# Almheiri, Marolf, Polchinski, Sully ## Black Holes: Complementarity or Firewalls? \[Links: [arXiv](https://arxiv.org/abs/1207.3123), [PDF](https://arxiv.org/pdf/1207.3123.pdf)\] \[Abstract: We argue that the following three statements cannot all be true: (i) [[0304 Hawking radiation|Hawking radiation]] is in a pure state, (ii) the information carried by the radiation is emitted from the region near the horizon, with low energy effective field theory valid beyond some microscopic distance from the horizon, and (iii) the infalling observer encounters nothing unusual at the horizon. Perhaps the most conservative resolution is that the infalling observer burns up at the horizon. Alternatives would seem to require novel dynamics that nevertheless cause notable violations of semiclassical physics at macroscopic distances from the horizon.\] ## Refs - this the original work of the [[0195 Firewall|firewall paradox]] # Balasubramanian, Bernamonti, Craps, Keranen, Keski-Vakkuri, Muller, Thorlacius, Vanhoof ## Thermalisation of the spectral function in strongly coupled 2d CFT \[Links: [arXiv](https://arxiv.org/abs/1212.6066), [PDF](https://arxiv.org/pdf/1212.6066.pdf)\] \[Abstract: Using Wigner transforms of Green functions, we discuss non-equilibrium generalizations of spectral functions and occupation numbers. We develop methods for computing time-dependent spectral functions in conformal field theories holographically dual to thin-shell AdS-Vaidya spacetimes.\] ## Refs - three ways of computing [[0103 Two-point functions]] on a time-dependent background 1. defining a new (non-standard) Euclidean continuation 2. using complex geodesics 3. splicing across the infalling shell ## Geodesic approximation - time ordered $\left\langle\mathcal{O}\left(x_{1}, t_{1}\right) \mathcal{O}\left(x_{2}, t_{2}\right)\right\rangle=\int \mathcal{D} \mathcal{P} e^{-\Delta L(\mathcal{P})}$ where $L(\mathcal{P}) \equiv \int_{\mathcal{P}} \sqrt{g_{\mu \nu} \frac{d x^{\mu}}{d \lambda} \frac{d x^{\nu}}{d \lambda}} \mathrm{d} \lambda$ - now in the heavy operator limit (large $\Delta$ in the exponent), the geodesics dominate: $\left\langle\mathcal{O}\left(x_{1}, t_{1}\right) \mathcal{O}\left(x_{2}, t_{2}\right)\right\rangle \sim \sum_{\text {geodesics }} e^{-\Delta \mathcal{L}}$ ## General problem with correlators in time-dependent spacetimes - timelike geodesics do not reach AdS boundary. which can be fixed if they have Euclidean continuation (Appendix B: correlators in AdS and BTZ via standard Wick rotation) - but time dependent spacetimes do not have Euclidean continuation ## Method I: Non-standard Euclidean continuation - take Vaidya to be a limit of a family of spacelike Vaidya - the shell is spacelike (thus unphysical) - each can be Wick rotated and thus the geodesic length calculated - $z=iv$ and $Z=iE$, where $E$ labels the family and $E\rightarrow\infty$ gives the Vaidya metric - result then Wick rotated back ## Method II: Complex geodesics - real timelike geodesic cannot reach boundary, but complex ones can ## Method III: Splicing - propagator from boundary before the shell to the shell, then from the shell to the boundary after the shell # Bardarson, Pollman. Moore ## Unbounded growth of entanglement in models of many-body localization \[Links: [arXiv](https://arxiv.org/abs/1202.5532), [PDF](https://arxiv.org/pdf/1202.5532.pdf)\] \[Abstract: An important and incompletely answered question is whether a closed quantum system of many interacting particles can be localized by disorder. The time evolution of simple (unentangled) initial states is studied ==numerically for a system of interacting spinless fermions in one dimension described by the random-field XXZ Hamiltonian==. Interactions induce a dramatic change in the [[0522 Entanglement dynamics|propagation of entanglement]] and a smaller change in the propagation of particles. For even weak interactions, when the system is thought to be in a many-body localized phase, entanglement shows neither localized nor diffusive behavior but grows without limit in an infinite system: interactions act as a singular perturbation on the localized state with no interactions. The significance for proposed atomic experiments is that local measurements will show a large but nonthermal entropy in the many-body localized state. This entropy develops slowly (approximately logarithmically) over a diverging time scale as in glassy systems.\] ## Refs - [[0522 Entanglement dynamics]]: this paper gives a 1d example with unbounded entanglement growth - [[0541 Thermalisation]]: this paper provides a case of non-thermalisation # Bianchi, Myers ## On the architecture of spacetime geometry \[Links: [arXiv](https://arxiv.org/abs/1212.5183), [PDF](https://arxiv.org/pdf/1212.5183.pdf)\] \[Abstract: \] ## Summary - conjectures that the leading term of [[0004 Black hole entropy|Gravitational entanglement entropy]] in quantum gravity is given by Bekenstein-Hawking entropy, for the region large enough and in a smooth spacetime - various lines of evidence from perturbative quantum gravity, simplified models of induced gravity and loop quantum gravity, as well as [[0001 AdS-CFT]] # Broedel, Dixon ## Color-kinematics duality and double-copy construction for amplitudes from higher-dimension operators \[Links: [arXiv](https://arxiv.org/abs/1208.0876), [PDF](https://arxiv.org/pdf/1208.0876.pdf)\] \[Abstract: We investigate [[0152 Colour-kinematics duality|color-kinematics duality]] for gauge-theory amplitudes produced by the pure nonabelian Yang-Mills action deformed by higher-dimension operators. For the operator denoted by $F^3$, the product of three field strengths, the existence of color-kinematic dual representations follows from string-theory monodromy relations. We provide explicit dual representations, and show how the double-copy construction of gravity amplitudes based on them is consistent with the [[0398 KLT relations|Kawai-Lewellen-Tye relations]]. It leads to the amplitudes produced by [[0554 Einstein gravity|Einstein gravity]] coupled to a dilaton field $\phi$, and deformed by operators of the form $\phi R^2$ and $R^3$. For operators with higher dimensions than $F^3$, such as $F^4$-type operators appearing at the next order in the low-energy expansion of bosonic and superstring theory, the situation is more complex. The color structure of some of the $F^4$ operators is incompatible with a simple color-kinematics duality based on structure constants $f^{abc}$, but even the color-compatible $F^4$ operators do not admit the duality. In contrast, the next term in the alpha-prime expansion of the [[0329 String effective action|superstring effective action]] --- a particular linear combination of $D^2 F^4$ and $F^5$-type operators --- does admit the duality, at least for amplitudes with up to six external gluons.\] # Casini, Huerta ## On the RG running of the entanglement entropy of a circle \[Links: [arXiv](https://arxiv.org/abs/1202.5650), [PDF](https://arxiv.org/pdf/1202.5650)\] \[Abstract: We show, using [[0218 Strong subadditivity|strong subadditivity]] and Lorentz covariance, that in three dimensional space-time the [[0301 Entanglement entropy|entanglement entropy]] of a circle is a concave function. This implies the decrease of the coefficient of the area term and the increase of the constant term in the entropy between the ultraviolet and infrared fixed points. This is in accordance with recent [[0257 Holographic RG flow|holographic c-theorems]] and with conjectures about the renormalization group flow of the partition function of a three sphere ([[0351 Irreversibility theorems|F-theorem]]). The irreversibility of the renormalization group flow in three dimensions would follow from the argument provided there is an intrinsic definition for the constant term in the entropy at fixed points. We discuss the difficulties in generalizing this result for spheres in higher dimensions.\] # Castro, Maloney ## The Wave Function of Quantum de Sitter \[Links: [arXiv](https://arxiv.org/abs/1209.5757), [PDF](https://arxiv.org/pdf/1209.5757)\] \[Abstract: We consider quantum general relativity in three dimensions with a [[0545 de Sitter quantum gravity|positive cosmological constant]]. The Hartle-Hawking wave function is computed as a function of metric data at asymptotic future infinity. The analytic continuation from Euclidean Anti-de Sitter space provides a natural integration contour in the space of metrics, allowing us -- with certain assumptions -- to compute the wave function exactly, including both perturbative and non-perturbative effects. The resulting wave function is a non-normalizable function of the conformal structure of future infinity which is infinitely peaked at geometries where $I^+$ becomes infinitely inhomogeneous. We interpret this as a non-perturbative instability of de Sitter space in three dimensional Einstein gravity.\] # Chiodaroli, D'Hoker, Gutperle ## Holographic duals of Boundary CFTs \[Links: [arXiv](https://arxiv.org/abs/1205.5303), [PDF](https://arxiv.org/pdf/1205.5303.pdf)\] \[Abstract: New families of regular half-[[0178 BPS|BPS]] solutions to 6-dimensional Type 4b supergravity with $m$ tensor multiplets are constructed exactly. Their space-time consists of $AdS_2 \times S^2$ warped over a Riemann surface with an arbitrary number of boundary components, and arbitrary genus. The solutions have an arbitrary number of asymptotic $AdS_3 \times S^3$ regions. In addition to strictly single-valued solutions to the supergravity equations whose scalars live in the coset $SO(5,m)/SO(5)\times SO(m)$, we also construct stringy solutions whose scalar fields are single-valued up to transformations under the U-duality group $SO(5,m;\mathbb{Z})$, and live in the coset $SO(5,m;\mathbb{Z})\backslash SO(5,m)/SO(5)\times SO(m)$. We argue that these Type 4b solutions are holographically dual to general classes of [[0065 Defect CFT|interface]] and [[0548 Boundary CFT|boundary CFTs]] arising at the juncture of the end-points of 1+1-dimensional bulk CFTs. We evaluate their corresponding holographic entanglement and boundary entropy, and discuss their brane interpretation. We conjecture that the solutions for which $\Sigma$ has handles and multiple boundaries correspond to the near-horizon limit of half-BPS webs of dyonic strings and three-branes.\] # Costa, Goncalves, Penedones ## Conformal Regge theory \[Links: [arXiv](https://arxiv.org/abs/1209.4355), [PDF](https://arxiv.org/pdf/1209.4355.pdf)\] \[Abstract: We generalize Regge theory to correlation functions in conformal field theories. This is done by exploring the analogy between Mellin amplitudes in [[0001 AdS-CFT|AdS/CFT]] and $S$-matrix elements. In the process, we develop the conformal partial wave expansion in Mellin space, elucidating the analytic structure of the partial amplitudes. We apply the new formalism to the case of four point correlation functions between protected scalar operators in $\mathcal{N}=4$ Super Yang Mills, in cases where the Regge limit is controlled by the leading twist operators associated to the pomeron-graviton Regge trajectory. At weak coupling, we are able to predict to arbitrary high order in the 't Hooft coupling the behaviour near $J=1$ of the [[0030 Operator product expansion|OPE]] coefficients $C_{OOJ}$ between the external scalars and the spin $J$ leading twist operators. At strong coupling, we use recent results for the anomalous dimension of the leading twist operators to improve current knowledge of the AdS graviton Regge trajectory - in particular, determining the next and next to next leading order corrections to the intercept. Finally, by taking the flat space limit and considering the Virasoro-Shapiro $S$-matrix element, we compute the strong coupling limit of the OPE coefficient $C_{LLJ}$ between two Lagrangians and the leading twist operators of spin $J$.\] # Hodges ## A simple formula for gravitational MHV amplitudes \[Links: [arXiv](https://arxiv.org/abs/1204.1930), [PDF](https://arxiv.org/pdf/1204.1930.pdf)\] \[Abstract: \] ## Summary - *provides* ==tree-level== gravity [[0061 Maximally helicity violating amplitudes]] amplitudes using as a determinant of a matrix ## Comments - useful for obtaining [[0078 Collinear limit]] like #atulsharma does ## Notations - $\phi_{j}^{i}=\frac{[i j]}{\langle i j\rangle}($for $i \neq j)$ - $\phi_{i}^{i}=-\sum_{j \neq i} \frac{[i j]\langle j x\rangle\langle j y\rangle}{\langle i j\rangle\langle i x\rangle\langle i y\rangle}$ - dependence of reference spinors $x$ and $y$ ensured by momentum conservation - square brackets mean antisymmetrisation without the $1/n!$ factor ## Motivation for the new expression - $\phi^i_i$ is (negative of) the universal soft factor (defined in [[2009#Nguyen, Spradlin, Volovich, Wen]]) ## Results - $\bar{M}_{n}(12 \ldots n)=(-1)^{n+1} \operatorname{sgn}(\alpha \beta) c_{\alpha(1) \alpha(2) \alpha(3)} c^{\beta(1) \beta(2) \beta(3)} \phi_{[\alpha(4)}^{\beta(4)} \phi_{\alpha(5)}^{\beta(5)} \ldots \phi_{\alpha(n)]}^{\beta(n)}$ ## Generalisations beyond MHV - [[CachazoGeyer2012]] - [[CachazoSkinner2012]] - [[Bullimore2012]] - [[Cachazo2013]] # Hollands, Wald ## Stability of BHs and black branes \[Links: [arXiv](https://arxiv.org/abs/1201.0463), [PDF](https://arxiv.org/pdf/1201.0463.pdf)\] \[Abstract: \] ## Refs - [[0414 Second order formalism]] ## Summary - a new criterion for dynamical [[0339 Stability of GR solutions|stability]] of black holes in $D\ge4$ w.r.t. ==axisymmetric perturbations== - *proves* that for any black brane corresponding to a thermodynamically unstable BH, sufficiently long wavelength perturbations can be found with $\mathcal{E}<0$ and vanishing linearised ADM quantities => dynamically unstable ## Criterion for dynamical stability - dynamical stability = positivity of canonical energy on a subspace of linearised solutions that have vanishing linearised ADM mass, momentum and angular momentum at infinity and satisfy certain gauge conditions at the horizon - canonical energy: $\mathcal{E}=\delta^{2} M-\sum_{A} \Omega_{A} \delta^{2} J_{A}-\frac{\kappa}{8 \pi} \delta^{2} A$ # Hubeny, Rangamani ## Causal holographic information \[Links: [arXiv](https://arxiv.org/abs/1204.1698), [PDF](https://arxiv.org/pdf/1204.1698.pdf)\] \[Abstract: \] ## Refs - later paper [[2014#Headrick, Hubeny, Lawrence, Rangamani]] ## Summary - defines *causal holographic information* as the area of the boundary of the causal wedge - agrees with entanglement entropy in all cases where one has a microscopic understanding of entanglement entropy <!-- - project [[race]] - appendix may contain calculation for finding the causal wedge ---> # Lucietti, Murata, Reall, Tanahashi ## On the horizon instability of an extreme Reissner-Nordström black hole \[Links: [arXiv](https://arxiv.org/abs/1212.2557), [PDF](https://arxiv.org/pdf/1212.2557.pdf)\] \[Abstract: Aretakis has proved that a massless scalar field has an instability at the horizon of an extreme Reissner-Nordström black hole. We show that a similar instability occurs also for a massive scalar field and for coupled linearized gravitational and electromagnetic perturbations. We present numerical results for the late time behaviour of massless and massive scalar fields in the extreme RN background and show that instabilities are present for initial perturbations supported outside the horizon, e.g. an ingoing wavepacket. For a massless scalar we show that the numerical results for the late time behaviour are reproduced by an analytic calculation in the near-horizon geometry. We relate Aretakis' conserved quantities at the future horizon to the [[0456 Newman-Penrose charges|Newman-Penrose conserved quantities]] at future null infinity.\] ## Refs - extends [[0340 Aretakis instability|Aretakis instability]] to gravitational and electromagnetic perturbations ## Comments - nice review of [[0340 Aretakis instability]], [[0460 Aretakis-NP charge connection]] (via [[0458 Couch-Torrence inversion isometry]]) # Lucietti, Reall ## Gravitational instability of an extreme Kerr black hole \[Links: [arXiv](https://arxiv.org/abs/1208.1437), [PDF](https://arxiv.org/pdf/1208.1437.pdf)\] \[Abstract: \] ## Summary - proves [[0340 Aretakis instability]] for gravitational perturbations (i.e. pure GR, no matter field) # Myers, Singh ## Entanglement Entropy for Singular Surfaces \[Links: [arXiv](https://arxiv.org/abs/1206.5225), [PDF](https://arxiv.org/pdf/1206.5225.pdf)\] \[Abstract: \] ## Summary - *studies* [[0387 Higher codimension defects]] as entanglement entropy for cusps # Nandan, Wen ## Generating All Tree Amplitudes in N=4 SYM by Inverse Soft Limit \[Links: [arXiv](https://arxiv.org/abs/1204.4841), [PDF](https://arxiv.org/pdf/1204.4841.pdf)\] \[Abstract: The idea of adding particles to construct amplitudes has been utilized in various ways in exploring the structure of scattering amplitudes. This idea is often called [[0515 Inverse soft construction|Inverse Soft]] Limit, namely it is the reverse mechanism of taking particles to be soft. We apply the Inverse Soft Limit to the tree-level amplitudes in $\mathcal{N}=4$ super Yang-Mills theory, which allows us to generate full tree-level superamplitudes by adding "soft" particles in a certain way. With the help from [[0058 BCFW|Britto-Cachazo-Feng-Witten]] recursion relations, a systematic and concrete way of adding particles is determined recursively. The amplitudes constructed solely by adding particles not only have manifest Yangian symmetry, but also make the soft limit transparent. The method of generating amplitudes by Inverse Soft Limit can also be generalized for constructing form factors.\] # Nozaki, Takayanagi, Ugajin ## Central Charges for BCFTs and Holography \[Links: [arXiv](https://arxiv.org/abs/1205.1573), [PDF](https://arxiv.org/pdf/1205.1573.pdf)\] \[Abstract: \] ## Refs - [[0181 AdS-BCFT]] - [[0011 Fefferman-Graham expansion|FG expansion]] ## Summary - *gives* perturbative gravity solutions (away from pure AdS) for arbitrary shapes of the boundary - standard FG breaks down for generic choices of BCFT boundaries # Osborn ## Conformal blocks for arbitrary spins in 2d \[Links: [arXiv](https://arxiv.org/abs/1205.1941), [PDF](https://arxiv.org/pdf/1205.1941.pdf)\] \[Abstract: \] ## Summary - [[0031 Conformal block]] for arbitrary spin - results are products of hypergeometric functions <!-- ## Use for [[asymp]] - want to repeat this for ccft not cft --> # Raju ## New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators \[Links: [arXiv](https://arxiv.org/abs/1201.6449), [PDF](https://arxiv.org/pdf/1201.6449)\] \[Abstract: We consider correlation functions of the stress-tensor or a conserved current in AdS$_{d+1}$/CFT$_d$ computed using the Hilbert or the Yang-Mills action in the bulk. We introduce new recursion relations to compute these correlators at tree level. These relations have an advantage over the [[0058 BCFW|BCFW]]-like relations described in [arXiv:1102.4724](https://arxiv.org/abs/1102.4724) and [arXiv:1011.0780](https://arxiv.org/abs/1011.0780) because they can be used in all dimensions including $d=3$. We also introduce a new method of extracting flat-space S-matrix elements from AdS/CFT correlators in momentum space. We show that the ($d+1$)-dimensional flat-space amplitude of gravitons or gluons can be obtained as the coefficient of a particular singularity of the $d$-dimensional correlator of the stress-tensor or a conserved current; this technique is valid even at loop-level in the bulk. Finally, we show that our recursion relations automatically generate correlators that are consistent with this observation: they have the expected singularity and the flat-space gluon or graviton amplitude appears as its coefficient.\] # Simmons-Duffin ## Projectors, Shadows, and Conformal Blocks \[Links: [arXiv](https://arxiv.org/abs/1204.3894), [PDF](https://arxiv.org/pdf/1204.3894.pdf)\] \[Abstract: \] ## Summary - presents a general method for computing [[0031 Conformal block]] of operators in *arbitrary* Lorentz representations (i.e. not just scalars) - uses the shadow formalism ## Comments - also works in Mellin space => works for CFTs dual to AdS theories - explains conformal blocks (ch.2) # Teschner, Vartanov ## 6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories \[Links: [arXiv](https://arxiv.org/abs/1202.4698), [PDF](https://arxiv.org/pdf/1202.4698.pdf)\] \[Abstract: We revisit the definition of the [[0597 6j symbol|6j-symbols]] from the modular double of $U_q(sl(2,R))$, referred to as b-6j symbols. Our new results are (i) the identification of particularly natural normalization conditions, and (ii) new integral representations for this object. This is used to briefly discuss possible applications to quantum hyperbolic geometry, and to the study of certain supersymmetric gauge theories. We show, in particular, that the b-6j symbol has leading semiclassical asymptotics given by the volume of a non-ideal tetrahedron. We furthermore observe a close relation with the problem to quantize natural Darboux coordinates for moduli spaces of flat connections on Riemann surfaces related to the Fenchel-Nielsen coordinates. Our new integral representations finally indicate a possible interpretation of the b-6j symbols as partition functions of three-dimensional $N=2$ supersymmetric gauge theories.\] # Wall ## Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy \[Links: [arXiv](https://arxiv.org/abs/1211.3494), [PDF](https://arxiv.org/pdf/1211.3494)\] \[Abstract: The covariant holographic entropy conjecture of AdS/CFT relates the entropy of a boundary region $R$ to the area of an extremal surface in the bulk spacetime. This extremal surface can be obtained by a maximin construction, allowing many new results to be proven. On manifolds obeying the null curvature condition, these extremal surfaces: i) always lie outside the causal wedge of $R$, ii) have less area than the bifurcation surface of the causal wedge, iii) move away from the boundary as $R$ grows, and iv) obey [[0218 Strong subadditivity|strong subadditivity]] and monogamy of mutual information. These results suggest that the information in R allows the bulk to be reconstructed all the way up to the extremal area surface. The maximin surfaces are shown to exist on spacetimes without horizons, and on black hole spacetimes with Kasner-like singularities.\]