# Afkhami-Jeddi ## Soft Graviton Theorem in Arbitrary Dimensions \[Links: [arXiv](https://arxiv.org/abs/1405.3533), [PDF](https://arxiv.org/pdf/1405.3533.pdf)\] \[Abstract: In this note we show that the recent conjecture proposed by [[2014#Cachazo, Strominger|Cachazo and Strominger]] holds at tree level in arbitrary dimensions. The proof makes crucial use of the fact that the sub-leading operator is defined using the total angular momentum operator. A key ingredient that makes the proof possible is the CHY formula for graviton amplitudes in arbitrary number of dimensions.\] ## Summary - generalisation of subleading soft graviton theorem to higher dimensions (and in YM) ## Refs - independent work [[2014#Schwab, Volovich]] # Almheiri, Dong, Harlow ## Bulk Locality and Quantum Error Correction in AdS/CFT \[Links: [arXiv](https://arxiv.org/abs/1411.7041), [PDF](https://arxiv.org/pdf/1411.7041.pdf)\] \[Abstract: We point out a connection between the emergence of bulk locality in [[0001 AdS-CFT|AdS/CFT]] and the theory of [[0146 Quantum error correction|quantum error correction]]. Bulk notions such as Bogoliubov transformations, location in the radial direction, and the holographic entropy bound all have natural CFT interpretations in the language of quantum error correction. We also show that the question of whether bulk operator reconstruction works only in the causal wedge or all the way to the extremal surface is related to the question of whether or not the quantum error correcting code realized by AdS/CFT is also a "quantum secret sharing scheme", and suggest a [[0054 Tensor network|tensor network]] calculation that may settle the issue. Interestingly, the version of quantum error correction which is best suited to our analysis is the somewhat nonstandard "operator algebra quantum error correction" of Beny, Kempf, and Kribs. Our proposal gives a precise formulation of the idea of "subregion-subregion" duality in AdS/CFT, and clarifies the limits of its validity.\] ## Bulk locality - bulk locality follows from [[0146 Quantum error correction|quantum error correcting]] properties of the approximate isometry $V: \mathcal{H}_{\text {bulk }} \rightarrow \mathcal{H}_{\text {Boundary }}$ # Almheiri, Polchinski ## Models of AdS$_2$ Backreaction and Holography \[Links: [arXiv](https://arxiv.org/abs/1402.6334), [PDF](https://arxiv.org/pdf/1402.6334)\] \[Abstract: We develop models of 1+1 dimensional dilaton gravity describing flows to AdS$_2$ from higher dimensional AdS and other spaces. We use these to study the effects of backreaction on holographic correlators. We show that this scales as a relevant effect at low energies, for compact transverse spaces. We also discuss effects of matter loops, as in the [[0279 CGHS model|CGHS model]].\] # Asplund, Bernamonti, Galli, Hartman ## Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches \[Links: [arXiv](https://arxiv.org/abs/1410.1392), [PDF](https://arxiv.org/pdf/1410.1392.pdf)\] \[Abstract: We consider the [[0301 Entanglement entropy|entanglement entropy]] in [[0003 2D CFT|2d conformal field theory]] in a class of excited states produced by the insertion of a heavy local operator. These include both high-energy eigenstates of the Hamiltonian and time-dependent local quenches. We compute the universal contribution from the stress tensor to the single interval [[0293 Renyi entropy|Renyi entropies]] and entanglement entropy, and conjecture that this dominates the answer in theories with a large [[0033 Central charge|central charge]] and a sparse spectrum of low-dimension operators. The resulting entanglement entropies agree precisely with holographic calculations in [[0002 3D gravity|three-dimensional gravity]]. High-energy eigenstates are dual to microstates of the [[0086 Banados-Teitelboim-Zanelli black hole|BTZ]] black hole, so the corresponding holographic calculation is a geodesic length in the black hole geometry; agreement between these two answers demonstrates that entanglement entropy thermalizes in individual microstates of holographic CFTs. For local quenches, the dual geometry is a highly boosted black hole or conical defect. On the CFT side, the rise in entanglement entropy after a quench is directly related to the monodromy of a Virasoro conformal block.\] # Balasubramanian, Chowdhury, Czech, de Boer ## Entwinement and the emergence of spacetime \[Links: [arXiv](https://arxiv.org/abs/1406.5859), [PDF](https://arxiv.org/pdf/1406.5859)\] \[Abstract: It is conventional to study the entanglement between spatial regions of a quantum field theory. However, in some systems entanglement can be dominated by "internal", possibly gauged, degrees of freedom that are not spatially organized, and that can give rise to gaps smaller than the inverse size of the system. In a holographic context, such small gaps are associated to the appearance of horizons and singularities in the dual spacetime. Here, we propose a concept of [[0282 Entwinement|entwinement]], which is intended to capture this fine structure of the wavefunction. Holographically, entwinement probes the entanglement shadow -- the region of spacetime not probed by the minimal surfaces that compute spatial entanglement in the dual field theory. We consider the simplest example of this scenario -- a 2d conformal field theory (CFT) that is dual to a conical defect in AdS3 space. Following our previous work, we show that spatial entanglement in the CFT reproduces spacetime geometry up to a finite distance from the conical defect. We then show that the interior geometry up to the defect can be reconstructed from entwinement that is sensitive to the discretely gauged, fractionated degrees of freedom of the CFT. Entwinement in the CFT is related to non-minimal geodesics in the conical defect geometry, suggesting a potential quantum information theoretic meaning for these objects in a holographic context. These results may be relevant for the reconstruction of black hole interiors from a dual field theory.\] # Balasubramanian, Hayden, Maloney, Marolf, Ross ## Multiboundary wormholes and holographic entanglement \[Links: [arXiv](https://arxiv.org/abs/1406.2663), [PDF](https://arxiv.org/pdf/1406.2663.pdf)\] \[Abstract: The [[0001 AdS-CFT|AdS/CFT]] correspondence relates quantum entanglement between boundary Conformal Field Theories and geometric connections in the dual asymptotically Anti-de Sitter space-time. We consider entangled states in the $n$-fold tensor product of a 1+1 dimensional CFT Hilbert space defined by the Euclidean path integral over a Riemann surface with $n$ holes. In one region of moduli space, the dual bulk state is a black hole with $n$ asymptotically AdS$_3$ regions connected by a common wormhole, while in other regions the bulk fragments into disconnected components. We study the entanglement structure and compute the wave function explicitly in the puncture limit of the Riemann surface in terms of CFT $n$-point functions. We also use AdS [[0007 RT surface|minimal surfaces]] to measure entanglement more generally. In some regions of the moduli space the entanglement is entirely [[0264 Multi-partite entanglement|multipartite]], though not of the GHZ type. However, even when the bulk is completely connected, in some regions of the moduli space the entanglement is almost entirely bipartite: significant entanglement occurs only between pairs of CFTs. We develop new tools to analyze intrinsically $n$-partite entanglement, and use these to show that for some wormholes with $n$ similar sized horizons there is intrinsic entanglement between at least $n-1$ parties, and that the distillable entanglement between the asymptotic regions is at least $(n+1)/2$ partite.\] ## Refs - [[0264 Multi-partite entanglement]] ## Summary - even for entirely connected bulk spacetime, the entanglement can be either multipartite (though never GHZ-like) or mostly bipartite - tuning the moduli to disconnect a single boundary from the 3-boundary wormhole leads not only to a small entropy for the disconnected CFT, but also to a sharp decrease in entropy for the two CFTs that remain connected ## Context - *time-symmetric* initial data - -> so vanishing extrinsic curvature - -> can find Euclidean solution with such a surface as the boundary (done by taking the "same" [[0099 Quotient method in AdS3|quotient]] of Euclidean AdS3) - alternatively, analytic continue # Bern, Davies, Nohle ## On loop corrections to subleading soft behaviour of gluons and gravitons [](https://arxiv.org/abs/1405.1015) ## Summary - *shows* that there is - 1-loop correction to subleading soft symmetry - 2-loop correction to subsubleading - no further - both ==gauge== and ==gravity== ## The method? - do the regularisation for UV and IR divergences at loop level - find the modified Ward identity from symmetries ## Infrared divergence - QCD $\left.A_{n}^{1-\mathrm{loop}}(1,2, \cdots, n)\right|_{\mathrm{div} .}=-\frac{1}{\epsilon^{2}} A_{n}^{\mathrm{tree}}(1,2, \cdots, n) \sigma_{n}^{\mathrm{YM}}$ (3.4) ## The result - leading - $A_{n}^{1-\text { loop }} \rightarrow S_{n \mathrm{YM}}^{(0)} A_{n-1}^{1 \text {-loop}}+S_{n \mathrm{YM}}^{(0) 1\text {-loop}} A_{n-1}^{\text {tree }}$ - $S_{n \mathrm{YM}}^{(0) 1\text{-}\mathrm{loop}}=-S_{n \mathrm{YM}}^{(0)} \frac{c_{\Gamma}}{\epsilon^{2}}\left(\frac{-\mu^{2} s_{(n-1) 1}}{s_{(n-1) n} s_{n 1}}\right)^{\epsilon} \frac{\pi \epsilon}{\sin (\pi \epsilon)}$=-S_{n}^{(0)} c_{\Gamma}\left(\frac{1}{\epsilon^{2}}+\frac{1}{\epsilon} \log \left(\frac{-\mu^{2} s_{(n-1) 1}}{\delta^{2} s_{(n-1) n} s_{n 1}}\right)+\frac{1}{2} \log ^{2}\left(\frac{-\mu^{2} s_{(n-1) 1}}{\delta^{2} s_{(n-1) n} s_{n 1}}\right)+\frac{\pi^{2}}{6}\right)+\mathcal{O}(\epsilon)$ - need to first expand in $\epsilon$ before taking the soft limit $\delta\rightarrow0$ - subleading - $\left.\left.A_{n}^{1-\mathrm{loop}}\right|_{\text {div. }} \rightarrow\left(\frac{1}{\delta^{2}} S_{n \mathrm{YM}}^{(0)}+\frac{1}{\delta} S_{n \mathrm{YM}}^{(1)}\right) A_{n-1}^{1\text{-}\mathrm{loop}}\right|_{\mathrm{div} .}+\left.\left(\frac{1}{\delta^{2}} S_{n \mathrm{YM}}^{(0) 1 \text {-loop }}+\frac{1}{\delta} S_{n \mathrm{YM}}^{(1) 1 \text {-loop}}\right) A_{n-1}^{\text {tree }}\right|_{\text {div. }}$ - infrared-finite parts also contribute non-trivially - identical helicity: no change - with one negative helicity: has non-trivial change - ==left to future work== # Bianchi, He, Huang, Wen ## More on Soft Theorems: Trees, Loops and Strings \[Links: [arXiv](https://arxiv.org/abs/1406.5155), [PDF](https://arxiv.org/pdf/1406.5155.pdf)\] \[Abstract: We study [[0009 Soft theorems|soft theorems]] in a broader context, addressing their fate at loop level and their universality in effective field theories and string theory. We argue that for gauge theories in the planar limit, loop-level soft gluon theorems can be made manifest already at the integrand level. In particular, we show that the planar integrand for $\mathcal{N}=4$ SYM satisfies the tree-level soft theorem to all orders in perturbation theory and provide strong evidence to this effect for integrands in $\mathcal{N}<4$ SYM. We consider soft theorems for non-supersymmetric Yang-Mills theories and gravity, and show the validity of integrand soft theorem, while loop corrections to the integrated soft theorems are intimately tied to the presence of conformal anomalies. We then address the question of universality of the soft theorems for various theories. In effective field theories with $F^3$ and $R^3$ interactions, the soft theorems are not modified. However for gravity theories with $R^2 \phi$ interactions, the sub-sub-leading order soft graviton theorem, which is beyond what is implied by the extended BMS symmetry, requires modifications at tree level for non-supersymmetric theories, and at loop level for $\mathcal{N}<5$ supergravity due to anomalies. Finally, for superstring amplitudes at finite $\alpha'$, via explicit calculation for lower-point examples as well as world-sheet OPE analysis for arbitrary multiplicity, we show that the superstring amplitudes satisfy the same soft theorem as its field-theory counterpart. This is no longer true for bosonic closed strings due to the presence of $R^2 \phi$ interactions.\] # Bousso, Casini, Fisher, Maldacena ## Proof of a quantum Bousso bound \[Links: [arXiv](https://arxiv.org/abs/1404.5635), [PDF](https://arxiv.org/pdf/1404.5635.pdf)\] \[Abstract: \] ## Summary - proves [[0171 Covariant entropy bound|Bousso bound]], assuming free fields, small gravitational backreaction; not assuming [[0480 Null energy condition|NEC]] # Cachazo, Strominger ## Evidence for a new soft graviton theorem \[Links: [arXiv](https://arxiv.org/abs/1404.4091), [PDF](https://arxiv.org/pdf/1404.4091.pdf)\] \[Abstract: \] ## Summary - subleading [[0009 Soft theorems]] for graviton - conjectured universal formula for the finite subleading term in the expansion about the soft soft limit - it is gauge invariant: follows from global angular momentum conservation - *verifies* the above for all tree-level graviton amplitudes using [[0058 BCFW]] recursion relation - *understands* it as a the [[0106 Ward identity]] of superrotation [[0032 Virasoro algebra|Virasoro]] symmetry ## How to prove subleading soft theorem 1. diagrammatic approach like Weinberg - but many sources of subleading corrections 2. explicitly constructing the generators of the conjectured symmetries and rewriting the invariance of S matrix in the form of a soft theorem - more complicated than leading 3. (this paper) testing it using BCFW - turns out all tree-level results pass (non-trivial) ## Subsubleading - $\mathcal{M}_{n+1}\left(k_{1}, k_{2}, \ldots k_{n}, q\right)=\left(S^{(0)}+S^{(1)}+S^{(2)}\right) \mathcal{M}_{n}\left(k_{1}, k_{2}, \ldots k_{n}\right)+\mathcal{O}\left(q^{2}\right)$ with $S^{(2)} \equiv-\frac{1}{2} \sum_{a=1}^{n} \frac{E_{\mu \nu}\left(q_{\rho} J_{a}^{\rho \mu}\right)\left(q_{\sigma} J_{a}^{\sigma \nu}\right)}{q \cdot k_{a}}$ - gauge invariance: simply because $J^{\mu\nu}$ is antisymmetric - for leading and subleading it was due to conservation laws - unlike subleading, there is no argument for why it is universal beyond tree-level # Camanho, Edelstein, Maldacena, Zhiboedov ## Causality constraints on corrections to the graviton 3-pt. coupling \[Links: [arXiv](https://arxiv.org/abs/1407.5597), [PDF](https://arxiv.org/pdf/1407.5597.pdf)\] \[Abstract: We consider [[0006 Higher-derivative gravity|higher derivative corrections]] to the graviton three-point coupling within a weakly coupled theory of gravity. Lorentz invariance allows further structures beyond the one present in the Einstein theory. We argue that these are constrained by [[0118 Causality constraints for gravity|causality]]. We devise a thought experiment involving a high energy scattering process which leads to causality violation if the graviton three-point vertex contains the additional structures. This violation cannot be fixed by adding conventional particles with spins $J \leq 2$. But, it can be fixed by adding an infinite tower of extra massive particles with higher spins, $J > 2$. In AdS theories this implies a constraint on the conformal anomaly coefficients $\left|{a - c \over c} \right| \lesssim {1 \over \Delta_{gap}^2}$ in terms of $\Delta_{gap}$, the dimension of the lightest single particle operator with spin $J > 2$. For inflation, or de Sitter-like solutions, it indicates the existence of massive higher spin particles if the gravity wave non-gaussianity deviates significantly from the one computed in the Einstein theory.\] ## Comment - if we view gravity as a low-energy effective theory with a UV cutoff of order $M_{pl}$ and we add higher derivative terms with dimensionless coefficients which are of order one, then we have nothing to say; in other words, the causality constraints only rule out much larger couplings ## Refs - [[0118 Causality constraints for gravity]] - [[0115 Superluminality]] ## Summary - causality constrain [[0006 Higher-derivative gravity|HDG]] terms - n.b. only consider second-order ones - a discussion on AdS using [[0117 Shockwave|shockwaves]] - used [[0091 Boundary causality]] - e.g. in Einstein-Gauss-Bonnet gravity, gravitons can experience a negative Shapiro time delay, i.e. a time advance. - this is established using singular shock wave geometries. later generalised to smooth geometries by [[PapalloReall2015]][](https://arxiv.org/abs/1508.05303) - conjecture $a\approx c$ for conformal anomaly coefficients - thought of as a [[0132 Causality constraints in CFT]] # Casali ## Soft sub-leading divergences in Yang-Mills amplitudes \[Links: [arXiv](https://arxiv.org/abs/1404.5551), [PDF](https://arxiv.org/pdf/1404.5551.pdf)\] \[Abstract: In this short note I show that the [[0009 Soft theorems|soft]] limit for colour-ordered tree-level Yang-Mills amplitudes contains a sub-leading divergent term analogous to terms found recently by [[2014#Cachazo, Strominger]] for tree-level gravity amplitudes.\] ## Summary - uses [[0515 Inverse soft construction|inverse soft recursion]] to derive a subleading [[0009 Soft theorems|soft gluon theorem]] # Costa, Goncalves, Penedones ## Spinning AdS Propagators \[Links: [arXiv](https://arxiv.org/abs/1404.5625), [PDF](https://arxiv.org/pdf/1404.5625)\] \[Abstract: We develop the embedding formalism to describe symmetric traceless tensors in Anti-de Sitter space. We use this formalism to construct the bulk-to-bulk propagator of massive spin $J$ fields and check that it has the expected short distance and massless limits. We also and a split representation for the [[0103 Two-point functions|bulk-to-bulk propagator]], by writing it as an integral over the boundary of the product of two bulk-to-boundary propagators. We exemplify the use of this representation with the computation of the conformal partial wave decomposition of [[0109 Witten diagrams|Witten diagrams]]. In particular, we determine the [[0079 Mellin transform|Mellin amplitude]] associated to AdS graviton exchange between minimally coupled scalars of general dimension, including the regular part of the amplitude.\] # Donos, Gauntlett (Jun) ## Thermoelectric DC conductivities from black hole horizons \[Links: [arXiv](https://arxiv.org/abs/1406.4742), [PDF](https://arxiv.org/pdf/1406.4742.pdf)\] \[Abstract: \] ## Refs - later work, using Navier-Stokes equation [[2015#Donos, Gauntlett]] ## Summary - works out [[0434 Diffusivity]] in general - *illustrates* the results for [[0447 Q-lattice]] BHs and BHs with linear massless axions ## Setup - action $S=\int d^{4} x \sqrt{-g}$\left[R-\frac{1}{2}\left[(\partial \phi)^{2}+\Phi_{1}(\phi)\left(\partial \chi_{1}\right)^{2}+\Phi_{2}(\phi)\left(\partial \chi_{2}\right)^{2}\right]\right.$\left.-V(\phi)-\frac{Z(\phi)}{4} F^{2}\right]$ - metric $d s^{2}=-U d t^{2}+U^{-1} d r^{2}+e^{2 V_{1}} d x_{1}^{2}+e^{2 V_{2}} d x_{2}^{2}$ - fields $A=a d t, \quad \chi_{1}=k_{1} x_{1}, \quad \chi_{2}=k_{2} x_{2}$ ## Thermal conductivity # Engelhardt, Wall ## Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime \[Links: [arXiv](https://arxiv.org/abs/1408.3203), [PDF](https://arxiv.org/pdf/1408.3203.pdf)\] \[Abstract: We propose that [[0145 Generalised area|holographic entanglement entropy]] can be calculated at arbitrary orders in the bulk Planck constant using the concept of a "[[0212 Quantum extremal surface|quantum extremal surface]]": a surface which extremizes the generalized entropy, i.e. the sum of [[0004 Black hole entropy|area]] and bulk [[0301 Entanglement entropy|entanglement entropy]]. At leading order in bulk quantum corrections, our proposal agrees with the formula of [[2013#Faulkner, Lewkowycz, Maldacena|Faulkner, Lewkowycz, and Maldacena]], which was derived only at this order; beyond leading order corrections, the two conjectures diverge. Quantum extremal surfaces lie outside the causal domain of influence of the boundary region as well as its complement, and in some spacetimes there are barriers preventing them from entering certain regions. We comment on the implications for [[0026 Bulk reconstruction|bulk reconstruction]].\] ## Summary - extends [[0007 RT surface|RT-HRT]] and [[2013#Faulkner, Lewkowycz, Maldacena]] ## RG flow - shifts matter terms to higher derivative term # Feige, Schwartz ## Hard-Soft-Collinear Factorization to All Orders \[Links: [arXiv](https://arxiv.org/abs/1403.6472), [PDF](https://arxiv.org/pdf/1403.6472.pdf)\] \[Abstract: We provide a precise statement of hard-soft-collinear factorization of scattering amplitudes and prove it to all orders in perturbation theory. [[0609 Soft factorisation|Factorization]] is formulated as the equality at leading power of scattering amplitudes in QCD with other amplitudes in QCD computed from a product of operator matrix elements. The equivalence is regulator independent and gauge independent. As the formulation relates amplitudes to the same amplitudes with additional [[0009 Soft theorems|soft]] or [[0078 Collinear limit|collinear]] particles, it includes as special cases the factorization of soft currents and collinear splitting functions from generic matrix elements, both of which are shown to be process independent to all orders. We show that the overlapping soft-collinear region is naturally accounted for by vacuum matrix elements of kinked Wilson lines. Although the proof is self-contained, it combines techniques developed for the study of pinch surfaces, scattering amplitudes, and effective field theory.\] # Goncalves, Penedones, Trevisani ## Factorisation of Mellin amplitudes \[Links: [arXiv](https://arxiv.org/abs/1410.4185), [PDF](https://arxiv.org/pdf/1410.4185.pdf)\] \[Abstract: \] ## Summary - introduce Mellin amplitudes for correlation functions of $k$ *scalar* operators and one with *spin* in *CFT* - Mellin amplitudes for scalar operators factorise in terms of lower point Mellin amplitudes -> [[0057 Multi-point function factorisation]] - study the flat limit ## Multiple OPE ![[GoncalvesPenedonesTrevisani2014_replace.png]] - $\mathcal{O}_{1}\left(x_{1}\right) \ldots \mathcal{O}_{k}\left(x_{k}\right)=\sum_{p} C_{\mu_{1} \ldots \mu_{J}}^{(1 \ldots k, p)}\left(x_{1}, \ldots, x_{k}, y, \partial_{y}\right) \mathcal{O}_{p}^{\mu_{1} \ldots \mu_{J}}(y)$ - the sum converges inside a $n$-point correlation function if there is a sphere centred at $y$ containing all $x_1,\dots,x_k$ and none of the remaining $n-k$ points ## What it is good for - For scattering amplitudes, understanding their factorization properties is the starting point for the construction of recursion relations (like [[0058 BCFW]]). Such recursion relations determine n-particle scattering amplitudes in terms of scattering amplitudes with a smaller number of particles. In some cases (gluons or gravitons), this can be iterated successively until all scattering amplitudes are fixed in terms of the 3-particle amplitudes. Our long term goal is to generalize this type of recursion relations for Mellin amplitudes # Haehl, Hartman, Marolf, Maxfield, Rangamani ## Topological aspects of generalized gravitational entropy \[Links: [arXiv](https://arxiv.org/abs/1412.7561), [PDF](https://arxiv.org/pdf/1412.7561.pdf)\] \[Abstract: \] ## Main question - when does [[2013#Lewkowycz, Maldacena|LM]] prescription enforce the homology constraint? ## Interpretation of the assumptions in LM - global interpretation - global topology outside of conical singularity is given by $S^1\times X$ for some $X$ (trivial bundle) - this implies homology constraint but too strong - local interpretation - locally $S^1\times X$ 1. $S^1$ fiber need not be defined away from this conical singularity 2. can be a **non-trivial bundle** - **result of this paper**: ==this also implies homology constraint as along as the family of geodesics exists at all $q\ge1$ and is an appropriate $\mathbb{Z}_q$ quotient of a smooth $q$-fold replica at all integer $q$== # Hartman, Keller, Stoica ## Universal Spectrum of 2d Conformal Field Theory in the Large $c$ Limit \[Links: [arXiv](https://arxiv.org/abs/1405.5137), [PDF](https://arxiv.org/pdf/1405.5137.pdf)\] \[Abstract: Two-dimensional conformal field theories exhibit a universal free energy in the high temperature limit $T \to \infty$, and a universal spectrum in the [[0406 Cardy formula|Cardy]] regime, $\Delta \to \infty$. We show that a much stronger form of universality holds in theories with a large central charge $c$ and a sparse light spectrum. In these theories, the free energy is universal at all values of the temperature, and the microscopic spectrum matches the Cardy entropy for all $\Delta \geq c/6$. The same is true of three-dimensional quantum gravity; therefore our results provide simple necessary and sufficient criteria for 2d CFTs to behave holographically in terms of the leading spectrum and thermodynamics. We also discuss several applications to CFT and gravity, including operator dimension bounds derived from the modular bootstrap, universality in symmetric orbifolds, and the role of non-universal 'enigma' saddlepoints in the thermodynamics of [[0002 3D gravity|3d gravity]].\] # Headrick, Hubeny, Lawrence, Rangamani ## Causality and HEE \[Links: [arXiv](https://arxiv.org/abs/1408.6300), [PDF](https://arxiv.org/pdf/1408.6300.pdf)\] \[Abstract: \] ## Refs - [[0252 Causality of HEE]] ## Summary - identify causality conditions for [[0145 Generalised area|HEE]] - prove: [[0480 Null energy condition|NEC]] => these conditions - introduces [[0219 Entanglement wedge reconstruction|entanglement wedge]] ## Restrictions 1. the bulk theory is two derivative ## Causality implications in CFT 1. same domain of dependence -> same entropy 2. fixing initial state, a perturbation contained in $D[\mathcal{A}] \cup D[\mathcal{A}^c]$ cannot affect EE ## Causality implications in bulk 1. same HRT surface for if boundary regions has same domain of dependence 2. HRT lies in causal shadow ## Proof: [[0480 Null energy condition|NEC]] => these conditions # He, Huang, Wen ## Loop corrections to soft theorems in gauge theories and gravity \[Links: [arXiv](https://arxiv.org/abs/1405.1410), [PDF](https://arxiv.org/pdf/1405.1410.pdf)\] \[Abstract: \] ## Summary - *finds* ==IR-finite== loop correction to subleading [[0009 Soft theorems]] for [[0107 Soft gluon symmetry]] ## 3.1 All plus gluons - both leading and subleading receive no loop correction ## 3.2 Single minus gluons # He, Mitra, Porfyriadis, Strominger ## New symmetries of massless QED \[Links: [arXiv](https://arxiv.org/abs/1407.3789), [PDF](https://arxiv.org/pdf/1407.3789.pdf)\] \[Abstract: \] ## Summary - [[0060 Asymptotic symmetry]] for ==massless QED== # Kapec, Lysov, Pasterski, Strominger ## Semiclassical Virasoro Symmetry of the Quantum Gravity S-Matrix \[Links: [arXiv](https://arxiv.org/abs/1406.3312), [PDF](https://arxiv.org/pdf/1406.3312.pdf)\] \[Abstract: It is shown that the tree-level S-matrix for quantum gravity in four-dimensional Minkowski space has a [[0032 Virasoro algebra|Virasoro symmetry]] which acts on the conformal sphere at null infinity.\] # Kelly, Maloney ## Poincare Series, 3D Gravity and CFT Spectroscopy \[Links: [arXiv](https://arxiv.org/abs/1407.6008), [PDF](https://arxiv.org/pdf/1407.6008)\] \[Abstract: [[0612 Modular invariance|Modular invariance]] strongly constrains the spectrum of states of two dimensional conformal field theories. By summing over the images of the modular group, we construct candidate CFT partition functions that are modular invariant and have positive spectrum. This allows us to efficiently extract the constraints on the CFT spectrum imposed by modular invariance, giving information on the spectrum that goes beyond the Cardy growth of the asymptotic density of states. Some of the candidate modular invariant partition functions we construct have gaps of size $(c-1)/12$, proving that gaps of this size and smaller are consistent with modular invariance. We also revisit the partition function of [[0002 3D gravity|pure Einstein gravity in AdS3]] obtained by summing over geometries, which has a spectrum with two unphysical features: it is continuous, and the density of states is not positive definite. We show that both of these can be resolved by adding corrections to the spectrum which are subleading in the semi-classical (large central charge) limit.\] # Kelly, Wall ## A holographic proof of the averaged null energy condition \[Links: [arXiv](https://arxiv.org/abs/1408.3566), [PDF](https://arxiv.org/pdf/1408.3566.pdf)\] \[Abstract: The [[0417 Averaged null energy condition|averaged null energy conditions]] (ANEC) states that, along a complete null curve, the negative energy fluctuations of a quantum field must be balanced by positive energy fluctuations. We use the [[0001 AdS-CFT|AdS/CFT]] correspondence to prove the ANEC for a class of strongly coupled conformal field theories in ==flat spacetime==. A violation of the ANEC in the field theory would lead to acausal propagation of signals in the bulk.\] # Kjall, Bardarson, Pollman ## Many-body localization in a disordered quantum Ising chain \[Links: [arXiv](https://arxiv.org/abs/1403.1568), [PDF](https://arxiv.org/pdf/1403.1568.pdf)\] \[Abstract: Many-body localization occurs in isolated quantum systems when Anderson localization persists in the presence of finite interactions. Despite strong evidence for the existence of a many-body localization transition a reliable extraction of the critical disorder strength is difficult due to a large drift with system size in the studied quantities. In this work we explore two entanglement properties that are promising for the study of the manybody localization transition: the variance of the half-chain [[0301 Entanglement entropy|entanglement entropy]] of exact eigenstates and the long time change in entanglement after a local quench from an exact eigenstate. We investigate these quantities in a ==disordered quantum Ising chain== and use them to estimate the critical disorder strength and its energy dependence. In addition, we analyze a [[0246 Spin glass|spin-glass]] transition at large disorder strength and provide evidence for it being a separate transition. We thereby give numerical support for a recently proposed phase diagram of many-body localization with localization protected quantum order [[2013#Huse, Nandkishore, Oganesyan, Pal, Sondhi|Huse et al. Phys. Rev. B 88, 014206 (2013)]].\] # Kubiznak, Mann ## Black Hole Chemistry \[Links: [arXiv](https://arxiv.org/abs/1404.2126), [PDF](https://arxiv.org/pdf/1404.2126.pdf)\] \[Abstract: The mass of a black hole has traditionally been identified with its energy. We describe a new perspective on black hole thermodynamics, one that identifies the mass of a black hole with chemical enthalpy, and the cosmological constant as thermodynamic pressure. This leads to an understanding of black holes from the viewpoint of chemistry, in terms of concepts such as Van der Waals fluids, reentrant phase transitions, and triple points. Both charged and rotating black holes exhibit novel chemical-type phase behaviour, hitherto unseen.\] ## Refs - [[0525 Black hole chemistry]] # Leichenauer ## Disrupting Entanglement of Black Holes \[Links: [arXiv](https://arxiv.org/abs/1405.7365), [PDF](https://arxiv.org/pdf/1405.7365.pdf)\] \[Abstract: We study entanglement in thermofield double states of strongly coupled CFTs by analyzing two-sided Reissner-Nordstrom solutions in AdS. The central object of study is the [[0300 Mutual information|mutual information]] between a pair of regions, one on each asymptotic boundary of the black hole. For large regions the mutual information is positive and for small ones it vanishes; we compute the critical length scale, which goes to infinity for extremal black holes, of the transition. We also generalize the butterfly effect of [[2013#Shenker, Stanford (Jun)]] to a wide class of charged black holes, showing that mutual information is disrupted upon perturbing the system and waiting for a time of order $\log E/\delta E$ in units of the temperature. We conjecture that the parametric form of this timescale is universal.\] ## Refs - [[0326 Charged BH in holography]] ## Summary - [[0300 Mutual information|mutual information]] between two regions on separate boundaries - large regions: positive - small: vanish - there is a length scale of transition - goes to infinity for extremal BHs - [[0008 Quantum chaos|quantum chaos]] perspective - [[0300 Mutual information|mutual information]] is disrupted by perturbation after a time scale $\log E/\delta E$ - conjectures that this is universal # Liu, Zayas, Yang ## Small Treatise on Spin-3/2 Fields and their Dual Spectral Functions \[Links: [arXiv](https://arxiv.org/abs/1401.0008), [PDF](https://arxiv.org/pdf/1401.0008.pdf)\] \[Abstract: In this work we systematically study various aspects of spin-3/2 fields in a curved background. We mostly focus on a minimally coupled massive spin-3/2 field in arbitrary dimensions, and solve the equation of motion either explicitly or numerically in AdS, Schwarzschild-AdS and Reissner-Nordström-AdS backgrounds. Although not the main focus of this work, we also make a connection with the gravitino equation of motion in gauged [[0332 Supergravity|supergravity]]. Motivated by the [[0001 AdS-CFT|AdS/CFT]] correspondence, we emphasize calculational improvements and technical details of the dual [[0536 Spectral function|spectral functions]]. We attempt to provide a coherent and comprehensive picture of the existing literature.\] ## Refs - [[0461 Fermions in AdS-CFT]] - [[0473 Retarded Green's function]] - [[0536 Spectral function]] # Lysov, Pasterski, Strominger ## Low's subleading soft theorem as a symmetry of QED ## Refs - an example of [[0009 Soft theorems]] ## Summary - ==massless QED== subleading soft theorem at ==tree-level== - interpret the subleading soft theorem as an infinitesimal symmetry of the S-matrix ## NOT a subgroup - unlike leading soft photon, leading and subleading soft graviton, this is not a subgroup of the original gauge symmetry - expect to be true also for subsubleading soft graviton ## Soft theorem -> symmetry - soft theorem $\left\langle z_{n+1}, \ldots\left|a_{-}^{o u t}(\vec{q}) \mathcal{S}\right| z_{1}, \ldots\right\rangle=\left(J^{(0)-}+J^{(1)-}\right)\left\langle z_{n+1}, \ldots|\mathcal{S}| z_{1}, \ldots\right\rangle+\mathcal{O}(\omega)$ - $J^{(0)-}=e \sum_{k} Q_{k} \frac{p_{k} \cdot \varepsilon^{-}}{p_{k} \cdot q} \sim \mathcal{O}\left(\omega^{-1}\right)$ - $J^{(1)-}=-i e \sum_{k} Q_{k} \frac{q_{\mu} \varepsilon_{\nu}^{-} J_{k}^{\mu \nu}}{p_{k} \cdot q} \sim \mathcal{O}\left(\omega^{0}\right)$ - eliminate the leading term - $\lim _{\omega \rightarrow 0}\left(1+\omega \partial_{\omega}\right)\left\langle z_{n+1}, \ldots\left|a_{-}^{o u t}(\vec{q}) \mathcal{S}\right| z_{1}, \ldots\right\rangle=J^{(1)-}\left\langle z_{n+1}, \ldots|\mathcal{S}| z_{1}, \ldots\right\rangle$ - symmetry - $\left\langle z_{n+1}, \ldots\left|\mathcal{Q}^{+} \mathcal{S}-\mathcal{S} \mathcal{Q}^{-}\right| z_{1}, \ldots\right\rangle=0$ - $\mathcal{Q}_{S}^{+}=-\frac{2}{e^{2}} \int d^{2} z d u u \partial_{u} A_{\bar{z}} D_{z}^{2} Y^{z}$ - $\mathcal{Q}_{S}^{-}=\frac{2}{e^{2}} \int d^{2} z d v v \partial_{v} A_{\bar{z}}^{-} D_{z}^{2} Y^{z}$ # Maxfield ## Entanglement entropy in three dimensional gravity \[Links: [arXiv](https://arxiv.org/abs/1412.0687), [PDF](https://arxiv.org/pdf/1412.0687.pdf)\] \[Abstract: \] ## Refs - helps with dealing with [[0002 3D gravity]] - use matrices to represent quotients ## Summary - shows how to calculate geodesic lengths by [[0099 Quotient method in AdS3]] - examples - rotating BTZ - $\mathbb{R} \mathbb{P}^{2}$ geon - some wormholes - spatial and temporal dependence worked out! - considers obtaining HRT from **analytic continuations** # Miao, Guo ## Holographic Entanglement Entropy for the Most General Higher Derivative Gravity \[Links: [arXiv](https://arxiv.org/abs/1411.5579), [PDF](https://arxiv.org/pdf/1411.5579.pdf)\] \[Abstract: The [[0145 Generalised area|holographic entanglement entropy]] for the most general [[0006 Higher-derivative gravity|higher derivative gravity]] is investigated. We find a new type of Wald entropy, which appears on entangling surface without the rotational symmetry and reduces to usual Wald entropy on Killing horizon. Furthermore, we obtain a formal formula of HEE for the most general higher derivative gravity and work it out exactly for some squashed cones. As an important application, we derive HEE for gravitational action with one derivative of the curvature when the extrinsic curvature vanishes. We also study some toy models with non-zero extrinsic curvature. We prove that our formula yields the correct universal term of entanglement entropy for 4d CFTs. Furthermore, we solve the puzzle raised by Hung, Myers and Smolkin that the logarithmic term of entanglement entropy derived from Weyl anomaly of CFTs does not match the holographic result even if the extrinsic curvature vanishes. We find that such mismatch comes from the 'anomaly of entropy' of the derivative of curvature. After considering such contributions carefully, we resolve the puzzle successfully. In general, we need to fix the splitting problem for the conical metrics in order to derive the holographic entanglement entropy. We find that, at least for Einstein gravity, the splitting problem can be fixed by using equations of motion. How to derive the splittings for higher derivative gravity is a non-trivial and open question. For simplicity, we ignore the splitting problem in this paper and find that it does not affect our main results.\] ## Summary - [[0145 Generalised area|HEE]] for $f(R, \nabla R,\dots)$ ## $2n$-derivative gravity - considers $S\left(g, R, \nabla R, \ldots, \nabla^{n-2} R\right)$, but the cone metric is chosen to be a "highest-order cone", meaning that only $\nabla^{n-2} R$ contributes to the anomaly term in this case # Ohmori, Tachikawa ## Physics at the entangling surface \[Links: [arXiv](https://arxiv.org/abs/1406.4167), [PDF](https://arxiv.org/pdf/1406.4167)\] \[Abstract: To consider the entanglement between the spatial region $A$ and its complement in a QFT, we need to assign a Hilbert space $\mathcal{H}_A$ to the region, by making a certain choice on the boundary $\partial A$. We argue that a small [[0548 Boundary CFT|physical boundary]] is implicitly inserted at the entangling surface. We investigate these issues in the context of [[0003 2D CFT|2d CFTs]], and show that we can indeed read off the Cardy states of the $c=1/2$ minimal model from the [[0301 Entanglement entropy|entanglement entropy]] of the critical Ising chain.\] # Roberts, Stanford ## Two-dimensional conformal field theory and the butterfly effect \[Links: [arXiv](https://arxiv.org/abs/1412.5123), [PDF](https://arxiv.org/pdf/1412.5123.pdf)\] \[Abstract: We study chaotic dynamics in two-dimensional conformal field theory through [[0482 Out-of-time-order correlator|out-of-time order thermal correlators]] of the form $\langle W(t)VW(t)V\rangle$. We reproduce bulk calculations similar to those of \[[[2013#Shenker, Stanford (Jun)|1]]\], by studying the large $c$ [[0032 Virasoro algebra|Virasoro]] identity block. The contribution of this block to the above correlation function begins to decrease exponentially after a delay of $\sim t_* - \frac{\beta}{2\pi}\log \beta^2E_w E_v$, where $t_*$ is the scrambling time $\frac{\beta}{2\pi}\log c$, and $E_w,E_v$ are the energy scales of the $W,V$ operators.\] ## Refs - simultaneous release [[JacksonMcGoughVerlinde2014]][](https://arxiv.org/abs/1412.5205) # Roberts, Stanford, Susskind ## Localized shocks \[Links: [arXiv](https://arxiv.org/abs/1409.8180), [PDF](https://arxiv.org/pdf/1409.8180.pdf)\] \[Abstract: We study products of precursors of spatially local operators, $W_{x_{n}}(t_{n}) ... W_{x_1}(t_1)$, where $W_x(t) = e^{-iHt} W_x e^{iHt}$. Using chaotic spin-chain numerics and [[0001 AdS-CFT|gauge/gravity duality]], we show that a single precursor fills a spatial region that grows linearly in $t$. In a lattice system, products of such operators can be represented using [[0054 Tensor network|tensor networks]]. In gauge/gravity duality, they are related to Einstein-Rosen bridges supported by localized [[0117 Shockwave|shock waves]]. We find a geometrical correspondence between these two descriptions, generalizing earlier work in the spatially homogeneous case.\] ## Summary - extension of [[2013#Shenker, Stanford (Jun)]] - [[0117 Shockwave]] calculation of [[0167 Butterfly velocity]] ## Related - [[0008 Quantum chaos]] - [[0117 Shockwave]] # Schwab, Volovich ## Subleading soft theorem in arbitrary dimension from scattering equations \[Links: [arXiv](https://arxiv.org/abs/1404.7749), [PDF](https://arxiv.org/pdf/1404.7749.pdf)\] \[Abstract: We investigate the new [[0009 Soft theorems|soft graviton theorem]] recently proposed in [[2014#Cachazo, Strominger]]. We use the [[0543 Cachazo-He-Yuan formalism|CHY formula]] to prove this universal formula for both Yang-Mills theory and gravity scattering amplitudes at tree level in arbitrary dimension.\] ## Summary - generalisation of subleading soft graviton theorem to higher dimensions (and in YM) ## Refs - independent work [[2014#Afkhami-Jeddi]] # Shenker, Stanford ## Stringy effects in scrambling \[Links: [arXiv](https://arxiv.org/abs/1412.6087), [PDF](https://arxiv.org/pdf/1412.6087.pdf)\] \[Abstract: In [1] we gave a precise holographic calculation of [[0008 Quantum chaos|chaos]] at the scrambling time scale. We studied the influence of a small perturbation, long in the past, on a two-sided correlation function in the thermofield double state. A similar analysis applies to squared commutators and other [[0482 Out-of-time-order correlator|out-of-time-order]] one-sided correlators [2-4]. The essential bulk physics is a high energy scattering problem near the horizon of an AdS black hole. The above papers used [[0554 Einstein gravity|Einstein gravity]] to study this problem; in the present paper we consider stringy and Planckian corrections. Elastic stringy corrections play an important role, effectively weakening and smearing out the development of chaos. We discuss their signature in the boundary field theory, commenting on the extension to weak coupling. Inelastic effects, although important for the evolution of the state, leave a parametrically small imprint on the correlators that we study. We briefly discuss ways to diagnose these small corrections, and we propose another correlator where inelastic effects are order one.\] ## Summary - scrambling time corrected in string theory - $t_*=\frac{\beta}{2 \pi}\left[1+\frac{d(d-1) \ell_s^2}{4 \ell_{A d S}^2}+\ldots\right] \log S$ - tree-level stringy corrections using techniques of [[2006#Brower, Polchinski, Strassler, Tan]] - inelastic effects have very little effect on the correlators ## The important formula Define$D\left(\left\{t_i, x_i\right\}\right)=\left\langle V_{x_1}\left(t_1\right) W_{x_2}\left(t_2\right) V_{x_3}\left(t_3\right) W_{x_4}\left(t_4\right)\right\rangle,$then $D\left(\left\{t_i, x_i\right\}\right)=\frac{a_0^4}{(4 \pi)^2} \int e^{i \delta\left(s,\left|x-x^{\prime}\right|\right)}\left[p_1^u \psi_1^*(p_1^u, x) \psi_3(p_1^u, x)\right][p_2^v \psi_2^*(p_2^v, x^{\prime}) \psi_4(p_2^v, x^{\prime})]$where $s=a_0 p_1^u p_2^v$, and the wavefunctions are Fourier transforms of bulk-to-boundary propagators along one of the horizons. # Stanford, Susskind ## Complexity and Shock Wave Geometries \[Links: [arXiv](https://arxiv.org/abs/1406.2678), [PDF](https://arxiv.org/pdf/1406.2678.pdf)\] \[Abstract: In this paper we refine a conjecture relating the time-dependent size of an Einstein-Rosen bridge to the computational [[0204 Quantum complexity|complexity]] of the of the dual quantum state. Our refinement states that the complexity is proportional to the spatial volume of the ERB. More precisely, up to an ambiguous numerical coefficient, we propose that the complexity is the regularized volume of the largest codimension one surface crossing the bridge, divided by $G_N l_{AdS}$. We test this conjecture against a wide variety of spherically symmetric [[0117 Shockwave|shock wave]] geometries in different dimensions. We find detailed agreement.\] # Strominger, Zhiboedov ## Gravitational Memory, BMS Supertranslations and Soft Theorems \[Links: [arXiv](https://arxiv.org/abs/1411.5745), [PDF](https://arxiv.org/pdf/1411.5745.pdf)\] \[Abstract: The transit of a gravitating radiation pulse past arrays of detectors stationed near future null infinity in the vacuum is considered. It is shown that the relative positions and clock times of the detectors before and after the radiation transit differ by a BMS supertranslation. An explicit expression for the supertranslation in terms of moments of the radiation energy flux is given. The relative spatial displacement found for a pair of nearby detectors reproduces the well-known and potentially measurable [[0287 Memory effect|gravitational memory effect]]. The displacement memory formula is shown to be equivalent to Weinberg's formula for soft graviton production.\] ## Refs - OG for relating [[0009 Soft theorems|soft theorems]] to [[0287 Memory effect|the memory effect]] # Susskind (Feb) ## Computational Complexity and Black Hole Horizons \[Links: [inspire](https://inspirehep.net/literature/1282259); paper: [arXiv](https://arxiv.org/abs/1402.5674), [PDF](https://arxiv.org/pdf/1402.5674.pdf); addendum: [arXiv](https://arxiv.org/abs/1403.5695), [PDF](https://arxiv.org/pdf/1403.5695.pdf)\] ## Refs - original proposal for the [[0204 Quantum complexity|complexity]] = volume conjecture