# Alba, Diab ## Constraining conformal field theories with a higher spin symmetry in d> 3 dimensions \[Links: [arXiv](https://arxiv.org/abs/1510.02535), [PDF](https://arxiv.org/pdf/1510.02535)\] \[Abstract: We study unitary conformal field theories with a unique stress tensor and at least one higher-spin conserved current in $d>3$ dimensions. We prove that every such theory contains an infinite number of [[0621 Higher-spin conserved currents in CFT|higher-spin conserved currents]] of arbitrarily high spin, and that Ward identities generated by the conserved charges of these currents imply that the correlators of the stress tensor and the conserved currents of the theory must coincide with one of the following three possibilities: a) a theory of $n$ free bosons (for some integer $n$), b) a theory of $n$ free fermions, or c) a theory of $n$ $(d-2)/2$-forms. For $d$ even, all three structures exist, but for $d$ odd, it may be the case that the third structure (c) does not; if it does exist, it is unclear what theory, if any, realizes it. This is a generalization of the result proved in three dimensions by Maldacena and Zhiboedov [[arXiv:1112.1016](https://arxiv.org/abs/1112.1016)]. This paper supersedes the previous paper by the authors [[arXiv:1307.8092](https://arxiv.org/abs/1307.8092)]\] # Banks, Donos, Gauntlett ## Thermoelectric DC conductivities and Stokes flows on black hole horizons \[Links: [arXiv](https://arxiv.org/abs/1507.00234), [PDF](https://arxiv.org/pdf/1507.00234.pdf)\] \[Abstract: We consider a general class of electrically charged black holes of Einstein-Maxwell-scalar theory that are holographically dual to conformal field theories at finite charge density which break translation invariance explicitly. We examine the linearised perturbations about the solutions that are associated with the thermoelectric DC conductivity. We show that there is a decoupled sector at the black hole horizon which must solve generalised Stokes equations for a charged fluid. By solving these equations we can obtain the DC conductivity of the dual field theory. For Q-lattices and one-dimensional lattices we solve the fluid equations to obtain closed form expressions for the DC conductivity in terms of the solution at the black hole horizon. We also determine the leading order DC conductivity for lattices that can be expanded as a perturbative series about translationally invariant solutions.\] ## Summary - solves for [[0434 Diffusivity]] from equations for Stokes flows on the horizon - breaks translation invariance # Barnich, Gonzalez, Maloney, Oblak ## One-loop partition function of three-dimensional flat gravity \[Links: [arXiv](https://arxiv.org/abs/), [PDF](https://arxiv.org/pdf/.pdf)\] \[Abstract: In this note we point out that the one-loop partition function of [[0002 3D gravity|three-dimensional flat gravity]], computed along the lines originally developed for the anti-de Sitter case, reproduces characters of the [[0064 BMS group|BMS3 group]].\] # Benini, Zaffaroni ## A topologically twisted index for three-dimensional supersymmetric theories \[Links: [arXiv](https://arxiv.org/abs/1504.03698), [PDF](https://arxiv.org/pdf/1504.03698.pdf)\] \[Abstract: We provide a general formula for the partition function of three-dimensional $\mathcal{N}=2$ gauge theories placed on $S^2 \times S^1$ with a topological twist along $S^2$, which can be interpreted as an index for chiral states of the theories immersed in background magnetic fields. The result is expressed as a sum over magnetic fluxes of the residues of a meromorphic form which is a function of the scalar zero-modes. The partition function depends on a collection of background magnetic fluxes and fugacities for the global symmetries. We illustrate our formula in many examples of 3d Yang-Mills-Chern-Simons theories with matter, including Aharony and Giveon-Kutasov dualities. Finally, our formula generalizes to $\Omega$-backgrounds, as well as two-dimensional theories on $S^2$ and four-dimensional theories on $S^2 \times T^2$. In particular this provides an alternative way to compute genus-zero A-model topological amplitudes and Gromov-Witten invariants.\] ## Refs - [[0546 Topologically twisted index]] # Bhattacharjee, Sarkar, Wall ## HEE increases in quadratic gravity \[Links: [arXiv](https://arxiv.org/abs/1504.04706), [PDF](https://arxiv.org/pdf/1504.04706.pdf)\] \[Abstract: \] ## Refs - precursers - $R^2$: [[JacobsonKangMyers1995]][](https://arxiv.org/abs/gr-qc/9503020) - GB: [[2013#Sarkar, Wall]] ## Comments - An earlier paper on [[0005 Black hole second law]] than [[2015#Wall (Essay)]]. - writes down a generalised expansion and its time derivative to leading order (or, generalised [[0408 Raychaudhuri equation]]) ## Summary 1. Require entropy increasing for linear perturbations -> fix ambiguities 2. Show that it does increase in a particular example (Vaidya-like). ## Method - write entropy as $S=(1 / 4) \int(1+\rho) \sqrt{\gamma} d^{D-2} x$ - define generalised expansion as $\Theta=\frac{d \rho}{d t}+\theta_{k}(1+\rho)$ - Then [[0408 Raychaudhuri equation]] becomes $\frac{d \Theta}{d t}-\kappa \Theta=-8 \pi T_{k k}+\nabla_{k} \nabla_{k} \rho-\rho R_{k k}+\beta H_{k k}$ plus explicit second order terms - task: show $E_{k k}=\nabla_{k} \nabla_{k} \rho-\rho R_{k k}+\beta H_{k k}=\mathcal{O}\left(\epsilon^{2}\right)$ - proof: in Vaidya spacetime only # Bousso, Fisher, Leichenauer, Wall ## A Quantum Focussing Conjecture \[Links: [arXiv](https://arxiv.org/abs/1506.02669), [PDF](https://arxiv.org/pdf/1506.02669.pdf)\] \[Abstract: \] ## Summary - OG of [[0405 Quantum null energy condition]] - proposes [[0405 Quantum null energy condition]] by studying the weak gravity limit of [[0243 Quantum focusing conjecture]] # Brown ## The decay of hot KK space \[Links: [arXiv](https://arxiv.org/abs/1408.5903), [PDF](https://arxiv.org/pdf/1408.5903.pdf)\] \[Abstract: \] ## Refs - talks about [[0168 Bubble of nothing]] as instantons ## Summary - 4 instabilities of ==KK space at non-zero temperature== (flat, not AdS) - nucleation of 4D BHs, 5D BHs, bubbles by quantum tunnelling, bubbles by thermal fluctuation - controlled by two Euclidean instantons - each have two inequivalent analytic continuations - thermodynamic instability of one related to mechanical instability of the other - high/low-temperature duality - related BH to bubbles of nothing ## Instanton A: Black string instanton - $d s^{2}=\left(1-\frac{R}{r}\right) d w^{2}+\frac{d r^{2}}{1-\frac{R}{r}}+r^{2}\left(d \theta^{2}+\cos ^{2} \theta d \phi^{2}\right)+d z^{2}$ - two ways to use it 1. continue $w$ to real time -> 5D black string/4D BH in real time 2. continue $z$ to real time -> thermal bubble of nothing ## Instanton B: BH instanton - $d s^{2}=\left(1-\frac{R^{2}}{r^{2}}\right) d w^{2}+\frac{d r^{2}}{1-\frac{R^{2}}{r^{2}}}+r^{2}\left(d \theta^{2}+\cos ^{2} \theta\left(d \phi^{2}+\sin ^{2} \phi d \psi^{2}\right)\right)$ - only exact in the limit of large $z$ - there are approximate solutions for finite but large $z$ - two ways to use it 1. continue $w$ to real time -> 5D BH 2. continue $\theta$ to real time -> quantum bubble of nothing # Brown, Roberts, Susskind, Swingle, Zhao (Sep) ## Complexity Equals Action \[Links: [arXiv](https://arxiv.org/abs/1509.07876), [PDF](https://arxiv.org/pdf/1509.07876.pdf)\] \[Abstract: We conjecture that the quantum [[0204 Quantum complexity|complexity]] of a holographic state is dual to the action of a certain spacetime region that we call a Wheeler-DeWitt patch. We illustrate and test the conjecture in the context of neutral, charged, and rotating black holes in AdS, as well as black holes perturbed with static shells and with [[0117 Shockwave|shock waves]]. This conjecture evolved from a previous conjecture that complexity is dual to spatial volume, but appears to be a major improvement over the original. In light of our results, we discuss the hypothesis that black holes are the fastest computers in nature.\] ## Refs - this is the original proposal of [[0204 Quantum complexity|complexity]] = action - longer paper [[2015#Brown, Roberts, Susskind, Swingle, Zhao (Dec)]] # Brown, Roberts, Susskind, Swingle, Zhao (Dec) ## Complexity, action, and black holes \[Links: [arXiv](https://arxiv.org/abs/1512.04993), [PDF](https://arxiv.org/pdf/1512.04993.pdf)\] \[Abstract: Our earlier paper "[[0204 Quantum complexity|Complexity]] Equals Action" conjectured that the quantum computational complexity of a holographic state is given by the classical action of a region in the bulk (the "Wheeler-DeWitt" patch). We provide calculations for the results quoted in that paper, explain how it fits into a broader (tensor) network of ideas, and elaborate on the hypothesis that black holes are the fastest computers in nature.\] ## Refs - longer version of [[2015#Brown, Roberts, Susskind, Swingle, Zhao (Sep)]] # Bueno, Myers, Witczak-Krempa ## Universality of corner entanglement in conformal field theories \[Links: [arXiv](https://arxiv.org/abs/1505.04804), [PDF](https://arxiv.org/pdf/1505.04804.pdf)\] \[Abstract: \] ## Summary - *finds*, in the context of [[0362 Entanglement surface with cusps]], that the function $a(\theta)/C_T$ where $a(\theta)$ is the contribution to the entanglement entropy as a function of the angle of the cusp and $C_T$ is the [[0033 Central charge]] of the CFT is quite universal and agrees exactly in the limit $\theta\to \pi$ # Campiglia ## Null to time-like infinity Green's functions for asymptotic symmetries in Minkowski spacetime \[Links: [arXiv](https://arxiv.org/abs/1509.01408), [PDF](https://arxiv.org/pdf/1509.01408.pdf)\] \[Abstract: We elaborate on the Green's functions that appeared in [1,2] when generalizing, from massless to massive particles, various equivalences between [[0009 Soft theorems|soft theorems]] and [[0106 Ward identity|Ward identities]] of [[0060 Asymptotic symmetry|large gauge symmetries]]. We analyze these Green's functions in considerable detail and show that they form a hierarchy of functions which describe 'boundary to bulk' propagators for large $U(1)$ gauge parameters, supertranslations and sphere vector fields respectively. As a consistency check we verify that the Green's functions associated to the large diffeomorphisms map the Poincare group at null infinity to the Poincare group at time-like infinity.\] # Campiglia, Laddha (May) ## Asymptotic symmetries of QED and Weinberg's soft photon theorem \[Links: [arXiv](https://arxiv.org/abs/1505.05346), [PDF](https://arxiv.org/pdf/1505.05346.pdf)\] \[Abstract: Various equivalences between so-called [[0009 Soft theorems|soft theorems]] which constrain scattering amplitudes and [[0106 Ward identity|Ward identities]] related to [[0060 Asymptotic symmetry|asymptotic symmetries]] have recently been established in gauge theories and gravity. So far these equivalences have been restricted to the case of massless matter fields, the reason being that the asymptotic symmetries are defined at null infinity. The restriction is however unnatural from the perspective of soft theorems which are insensitive to the masses of the external particles. In this work we remove the aforementioned restriction in the context of scalar QED. Inspired by the radiative phase space description of massless fields at null infinity, we introduce a manifold description of time-like infinity on which the asymptotic phase space for [[0256 Massive particles in CCFT|massive fields]] can be defined. The "angle dependent" large gauge transformations are shown to have a well defined action on this phase space, and the resulting Ward identities are found to be equivalent to Weinberg's soft photon theorem.\] # Campiglia, Laddha (Sep) ## Asymptotic symmetries of gravity and soft theorems for massive particles \[Links: [arXiv](https://arxiv.org/abs/1509.01406), [PDF](https://arxiv.org/pdf/1509.01406.pdf)\] \[Abstract: The existing equivalence between (generalized) [[0064 BMS group|BMS]] [[0106 Ward identity|Ward identities]] with leading and subleading [[0009 Soft theorems|soft graviton theorems]] is extended to the case where the scattering particles are massive scalars. By extending the action of generalized BMS group off null infinity at late times, we show that there is a natural action of such group not only on the radiative data at null infinity but also on the scattering data of the massive scalar field. This leads to a formulation of Ward identities associated to the generalized BMS group when the scattering states are [[0256 Massive particles in CCFT|massive]] scalars or massless gravitons and we show that these Ward identities are equivalent to the leading and subleading soft graviton theorems.\] # Casini, Liu, Mezei ## Spread of entanglement and causality \[Links: [arXiv](https://arxiv.org/abs/1509.05044), [PDF](https://arxiv.org/pdf/1509.05044.pdf)\] \[Abstract: We investigate causality constraints on the time evolution of [[0301 Entanglement entropy|entanglement entropy]] after a global quench in relativistic theories. We first provide a general proof that the so-called tsunami velocity is bounded by the speed of light. We then generalize the free particle streaming model of [arXiv:cond-mat/0503393](https://arxiv.org/abs/cond-mat/0503393) to general dimensions and to an arbitrary entanglement pattern of the initial state. In more than two spacetime dimensions the spread of entanglement in these models is highly sensitive to the initial entanglement pattern, but we are able to prove an upper bound on the normalized rate of growth of entanglement entropy, and hence the tsunami velocity. The bound is smaller than what one gets for quenches in holographic theories, which highlights the importance of interactions in the spread of entanglement in many-body systems. We propose an interacting model which we believe provides an upper bound on the spread of entanglement for interacting relativistic theories. In two spacetime dimensions with multiple intervals, this model and its variations are able to reproduce intricate results exhibited by holographic theories for a significant part of the parameter space. For higher dimensions, the model bounds the tsunami velocity at the speed of light. Finally, we construct a geometric model for entanglement propagation based on a [[0054 Tensor network|tensor network]] construction for global quenches.\] ## Summary - proves a bound on [[0327 Entanglement velocity|entanglement velocity]] using [[0300 Mutual information|mutual information]] - constructs a [[0518 Quasiparticle model|quasiparticle model]] for entanglement # Cheung, Shen, Trnka ## Simple Recursion Relations for General Field Theories \[Links: [arXiv](https://arxiv.org/abs/1502.05057), [PDF](https://arxiv.org/pdf/1502.05057.pdf)\] \[Abstract: [[0551 On-shell recursion relations|On-shell methods]] offer an alternative definition of quantum field theory at tree-level, replacing Feynman diagrams with recursion relations and interaction vertices with a handful of seed scattering amplitudes. In this paper we determine the simplest recursion relations needed to construct a general four-dimensional quantum field theory of massless particles. For this purpose we define a covering space of recursion relations which naturally generalizes all existing constructions, including those of [[0058 BCFW|BCFW]] and Risager. The validity of each recursion relation hinges on the large momentum behavior of an $n$-point scattering amplitude under an $m$-line momentum shift, which we determine solely from dimensional analysis, Lorentz invariance, and locality. We show that all amplitudes in a renormalizable theory are 5-line constructible. Amplitudes are 3-line constructible if an external particle carries spin or if the scalars in the theory carry equal charge under a global or gauge symmetry. Remarkably, this implies the 3-line constructibility of all gauge theories with fermions and complex scalars in arbitrary representations, all supersymmetric theories, and the standard model. Moreover, all amplitudes in non-renormalizable theories without derivative interactions are constructible; with derivative interactions, a subset of amplitudes is constructible. We illustrate our results with examples from both renormalizable and non-renormalizable theories. Our study demonstrates both the power and limitations of recursion relations as a self-contained formulation of quantum field theory.\] # Crossley, Glorioso, Liu ## Effective field theory of dissipative fluids \[Links: [arXiv](https://arxiv.org/abs/1511.03646), [PDF](https://arxiv.org/pdf/1511.03646.pdf)\] \[Abstract: We develop an effective field theory for dissipative fluids which governs the dynamics of long-lived gapless modes associated with conserved quantities. The resulting theory gives a path integral formulation of fluctuating [[0429 Hydrodynamics|hydrodynamics]] which systematically incorporates nonlinear interactions of noises. The dynamical variables are mappings between a "fluid spacetime" and the physical spacetime and an essential aspect of our formulation is to identify the appropriate symmetries in the fluid spacetime. The theory applies to nonlinear disturbances around a general density matrix. For a thermal density matrix, we require an additional $Z_2$ symmetry, to which we refer as the local [[0521 KMS condition|KMS condition]]. This leads to the standard constraints of hydrodynamics, as well as a nonlinear generalization of the Onsager relations. It also leads to an emergent supersymmetry in the classical statistical regime, and a higher derivative deformation of supersymmetry in the full quantum regime.\] ## Refs - part II: [[2017#Glorioso, Crossley, Liu]] # D'Alessio, Kafri, Polkovnikov, Rigol (Review) ## From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics \[Links: [arXiv](https://arxiv.org/abs/1509.06411), [PDF](https://arxiv.org/pdf/1509.06411.pdf)\] \[Abstract: This review gives a pedagogical introduction to the [[0040 Eigenstate thermalisation hypothesis|eigenstate thermalization hypothesis (ETH)]], its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from [[0008 Quantum chaos|quantum chaos]] and [[0579 Random matrix theory|random matrix theory]] (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We introduce the concept of the generalized Gibbs ensemble, and discuss its connection with ideas of prethermalization in weakly interacting systems.\] # Dong, Miao ## Generalised gravitational entropy from total derivative action \[Links: [arXiv](https://arxiv.org/abs/1510.04273), [PDF](https://arxiv.org/pdf/1510.04273.pdf)\] \[Abstract: \] ## Summary - [[0145 Generalised area|HEE]] vanishes for total derivative actions - if they do not, HEE would not agree with field-theoretic universal terms found by [[AstanehPatrushevSolodukhin201411]] and [[AstanehPatrushevSolodukhin201412]] ## Two methods - Dong v.s. APS method - "the main differences between the two methods are whether the regularized cone approaches the singular cone away from the conical singularity, and whether the on-shell action of the singular cone is properly subtracted from that of the regularized cone" ## Wald formalism - Killing horizon - total derivative terms vanish by definition: the prescription involve total derivatives ## 2 Replica method - three methods 1. no regularisation; use total derivative to reduce to boundary 2. regularisation 3. regularisation but use bruteforce 1. direct method - actually clear if we go to integer $n$ - completely regular in parent space so no boundary term at IR; so zero 2. boundary method 3. bulk method ## 3 General higher derivative theories - gave a few examples - there is a ==generalised== Wald entropy that comes from delta functions - find that the generalised entropy cancels with the anomaly term ## 4 APS method - find non-zero entropy - problem: the regularised cone does not approach the singular cone away from the conical singularity ## 5 Why is APS wrong 1. it does not agree with universal term in the field-theoretic calculation 2. [[0082 Generalised second law]] would be violated # Donnay, Giribet, Gonzales, Pino ## Supertranslations and Superrotations at the Black Hole Horizon \[Links: [arXiv](https://arxiv.org/abs/1511.08687), [PDF](https://arxiv.org/pdf/1511.08687.pdf)\] \[Abstract: We show that the [[0060 Asymptotic symmetry|asymptotic symmetries]] close to nonextremal black hole horizons are generated by an extension of supertranslations. This group is generated by a semidirect sum of [[0032 Virasoro algebra|Virasoro]] and Abelian currents. The charges associated with the asymptotic Killing symmetries satisfy the same algebra. When considering the special case of a stationary black hole, the zero mode charges correspond to the angular momentum and the entropy at the horizon.\] ## Refs - [[0561 Near-horizon symmetry]] # Donos, Gauntlett ## Navier-Stokes Equations on Black Hole Horizons and DC Thermoelectric Conductivity \[Links: [arXiv](https://arxiv.org/abs/1506.01360), [PDF](https://arxiv.org/pdf/1506.01360.pdf)\] \[Abstract: \] ## Refs - earlier work, from EOMs directly: [[2014#Donos, Gauntlett (Jun)]] ## Summary - *obtains* Navier-Stokes equations on BH horizons - *obtains* thermoelectric conductivity by solving linearised, time-independent and forced Navier-Stokes equations on the BH horizon for an incompressible and charged fluid ## Theory - $S=\int d^{4} x \sqrt{-g}\left(R+6-\frac{1}{4} F^{2}\right)$ - focus on charged static black holes ## Thermoelectric conductivities - $\left(\begin{array}{c}\bar{J}^{i} \\ \bar{Q}^{i}\end{array}\right)=\left(\begin{array}{cc}\sigma^{i j} & T \alpha^{i j} \\ T \bar{\alpha}^{i j} & T \bar{\kappa}^{i j}\end{array}\right)\left(\begin{array}{c}E_{j} \\ \zeta_{j}\end{array}\right)$ # Giddings ## Hilbert space structure in quantum gravity: an algebraic perspective \[Links: [arXiv](https://arxiv.org/abs/1503.08207), [PDF](https://arxiv.org/pdf/1503.08207)\] \[Abstract: If quantum gravity respects the principles of quantum mechanics, suitably generalized, it may be that a more viable approach to the theory is through identifying the relevant quantum structures rather than by quantizing classical spacetime. This viewpoint is supported by difficulties of such quantization, and by the apparent lack of a fundamental role for locality. In finite or discrete quantum systems, important structure is provided by tensor [[0046 Non-factorisation of Hilbert space in QG|factorizations of the Hilbert space]]. However, even in local quantum field theory properties of the generic type III [[0415 Von Neumann algebra|von Neumann algebras]] and of long range gauge fields indicate that factorization of the Hilbert space is problematic. Instead it is better to focus on the structure of the algebra of observables, and in particular on its subalgebras corresponding to regions. This paper suggests that study of analogous algebraic structure in gravity gives an important perspective on the nature of the quantum theory. Significant departures from the subalgebra structure of local quantum field theory are found, working in the correspondence limit of long-distances/low-energies. Particularly, there are obstacles to identifying commuting algebras of localized operators. In addition to suggesting important properties of the algebraic structure, this and related observations pose challenges to proposals of a fundamental role for entanglement.\] # Harlow ## Wormholes, Emergent Gauge Fields, and the Weak Gravity Conjecture \[Links: [arXiv](https://arxiv.org/abs/1510.07911), [PDF](https://arxiv.org/pdf/1510.07911.pdf)\] \[Abstract: This paper revisits the question of reconstructing bulk gauge fields as boundary operators in [[0001 AdS-CFT|AdS/CFT]]. In the presence of the wormhole dual to the thermofield double state of two CFTs, the existence of bulk gauge fields is in some tension with the microscopic tensor [[0514 Lorentzian factorisation problem|factorization]] of the Hilbert space. I explain how this tension can be resolved by splitting the gauge field into charged constituents, and I argue that this leads to a new argument for the "principle of completeness", which states that the charge lattice of a gauge theory coupled to gravity must be fully populated. I also claim that it leads to a new motivation for (and a clarification of) the "[[0177 Weak gravity conjecture|weak gravity conjecture]]", which I interpret as a strengthening of this principle. This setup gives a simple example of a situation where describing low-energy bulk physics in CFT language requires knowledge of high-energy bulk physics. This contradicts to some extent the notion of "effective conformal field theory", but in fact is an expected feature of the resolution of the black hole information problem. An analogous factorization issue exists also for the gravitational field, and I comment on several of its implications for reconstructing black hole interiors and the emergence of spacetime more generally.\] ## Refs - [[0514 Lorentzian factorisation problem]] ## Related - [[0249 Factorisation problem]] - [[0046 Non-factorisation of Hilbert space in QG]] # Hartman, Jain, Kundu ## Causality constraints in CFT \[Links: [arXiv](https://arxiv.org/abs/1509.00014), [PDF](https://arxiv.org/pdf/1509.00014.pdf)\] \[Abstract: Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In $d$-dimensional conformal field theory, we show how such constraints are encoded in crossing symmetry of Euclidean correlators, and derive analogous constraints directly from the conformal bootstrap (analytically). The bootstrap setup is a Lorentzian four-point function corresponding to propagation through a shockwave. Crossing symmetry fixes the signs of certain log terms that appear in the conformal block expansion, which constrains the interactions of low-lying operators. As an application, we use the bootstrap to rederive the well known sign constraint on the $(\partial \phi)^4$ coupling in effective field theory, from a dual CFT. We also find constraints on theories with [[0421 Higher-spin gravity|higher spin]] conserved currents. Our analysis is restricted to scalar correlators, but we argue that similar methods should also impose nontrivial constraints on the interactions of spinning operators.\] ## Summary - causality constraints are encoded in the crossing symmetry of Euclidean correlators ## Remarks - different from [[2006#Cornalba, Costa, Penedones, Schiappa (a)]] and [[2006#Cornalba, Costa, Penedones, Schiappa (b)]] - here do not assume finite $N$ - here [[0117 Shockwave|shockwave]] is on the boundary CFT - comparison to gravitational [[0117 Shockwave|shockwaves]] in AdS/CFT is in a later paper ## Related topics - [[0118 Causality constraints for gravity]] (although this paper is really for CFT) # He, Mitra, Strominger ## 2D Kac-Moody symmetry of 4D Yang-Mills theory \[Links: [arXiv](https://arxiv.org/abs/1503.02663), [PDF](https://arxiv.org/pdf/1503.02663.pdf)\] \[Abstract: \] ## Refs - explains [[0069 Kac-Moody algebra]] in YM - parent [[0010 Celestial holography]] ## Summary - shows that soft gluon theorem for massless theories at semi-classical level to be Wald identity of a holomorphic 2d G-Kac-Moody symmetry acting on correlation functions - shows that [[0060 Asymptotic symmetry]] of 4D YM is equivalent to the 2D **level-0** [[0069 Kac-Moody algebra]] ## Order of limits - the double soft limit involving one positive and one negative helicity is ambiguous -> need to specify the order - here our prescription is to always take positive one first -> we get a holomorphic [[0069 Kac-Moody algebra]] but not a second one from negative helicity soft gluons ## Currents - taking positive helicity gluon to zero first: $J_{z}^{a} J_{\bar{w}}^{b} \sim-\frac{i f^{a b c}}{z-w} J_{\bar{w}}^{c}$ - negative first: $J_{z}^{a} J_{\bar{w}}^{b} \sim-\frac{i f^{a b c}}{\bar{z}-\bar{w}} J_{z}^{c}$ - prescription - always take positive to zero first ## Issues and future directions - IR divergences - last paragraph of p2 - IR divergences give universal corrections - see e.g. [[FeigeSchwartz2014]] - relation to [[0089 Chern-Simons theory]] - end of section 5 - in CS there is also the situation that only one of holomorphic and anti-holomorphic can exist due to boundary conditions eliminating one of them - would be curious to see what happens if such a topological term is added too our theory ## 3. Deriving the algebra from bulk 1. temporal gauge: $\mathcal{A}_{u}=0$ near $\mathscr{I}^+$ and $\mathcal{A}_{v}=0$ near $\mathscr{I}^-$ 2. do expansion - $\mathcal{A}_{z}(r, u, z, \bar{z})=A_{z}(u, z, \bar{z})+\mathcal{O}(1 / r)$ - $\mathcal{A}_{r}(r, u, z, \bar{z})=\frac{1}{r^{2}} A_{r}(u, z, \bar{z})+\mathcal{O}\left(1 / r^{3}\right)$ - reason: fall off is chosen to make the charges finite 3. constrain field configurations at far past and future - need them to be vacuum - $\left.F_{u r}\right|_{\mathscr{I}_{+}^{+}}=\left.F_{u z}\right|_{\mathscr{I}_{+}^{+}}=\left.F_{z \bar{z}}\right|_{\mathscr{I}_{+}^{+}}=0$ - => $\left.\mathcal{U}_{z} \equiv A_{z}\right|_{\mathscr{I}_{+}^{+}}=i \mathcal{U} \partial_{z} \mathcal{U}^{-1}$, i.e. pure gauge - now residual gauge is generated by an arbitrary function $\epsilon(z,\bar z)$ on $S^2$ -> large gauge transformations - $\mathcal{U} \rightarrow g \mathcal{U}$ under finite large gauge transformations 4. matching at $i^0$ to define scattering amplitude - $\left.A_{z}\right|_{\mathscr{I}_{-}^{+}}=\left.B_{z}\right|_{\mathscr{I}_{+}^{-}}$ - this is preserved by $\varepsilon(z, \bar{z})=\varepsilon^{-}(z, \bar{z})$ where LHS is one defined for future and RHS for past ## 4. Holomorphic soft gluon current ###### 4.1 Soft gluon theorem - reviews leading [[0107 Soft gluon symmetry|soft gluon theorem]] ###### 4.2 Kac-Moody symmetry - derives a relation for $J_{\mathcal{C}}(\varepsilon) \equiv \oint_{\mathcal{C}} \frac{d z}{2 \pi i} \operatorname{tr}\left[\varepsilon J_{z}\right]$ - $\left\langle J_{\mathcal{C}}(\varepsilon) J_{w} O_{1} \cdots O_{n}\right\rangle_{U=1}=\sum_{k \in \mathcal{C}}\left\langle J_{w} O_{1} \cdots \varepsilon_{k}\left(z_{k}\right) O_{k} \cdots O_{n}\right\rangle_{U=1}+\left\langle\varepsilon(w) J_{w} O_{1} \cdots O_{n}\right\rangle_{U=1}$ - then realises that this is just the [[0106 Ward identity]] for a holomorphic [[0069 Kac-Moody algebra]] at level 0 ###### 4.3 Asymptotic symmetry - action of the AS: $O_{k}\left(z_{k}, \bar{z}_{k}\right) \rightarrow U_{k}\left(z_{k}, \bar{z}_{k}\right) O_{k}\left(z_{k}, \bar{z}_{k}\right)$ where $U_k$ acts in the representation of $O_k$ - then the general S-correlator is related to that of $U=1$ by - $\langle J_{z}^{a} O_{1}^{i_{1}} \cdots\rangle_{U}=U(z, \bar{z})^{a b} U_{1}\left(z_{1}, \bar{z}_{1}\right)^{i_{1} j_{1}} \cdots\langle J_{z}^{b} O_{1}^{j_{1}} \cdots\rangle_{U=1}$ - linearise it with $U(z, \bar{z})=1+i \varepsilon(z)+\cdots$ which is holomorphic inside $\mathcal{C}$ and vanishes outside - $\delta_{\varepsilon}\left\langle O_{1} \cdots O_{n}\right\rangle_{U=1}=i \sum_{k \in \mathcal{C}}\left\langle O_{1} \cdots \varepsilon_{k}\left(z_{k}\right) O_{k} \cdots O_{n}\right\rangle_{U=1}$ ## 5. Anti-holomorphic current - the negative helicity soft gluon currents do not generate a second [[0069 Kac-Moody algebra|KM symmetry]] which is anti-holomorphic - **issue**: in the double soft limit, the orders of limits do not commute - -> need a prescription - **prescription**: always take positive helicity soft first - -> then the $J^a_\bar{z}$ transforms in the adjoint and is not an independent symmetry - alternative prescriptions - take negative soft first: then only an anti-holomorphic symmetry - treat both symmetrically: no symmetry # Hellerman, Orlando, Reffert, Watanabe ## On the CFT Operator Spectrum at Large Global Charge \[Links: [arXiv](https://arxiv.org/abs/1505.01537), [PDF](https://arxiv.org/pdf/1505.01537)\] \[Abstract: We calculate the anomalous dimensions of operators with large global charge $J$ in certain strongly coupled conformal field theories in three dimensions, such as the $O(2)$ model and the supersymmetric fixed point with a single chiral superfield and a $W = \Phi^3$ superpotential. Working in a $1/J$ expansion, we find that the large-$J$ sector of both examples is controlled by a conformally invariant effective Lagrangian for a Goldstone boson of the global symmetry. For both these theories, we find that the lowest state with charge $J$ is always a scalar operator whose dimension $\Delta_J$ satisfies the sum rule $J^2 \Delta_J - \left( \tfrac{J^2}{2} + \tfrac{J}{4} + \tfrac{3}{16} \right) \Delta_{J-1} - \left( \tfrac{J^2}{2} - \tfrac{J}{4} + \tfrac{3}{16} \right) \Delta_{J+1} = 0.035147$ up to corrections that vanish at large $J$. The spectrum of low-lying excited states is also calculable explicitly: For example, the second-lowest primary operator has spin two and dimension $\Delta\ll J + \sqrt{3}$. In the supersymmetric case, the dimensions of all half-integer-spin operators lie above the dimensions of the integer-spin operators by a gap of order $J^{1/2}$. The propagation speeds of the Goldstone waves and heavy fermions are $\frac{1}{\sqrt{2}}$ and $\pm \frac{1}{2}$ times the speed of light, respectively. These values, including the negative one, are necessary for the consistent realization of the superconformal symmetry at large $J$.\] # Hijano, Kraus, Perlmutter, Snively ## Witten diagrams revisited: the AdS geometry of conformal blocks \[Links: [arXiv](https://arxiv.org/abs/1508.00501), [PDF](https://arxiv.org/pdf/1508.00501.pdf)\] \[Abstract: \] ## Comments - reviews [[0109 Witten diagrams]] - [[0185 Dual of conformal blocks]] # Jacobson ## Entanglement Equilibrium and the Einstein Equation \[Links: [arXiv](https://arxiv.org/abs/), [PDF](https://arxiv.org/pdf/.pdf)\] \[Abstract: A link between the semiclassical Einstein equation and a maximal vacuum entanglement hypothesis is established. The hypothesis asserts that [[0301 Entanglement entropy|entanglement entropy]] in small geodesic balls is maximized at fixed volume in a locally maximally symmetric vacuum state of geometry and quantum fields. A qualitative argument suggests that the Einstein equation implies validity of the hypothesis. A more precise argument shows that, for first-order variations of the local vacuum state of conformal quantum fields, the vacuum entanglement is stationary if and only if the Einstein equation holds. For nonconformal fields, the same conclusion follows modulo a conjecture about the variation of entanglement entropy.\] # Kapec, Pate, Strominger ## New Symmetries of QED \[Links: [arXiv](https://arxiv.org/abs/1506.02906), [PDF](https://arxiv.org/pdf/1506.02906.pdf)\] \[Abstract: The soft photon theorem in $U(1)$ gauge theories with only massless charged particles has recently been shown to be the Ward identity of an infinite-dimensional [[0060 Asymptotic symmetry|asymptotic symmetry]] group. This symmetry group is comprised of gauge transformations which approach angle-dependent constants at null infinity. In this paper, we extend the analysis to all $U(1)$ theories, including those with [[0256 Massive particles in CCFT|massive]] charged particles such as QED.\] # Klose, McLoughlin, Nandan, Plefka, Travaglini ## Double-Soft Limits of Gluons and Gravitons \[Links: [arXiv](https://arxiv.org/abs/1504.05558), [PDF](https://arxiv.org/pdf/1504.05558.pdf)\] \[Abstract: \] ## Summary - double [[0009 Soft theorems]] ## Results - (eq.39) $\mathrm{DSL}^{(0)}\left(n+2,1^{+}, 2^{+}, 3\right)=\frac{\langle n+23\rangle}{\langle n+21\rangle\langle 12\rangle\langle 23\rangle}=S^{(0)}\left(n+2,1^{+}, 2\right) S^{(0)}\left(n+2,2^{+}, 3\right)$ - (eq.40) $\mathrm{DSL}^{(0)}\left(n+2,1^{+}, 2^{-}, 3\right)=\frac{1}{\left\langle n+2\left|q_{12}\right| 3\right]}\left[\frac{1}{2 k_{n+2} \cdot q_{12}} \frac{[n+2\,3]\langle n+22\rangle^{3}}{\langle 12\rangle\langle n+21\rangle}-\frac{1}{2 k_{3} \cdot q_{12}} \frac{\langle n+2\,3\rangle[31]^{3}}{[12][23]}\right]$ - two terms! -> no factorisation # Maldacena, Shenker, Stanford ## A bound on chaos \[Links: [arXiv](https://arxiv.org/abs/1503.01409), [PDF](https://arxiv.org/pdf/1503.01409.pdf)\] \[Abstract: We conjecture a sharp [[0474 Chaos bound|bound]] on the rate of growth of [[0008 Quantum chaos|chaos]] in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an [[0482 Out-of-time-order correlator|out-of-time-order correlation function]] closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with [[0466 Lyapunov exponent|Lyapunov exponent]] $\lambda_L \le 2 \pi k_B T/\hbar$. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.\] ## Summary - *conjectures* that chaos cannot propagate faster than exponentially, with Lyapunov exponent $\lambda_{L} \leq 2 \pi k_{B} T / \hbar$ ## Assumptions - simple $V$ and $W$ (in $C(t)=-\left\langle[W(t), V(0)]^{2}\right\rangle$) - describable by a sum of terms of order $O(1)$ - assumed to have zero thermal one-point function - analyticity - there is a large hierarchy between the dissipation time and the scrambling time - certain time-ordered correlation functions should approximately factorize - justified with some classes of physical systems with many d.o.f. ## Related topics - [[0008 Quantum chaos]] - [[0474 Chaos bound]] # Maldacena, Simmons-Duffin, Zhiboedov ## Looking for a bulk point \[Links: [arXiv](https://arxiv.org/abs/1509.03612), [PDF](https://arxiv.org/pdf/1509.03612.pdf)\] \[Abstract: We consider Lorentzian correlators of local operators. In perturbation theory, singularities occur when we can draw a position-space Landau diagram with null lines. In theories with gravity duals, we can also draw Landau diagrams in the bulk. We argue that certain singularities can arise only from bulk diagrams, not from boundary diagrams. As has been previously observed, these singularities are a clear diagnostic of bulk locality. We analyze some properties of these perturbative singularities and discuss their relation to the OPE and the dimensions of double-trace operators. In the exact nonperturbative theory, we expect no singularity at these locations. We prove this statement in 1+1 dimensions by CFT methods.\] ## Summary - argue that certain singularities can only arise from bulk null lines, not boundary ones -> [[0128 Bulk point singularity]] - show that non-perturbative theory has no singularity at those locations - still do at boundary null lines - prove in 1+1 - using CFT methods ## Beyond perturbation - finite string - expect no singularity - $D$-instantons - singular but cannot trust below a scale between Planck and string - finite $G_N$ # Marolf, Maxfield, Peach, Ross ## Hot multiboundary wormholes from bipartite entanglement \[Links: [arXiv](https://arxiv.org/abs/1506.04128), [PDF](https://arxiv.org/pdf/1506.04128.pdf)\] \[Abstract: \] ## Refs - [[2014#Balasubramanian, Hayden, Maloney, Marolf, Ross]] - earlier work, also on multiboundary wormhole and entanglement - [[0264 Multi-partite entanglement]] - although this paper is the mostly bipartite case # Miao ## Universal Terms of Entanglement Entropy for 6d CFTs \[Links: [arXiv](https://arxiv.org/abs/1503.05538), [PDF](https://arxiv.org/pdf/1503.05538.pdf)\] \[Abstract: \] # Moore ## Computation Of Some Zamolodchikov Volumes, With An Application \[Links: [arXiv](https://arxiv.org/abs/1508.05612), [PDF](https://arxiv.org/pdf/1508.05612.pdf)\] \[Abstract: We compute the Zamolodchikov volumes of some moduli spaces of conformal field theories with target spaces K3, T4, and their symmetric products. As an application we argue that sequences of conformal field theories, built from products of such symmetric products, almost never have a holographic dual with weakly coupled gravity.\] # Papallo, Reall ## Graviton time delay and a speed limit for small black holes in Einstein-Gauss-Bonnet theory \[Links: [arXiv](https://arxiv.org/abs/1508.05303), [PDF](https://arxiv.org/pdf/1508.05303)\] \[Abstract: [[2014#Camanho, Edelstein, Maldacena, Zhiboedov|Camanho, Edelstein, Maldacena and Zhiboedov]] have shown that gravitons can experience a negative Shapiro [[0118 Causality constraints for gravity|time delay]], i.e. a time advance, in Einstein-Gauss-Bonnet theory. They studied gravitons propagating in singular "[[0117 Shockwave|shock-wave]]" geometries. We study this effect for gravitons propagating in smooth black hole spacetimes. For a small enough black hole, we find that gravitons of appropriate polarisation, and small impact parameter, can experience time advance. Such gravitons can also exhibit a deflection angle less than $\pi$, characteristic of a repulsive short-distance gravitational interaction. We discuss problems with the suggestion that the time advance can be used to build a "time machine". In particular, we argue that a small black hole cannot be boosted to a speed arbitrarily close to the speed of light, as would be required in such a construction.\] # Pastawski, Yoshida, Harlow, Preskill ## Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence \[Links: [arXiv](https://arxiv.org/abs/1503.06237), [PDF](https://arxiv.org/pdf/1503.06237.pdf)\] \[Abstract: We propose a family of exactly solvable toy models for the [[0001 AdS-CFT|AdS/CFT]] correspondence based on a novel construction of [[0146 Quantum error correction|quantum error-correcting codes]] with a [[0054 Tensor network|tensor network]] structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the [[0007 RT surface|Ryu-Takayanagi formula]] and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindler-wedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed by [[2014#Almheiri, Dong, Harlow|Almheiri et. al]] in [arXiv:1411.7041](https://arxiv.org/abs/1411.7041).\] # Pasterski, Strominger, Zhiboedov ## New Gravitational Memories \[Links: [arXiv](https://arxiv.org/abs/1502.06120), [PDF](https://arxiv.org/pdf/1502.06120.pdf)\] \[Abstract: The conventional gravitational memory effect is a relative displacement in the position of two detectors induced by radiative energy flux. We find a new type of gravitational 'spin memory' in which beams on clockwise and counterclockwise orbits acquire a relative delay induced by radiative angular momentum flux. It has recently been shown that the displacement memory formula is a Fourier transform in time of Weinberg's soft graviton theorem. Here we see that the spin memory formula is a Fourier transform in time of the recently-discovered subleading soft graviton theorem.\] ## Refs - [[0287 Memory effect]] # Prabhu ## The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom \[Links: [arXiv](https://arxiv.org/abs/1511.00388), [PDF](https://arxiv.org/pdf/1511.00388.pdf)\] \[Abstract: We derive the first law of [[0127 Black hole thermodynamics|black hole mechanics]] for physical theories based on a local, covariant and gauge-invariant Lagrangian where the dynamical fields transform non-trivially under the action of internal gauge transformations. The theories of interest include General Relativity formulated in terms of tetrads, Einstein-Yang-Mills theory and Einstein-Dirac theory. Since the dynamical fields of these theories have gauge freedom, we argue that there is no group action of diffeomorphisms of spacetime on such dynamical fields. In general, such fields cannot even be represented as smooth, globally well-defined tensor fields on spacetime. Consequently the derivation of the first law by Iyer-Wald cannot be used directly. We show how such theories can be formulated on a principal bundle and that there is a natural action of automorphisms of the bundle on the fields. These bundle automorphisms encode both spacetime diffeomorphisms and gauge transformations. Using this reformulation we define the Noether charge associated to an infinitesimal automorphism and the corresponding notion of stationarity and axisymmetry of the dynamical fields. We define certain potentials and charges at the horizon of a black hole so that the potentials are constant on the bifurcate Killing horizon, giving a generalised zeroth law for bifurcate Killing horizons. We identify the gravitational potential and perturbed charge as the temperature and perturbed entropy of the black hole which gives an explicit formula for the perturbed entropy analogous to the [[0559 Wald entropy|Wald entropy]] formula. We obtain a general first law of black hole mechanics for such theories. The first law relates the perturbed Hamiltonians at spatial infinity and the horizon, and the horizon contributions take the form of a potential times perturbed charge term. We also comment on the ambiguities in defining a prescription for the total entropy for black holes.\] # Ribault, Santachiara ## Liouville theory with a central charge less than one \[Links: [arXiv](https://arxiv.org/abs/1503.02067), [PDF](https://arxiv.org/pdf/1503.02067)\] \[Abstract: We determine the spectrum and correlation functions of [[0562 Liouville theory|Liouville theory]] with a central charge less than (or equal) one. This completes the definition of [[0562 Liouville theory|Liouville theory]] for all complex values of the central charge. The spectrum is always spacelike, and there is no consistent [[0622 Timelike Liouville|timelike Liouville theory]]. We also study the non-analytic conformal field theories that exist at rational values of the central charge. Our claims are supported by numerical checks of [[0021 Crossing symmetry|crossing symmetry]]. We provide Python code for computing Virasoro conformal blocks, and correlation functions in Liouville theory and (generalized) minimal models.\] # Stieberger, Taylor (Feb) ## Graviton amplitudes from collinear limits of gauge amplitudes \[Links: [arXiv](https://arxiv.org/abs/1502.00655), [PDF](https://arxiv.org/pdf/1502.00655.pdf)\] \[Abstract: \] ## Comments - The graviton amplitude is the full amplitude, not a collinear limit. - $2N-2$ gluons are paired up and have the same momentum with each other while 2 gluons are taken collinear and cause a divergent pole. ## Main result The main result is that the $N$-graviton amplitude in GR can be written as a collinear limit of $2N$ gluon in YM as follows: For positive helicity graviton: $\eqalign{A_{E}[k_{1},&\lambda_{1};\dots;k_{N{-}1},\lambda_{N{-}1};k_{N}=p+q,\lambda_{N}=+2]=\lim_{[pq]\to 0}\bigg{(}{1\over 2x}\bigg{)}^{4}{[pq]\over\langle pq\rangle}s_{pq}^{2}\cr&\times\sum_{\pi,\rho\in S_{N-3}}S[\pi|\rho]\ A[p,N{-}1,1,\pi(2,3,\ldots,N{-}2),1,\rho(2,\ldots,N{-}2),N{-}1,q]}$For negative helicity graviton: $\eqalign{A_{E}[k_{1},&\lambda_{1};\dots;k_{N{-}1},\lambda_{N{-}1};k_{N}=p+q,% \lambda_{N}=-2]=\lim_{\langle pq\rangle\to 0}\big{(}2x\big{)}^{4}{\langle pq% \rangle\over[pq]}s_{pq}^{2}\cr&\times\sum_{\pi,\rho\in S_{N-3}}S[\pi|\rho]\ A[% p,N{-}1,1,\pi(2,3,\ldots,N{-}2),1,\rho(2,\ldots,N{-}2),N{-}1,q]}$ ## Idea of proof The idea is to look for the triple pole in $2N$ YM scattering, at which the amplitude factors into two factors which precise match to GR graviton amplitudes due to [[0067 Double copy]]. # Stieberger, Taylor (Aug) ## Subleading terms in the collinear limit of YM amplitudes \[Links: [arXiv](https://arxiv.org/abs/1508.01116), [PDF](https://arxiv.org/pdf/1508.01116.pdf)\] \[Abstract: For two massless particles $i$ and $j$, the [[0078 Collinear limit|collinear limit]] is a special kinematic configuration in which the particles propagate with parallel four-momentum vectors, with the total momentum $P$ distributed as $p_i=xP$ and $p_j=(1-x)P$, so that $s_{ij}=(p_i+p_j)^2=P^2=0$. In Yang-Mills theory, if $i$ and $j$ are among $N$ gauge bosons participating in a scattering process, it is well known that the partial amplitudes associated to the (single trace) group factors with adjacent $i$ and $j$ are singular in the collinear limit and factorize at the leading order into $(N-1)$-particle amplitudes times the universal, $x$-dependent Altarelli-Parisi factors. We give a precise definition of the collinear limit and show that at the tree level, the subleading, non-singular terms are related to the amplitudes with a single graviton inserted instead of two collinear gauge bosons. To that end, we argue that in one-graviton Einstein-Yang-Mills amplitudes, the graviton with momentum $P$ can be replaced by a pair of collinear gauge bosons carrying arbitrary momentum fractions $xP$ and $(1-x)P$.\] ## Refs - [[0010 Celestial holography]] # Verlinde ## Poking Holes in AdS/CFT: Bulk Fields from Boundary States \[Links: [arXiv](https://arxiv.org/abs/1505.05069), [PDF](https://arxiv.org/pdf/1505.05069)\] \[Abstract: We propose an intrinsic CFT definition of local bulk operators in [[0073 AdS3-CFT2|AdS3/CFT2]] in terms of twisted Ishibashi boundary states. The bulk field $\Phi(X)$ creates a [[0620 Non-orientable CFT|cross cap]], a circular hole with opposite edge points identified, in the CFT space-time. The size of the hole is parameterized by the holographic radial coordinate $y$. Our definition is state-independent, non-perturbative, and does not presume or utilize a semi-classical bulk geometry. We argue that, at large [[0033 Central charge|central charge]], the matrix element between highly excited states satisfies the bulk wave equation in the AdS black hole background.\] # Wall (Essay) ## A second law for higher curvature gravity \[Links: [arXiv](https://arxiv.org/abs/1504.08040), [PDF](https://arxiv.org/pdf/1504.08040.pdf)\] \[Abstract: The [[0005 Black hole second law|Second Law of black hole thermodynamics]] is shown to hold for arbitrarily complicated theories of [[0006 Higher-derivative gravity|higher curvature gravity]], so long as we allow only linearized perturbations to stationary black holes. Some ambiguities in [[0019 Covariant phase space|Wald's Noether charge method]] are resolved. The increasing quantity turns out to be the same as the [[0145 Generalised area|holographic entanglement entropy]] calculated by [[2013#Dong]]. It is suggested that only the linearization of the higher-curvature Second Law is important, when consistently truncating a UV-complete quantum gravity theory.\] ## Summary - [[0005 Black hole second law|black hole second law]] and [[0082 Generalised second law|generalised second law]] holds for linearised pert. to stationary BH for higher derivative gravity - suggest that only linear perturbation important - an increasing quantity is found to exist, and *then* we compare it with entropy and find that it is the correct entropy in [[2013#Dong]] for $f$(Riemann) - this may not be true for more general theories ## Refs - extension of - ref.5 [[2013#Sarkar, Wall]]: f(Lovelock) - ref.6 [[2015#Bhattacharjee, Sarkar, Wall]]: quadratic gravity - extended to - [[2019#Bhattacharya, Bhattacharyya, Dinda, Kundu]]: 4-deriv - [[2021#Bhattacharyya, Dhivakar, Dinda, Kundu, Patra, Roy]]: arbitrary theory - [[2022#Hollands, Kovacs, Reall]]: quadratic in perturbation ## Typos - eq.2: $H_{\mu\nu}=\frac{2}{\sqrt{-g}} \frac{\delta (\sqrt {-g} L_g)}{\delta g^{\mu \nu}}$ ## Proof 1. pick gauge on horizon so that any tensor with weight $n$ always has at least $n$ $v$-derivatives acting on it. 2. define Killing weight: no. $v$-indices minus no. $u$-indices 3. identify JKM ambiguities as boost invariant products of weights n and -n. - (key) write delta $H_{vv}=T_{vv}$ as the second deriv. of some quantity (wrt. v) - n.b. $H_{vv}$ is well-defined without further requirements. the only thing is to write it in a desired form - $\delta H_{vv}=\sum_{n\ge0} X^{(-n)}\delta Y^{(2+n)}$. $n\ge0$ because $X$ with positive weight would be divergent on the horizon by Killing symmetry - (key) show that $\iota$ is exact - easy for [[0018 JKM ambiguity]]: since $Y^{(n>0)}=0$ on background horizon, so $X^{(-n)}\delta Y^{(n)}=\delta(X^{(-n)}Y^{(n)})$ ## A review (in [[2019#Bhattacharya, Bhattacharyya, Dinda, Kundu]]) - define generalised expansion $\partial_{v} S=\int_{\mathcal{H}_{v}} \partial_{v}\left(\sqrt{h}\left(1+s_{n}\right)\right) \equiv \int_{\mathcal{H}_{v}} \sqrt{h} \vartheta$ - then $\partial_{v} \vartheta=\partial_{v} \vartheta_{E}+\partial_{v}\left(\frac{1}{\sqrt{h}} \partial_{v}\left(\sqrt{h} s_{n}\right)\right)$=-T_{v v}+E_{v v}^{\mathrm{HD}}+\partial_{v}\left(\frac{1}{\sqrt{h}} \partial_{v}\left(\sqrt{h} s_{n}\right)\right)+\mathcal{O}\left(\epsilon^{2}\right)$ - now it is sufficient to show $E_{v v}^{\mathrm{HD}} \|_{\text {offshell }}=\partial_{v}\left(\frac{1}{\sqrt{h}} \partial_{v}(\sqrt{h} \varsigma)\right)+\mathcal{O}\left(\epsilon^{2}\right)$ - so that $\partial_{v} \vartheta=-T_{vv}+O(\epsilon^2)<0$ and since $\vartheta\to0$ , $\vartheta>0$. QED. ## Gauge choice and its implications - $g_{vv}=g_{vi}=g_{vv,u}=0,\; g_{uv}=1$ at $u=0$ - $g_{uu}=g_{ui}=0$ everywhere - derivation - see [[2016#Bhattacharyya, Haehl, Kundu, Loganayagam, Rangamani]] and [[2021#Bhattacharyya, Dhivakar, Dinda, Kundu, Patra, Roy]] - In this gauge, any tensor with with positive weight $n$ always has at least $n$ $v$-derivatives acting on it. If one wants to include vectors, one may need to do some gauge fixing for the vector fields in order to have this property. ## Explicit calculation for f(Riem) - $H_{vv}$ can only arise from varying $g^{ab}$ or $R_{abcd}$ wrt $g^{vv}$ - only the latter can produce derivatives of Riemann -> anomaly term ## From [[0005 Black hole second law|second law]] to [[0082 Generalised second law|GSL]] add matter fields; semiclassical; Ref 5, 22 ## Questions Q1. If theory has gravitational fields with spin, need gauge symmetry too. What are these? - In general there could be many higher spins, and sometimes gravity fields and matter fields mix in a way so it is not obvious how to separate the two. - E.g. [[0013 Vasiliev theory|Vasiliev theory]] has very large gauge symmetry, and diffeo is just one slice of that - the gauge symmetry is supposed to help the higher spin fields obtain a similar property to the metric: the relation between weight and number of $v$ derivatives # Winter ## Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints \[Links: [arXiv](https://arxiv.org/abs/1507.07775), [PDF](https://arxiv.org/pdf/1507.07775.pdf)\] \[Abstract: We present a bouquet of continuity bounds for quantum entropies, falling broadly into two classes: First, a tight analysis of the Alicki-Fannes continuity bounds for the conditional [[0301 Entanglement entropy|von Neumann entropy]], reaching almost the best possible form that depends only on the system dimension and the trace distance of the states. Almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators. As applications we give new proofs, with tighter bounds, of the asymptotic continuity of the [[0199 Relative entropy|relative entropy]] of entanglement, $E_R$, and its regularization $E^\infty_R$, as well as of the entanglement of formation, $E_F$. Using a novel "quantum coupling" of density operators, which may be of independent interest, we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation, aka entanglement cost, $E_C=E^\infty_F$. Second, analogous continuity bounds for the von Neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint, most importantly systems of multiple quantum harmonic oscillators. While without an energy bound the entropy is discontinuous, it is well-known to be continuous on states of bounded energy. However, a quantitative statement to that effect seems not to have been known. Here, under some regularity assumptions on the Hamiltonian, we find that, quite intuitively, the Gibbs entropy at the given energy roughly takes the role of the Hilbert space dimension in the finite-dimensional Fannes inequality.\] ## Summary - *reviews* bounds on the [[0275 Discontinuity of von Neumann entropy|discontinuity of von Neumann entropy]] for finite dimensional Hilbert space - *gives* quantitative statement about the [[0275 Discontinuity of von Neumann entropy|discontinuity of von Neumann entropy]] for *infinite* dimensional space