# Afkhami-Jeddi, Colville, Hartman, Maloney, Perlmutter ## Constraints on Higher Spin CFT$_2$ \[Links: [arXiv](https://arxiv.org/abs/1707.07717), [PDF](https://arxiv.org/pdf/1707.07717.pdf)\] \[Abstract: We derive constraints on two-dimensional conformal field theories with higher spin symmetry due to [[0035 Unitarity of CFT|unitarity]], [[0612 Modular invariance|modular invariance]], and [[0132 Causality constraints in CFT|causality]]. We focus on CFTs with $\mathcal{W}_N$ symmetry in the "irrational" regime, where $c>N-1$ and the theories have an infinite number of higher-spin primaries. The most powerful constraints come from positivity of the Kac matrix, which (unlike the Virasoro case) is non-trivial even when $c>N-1$. This places a lower bound on the dimension of any non-vacuum higher-spin primary state, which is linear in the [[0033 Central charge|central charge]]. At large $c$, this implies that the dual holographic theories of gravity in AdS$_3$, if they exist, have no local, perturbative degrees of freedom in the semi-classical limit.\] # Afkhami-Jeddi, Hartman, Kundu, Tajdini ## Shockwaves from the Operator Product Expansion \[Links: [arXiv](https://arxiv.org/abs/1709.03597), [PDF](https://arxiv.org/pdf/1709.03597.pdf)\] \[Abstract: We clarify and further explore the CFT [[0129 Dual of shockwaves|dual of shockwave]] geometries in Anti-de Sitter. The shockwave is dual to a CFT state produced by a heavy local operator inserted at a complex point. It can also be created by light operators, smeared over complex positions. We describe the dictionary in both cases, and compare to various calculations, old and new. In CFT, we analyze the [[0030 Operator product expansion|operator product expansion]] in the Regge limit, and find that the leading contribution is exactly the shockwave operator, $\int du h_{uu}$, localized on a bulk geodesic. For heavy sources this is a simple consequence of [[0028 Conformal symmetry|conformal invariance]], but for light operators it involves a smearing procedure that projects out certain double-trace contributions to the OPE. We revisit [[0132 Causality constraints in CFT|causality constraints]] in large-$N$ CFT from this perspective, and show that the [[0474 Chaos bound|chaos bound]] in CFT coincides with a bulk condition proposed by [[2016#Engelhardt, Fischetti]]. In particular states, this reproduces known constraints on CFT 3-point couplings, and confirms some assumptions about double-trace operators made in previous work.\] ## Summary - describes the dictionary in the second and third CFT description of a [[0117 Shockwave|shockwave]], i.e. heavy operator and complex position - analyses the [[0030 Operator product expansion|OPE]] in the ==Regge limit== - find the ==leading== contribution to be the shockwave operator $\int \mathrm{d}u\,h_{uu}$ - causality constraint in ==large-$N$== CFT - [[0474 Chaos bound|chaos bound]] (CFT) <-> Engelhardt-Fischetti ([[2016#Engelhardt, Fischetti]]) condition (bulk) ## Why OPE - (correlator and OPEs ultimately contain the same info.) 1. analogous results in the lightcone limit were useful 2. makes clear the match between bulk and boundary without doing calculations - can extend this match to more general quantum states 3. generalise existing constraints to higher-point functions and multiple shocks ## Refs - earlier paper [[2016#Afkhami-Jeddi, Hartman, Kundu, Tajdini]] # Akers, Chandrasekaran, Leichenauer, Levine, Shahbazi-Moghaddam ## QNEC, EWN, and QFC \[Links: [arXiv](https://arxiv.org/abs/1706.04183), [PDF](https://arxiv.org/pdf/1706.04183.pdf)\] \[Abstract: We study the consequences of [[0142 Entanglement wedge nesting|Entanglement Wedge Nesting]] for CFTs with holographic duals. The CFT is formulated on an arbitrary curved background, and we include the effects of curvature-squared couplings in the bulk. In this setup we find necessary and sufficient conditions for Entanglement Wedge Nesting to imply the [[0405 Quantum null energy condition|Quantum Null Energy Condition]] in $d\leq 5$, extending its earlier holographic proofs. We also show that the [[0243 Quantum focusing conjecture|Quantum Focusing Conjecture]] yields the Quantum Null Energy Condition as its nongravitational limit under these same conditions.\] ## Summary - *finds* necessary and sufficient conditions for [[0142 Entanglement wedge nesting|EWN]] to imply [[0405 Quantum null energy condition]]${}_{\partial}$ when: - CFT on arbitrary curved background - bulk includes [[0006 Higher-derivative gravity]] to curvature squared order - *shows* [[0243 Quantum focusing conjecture]] implies [[0405 Quantum null energy condition]] in the $G_N\to0$ limit ## Refs - extension of [[2016#Akers, Koeller, Leichenauer, Levine]] # Balakrishnan, Faulkner, Khandker, Wang ## A general proof of the QNEC \[Links: [arXiv](https://arxiv.org/abs/1706.09432), [PDF](https://arxiv.org/pdf/1706.09432.pdf)\] \[Abstract: We prove a conjectured lower bound on $\left< T_{--}(x) \right>_\psi$ in any state $\psi$ of a relativistic QFT dubbed the [[0405 Quantum null energy condition|Quantum Null Energy Condition]] (QNEC). The bound is given by the second order shape deformation, in the null direction, of the geometric entanglement entropy of an entangling cut passing through $x$. Our proof involves a combination of the two independent methods that were used recently to prove the weaker [[0417 Averaged null energy condition|Averaged Null Energy Condition]] (ANEC). In particular the properties of [[0416 Modular Hamiltonian|modular Hamiltonians]] under shape deformations for the state $\psi$ play an important role, as do causality considerations. We study the two point function of a "probe" operator $\mathcal{O}$ in the state $\psi$ and use a lightcone limit to evaluate this correlator. Instead of causality in time we consider *causality in modular time* for the modular evolved probe operators, which we constrain using Tomita-Takesaki theory as well as certain generalizations pertaining to the theory of modular inclusions. The QNEC follows from very similar considerations to the derivation of the chaos bound and the causality sum rule. We use a kind of defect [[0030 Operator product expansion|Operator Product Expansion]] to apply the replica trick to these modular flow computations, and the displacement operator plays an important role. Our approach was inspired by the AdS/CFT proof of the QNEC which follows from properties of the [[0007 RT surface|Ryu-Takayanagi (RT) surface]] near the boundary of AdS, combined with the requirement of [[0142 Entanglement wedge nesting|entanglement wedge nesting]]. Our methods were, as such, designed as a precise probe of the RT surface close to the boundary of a putative gravitational/stringy dual of *any* QFT with an interacting UV fixed point. We also prove a higher spin version of the QNEC.\] ## Summary - *proves* [[0405 Quantum null energy condition|QNEC]] - *using* (1) properties of [[0416 Modular Hamiltonian]] under shape deformation and (2) causality considerations - *proves* also a higher spin version of [[0405 Quantum null energy condition|QNEC]] # Banerjee, Banerjee, Bhatkar, Jain ## Conformal structure of massless scalar amplitudes beyond tree levels \[Links: [arXiv](https://arxiv.org/abs/1711.06690), [PDF](https://arxiv.org/pdf/1711.06690.pdf)\] \[Abstract: We show that the one-loop on-shell four-point scattering amplitude of massless $\phi^4$ scalar field theory in 4D Minkowski space time, when [[0079 Mellin transform|Mellin transformed]] to the [[0022 Celestial sphere|Celestial sphere]] at infinity, transforms covariantly under the global conformal group ($SL(2,C)$) on the sphere. The unitarity of the four-point scalar amplitudes is recast into this Mellin basis. We show that the same conformal structure also appears for the two-loop Mellin amplitude. Finally we comment on some universal structure for all loop four-point Mellin amplitudes specific to this theory.\] ## Refs - [[0260 Celestial loops]] ## Results - one-loop on-shell 4-point amplitude - momentum space $A=-i(2 \pi)^{4} \delta^{4}\left(\Sigma p_{i}\right)\left[\lambda_{R}-\frac{i \pi \lambda_{R}^{2}}{32 \pi^{2}}+\frac{\lambda_{R}^{2}}{32 \pi^{2}}\left(\ln \frac{s}{\mu^{2}}+\ln \frac{|t|}{\mu^{2}}+\ln \frac{|u|}{\mu^{2}}\right)\right]$ - Mellin space $\tilde{\mathcal{T}}_{2}=\frac{i \lambda_{R}^{2}}{4}\left(\frac{2}{\mu}\right)^{-i \Lambda}\left[6 \pi^{3} \delta^{\prime}(\Lambda)+\pi^{4} \delta(\Lambda)\right] \delta(|z-\bar{z}|)\left(\prod_{i<j}^{4}\left|z_{i j}\right|^{h / 3-h_{i}-h_{j}}\left|\bar{z}_{i j}\right|^{\bar{h} / 3-\bar{h}_{i}-\bar{h}_{j}}\right)[z(z-1)]^{2 / 3}$ - two-loop - $\tilde{\mathcal{T}}_{3} \sim \lambda_{R}^{3} \delta^{\prime \prime}(\Lambda) \delta(|z-\bar{z}|) \times\left(\prod_{i<j}^{4}\left|z_{i j}\right|^{h / 3-h_{i}-h_{j}}\left|\bar{z}_{i j}\right|^{h / 3-h_{i}-h_{j}}\right)$\left(\frac{2}{\mu}\right)^{-i \Lambda}[z(z-1)]^{2 / 3}\left\{z^{-i \frac{\Lambda}{3}}(z-1)^{i \frac{\Lambda}{6}}+z^{i \frac{\Lambda}{6}}(z-1)^{i \frac{\Lambda}{6}}+(z-1)^{-i \frac{\Lambda}{3}} z^{i \frac{\Lambda}{6}}\right\}$ # Bao, Ooguri ## On distinguishability of black hole microstates \[Links: [arXiv](https://arxiv.org/abs/1705.07943), [PDF](https://arxiv.org/pdf/1705.07943.pdf)\] \[Abstract: We use the [[0268 Holevo information|Holevo information]] to estimate distinguishability of microstates of a black hole in anti-de Sitter space by measurements one can perform on a subregion of a Cauchy surface of the dual conformal field theory. We find that microstates are not distinguishable at all until the subregion reaches a certain size and that perfect distinguishability can be achieved before the subregion covers the entire Cauchy surface. We will compare our results with expectations from the [[0219 Entanglement wedge reconstruction|entanglement wedge reconstruction]], [[0054 Tensor network|tensor network]] models, and the [[0211 Bit thread|bit threads]] interpretation of the [[0007 RT surface]].\] ## Summary - reviews [[0248 Black hole microstates]] # Blake, Davison, Sachdev ## Thermal diffusivity and chaos in metals without quasiparticles \[Links: [arXiv](https://arxiv.org/abs/1705.07896), [PDF](https://arxiv.org/pdf/1705.07896.pdf)\] \[Abstract: \] ## Summary - identifies the thermal diffusivity as the correct quantity related to [[0008 Quantum chaos]] - expresses [[0434 Diffusivity]] solely in terms of the near-horizon geometry ## Equations - thermal diffusivity - $D_{T} \equiv \frac{\kappa}{c_{\rho}}$ - thermodynamic specific heat at fixed density: - $c_{\rho}=T(\partial s / \partial T)_{\rho}$ - fixed solely by near-horizon geometry - the open circuit thermal conductivity is given by - $\kappa=\left.4 \pi \frac{f^{\prime} h^{d-2}}{\left(f^{\prime} h^{d / 2-1}\right)^{\prime}}\right|_{r_{0}}$ - also solely fixed by near-horizon geometry *after* using some Einstein equation - then taking the ratio gives a diffusivity expression solely dependent on near-horizon geometry - next, using *known* expressions for $v_B$ and $\tau_L$ (for Einstein gravity), one obtains - $\left.D_{T} \sim \frac{f^{\prime} h^{d / 2-1}}{\left(f^{\prime} h^{d / 2-1}\right)^{\prime}} \frac{h^{\prime}}{h}\right|_{r_{0}} v_{B}^{2} \tau_{L}$ - finally, if one considers Lifshitz / hyperscaling-violating geometries in the infrared - $D_{T}=\frac{z}{2 z-2} v_{B}^{2} \tau_{L}$ - $z$ is the dynamical critical exponent of the infrared fixed point ## Important insight - $\kappa$ depends on the matter fields, but this dependence can be removed by using the equations of motion for the background geometry - insight: the only way the matter fields appear in the thermal conductivity is through components of the stress tensor, and hence we are always able to eliminate them in favour of the geometry using the Einstein equations (not true for e.g. electrical conductivity etc) # Bzowski, McFadden, Skenderis ## Renormalised 3-point functions of stress tensors and conserved currents in CFT \[Links: [arXiv](https://arxiv.org/abs/1711.09105), [PDF](https://arxiv.org/pdf/1711.09105)\] \[Abstract: We present a complete momentum-space prescription for the renormalisation of tensorial correlators in conformal field theories. Our discussion covers all [[0633 CFT correlators|3-point functions]] of stress tensors and conserved currents in arbitrary spacetime dimensions. In dimensions three and four, we give explicit results for the renormalised correlators, the anomalous [[0106 Ward identity|Ward identities]] they obey, and the [[0306 Weyl anomaly|conformal anomalies]]. For the stress tensor 3-point function in four dimensions, we identify the specific evanescent tensorial structure responsible for the type A Euler anomaly, and show this anomaly has the form of a double copy of the chiral anomaly.\] # Cardoso, Franzin, Maselli, Pani, Raposo ## Testing strong-field gravity with tidal Love numbers \[Links: [arXiv](https://arxiv.org/abs/1701.01116), [PDF](https://arxiv.org/pdf/1701.01116.pdf)\] \[Abstract: The [[0581 Tidal Love numbers|tidal Love numbers]] (TLNs) encode the deformability of a self-gravitating object immersed in a tidal environment and depend significantly both on the object's internal structure and on the dynamics of the gravitational field. An intriguing result in classical [[0554 Einstein gravity|general relativity]] is the vanishing of the TLNs of black holes. We extend this result in three ways, aiming at testing the nature of compact objects: (i) we compute the TLNs of exotic compact objects, including different families of boson stars, gravastars, wormholes, and other toy models for quantum corrections at the horizon scale. In the black-hole limit, we find a universal logarithmic dependence of the TLNs on the location of the surface; (ii) we compute the TLNs of black holes beyond vacuum general relativity, including Einstein-Maxwell, [[0317 Brans-Dicke theory|Brans-Dicke]] and Chern-Simons gravity; (iii) We assess the ability of present and future gravitational-wave detectors to measure the TLNs of these objects, including the first analysis of TLNs with LISA. Both LIGO, ET and LISA can impose interesting constraints on boson stars, while LISA is able to probe even extremely compact objects. We argue that the TLNs provide a smoking gun of new physics at the horizon scale, and that future gravitational-wave measurements of the TLNs in a binary inspiral provide a novel way to test black holes and general relativity in the strong-field regime.\] # Cardy, Maloney, Maxfield ## A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance \[Links: [arXiv](https://arxiv.org/abs/1705.05855), [PDF](https://arxiv.org/pdf/1705.05855.pdf)\] \[Abstract: We derive an asymptotic formula for [[0030 Operator product expansion|operator product expansion]] coefficients of heavy operators in two dimensional conformal field theory. This follows from [[0612 Modular invariance|modular invariance]] of the genus two partition function, and generalises the asymptotic formula for the density of states from torus modular invariance. The resulting formula is universal, depending only on the [[0033 Central charge|central charge]], but involves the asymptotic behaviour of genus two [[0031 Conformal block|conformal blocks]]. We use monodromy techniques to compute the asymptotics of the relevant blocks at large central charge to determine the behaviour explicitly.\] # Caron-Huot ## Analyticity in Spin in Conformal Theories \[Links: [arXiv](https://arxiv.org/abs/1703.00278), [PDF](https://arxiv.org/pdf/1703.00278.pdf)\] \[Abstract: Conformal theory correlators are characterized by the spectrum and three-point functions of local operators. We present a formula which extracts this data as an analytic function of spin. In analogy with a classic formula due to Froissart and Gribov, it is sensitive only to an "imaginary part" which appears after analytic continuation to Lorentzian signature, and it converges thanks to recent bounds on the high-energy Regge limit. At large spin, substituting in cross-channel data, the formula yields $1/J$ expansions with controlled errors. In large-$N$ theories, the imaginary part is saturated by single-trace operators. For a sparse spectrum, it manifests the suppression of bulk higher-derivative interactions that constitutes the signature of a local gravity dual in Anti-de-Sitter space.\] # Chen, Fitzpatrick, Kaplan, Li ## The AdS${}_3$ propagator and the fate of locality \[Links: [arXiv](https://arxiv.org/abs/1712.02351), [PDF](https://arxiv.org/pdf/1712.02351.pdf)\] \[Abstract: \] ## Summary - study the propagator $\langle\phi\phi\rangle$ for a proto-field $\phi$ in ==AdS${}_3$== - the propagator is fine at long distances but has UV/IR mixing in it perturbative expansion in $G_N=\frac{3}{2c}$ - locality breakdown manifest as singularities or branch cuts at spacelike separation arising from non-perturbative QG effects ## Refs - based on earlier definition of a proto-field in [[AnandChenFitzpatrickKaplanLi2017]] # Cho, Collier, Yin ## Genus Two Modular Bootstrap \[Links: [arXiv](https://arxiv.org/abs/1705.05865), [PDF](https://arxiv.org/pdf/1705.05865.pdf)\] \[Abstract: We study the [[0032 Virasoro algebra|Virasoro]] [[0031 Conformal block|conformal block]] decomposition of the genus two partition function of a two-dimensional CFT by expanding around a $\mathbb{Z}_3$-invariant Riemann surface that is a three-fold cover of the Riemann sphere branched at four points, and explore constraints from genus two [[0612 Modular invariance|modular invariance]] and [[0035 Unitarity of CFT|unitarity]]. In particular, we find 'critical surfaces' that constrain the structure constants of a CFT beyond what is accessible via the crossing equation on the sphere.\] # Choi, Akhoury ## BMS Supertranslation Symmetry Implies Faddeev-Kulish Amplitudes \[Links: [arXiv](https://arxiv.org/abs/1712.04551), [PDF](https://arxiv.org/pdf/1712.04551.pdf)\] \[Abstract: We show explicitly that, among the scattering amplitudes constructed from eigenstates of the BMS supertranslation charge, the ones that conserve this charge, are equal to those constructed from [[0272 Faddeev-Kulish|Faddeev-Kulish]] states. Thus, Faddeev-Kulish states naturally arise as a consequence of the asymptotic symmetries of perturbative gravity and all charge conserving amplitudes are infrared finite. In the process we show an important feature of the Faddeev-Kulish clouds dressing the external hard particles: these clouds can be moved from the incoming states to the outgoing ones, and vice-versa, without changing the infrared finiteness properties of S matrix elements. We also apply our discussion to the problem of the decoherence of momentum configurations of hard particles due to soft boson effects.\] # Chu, Miao, Guo (Long) ## On new proposal for holographic BCFT \[Links: [arXiv](https://arxiv.org/abs/1701.07202), [PDF](https://arxiv.org/pdf/1701.07202.pdf)\] \[Abstract: \] ## Refs - extension of short paper [[ChuMiaoGuo2017short]] ## Summary - *obtains* boundary [[0306 Weyl anomaly]] - [[0209 Holographic renormalisation]] for [[0181 AdS-BCFT]] - *proposes* to use mixed BC on the branes # Cotler, Hunter-Jones, Liu, Yoshida ## Chaos, Complexity, and Random Matrices \[Links: [arXiv](https://arxiv.org/abs/1706.05400), [PDF](https://arxiv.org/pdf/1706.05400.pdf)\] \[Abstract: [[0008 Quantum chaos|Chaos]] and [[0204 Quantum complexity|complexity]] entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute [[0482 Out-of-time-order correlator|out-of-time-ordered correlation functions]] (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an $\mathcal{O}(1)$ scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce $k$-invariance as a precise definition of what it means for the dynamics of a quantum system to be described by random matrix theory. We envision that the dynamical onset of approximate $k$-invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by [[0579 Random matrix theory|random matrix theory]].\] # Das, Datta, Pal (Jun) ## Charged structure constants from modularity \[Links: [arXiv](https://arxiv.org/abs/1706.04612), [PDF](https://arxiv.org/pdf/1706.04612.pdf)\] \[Abstract: We derive a universal formula for the average heavy-heavy-light structure constants for 2d CFTs with non-vanishing $u(1)$ charge. The derivation utilizes the modular properties of one-point functions on the torus. Refinements in N=2 SCFTs, show that the resulting [[0406 Cardy formula|Cardy]]-like formula for the structure constants has precisely the same shifts in the [[0033 Central charge|central charge]] as that of the thermodynamic entropy found earlier. This analysis generalizes the recent results by [[2016#Kraus, Maloney|Kraus and Maloney]] for CFTs with an additional global $u(1)$ symmetry. Our results at large central charge are also shown to match with computations from the holographic dual, which suggest that the averaged CFT three-point coefficient also serves as an useful probe of detecting black hole hair.\] # Das, Datta, Pal (Dec) ## Modular crossings, OPE coefficients and black holes \[Links: [arXiv](https://arxiv.org/abs/1712.01842), [PDF](https://arxiv.org/pdf/1712.01842.pdf)\] \[Abstract: In (1+1)-d CFTs, the 4-point function on the plane can be mapped to the pillow geometry and thereby crossing symmetry gets translated into a modular property. We use these [[0612 Modular invariance|modular]] features to derive a universal asymptotic formula for [[0030 Operator product expansion|OPE]] coefficients in which one of the operators is averaged over heavy primaries. The coarse-grained heavy channel then reproduces features of the gravitational 2-to-2 S-matrix which has black holes as their intermediate states.\] # Dong, Lewkowycz ## Entropy, Extremality, Euclidean Variations, and the Equations of Motion \[Links: [arXiv](https://arxiv.org/abs/1705.08453), [PDF](https://arxiv.org/pdf/1705.08453.pdf)\] \[Abstract: We study the Euclidean gravitational path integral computing the [[0293 Renyi entropy|Rényi entropy]] and analyze its behavior under small variations. We argue that, in Einstein gravity, the extremality condition can be understood from the variational principle at the level of the action, without having to solve explicitly the equations of motion. This set-up is then generalized to arbitrary theories of gravity, where we show that the respective [[0145 Generalised area|entanglement entropy functional]] needs to be extremized. We also extend this result to all orders in Newton’s constant $G_N$ , providing a derivation of quantum extremality. Understanding quantum extremality for mixtures of states provides a generalization of the dual of the boundary [[0416 Modular Hamiltonian|modular Hamiltonian]] which is given by the bulk modular Hamiltonian plus the area operator, evaluated on the so-called modular extremal surface. This gives a bulk prescription for computing the relative entropies to all orders in $G_N$. We also comment on how these ideas can be used to derive an integrated version of the equations of motion, linearized around arbitrary states.\] ## Summary - *argues* that the extremality condition follows from variational principle at the level of the action, not needing to solve EOM - true for arbitrary theories of gravity - and for all orders in $G_N$ (quantum extremality) - *derives* a quantum [[0048 JLMS|JLMS]] formula ## qJLMS - $S_{\mathrm{rel}}(\rho \mid \sigma)=\left\langle A_{\mathrm{gen}}^{X_{\sigma}}+K_{\mathrm{bulk}, \sigma}^{X_{\sigma}}\right\rangle_{\rho}-\left\langle A_{\mathrm{gen}}^{X_{\rho}}+K_{\mathrm{bulk}, \rho}^{X_{\rho}}\right\rangle_{\rho}$ - $\left\langle K_{R, \sigma}\right\rangle_{\rho}=\operatorname{ext}_{X \sim R}\left[\left\langle A_{\text {gen }}^{X}\right\rangle_{\rho}+\left\langle K_{\text {bulk }, \sigma}^{X}\right\rangle_{\rho}\right]$ ## Comments - in A.1, $z$ and $\bar z$ are defined to cover the covering space (a choice) - solving EoM only affects higher order in $K$ than 2nd order # Emparan, Fernandez-Pique, Luna ## Geometric polarization of plasmas and Love numbers of AdS black branes \[Links: [arXiv](https://arxiv.org/abs/1707.02777), [PDF](https://arxiv.org/pdf/1707.02777.pdf)\] \[Abstract: We use [[0001 AdS-CFT|AdS/CFT]] holography to study how a strongly-coupled plasma polarizes when the geometry where it resides is not flat. We compute the linear-response polarization coefficients, which are directly related to the static two-point correlation function of the stress-energy tensor. In the gravitational dual description, these parameters correspond to the tidal deformation coefficients---the [[0581 Tidal Love numbers|Love numbers]]---of a black brane. We also compute the coefficients of static electric polarization of the plasma.\] ## Summary - computes stress-tensor two-point function $\langle T_{\alpha\beta}T_{\rho\sigma}\rangle$ for $\omega=0$ and general $k$, which is dual to the static tidal Love numbers - analytically for small and large $k$; numerically for all $k$ - where $\left\langle T_{\alpha \beta} T_{\rho \sigma}\right\rangle=-\frac{2}{\sqrt{-\gamma}} \frac{\delta\left\langle T_{\alpha \beta}\right\rangle}{\delta \gamma^{\rho \sigma}}$ # Engelhardt, Wall ## Decoding the Apparent Horizon: A Coarse-Grained Holographic Entropy \[Links: [arXiv](https://arxiv.org/abs/1706.02038), [PDF](https://arxiv.org/pdf/1706.02038.pdf)\] \[Abstract: \] ## Summary - proves that the area of [[0226 Apparent horizon]] is the coarse-grained entropy (over interior) - identifies the boundary dual to this entropy - explains why it has a [[0005 Black hole second law]] (both bulk and boundary entropies) # Faulkner, Haehl, Hijano, Parrikar, Rabideau, van Raamsdonk ## Nonlinear Gravity from Entanglement in Conformal Field Theories \[Links: [arXiv](https://arxiv.org/abs/1705.03026), [PDF](https://arxiv.org/pdf/1705.03026.pdf)\] \[Abstract: In this paper, we demonstrate the [[0302 Gravity from entanglement|emergence of nonlinear gravitational equations]] directly from the physics of a broad class of conformal field theories. We consider CFT excited states defined by adding sources for scalar primary or stress tensor operators to the Euclidean path integral defining the vacuum state. For these states, we show that up to second order in the sources, the entanglement entropy for all ball-shaped regions can always be represented geometrically (via [[0007 RT surface|the Ryu-Takayanagi formula]]) by an asymptotically AdS geometry. We show that such a geometry necessarily satisfies Einstein's equations perturbatively up to second order, with a stress energy tensor arising from matter fields associated with the sourced primary operators. We make no assumptions about [[0001 AdS-CFT|AdS/CFT]] duality, so our work serves as both a consistency check for the AdS/CFT correspondence and a direct demonstration that spacetime and gravitational physics can emerge from the description of entanglement in conformal field theories.\] # Faulkner, Lewkowycz ## Bulk locality from modular flow \[Links: [arXiv](https://arxiv.org/abs/1704.05464), [PDF](https://arxiv.org/pdf/1704.05464.pdf)\] \[Abstract: We study the [[0026 Bulk reconstruction|reconstruction of bulk operators]] in the [[0219 Entanglement wedge reconstruction|entanglement wedge]] in terms of low energy operators localized in the respective boundary region. To leading order in $N$, the dual boundary operators are constructed from the [[0416 Modular Hamiltonian|modular flow]] of single trace operators in the boundary subregion. The appearance of modular evolved boundary operators can be understood due to the equality between bulk and boundary modular flows and explicit formulas for bulk operators can be found with a complete understanding of the action of bulk modular flow, a difficult but in principle solvable task.\] # Fu, Koeller, Marolf (May) ## Violating QFC and quantum covariant entropy bound in $d>5$ dimensions \[Links: [arXiv](https://arxiv.org/abs/1705.03161), [PDF](https://arxiv.org/pdf/1705.03161.pdf)\] \[Abstract: \] ## Refs - disproven in [[Leichenauer2017]] ## Summary - *notes* that integrating out massive fields generally induce a GB term - *shows* that QFC can be violated in the presence of GB [[0006 Higher-derivative gravity]] correction - *shows* that [[0171 Covariant entropy bound|Bousso bound]] can be violated too # Fu, Marolf ## Bare QNEC \[Links: [arXiv](https://arxiv.org/abs/1711.02330), [PDF](https://arxiv.org/pdf/1711.02330.pdf)\] \[Abstract: \] ## Summary - *argues* that bare (unrenormalised) quantities respect a [[0405 Quantum null energy condition]] ## Motivation - [[0405 Quantum null energy condition]] was proposed as a weak gravity limit of [[0243 Quantum focusing conjecture]] which is scheme independent, but there can situations where it does not derive from [[0243 Quantum focusing conjecture]] ([[2017#Fu, Koeller, Marolf (May)]]) and therefore it makes sense to consider general situations where it is not scheme-independent # Gadde ## In search of conformal theories \[Links: [arXiv](https://arxiv.org/abs/1702.07362), [PDF](https://arxiv.org/pdf/1702.07362.pdf)\] \[Abstract: \] ## Summary - find infinite solutions to conformal crossing equation - this paper writes the solutions to the cross equation that is consistent with [[0055 Principal series representation]] for the [[0028 Conformal symmetry|(Euclidean) conformal group]] $SO(d+1,1)$ - formulate crossing equation in a nice way - makes conformal symmetry more transparent - allows for generalisation to any Lie group (i.e. not just the conformal group) <!-- - use to [[asymp]] - (Stephen) If BMS groups non-compact as their conformal cousins, it could help us. ---> # Geiller (Mar) ## Edge modes and corner ambiguities in 3d Chern-Simons theory and gravity \[Links: [arXiv](https://arxiv.org/abs/1703.04748), [PDF](https://arxiv.org/pdf/1703.04748.pdf)\] \[Abstract: \] # Glorioso, Crossley, Liu ## Effective field theory for dissipative fluids (II): classical limit, dynamical KMS symmetry and entropy current \[Links: [arXiv](https://arxiv.org/abs/1701.07817), [PDF](https://arxiv.org/pdf/1701.07817.pdf)\] \[Abstract: In this paper we further develop the fluctuating [[0429 Hydrodynamics|hydrodynamics]] proposed in [[2015#Crossley, Glorioso, Liu|arXiv:1511.03646]] in a number of ways. We first work out in detail the classical limit of the hydrodynamical action, which exhibits many simplifications. In particular, this enables a transparent formulation of the action in physical spacetime in the presence of arbitrary external fields. It also helps to clarify issues related to field redefinitions and frame choices. We then propose that the action is invariant under a $Z_2$ symmetry to which we refer as the dynamical KMS symmetry. The dynamical KMS symmetry is physically equivalent to the previously proposed local [[0521 KMS condition|KMS condition]] in the classical limit, but is more convenient to implement and more general. It is applicable to any states in local equilibrium rather than just thermal density matrix perturbed by external background fields. Finally we elaborate the formulation for a conformal fluid, which contains some new features, and work out the explicit form of the entropy current to second order in derivatives for a neutral conformal fluid.\] ## Refs - followup to [[2015#Crossley, Glorioso, Liu]] # Godazgar, Godazgar, Pope ## Aretakis Charges and Asymptotic Null Infinity \[Links: [arXiv](https://arxiv.org/abs/1707.09804), [PDF](https://arxiv.org/pdf/1707.09804.pdf)\] \[Abstract: \] ## Summary - establish relations between [[0340 Aretakis instability]] and [[0456 Newman-Penrose charges]] in ==4D and for massless scalar== perturbations - the [[0340 Aretakis instability]] of one BH is related to the [[0456 Newman-Penrose charges]] of another, and vice versa # Gralla, Zimmerman ## Critical Exponents of Extremal Kerr Perturbations \[Links: [arXiv](https://arxiv.org/abs/1711.00855), [PDF](https://arxiv.org/pdf/1711.00855.pdf)\] \[Abstract: We show that scalar, electromagnetic, and gravitational perturbations of extremal Kerr black holes are asymptotically self-similar under the near-horizon, late-time scaling symmetry of the background metric. This accounts for the [[0340 Aretakis instability|Aretakis instability]] (growth of transverse derivatives) as a critical phenomenon associated with the emergent symmetry. We compute the critical exponent of each mode, which is equivalent to its decay rate. It follows from symmetry arguments that, despite the growth of transverse derivatives, all generally covariant scalar quantities decay to zero.\] # Grozdanov, Schalm, Scopelliti ## Black hole scrambling from hydrodynamics \[Links: [arXiv](https://arxiv.org/abs/1710.00921), [PDF](https://arxiv.org/pdf/1710.00921.pdf)\] \[Abstract: We argue that the gravitational shock wave computation used to extract the scrambling rate in strongly coupled quantum theories with a holographic dual is directly related to probing the system's hydrodynamic sound modes. The information recovered from the [[0117 Shockwave|shock wave]] can be reconstructed in terms of purely diffusion-like, linearized gravitational waves at the horizon of a single-sided black hole with specific regularity-enforced imaginary values of frequency and momentum. In two-derivative bulk theories, this horizon "diffusion" can be related to late-time momentum diffusion via a simple relation, which ceases to hold in higher-derivative theories. We then show that the same values of imaginary frequency and momentum follow from a dispersion relation of a hydrodynamic sound mode. The frequency, momentum and group velocity give the holographic [[0466 Lyapunov exponent|Lyapunov exponent]] and the [[0167 Butterfly velocity|butterfly velocity]]. Moreover, at this special point along the sound dispersion relation curve, the residue of the retarded longitudinal stress-energy tensor two-point function vanishes. This establishes a direct link between a hydrodynamic sound mode at an analytically continued, imaginary momentum and the holographic butterfly effect. Furthermore, our results imply that infinitely strongly coupled, large-$N_c$ holographic theories exhibit properties similar to classical dilute gasses; there, late-time equilibration and early-time scrambling are also controlled by the same dynamics.\] ## Refs - OG for [[0179 Pole skipping|pole skipping]] # Haehl, Hijano, Parrikar, Rabideau ## Higher Curvature Gravity from Entanglement in Conformal Field Theories \[Links: [arXiv](https://arxiv.org/abs/1712.06620), [PDF](https://arxiv.org/pdf/1712.06620.pdf)\] \[Abstract: By generalizing different recent works to the context of [[0006 Higher-derivative gravity|higher curvature gravity]], we provide a unifying framework for three related results: (i) If an asymptotically AdS spacetime computes the entanglement entropies of ball-shaped regions in a CFT using a generalized [[0007 RT surface|Ryu-Takayanagi formula]] up to second order in state deformations around the vacuum, then the spacetime satisfies the correct gravitational equations of motion up to second order around AdS; (ii) The [[0145 Generalised area|holographic dual of entanglement entropy]] in higher curvature theories of gravity is given by Wald entropy plus a particular correction term involving extrinsic curvatures; (iii) CFT [[0199 Relative entropy|relative entropy]] is dual to gravitational canonical energy (also in higher curvature theories of gravity). Especially for the second point, our novel derivation of this previously known statement does not involve the Euclidean replica trick.\] ## Refs - [[0302 Gravity from entanglement]] ## Comments - obtains the $K^2$ terms in [[0145 Generalised area|HEE]] for [[0006 Higher-derivative gravity|higher-derivative gravity]] using a new method involving no replica trick # Hajian, Sheikh-Jabbari, Yavartanoo ## Extreme Kerr black hole microstates with horizon fluff \[Links: [arXiv](https://arxiv.org/abs/1708.06378), [PDF](https://arxiv.org/pdf/1708.06378.pdf)\] \[Abstract: \] # He, Kapec, Raclariu, Strominger ## Loop-Corrected Virasoro Symmetry of 4D Quantum Gravity \[Links: [arXiv](https://arxiv.org/abs/1701.00496), [PDF](https://arxiv.org/pdf/1701.00496.pdf)\] \[Abstract: \] # Horowitz, Shaghoulian ## Detachable circles and temperature-inversion dualities for CFT${}_d$ ## Summary - *relates* $S^1\times S^{d-1}$ to $S^{1} \times \mathcal{H}^{d-1} / \mathbb{Z}$ via Weyl transformation - *deduces* from [[0012 Hawking-Page transition]] a confining phase transition at finite temperature for gauge theories on hyperbolic space - *provides* examples of smooth bulk solutions that asymptote to conical singularities on AdS boundary - *discusses* Eguchi-Kawai mechanism and high-temperature/low-temperature duality ## Refs - previous work [[Shaghoulian201612]][](https://arxiv.org/abs/1612.05257) ## The trick The trick is to detach a circle # Jian, Yao ## Solvable Sachdev-Ye-Kitaev models in higher dimensions: from diffusion to many-body localization \[Links: [arXiv](https://arxiv.org/abs/1703.02051), [PDF](https://arxiv.org/pdf/1703.02051.pdf)\] \[Abstract: Many aspects of many-body localization (MBL) transitions remain elusive so far. Here, we propose a higher-dimensional generalization of the [[0201 Sachdev-Ye-Kitaev model|Sachdev-Ye-Kitaev (SYK) model]] and show that it exhibits a MBL transition. The model on a bipartite lattice has $N$ Majorana fermions with SYK interactions on each site of the $A$ sublattice and $M$ free Majorana fermions on each site the of $B$ sublattice, where $N$ and $M$ are large and finite. For $r\equiv M/N\!<\!r_c=1$, it describes a diffusive metal exhibiting maximal [[0008 Quantum chaos|chaos]]. Remarkably, its diffusive constant $D$ vanishes $[D\propto(r_c-r)^{1/2}]$ as $r\rightarrow r_c$, implying a dynamical transition to a MBL phase. It is further supported by numerical calculations of level statistics which changes from Wigner-Dyson ($r<r_c$) to Poisson ($r>r_c$) distributions. Note that no subdiffusive phase intervenes between diffusive and MBL phases. Moreover, the critical exponent $\nu=0$, violating the Harris criterion. Our higher-dimensional SYK model may provide a promising arena to explore exotic MBL transitions.\] # Kapec, Perry, Raclariu, Strominger ## Infrared Divergences in QED, Revisited \[Links: [arXiv](https://arxiv.org/abs/1705.04311), [PDF](https://arxiv.org/pdf/1705.04311.pdf)\] \[Abstract: Recently it has been shown that the vacuum state in QED is infinitely degenerate. Moreover a transition among the degenerate vacua is induced in any nontrivial scattering process and determined from the associated soft factor. Conventional computations of scattering amplitudes in QED do not account for this vacuum degeneracy and therefore always give zero. This vanishing of all conventional QED amplitudes is usually attributed to [[0295 Infrared divergences in scattering amplitude|infrared divergences]]. Here we show that if these vacuum transitions are properly accounted for, the resulting amplitudes are nonzero and infrared finite. Our construction of finite amplitudes is mathematically equivalent to, and amounts to a physical reinterpretation of, the 1970 construction of [[0272 Faddeev-Kulish|Faddeev and Kulish]].\] # Kitaev, Suh ## The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual \[Links: [arXiv](https://arxiv.org/abs/1711.08467), [PDF](https://arxiv.org/pdf/1711.08467.pdf)\] \[Abstract: We give an exposition of the [[0201 Sachdev-Ye-Kitaev model|SYK model]] with several new results. A non-local correction to the Schwarzian effective action is found. The same action is obtained by integrating out the bulk degrees of freedom in a certain variant of dilaton gravity. We also discuss general properties of [[0482 Out-of-time-order correlator|out-of-time-order correlators]].\] # Kitaev, Yoshida ## Efficient decoding for the Hayden-Preskill protocol \[Links: [arXiv](https://arxiv.org/abs/1710.03363), [PDF](https://arxiv.org/pdf/1710.03363.pdf)\] \[Abstract: We present two particular decoding procedures for reconstructing a quantum state from the Hawking radiation in the [[2007#Hayden, Preskill|Hayden-Preskill]] thought experiment. We work in an idealized setting and represent the black hole and its entangled partner by $n$ EPR pairs. The first procedure teleports the state thrown into the black hole to an outside observer by post-selecting on the condition that a sufficient number of EPR pairs remain undisturbed. The probability of this favorable event scales as $1/d_{A}^2$, where $d_A$ is the Hilbert space dimension for the input state. The second procedure is deterministic and combines the previous idea with Grover's search. The decoding complexity is $\mathcal{O}(d_{A}\mathcal{C})$ where $\mathcal{C}$ is the size of the quantum circuit implementing the unitary evolution operator U of the black hole. As with the original (non-constructive) decoding scheme, our algorithms utilize scrambling, where the decay of [[0482 Out-of-time-order correlator|out-of-time-order correlators]] (OTOCs) guarantees faithful state recovery.\] # Kusuki, Takayanagi ## Renyi Entropy for Local Quenches in 2D CFTs from Numerical Conformal Blocks \[Links: [arXiv](https://arxiv.org/abs/1711.09913), [PDF](https://arxiv.org/pdf/1711.09913.pdf)\] \[Abstract: We study the time evolution of [[0293 Renyi entropy|Renyi entanglement entropy]] for locally excited states in two dimensional large [[0033 Central charge|central charge]] CFTs. It generically shows a logarithmical growth and we compute the coefficient of $\log t$ term. Our analysis covers the entire parameter regions with respect to the replica number $n$ and the conformal dimension $h_O$ of the primary operator which creates the excitation. We numerically analyse relevant vacuum conformal blocks by using Zamolodchikov's recursion relation. We find that the behavior of the conformal blocks in two dimensional CFTs with a central charge $c$, drastically changes when the dimensions of external primary states reach the value $c/32$. In particular, when $h_O\geq c/32$ and $n\geq 2$, we find a new universal formula $\Delta S^{(n)}_A\simeq \frac{nc}{24(n-1)}\log t$. Our numerical results also confirm existing analytical results using the HHLL approximation.\] # Laddha, Mitra ## Asymptotic Symmetries and Subleading Soft Photon Theorem in Effective Field Theories \[Links: [arXiv](https://arxiv.org/abs/1709.03850), [PDF](https://arxiv.org/pdf/1709.03850.pdf)\] \[Abstract: \] ## Summary - shows [[0009 Soft theorems|subleading soft photon theorem]] in EFT containing photons and an arbitrary spectrum of massless particles # Marolf, Parrikar, Rabideau, Rad, van Raamsdonk ## From Euclidean Sources to Lorentzian Spacetimes in Holographic Conformal Field Theories \[Links: [arXiv](https://arxiv.org/abs/1709.10101), [PDF](https://arxiv.org/pdf/1709.10101.pdf)\] \[Abstract: We consider states of [[0001 AdS-CFT|holographic]] conformal field theories constructed by adding sources for local operators in the Euclidean path integral, with the aim of investigating the extent to which arbitrary bulk coherent states can be represented by such Euclidean path-integrals in the CFT. We construct the associated dual Lorentzian spacetimes perturbatively in the sources. Extending earlier work, we provide explicit formulae for the Lorentzian fields to first order in the sources for general scalar field and metric perturbations in arbitrary dimensions. We check the results by holographically computing the Lorentzian one-point functions for the sourced operators and comparing with a direct CFT calculation. We present evidence that at the linearized level, arbitrary bulk initial data profiles can be generated by an appropriate choice of Euclidean sources. However, in order to produce initial data that is very localized, the amplitude must be taken small at the same time otherwise the required sources diverge, invalidating the perturbative approach.\] ## Refs - [[0042 Schwinger-Keldysh techniques]] # Mefford, Shaghoulian, Shyani ## Sparseness bounds on local operators in holographic CFT$_d$ \[Links: [arXiv](https://arxiv.org/abs/1711.03122), [PDF](https://arxiv.org/pdf/1711.03122.pdf)\] \[Abstract: We use the thermodynamics of anti-de Sitter gravity to derive sparseness bounds on the spectrum of local operators in [[0122 Holographic CFT|holographic conformal field theories]]. The simplest such bound is $\rho(\Delta) \lesssim \exp\left(\frac{2\pi\Delta}{d-1}\right)$ for CFT$_d$. Unlike the case of $d=2$, this bound is strong enough to rule out weakly coupled holographic theories. We generalize the bound to include spins $J_i$ and $U(1)$ charge $Q$, obtaining bounds on $\rho(\Delta, J_i, Q)$ in $d=3$ through $6$. All bounds are saturated by black holes at the [[0012 Hawking-Page transition|Hawking-Page transition]] and vanish beyond the corresponding [[0178 BPS|BPS]] bound.\] # Mertens, Turiaci, Verlinde ## Solving the Schwarzian via the Conformal Bootstrap \[Links: [arXiv](https://arxiv.org/abs/1705.08408), [PDF](https://arxiv.org/pdf/1705.08408.pdf)\] \[Abstract: We obtain exact expressions for a general class of correlation functions in the 1D quantum mechanical model described by the Schwarzian action, that arises as the low energy limit of the [[0201 Sachdev-Ye-Kitaev model|SYK]] model. The answer takes the form of an integral of a momentum space amplitude obtained via a simple set of diagrammatic rules. The derivation relies on the precise equivalence between the 1D Schwarzian theory and a suitable large $c$ limit of 2D Virasoro CFT. The mapping from the 1D to the 2D theory is similar to the construction of kinematic space. We also compute the [[0482 Out-of-time-order correlator|out-of-time ordered four point function]]. The momentum space amplitude in this case contains an extra factor in the form of a crossing kernel, or R-matrix, given by a 6j-symbol of $SU(1,1)$. We argue that the R-matrix describes the gravitational scattering amplitude near the horizon of an AdS${}_2$ black hole. Finally, we discuss the generalization of some of our results to ${\cal N}=1$ and ${\cal N}=2$ supersymmetric Schwarzian QM.\] # Miao, Chu, Guo (Short) ## A New Proposal for Holographic BCFT \[Links: [arXiv](https://arxiv.org/abs/1701.04275), [PDF](https://arxiv.org/pdf/1701.04275)\] \[Abstract: We propose a new holographic dual of conformal field theory defined on a manifold with boundaries, i.e. [[0548 Boundary CFT|BCFT]]. Our proposal can apply to general boundaries and agrees with [arXiv:1105.5165](https://arxiv.org/abs/1105.5165) for the special case of a disk and half plane. Using the new proposal of [[0181 AdS-BCFT|AdS/BCFT]], we successfully obtain the expected boundary Weyl anomaly and the obtained boundary central charges satisfy naturally a c-like theorem holographically. We also investigate the [[0007 RT surface|holographic entanglement entropy]] of BCFT and find that the minimal surface must be normal to the bulk spacetime boundaries when they intersect. Interestingly, the entanglement entropy depends on the boundary conditions of BCFT and the distance to the boundary. The entanglement wedge has an interesting phase transition which is important for the self-consistency of AdS/BCFT.\] # Mikhaylov ## Teichmuller TQFT vs Chern-Simons Theory \[Links: [arXiv](https://arxiv.org/abs/1710.04354), [PDF](https://arxiv.org/pdf/1710.04354.pdf)\] \[Abstract: Teichmüller TQFT is a unitary 3d topological theory whose Hilbert spaces are spanned by [[0562 Liouville theory|Liouville]] [[0031 Conformal block|conformal blocks]]. It is related but not identical to $PSL(2,R)$ [[0089 Chern-Simons theory|Chern-Simons theory]]. To physicists, it is known in particular in the context of 3d-3d correspondence and also in the holographic description of [[0032 Virasoro algebra|Virasoro]] conformal blocks. We propose that this theory can be defined by an analytically-continued Chern-Simons path-integral with an unusual integration cycle. On hyperbolic three-manifolds, this cycle is singled out by the requirement of invertible vielbein. Mathematically, our proposal translates a known conjecture by Andersen and Kashaev into a conjecture about the Kapustin-Witten equations. We further explain that Teichmüller TQFT is dual to complex $SL(2,C)$ Chern-Simons theory at integer level $k=1$, clarifying some puzzles previously encountered in the 3d-3d correspondence literature. We also present a new simple derivation of complex Chern-Simons theories from the 6d (2,0) theory on a lens space with a transversely-holomorphic foliation.\] # Nahum, Vijay, Haah ## Operator Spreading in Random Unitary Circuits \[Links: [arXiv](https://arxiv.org/abs/1705.08975), [PDF](https://arxiv.org/pdf/1705.08975.pdf)\] \[Abstract: Random quantum circuits yield minimally structured models for [[0008 Quantum chaos|chaotic quantum dynamics]], able to capture for example universal properties of [[0522 Entanglement dynamics|entanglement growth]]. We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries. We study both 1+1D and higher dimensions, and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems. In 1+1D, we demonstrate that the [[0482 Out-of-time-order correlator|out-of-time-order correlator (OTOC)]] satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC, and the [[0167 Butterfly velocity|butterfly speed]] $v_{B}$. We find that in 1+1D the 'front' of the OTOC broadens diffusively, with a width scaling in time as $t^{1/2}$. We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front. Turning to higher D, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a classical droplet growth problem. This implies that the width of the front of the averaged OTOC scales as $t^{1/3}$ in 2+1D and $t^{0.24}$ in 3+1D ([[0524 Kardar-Parisi-Zhang equation|KPZ]] exponents). We support our analytic argument with simulations in 2+1D. We point out that, in a lattice model, the late time shape of the spreading operator is in general not spherical. However when full spatial rotational symmetry is present in 2+1D, our mapping implies an exact asymptotic form for the OTOC in terms of the Tracy-Widom distribution. For an alternative perspective on the OTOC in 1+1D, we map it to the partition function of an Ising-like model. As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly. We also use this mapping to give exact results for entanglement growth in 1+1D circuits.\] ## Refs - [[0523 Random circuit model]] - [[0522 Entanglement dynamics]] # Nande, Pate, Strominger ## Soft Factorization in QED from 2D Kac-Moody Symmetry \[Links: [arXiv](https://arxiv.org/abs/1705.00608), [PDF](https://arxiv.org/pdf/1705.00608.pdf)\] \[Abstract: The soft factorization theorem for 4D abelian gauge theory states that the $\mathcal{S}$-matrix factorizes into soft and hard parts, with the universal soft part containing all [[0009 Soft theorems|soft]] and [[0078 Collinear limit|collinear]] poles. Similarly, correlation functions on the sphere in a 2D CFT with a $U(1)$ [[0069 Kac-Moody algebra|Kac-Moody]] current algebra factorize into current algebra and non-current algebra factors, with the current algebra factor fully determined by its pole structure. In this paper, we show that these 4D and 2D factorizations are mathematically the same phenomena. The soft 't Hooft-Wilson lines and soft photons are realized as a complexified 2D current algebra on the [[0022 Celestial sphere|celestial sphere]] at null infinity. The current algebra level is determined by the cusp anomalous dimension. The associated complex $U(1)$ boson lives on a torus whose modular parameter is $\tau =\frac {2\pi i }{e^2}+\frac{\theta}{ 2 \pi}$. The correlators of this 2D current algebra fully reproduce the known soft part of the 4D $\mathcal{S}$-matrix, as well as a conjectured generalization involving magnetic charges.\] # Pasterski, Shao ## Conformal basis for flat space amplitudes \[Links: [arXiv](https://arxiv.org/abs/1705.01027), [PDF](https://arxiv.org/pdf/1705.01027.pdf)\] \[Abstract: \] ## Summary - solutions of KG, Maxwell and linearised Einstein in conformal basis - both massive and massless discussed # Pasterski, Shao, Strominger (Jan) ## Flat space amplitudes and conformal symmetry of the celestial sphere \[Links: [arXiv](https://arxiv.org/abs/1701.00049), [PDF](https://arxiv.org/pdf/1701.00049.pdf)\] \[Abstract: The four-dimensional (4D) Lorentz group $SL(2,\mathbb{C})$ acts as the two-dimensional (2D) global conformal group on the celestial sphere at infinity where asymptotic 4D scattering states are specified. Consequent similarities of 4D flat space amplitudes and 2D correlators on the conformal sphere are obscured by the fact that the former are usually expressed in terms of asymptotic wavefunctions which transform simply under spacetime translations rather than the Lorentz $SL(2,\mathbb{C})$. In this paper we construct on-shell [[0256 Massive particles in CCFT|massive]] scalar wavefunctions in 4D Minkowski space that transform as $SL(2,\mathbb{C})$ conformal primaries. Scattering amplitudes of these wavefunctions are $SL(2,\mathbb{C})$ covariant by construction. For certain mass relations, we show explicitly that their three-point amplitude reduces to the known unique form of a 2D CFT primary three-point function and compute the coefficient. The computation proceeds naturally via Witten-like diagrams on a hyperbolic slicing of Minkowski space and has a holographic flavor.\] ## Summary - Map the flat space *plane wave basis* to a *conformal primary wave function basis* on the celestial sphere - => can relate 4D scattering amplitudes to 2d correlators (celestial amplitudes) ([[2019#Law, Zlotnikov]]) - done for ==massive scalars== - transform as $SL(2,\mathbb{C})$ conformal primaries ## Refs - Following this paper, explicit examples of amplitudes were done: ([[2019#Law, Zlotnikov]]) - scalar scattering - gluon scattering - string/graviton scattering - [[0037 Modifications of PSS]] ## The mapping - momentum state $e^{i \omega q \cdot X}$ - basis $p^{\mu}=\omega q^{\mu}(w, \bar{w}), \ell$ labelled by $\omega$ and $\ell$ - $w=\frac{p^{1}+i p^{2}}{p^{3}+p^{0}}$ - the transform $\int_{0}^{\infty} d w w^{\Delta-1}$ - the transformed $\frac{\mathcal{N}(\Delta)}{(-q \cdot X)^{\Delta}} \equiv \Phi^{\Delta}$ - $\Delta\in 1+i \mathbb{R}, J=\ell$, labelled by $\Delta$ and $J$ # Qi, Yang ## Butterfly velocity and bulk causal structure \[Links: [arXiv](https://arxiv.org/abs/1705.01728), [PDF](https://arxiv.org/pdf/1705.01728.pdf)\] \[Abstract: The [[0167 Butterfly velocity|butterfly velocity]] was recently proposed as a characteristic velocity of [[0008 Quantum chaos|chaos]] propagation in a local system. Compared to the [[0322 Lieb-Robinson bound|Lieb-Robinson velocity]] that bounds the propagation speed of all perturbations, the butterfly velocity, studied in thermal ensembles, is an "effective" Lieb-Robinson velocity for a subspace of the Hilbert space defined by the microcanonical ensemble. In this paper, we generalize the concept of butterfly velocity beyond the thermal case to a large class of other subspaces. Based on [[0001 AdS-CFT|holographic duality]], we consider the code subspace of low energy excitations on a classical background geometry. Using local reconstruction of bulk operators, we prove a general relation between the boundary butterfly velocities (of different operators) and the bulk causal structure. Our result has implications in both directions of the bulk-boundary correspondence. Starting from a boundary theory with a given Lieb-Robinson velocity, our result determines an upper bound of the bulk light cone starting from a given point. Starting from a bulk space-time geometry, the butterfly velocity can be explicitly calculated for all operators that are the local reconstructions of bulk local operators. If the bulk geometry satisfies Einstein equation and the [[0480 Null energy condition|null energy condition]], for rotation symmetric geometries we prove that infrared operators always have a slower butterfly velocity that the ultraviolet one. For asymptotic AdS geometries, this also implies that the butterfly velocities of all operators are upper bounded by the speed of light. We further prove that the butterfly velocity is equal to the speed of light if the causal wedge of the boundary region coincides with its entanglement wedge. Finally, we discuss the implication of our result to geometries that are not asymptotically AdS, and in particular, obtain constraints that must be satisfied by a dual theory of flat space gravity.\] # Song, Jian, Balents ## A strongly correlated metal built from Sachdev-Ye-Kitaev models \[Links: [arXiv](https://arxiv.org/abs/1705.00117), [PDF](https://arxiv.org/pdf/1705.00117.pdf)\] \[Abstract: Strongly correlated metals comprise an enduring puzzle at the heart of condensed matter physics. Commonly a highly renormalized heavy Fermi liquid occurs below a small coherence scale, while at higher temperatures a broad incoherent regime pertains in which quasi-particle description fails. Despite the ubiquity of this phenomenology, strong correlations and quantum fluctuations make it challenging to study. The Sachdev-Ye-Kitaev(SYK) model describes a 0+1D quantum cluster with random all-to-all *four*-fermion interactions among $N$ Fermion modes which becomes exactly solvable as $N\rightarrow \infty$, exhibiting a zero-dimensional non-Fermi liquid with emergent conformal symmetry and complete absence of quasi-particles. Here we study a lattice of complex-fermion SYK dots with random inter-site *quadratic* hopping. Combining the imaginary time path integral with *real* time path integral formulation, we obtain a heavy Fermi liquid to incoherent metal crossover in full detail, including thermodynamics, low temperature Landau quasiparticle interactions, and both electrical and thermal conductivity at all scales. We find linear in temperature resistivity in the incoherent regime, and a Lorentz ratio $L\equiv \frac{\kappa\rho}{T}$ varies between two universal values as a function of temperature. Our work exemplifies an analytically controlled study of a strongly correlated metal.\] # Sorce, Wald ## Gedanken Experiments to Destroy a Black Hole II: Kerr-Newman Black Holes Cannot be Over-Charged or Over-Spun \[Links: [arXiv](https://arxiv.org/abs/1707.05862), [PDF](https://arxiv.org/pdf/1707.05862.pdf)\] \[Abstract: \] ## Summary - *shows* that you cannot overcharge or overspin a near extremal Kerr-Newman BH - allowing arbitrary matter satisfying [[0480 Null energy condition|NEC]] (except for EM fields) - *performs* a ==second-order== analysis to see the effects on the final mass, including self-force effects - *obtains* a formula for $\delta^2 M$ without explicitly computing self-force or finite size effects ## Refs - [[Hubeny1998]][](https://arxiv.org/abs/gr-qc/9808043) suggested that you can overcharge a near extremal RN BH by neglecting the back reaction; but to test that one needs to compute everything to second order ## Fundamental identity - starting with $\mathcal{J}_{X}=\mathbf{C}_{X}+d \mathbf{Q}_{X}$ (eq.16) - taking a field derivative, the LHS can be written as $\frac{d \mathcal{J}_{X}}{d \lambda}=-\iota_{X}\left(\mathbf{E}(\phi) \cdot \frac{d \phi}{d \lambda}\right)+\omega\left(\phi ; \frac{d \phi}{d \lambda}, \mathscr{L}_{X} \phi\right)$+d\left[\iota_{X} \boldsymbol{\theta}\left(\phi, \frac{d \phi}{d \lambda}\right)\right]$ - then take derivative on RHS and rearrange, we get $d\left[\frac{d \mathbf{Q}_{X}}{d \lambda}-\iota_{X} \boldsymbol{\theta}\left(\phi, \frac{d \phi}{d \lambda}\right)\right]=\boldsymbol{\omega}\left(\phi ; \frac{d \phi}{d \lambda}, \mathscr{L}_{X} \phi\right)-\frac{d \mathbf{C}_{X}}{d \lambda}$-\iota_{X}\left(\mathbf{E}(\phi) \cdot \frac{d \phi}{d \lambda}\right)$ ## Second variation formula - differentiating the fundamental identity wrt $\lambda$, we get $d\left[\delta^{2} \mathbf{Q}_{\xi}-\iota_{\xi} \delta \boldsymbol{\theta}(\phi, \delta \phi)\right]=\boldsymbol{\omega}\left(\phi ; \delta \phi, \mathscr{L}_{\xi} \delta \phi\right)-\delta^{2} \mathbf{C}_{\xi}$-\iota_{\xi}(\delta \mathbf{E} \cdot \delta \phi)$ - integrating over $\Sigma$ gives $\mathcal{E}_{\Sigma}(\phi ; \delta \phi)=\int_{\partial \Sigma}\left[\delta^{2} \mathbf{Q}_{\xi}-\iota_{\xi} \delta \boldsymbol{\theta}(\phi, \delta \phi)\right]+\int_{\Sigma} \delta^{2} \mathbf{C}_{\xi}$+\int_{\Sigma} \iota_{\xi}(\delta \mathbf{E} \cdot \delta \phi)$ - the first term is evaluated at both the horizon and the infinity; - at infinity, it looks like $\int_{\infty}\left[\delta^{2} \mathbf{Q}_{\xi}-\iota_{\xi} \delta \boldsymbol{\theta}(\phi, \delta \phi)\right]=\delta^{2} M-\Omega_{H} \delta^{2} J$ - at bifurcate horizon: - first order: $\int_{B}\left[\delta \mathbf{Q}_{\xi}^{G R}-\iota_{\xi} \boldsymbol{\theta}^{G R}(\phi, \delta \phi)\right]=\int_{B} \delta \mathbf{Q}_{\xi}^{G R}=\frac{\kappa}{8 \pi} \delta A_{B}$ and $\int_{B}\left[\delta \mathbf{Q}_{\xi}^{E M}-\iota_{\xi} \boldsymbol{\theta}^{E M}(\phi, \delta \phi)\right]=\frac{1}{8 \pi} \Phi_{H} \int_{B} \delta\left(\epsilon_{a b c d} F^{c d}\right)$=\Phi_{H} \delta Q_{B}$ - 2nd order: should just take another derivative, so we get $\delta^2 A$ and $\delta^2 Q_B$ - all together: $\delta^{2} M-\Omega_{H} \delta^{2} J-\Phi_{H} \delta^{2} Q_{B}-\frac{\kappa}{8 \pi} \delta^{2} A_{B}$=\mathcal{E}_{\Sigma}(\phi ; \delta \phi)-\int_{\Sigma} \iota_{\xi}(\delta \mathbf{E}(\phi) \cdot \delta \phi)-\int_{\Sigma} \delta^{2} \mathbf{C}_{\xi}$ ## Choice of Cauchy surface ![[SorceWald2017_fig2.png|400]] # Speranza ## Local phase space and edge modes for diffeomorphism-invariant theories \[Links: [arXiv](https://arxiv.org/abs/1706.05061), [PDF](https://arxiv.org/pdf/1706.05061.pdf)\] \[Abstract: \] ## Remarks - great intro to [[0019 Covariant phase space]] and [[0044 Extended phase space]] ## Summary - a general construction for edge mode symplectic structure - i.e. any diffeo-inv theory - for surface invariant theories, the algebra is universal for all diffeo-inv theories - Diff(S) x SL(2,R) (x normal shearing) - if the boundary are chosen such that surface translation are symmetries, the algebra acquires a central extension ## [[0018 JKM ambiguity]] - talks about how each ambiguity affects the symplectic form and charge - the $\theta$ ambiguity: - $\theta \rightarrow \theta + d\beta$ - affects the charge - can have arbitrary derivatives - fixed by [[2015#Wall (Essay)]] or [[2013#Dong]] - fixing by [[2019#Harlow, Wu]] prescription - does not give the anomaly term - (by talking to Jie and looking at [[2018#Jiang, Zhang]]) ## [[0066 De Rham cohomology]] - for some Lagrangians, $d\Omega=0$ but $\Omega$ is not exact => non-trivial cohomology - #idea what is the consequence? investigate the topology of solutions space? - maybe do it in a low dimension solution space first. e.g. 3d gravity -> read [[2020#Maxfield, Turiaci]] ## Relation to [[1993#Wald]] method - also see [[2017#Geiller (Mar)]] # Strominger (Lectures) ## Lectures on the Infrared Structure of Gravity and Gauge Theory \[Links: [arXiv](https://arxiv.org/abs/1703.05448), [PDF](https://arxiv.org/pdf/1703.05448.pdf)\] \[Abstract: This is a redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of [[0009 Soft theorems|soft theorems]], the [[0287 Memory effect|memory effect]] and [[0060 Asymptotic symmetry|asymptotic symmetries]] in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. The lectures may be viewed online at [this https URL](https://goo.gl/3DJdOr). Please send typos or corrections to strominger@physics.[this http URL](http://harvard.edu/).\] # Yang, Ruan ## Comments on Joint Terms in Gravitational Action \[Links: [arXiv](https://arxiv.org/abs/1704.03232), [PDF](https://arxiv.org/pdf/1704.03232.pdf)\] \[Abstract: This paper compares three different methods about computing [[0102 Hayward term|joint terms]] in on-shell action of gravity, which are identifying the joint term by the variational principle in Dirichlet boundary condition, treating the joint term as the limit contribution of smooth boundary and finding the joint term by local $SO(1,d-1)$ transformation. In general metric gravitational theory, we show that the differences between these joint terms are some variational invariants under fixed boundary condition. We also give an explicit condition to judge the existence of joint term determined by [[0138 Variational principle|variational principle]] and apply it into [[0554 Einstein gravity|General Relativity]] as an example.\] # 1710.05835 ## A gravitational wave standard siren measurement of the Hubble constant \[Links: [arXiv](https://arxiv.org/abs/1710.05835), [PDF](https://arxiv.org/pdf/1710.05835.pdf)\] \[Abstract: \] ## Summary - estimates Hubble constant to be $70.0_{-8.0}^{+12.0} \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$ - using GW as standard siren to measure the distance - and using EM waves to measure redshift ## Refs - a simpler method described in [[0239 Hubble constant measurement from gravitational waves]]