \[Links: [arXiv](https://arxiv.org/abs/1603.04426), [PDF](https://arxiv.org/pdf/1603.04426)\] \[Abstract: We consider the holographic computation of [[0003 2D CFT|two dimensional conformal field theory]] partition functions on [[0620 Non-orientable CFT|non-orientable surfaces]]. We classify the three dimensional geometries that give bulk saddle point contributions to the partition function, and find that there are fewer saddles than in the orientable case. For example, for the Klein bottle there is a single smooth saddle and a single additional saddle with an orbifold singularity. We argue that one must generally include singular bulk saddle points in order to reproduce the CFT results. We also discuss loop corrections to these partition functions for the Klein bottle.\] # McLoughlin, Nandan ## Multi-soft gluon limits and extended current algebras at null-infinity \[Links: [arXiv](https://arxiv.org/abs/1610.03841), [PDF](https://arxiv.org/pdf/1610.03841.pdf)\] \[Abstract: \] ## Refs - current algebra for [[0069 Kac-Moody algebra]] ## Notation - $\mathcal{J}$ for the 4d expression - $J$ for the 2d expression ## Mode expansion - $\begin{aligned}\left(\hat{L}_{-n} \Phi\right)(u) &=\oint \frac{d z}{2 \pi i} \frac{1}{(z-u)^{n-1}} T(z) \Phi(u) \\\left(\hat{J}_{-n}^{a} \Phi\right)(u) &=\oint \frac{d z}{2 \pi i} \frac{1}{(z-u)^{n}} J^{a}(z) \Phi(u) \end{aligned}$ ## Anti-holomorphic - $J^{a}\left(z_{1}\right) \bar{J}^{b}\left(z_{2}\right) \sim i {f^{a b}}_c\left[d_{1} \frac{\bar{J}^{c}\left(z_{2}\right)}{z_{12}}-d_{2} \frac{J^{c}\left(z_{2}\right)}{\bar{z}_{12}}-d_{2} \frac{z_{12}}{\bar{z}_{12}} \partial J^{c}\left(z_{2}\right)\right]$ ## Subleading - $J^{a}\left(z_{1}\right) J_{\mathrm{sub}}^{b}\left(z_{2} ; \omega_{z_{2}}\right) \sim \frac{i c_{1}^{s} {f^{a b}}_c J_{\mathrm{sub}}^{c}\left(z_{2} ; \omega_{z_{2}}\right)}{z_{12}}$ - $J_{\mathrm{sub}}^{a}\left(z_{1} ; \omega_{z_{1}}\right) J_{\mathrm{sub}}^{b}\left(z_{2} ; \omega_{z_{2}}\right) \sim \frac{i {f^{a b}}_c J_{\mathrm{sub}}^{c}\left(z_{2} ; c_{2}^{s s} \omega_{z_{1}}+c_{1}^{s s} \omega_{z_{2}}\right)}{z_{12}}$ - $\bar{J}^{a}\left(z_{1}\right) J_{\mathrm{sub}}^{b}\left(z_{2} ; \omega_{z_{2}}\right) \sim i {f^{a b}}_{c} \frac{\bar{J}_{\mathrm{sub}}^{c}\left(z_{2} ; \omega_{z_{2}}\right)}{\bar{z}_{12}}$ - $J_{\mathrm{sub}}^{a}\left(z_{1} ; \omega_{z_{1}}\right) \bar{J}_{\mathrm{sub}}^{b}\left(z_{2} ; \omega_{z_{2}}\right) \sim i {f^{a b}}_c \left[\frac{J_{\mathrm{sub}}^{c}\left(z_{2} ; d_{2}^{s} \omega_{z_{2}}\right)}{\bar{z}_{12}}-\frac{\bar{J}_{\mathrm{sub}}^{c}\left(z_{2} ; d_{1}^{s} \omega_{z_{1}}\right)}{z_{12}}\right.$ $\left.+\frac{\bar{z}_{12}}{z_{12}} \bar{\partial} \bar{J}_{\mathrm{sub}}^{c}\left(z_{2} ; d_{1}^{s} \omega_{z_{1}}\right)+2 \frac{z_{12}}{\bar{z}_{12}} \partial J_{\mathrm{sub}}^{c}\left(z_{2} ; d_{2}^{s} \omega_{z_{2}}\right)\right]$ # Mezei ## On entanglement spreading from holography \[Links: [arXiv](https://arxiv.org/abs/1612.00082), [PDF](https://arxiv.org/pdf/1612.00082.pdf)\] \[Abstract: A global quench is an interesting setting where we can study thermalization of subsystems in a pure state. We investigate [[0301 Entanglement entropy|entanglement entropy]] (EE) growth in global quenches in holographic field theories and relate some of its aspects to quantities characterizing [[0008 Quantum chaos|chaos]]. More specifically we obtain four key results: 1. We prove holographic bounds on the entanglement velocity $v_E$ and the [[0167 Butterfly velocity|butterfly effect speed]] $v_B$ that arises in the study of chaos. 2. We obtain the EE as a function of time for large spherical entangling surfaces analytically. We show that the EE is insensitive to the details of the initial state or quench protocol. 3. In a thermofield double state we determine analytically the two-sided [[0300 Mutual information|mutual information]] between two large concentric spheres separated in time. 4. We derive a bound on the rate of growth of EE for arbitrary shapes, and develop an expansion for EE at early times. In a companion paper [[2016#Mezei, Stanford]], we put these results in the broader context of EE growth in chaotic systems: we relate EE growth to the chaotic spreading of operators, derive bounds on EE at a given time, and compare the holographic results to spin chain numerics and toy models. In this paper, we perform holographic calculations that provide the basis of arguments presented in that paper.\] ## Summary - gives a bound on the [[0167 Butterfly velocity|butterfly velocity]] - studies arbitrary shapes, insensitivity to initial states, [[0300 Mutual information|mutual information]] etc # Mezei, Stanford ## On entanglement spreading in chaotic systems \[Links: [arXiv](https://arxiv.org/abs/1608.05101), [PDF](https://arxiv.org/pdf/1608.05101.pdf)\] \[Abstract: We discuss the time dependence of subsystem entropies in interacting quantum systems. As a model for the time dependence, we suggest that the entropy is as large as possible given two constraints: one follows from the existence of an emergent light cone, and the other is a conjecture associated to the "[[0327 Entanglement velocity|entanglement velocity]]" $v_E$. We compare this model to new holographic and spin chain computations, and to an operator growth picture. Finally, we introduce a second way of computing the emergent light cone speed in holographic theories that provides a boundary dynamics explanation for a special case of [[0219 Entanglement wedge reconstruction|entanglement wedge subregion duality]] in AdS/CFT.\] ## Refs - [[0167 Butterfly velocity]] ## Shockwave results - $\mathcal{L}=R+\Lambda_{\mathrm{GB}}\left(R_{\mu \nu \rho \sigma} R^{\mu \nu \rho \sigma}-4 R_{\mu \nu} R^{\mu \nu}+R^{2}\right)+\Lambda_{1} R^{2}+\Lambda_{2} R_{\mu \nu} R^{\mu \nu}$ - $h(x)\sim e^{-\mu|x|}$ - $\left(1+C_{1}\left(\Lambda_{1}, \Lambda_{2}\right)\right) \mu^{2}-\frac{B^{\prime}(0)}{A(0)}\left(\frac{d-1}{2}+C_{2}\left(\Lambda_{1}, \Lambda_{2}\right)\right)+\Lambda_{2} \mu^{4}=0$ - independent of $\Lambda_{GB}$: GB does not receive correction (in fact Lovelock in general receives no corrections: [[2022#Dong, Wang, Weng, Wu]]) # Mozaffar, Mollabashi, Sheikh-Jabbari, Vahidinia ## HEE, field redefinition invariance and higher derivative gravity \[Links: [arXiv](https://arxiv.org/abs/1603.05713), [PDF](https://arxiv.org/pdf/1603.05713.pdf)\] \[Abstract: It is established that physical observables in local quantum field theories should be invariant under invertible field redefinitions. It is then expected that this statement should be true for the entanglement entropy and moreover that, via the gauge/gravity correspondence, the recipe for computing entanglement entropy holographically should also be invariant under local field redefinitions in the gravity side. We use this fact to fix the recipe for computing [[0007 RT surface|holographic entanglement entropy]] (HEE) for $f(R,R_{\mu\nu})$ theories which could be mapped to Einstein gravity. An outcome of our prescription is that the surfaces that minimize the corresponding HEE functional for $f(R,R_{\mu\nu})$ theories always have vanishing trace of extrinsic curvature and that the HEE may be evaluated using the Wald entropy functional. We show that similar results follow from the FPS and [[2013#Dong|Dong]] HEE functionals, for Einstein manifold backgrounds in $f(R,R_{\mu\nu})$ theories.\] ## Remarks - did not obtain the correct results for $f(R_{ab})$. see below ## Summary - use field redefinition to find [[0145 Generalised area]] for $f(R, R_{ab})$ - result: the surface that minimises $S_{EE}$ always has $K=0$, and entropy is given by Wald entropy ## Field redefinition invariance - field redefinition should be viewed as a **change of basis** in the Hilbert space, and physical observables should be invariant under a change of basis - as long as it does not change the partition between two subregions, field redefinition should not change the entanglement entropy, which is a physical (though non-local) observable - boundary side: - on the boundary side it is also a change of basis, although there it is something complicated - we do not worry about the boundary, because we will change the basis back in the bulk and apply to the same boundary basis! ## Vanishing of K - see Sec. A.2 and Sec. B - they argue that $K_i=0$ is always a solution, and other solutions may not give the right answer - -> this is baseless and I believe it is not true ## Results - $S_{E E}=\frac{1}{4 G_{d+1}} \int_{\gamma_{A}} \sqrt{|g| \operatorname{det}\left(\mathcal{F}^{\mu \nu}\right) \operatorname{det}\left(\mathcal{F}_{a b}^{-1}\right)}$ where $\mathcal{F}^{\mu \nu} \equiv \frac{\partial f}{\partial R_{\mu \nu}}$ - to proceed further, have to either do perturbative higher derivative terms or restrict the special solutions - perturbative $S_{\mathrm{EE}}=\frac{1}{4 G_{d+1}} \int_{\gamma_{A}} d^{d-1} \xi \sqrt{\gamma}\left[1+\frac{\lambda}{2} \tilde{\mathcal{F}}^{\mu \nu} n_{\mu}^{(i)} n_{\nu}^{(i)}\right]$ - restrict to Einstein manifolds $S_{\mathrm{EE}}=\frac{\mathcal{X}}{4 G_{d+1}} \int_{\gamma_{A}} d^{d-1} \xi \sqrt{\gamma}$ where $\mathcal{X}=\frac{(d+1) f}{2 R}$ # Nahum, Ruhman, Vijay, Haah ## Quantum Entanglement Growth Under Random Unitary Dynamics \[Links: [arXiv](https://arxiv.org/abs/1608.06950), [PDF](https://arxiv.org/pdf/1608.06950.pdf)\] \[Abstract: Characterizing how entanglement grows with time in a many-body system, for example after a quantum quench, is a key problem in non-equilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time-dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the 'entanglement tsunami' in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated [[0524 Kardar-Parisi-Zhang equation|Kardar-Parisi-Zhang (KPZ)]] equation. The mean entanglement grows linearly in time, while fluctuations grow like $(\text{time})^{1/3}$ and are spatially correlated over a distance $\propto (\text{time})^{2/3}$. We derive KPZ universal behaviour in three complementary ways, by mapping random entanglement growth to: (i) a stochastic model of a growing surface; (ii) a 'minimal cut' picture, reminiscent of the [[0007 RT surface|Ryu-Takayanagi formula]] in holography; and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple 'minimal cut' picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the 'velocity' of entanglement growth in the 1D 'entanglement tsunami'. We show that in higher dimensions, noisy entanglement evolution maps to the well-studied problem of pinning of a membrane or domain wall by disorder.\] ## Refs - [[0522 Entanglement dynamics]] # Nakayama, Ooguri ## Bulk Local States and Crosscaps in Holographic CFT \[Links: [arXiv](https://arxiv.org/abs/1605.00334), [PDF](https://arxiv.org/pdf/1605.00334)\] \[Abstract: In a weakly coupled gravity theory in the anti-de Sitter space, local states in the bulk are linear superpositions of Ishibashi states for a [[0620 Non-orientable CFT|crosscap]] in the dual conformal field theory. The superposition structure can be constrained either by the microscopic causality in the bulk gravity or the bootstrap condition in the boundary conformal field theory. We show, contrary to some expectation, that these two conditions are not compatible to each other in the weak gravity regime. We also present an evidence to show that bulk local states in three dimensions are not organized by the Virasoro symmetry.\] # Perlmutter ## Bounding the Space of Holographic CFTs with Chaos \[Links: [arXiv](https://arxiv.org/abs/1602.08272), [PDF](https://arxiv.org/pdf/1602.08272.pdf)\] \[Abstract: Thermal states of quantum systems with many degrees of freedom are subject to a [[0474 Chaos bound|bound]] on the rate of onset of [[0008 Quantum chaos|chaos]], including a bound on the Lyapunov exponent, $\lambda_L\leq 2\pi /\beta$. We harness this bound to constrain the space of putative holographic CFTs and their would-be dual theories of AdS gravity. First, by studying [[0482 Out-of-time-order correlator|out-of-time-order four-point functions]], we discuss how $\lambda_L=2\pi/\beta$ in ordinary two-dimensional holographic CFTs is related to properties of the OPE at strong coupling. We then rule out the existence of unitary, sparse two-dimensional CFTs with large [[0033 Central charge|central charge]] and a set of higher spin currents of bounded spin; this implies the inconsistency of weakly coupled AdS$_3$ [[0421 Higher-spin gravity|higher spin gravities]] without infinite towers of gauge fields, such as the $SL(N)$ theories. This fits naturally with the structure of higher-dimensional gravity, where finite towers of higher spin fields lead to [[0118 Causality constraints for gravity|acausality]]. On the other hand, unitary CFTs with classical $W_{\infty}[\lambda]$ symmetry, dual to 3D [[0013 Vasiliev theory|Vasiliev]] or hs$[\lambda]$ higher spin gravities, do not violate the chaos bound, instead exhibiting no chaos: $\lambda_L=0$. Independently, we show that such theories violate unitarity for $|\lambda|>2$. These results encourage a tensionless string theory interpretation of the 3D [[0013 Vasiliev theory|Vasiliev theory]]. We also perform some CFT calculations of chaos in Rindler space in various dimensions.\] # Polchinski, Rosenhaus ## The Spectrum in the Sachdev-Ye-Kitaev Model \[Links: [arXiv](https://arxiv.org/abs/1601.06768), [PDF](https://arxiv.org/pdf/1601.06768.pdf)\] \[Abstract: \] # Porto ## The Tune of Love and the Nature(ness) of Spacetime \[Links: [arXiv](https://arxiv.org/abs/1606.08895), [PDF](https://arxiv.org/pdf/1606.08895.pdf)\] \[Abstract: The [[0131 Information paradox|black hole information paradox]] is among the most outstanding puzzles in physics. I argue here there is yet another black hole quandary which, in light of the recent direct detection of gravitational waves by Advanced LIGO, reveals a new window to probe the nature of spacetime in the forthcoming era of 'precision gravity.'\] # Roberts, Swingle ## Lieb-Robinson and the butterfly effect \[Links: [arXiv](https://arxiv.org/abs/1603.09298), [PDF](https://arxiv.org/pdf/1603.09298.pdf)\] \[Abstract: As experiments are increasingly able to probe the quantum dynamics of systems with many degrees of freedom, it is interesting to probe fundamental bounds on the dynamics of quantum information. We elaborate on the relationship between one such bound---[[0322 Lieb-Robinson bound|the Lieb-Robinson bound]]---and the butterfly effect in strongly-coupled quantum systems. The butterfly effect implies the ballistic growth of local operators in time, which can be quantified with [[0167 Butterfly velocity|the "butterfly" velocity]] $v_B$. Similarly, the Lieb-Robinson velocity places a state independent ballistic upper bound on the size of time evolved operators in non-relativistic lattice models. Here, we argue that $v_B$ is a state-dependent effective Lieb-Robinson velocity. We study the butterfly velocity in a wide variety of quantum field theories using holography and compare with free particle computations to understand the role of strong coupling. We find that, depending on the way length and time scale, $v_B$ acquires a temperature dependence and decreases towards the IR. We also comment on experimental prospects and on the relationship between the butterfly velocity and signaling.\] ## Summary - *argues* that [[0167 Butterfly velocity|the butterfly velocity]] is a state-dependent effective Lieb-Robinson velocity (which is state independent) # Shi, Mei ## Extended Symmetries at Black Hole Horizons in Generic Dimensions \[Links: [arXiv](https://arxiv.org/abs/1611.09491), [PDF](https://arxiv.org/pdf/1611.09491.pdf)\] \[Abstract: Recently it has been shown that there is asymptotic [[0064 BMS group|BMS]]-like [[0561 Near-horizon symmetry|symmetry associated with the near-horizon geometry]] of black holes in three and four dimensions. In this paper, we show that the presence of such BMS-like symmetry is a ubiquitous feature for black holes in generic dimensions. For black holes in $D$ dimensions, the symmetry contains $2$ supertranslations and $D-2$ generalized superrotations. The superrotations are found to generate a generalized Witt-like algebra that was previously noticed in a rather different construction. In the case of stationary and axisymmetric black holes, we calculate the surface charges and show that the zero-mode charges are intimately related to the [[0004 Black hole entropy|entropy]] and angular momenta of the black hole.\] # Simmons-Duffin (Lectures) ## TASI Lectures on the Conformal Bootstrap \[Links: [arXiv](https://arxiv.org/abs/1602.07982), [PDF](https://arxiv.org/pdf/1602.07982.pdf)\] \[Abstract: These notes are from courses given at TASI and the Advanced Strings School in summer 2015. Starting from principles of quantum field theory and the assumption of a traceless stress tensor, we develop the basics of conformal field theory, including conformal Ward identities, radial quantization, reflection positivity, the operator product expansion, and conformal blocks. We end with an introduction to numerical [[0036 Conformal bootstrap|bootstrap]] methods, focusing on the 2d and 3d Ising models.\] # Taylor, Woodhead ## Renormalized entanglement entropy \[Links: [arXiv](https://arxiv.org/abs/1604.06808), [PDF](https://arxiv.org/pdf/1604.06808.pdf)\] \[Abstract: We develop a renormalization method for [[0145 Generalised area|HEE]] based on area renormalization of entangling surfaces. The renormalized entanglement entropy is derived for entangling surfaces in asymptotically locally anti-de Sitter spacetimes in general dimensions and for entangling surfaces in four dimensional [[0209 Holographic renormalisation|holographic renormalization group flows]]. The renormalized entanglement entropy for disk regions in AdS4 spacetimes agrees precisely with the holographically renormalized action for AdS4 with spherical slicing and hence with the F quantity, in accordance with the Casini-Huerta-Myers map. We present a generic class of holographic RG flows associated with deformations by operators of dimension $3/2<\Delta<5/2$ for which the F quantity increases along the RG flow, hence violating the strong version of the F theorem. We conclude by explaining how the renormalized entanglement entropy can be derived directly from the renormalized partition function using the replica trick i.e. our renormalization method for the entanglement entropy is inherited directly from that of the partition function. We show explicitly how the entanglement entropy counterterms can be derived from the standard holographic renormalization counterterms for asymptotically locally anti-de Sitter spacetimes.\] ## Summary - combines [[0209 Holographic renormalisation|holographic renormalisation]] and [[0145 Generalised area|HEE]] - *derives* a formula for the counterterm needed to renormalise the [[0145 Generalised area|HEE]] ## Einstein case - in AdS${}_4$, the counterterm at replica number $n$ is given by $I_{\mathrm{ct}}(n)=\frac{1}{4 \pi G_{4}} \int_{\partial \mathcal{M}_{n}} d^{3} x \sqrt{h_{n}}\left(-\frac{1}{2} K_{n}+1+\frac{1}{4} R_{n}\right)$ where the first term is the usual GHY term - This localised to a 1D CT at the end of the entanglement surface: $S_{\mathrm{ct}}=-\frac{1}{4 G_{4}} \int_{\partial \Sigma} d x \sqrt{\tilde{\gamma}}$. This can be derived from using the quotient method. - n.b. $K$ term does not contribute because it does not diverge at conical singularity - In higher dim., the CT looks more complicated: $I_{\mathrm{ct}}(1)=\frac{1}{16 \pi G_{D+2}} \int_{\partial \mathcal{M}} d^{D+1} x \sqrt{h}\left(2 D+\frac{1}{(D-1)} R\right.$\left.+\frac{1}{(D-3)(D-1)^{2}}\left(R_{a b} R^{a b}-\frac{D+1}{4 D} R^{2}\right)+\cdots\right)$. Then the localised counterterm looks like Dong entropy for these higher derivative terms. I.e. $\int_{\partial \mathcal{M}_{n}} d^{D+1} x \sqrt{h_{n}} R_{n}^{2}=n \int_{\partial \mathcal{M}} d^{D+1} x \sqrt{h} R^{2}$+8 \pi(1-n) \int_{\partial \Sigma} d^{D-1} x \sqrt{\tilde{\gamma}} R$ etc. ## Higher derivative case - only considered GB - action counter term is known for GB perturbatively: $I_{c t}=a_{2} \int d^{4} x \sqrt{h} R$ - FG not known but this paper chose a class of allowed solutions: $d s^{2}=l^{2}(\lambda)\left(\frac{d \rho^{2}}{4 \rho^{2}}+\frac{1}{\rho} d x \cdot d x\right)$ # Turiaci, Verlinde ## On CFT and Quantum Chaos \[Links: [arXiv](https://arxiv.org/abs/1603.03020), [PDF](https://arxiv.org/pdf/1603.03020.pdf)\] \[Abstract: We make three observations that help clarify the relation between CFT and [[0008 Quantum chaos|quantum chaos]]. We show that any 1+1-D system in which conformal symmetry is non-linearly realized exhibits two main characteristics of chaos: maximal [[0466 Lyapunov exponent|Lyapunov behavior]] and a spectrum of Ruelle resonances. We use this insight to identify a lattice model for quantum chaos, built from parafermionic spin variables with an equation of motion given by a Y-system. Finally we point to a relation between the spectrum of Ruelle resonances of a CFT and the analytic properties of OPE coefficients between light and heavy operators. In our model, this spectrum agrees with the [[0325 Quasi-normal modes|quasi-normal modes]] of the [[0086 Banados-Teitelboim-Zanelli black hole|BTZ black hole]].\] # Zimmerman ## Horizon instability of extremal Reissner-Nordström black holes to charged perturbations \[Links: [arXiv](https://arxiv.org/abs/1612.03172), [PDF](https://arxiv.org/pdf/1612.03172.pdf)\] \[Abstract: We investigate the stability of highly charged Reissner-Nordström black holes to charged scalar perturbations. We show that the near-horizon region exhibits a transient instability which becomes the [[0340 Aretakis instability|Aretakis instability]] in the extremal limit. The rates we obtain match the enhanced rates for nonaxisymmetric perturbations of the near-extremal and extremal Kerr solutions. The agreement is shown to arise from a shared near-horizon symmetry of the two scenarios.\] ## Refs - Kerr version: [[2016#Gralla, Zimmerman, Zimmerman]]