# Amsel, Horowitz, Marolf, Roberts (Jun, a) ## Uniqueness of Extremal Kerr and Kerr-Newman Black Holes \[Links: [arXiv](https://arxiv.org/abs/0906.2367), [PDF](https://arxiv.org/pdf/0906.2367.pdf)\] \[Abstract: \] ## Refs - [[0455 Black hole uniqueness theorems]] - [[AmselHorowitzMarolfRoberts200906b]] shows that all solutions with vanishing charges for all times are diffeomorphic to NHEK # Arkani-Hamed, Cachazo, Cheung, Kaplan ## A Duality For The S Matrix \[Links: [arXiv](https://arxiv.org/abs/0907.5418), [PDF](https://arxiv.org/pdf/0907.5418.pdf)\] \[Abstract: We propose a dual formulation for the S Matrix of N = 4 SYM. The dual provides a basis for the "leading singularities" of scattering amplitudes to all orders in perturbation theory, which are sharply defined, IR safe data that uniquely determine the full amplitudes at tree level and 1-loop, and are conjectured to do so at all loop orders. The scattering amplitude for $n$ particles in the sector with $k$ negative helicity gluons is associated with a simple integral over the space of $k$ planes in $n$ dimensions, with the action of parity and cyclic symmetries manifest. The residues of the integrand compute a basis for the leading singularities. A given leading singularity is associated with a particular choice of integration contour, which we explicitly identify at tree level and 1-loop for all NMHV amplitudes as well as the 8 particle NNMHV amplitude. We also identify a number of 2-loop leading singularities for up to 8 particles. There are a large number of relations among residues which follow from the multi-variable generalization of Cauchy's theorem known as the "global residue theorem". These relations imply highly non-trivial identities guaranteeing the equivalence of many different representations of the same amplitude. They also enforce the cancellation of non-local poles as well as consistent infrared structure at loop level. Our conjecture connects the physics of scattering amplitudes to a particular subvariety in a Grassmannian; space-time locality is reflected in the topological properties of this space.\] # Azeyanagi, Compere, Ogawa, Tachikawa, Terashima ## Higher-Derivative Corrections to the Asymptotic Virasoro Symmetry of 4d Extremal Black Holes \[Links: [arXiv](https://arxiv.org/abs/0903.4176), [PDF](https://arxiv.org/pdf/0903.4176.pdf)\] \[Abstract: We study the asymptotic [[0032 Virasoro algebra|Virasoro symmetry]] which acts on the near-horizon region of extremal four-dimensional black hole solutions of gravity theories with [[0006 Higher-derivative gravity|higher-derivative corrections]], following the recently proposed [[0520 Kerr-CFT correspondence|Kerr/CFT correspondence]]. We demonstrate that its [[0033 Central charge|central charge]] correctly reproduces the entropy formula of Iyer-Wald, once the boundary terms in the symplectic structure are carefully chosen.\] # Berti, Cardoso, Starinets (Review) ## Quasinormal modes of black holes and black branes \[Links: [arXiv](https://arxiv.org/abs/0905.2975), [PDF](https://arxiv.org/pdf/0905.2975.pdf)\] \[Abstract: [[0325 Quasi-normal modes|Quasinormal modes]] are eigenmodes of dissipative systems. Perturbations of classical gravitational backgrounds involving black holes or branes naturally lead to quasinormal modes. The analysis and classification of the quasinormal spectra requires solving non-Hermitian eigenvalue problems for the associated linear differential equations. Within the recently developed gauge-gravity duality, these modes serve as an important tool for determining the near-equilibrium properties of strongly coupled quantum field theories, in particular their transport coefficients, such as viscosity, conductivity and diffusion constants. In astrophysics, the detection of quasinormal modes in gravitational wave experiments would allow precise measurements of the mass and spin of black holes as well as new tests of general relativity. This review is meant as an introduction to the subject, with a focus on the recent developments in the field.\] # Binnington, Poisson ## Relativistic theory of tidal Love numbers \[Links: [arXiv](https://arxiv.org/abs/0906.1366), [PDF](https://arxiv.org/pdf/0906.1366.pdf)\] \[Abstract: In Newtonian gravitational theory, a [[0581 Tidal Love numbers|tidal Love number]] relates the mass multipole moment created by tidal forces on a spherical body to the applied tidal field. The Love number is dimensionless, and it encodes information about the body's internal structure. We present a relativistic theory of Love numbers, which applies to compact bodies with strong internal gravities; the theory extends and completes a recent work by Flanagan and Hinderer, which revealed that the tidal Love number of a neutron star can be measured by Earth-based gravitational-wave detectors. We consider a spherical body deformed by an external tidal field, and provide precise and meaningful definitions for electric-type and magnetic-type Love numbers; and these are computed for polytropic equations of state. The theory applies to black holes as well, and we find that the relativistic Love numbers of a nonrotating black hole are all zero.\] # Cubrovic, Zaanen, Schalm ## String Theory, Quantum Phase Transitions and the Emergent Fermi-Liquid \[Links: [arXiv](https://arxiv.org/abs/0904.1993), [PDF](https://arxiv.org/pdf/0904.1993.pdf)\] \[Abstract: A central problem in quantum condensed matter physics is the critical theory governing the zero temperature quantum phase transition between strongly renormalized Fermi-liquids as found in heavy fermion intermetallics and possibly high $T_c$ superconductors. We present here results showing that the mathematics of string theory is capable of describing such fermionic quantum critical states. Using the [[0001 AdS-CFT|Anti-de-Sitter/Conformal Field Theory]] (AdS/CFT) correspondence to relate fermionic quantum critical fields to a gravitational problem, we compute the spectral functions of fermions in the field theory. By increasing the fermion density away from the relativistic quantum critical point, a state emerges with all the features of the Fermi-liquid.\] ## Refs - [[0535 Holographic Fermi surface]] # Damour, Nagar ## Relativistic tidal properties of neutron stars \[Links: [arXiv](https://arxiv.org/abs/0906.0096), [PDF](https://arxiv.org/pdf/0906.0096.pdf)\] \[Abstract: We study the various linear responses of neutron stars to external relativistic tidal fields. We focus on three different tidal responses, associated to three different tidal coefficients: (i) a gravito-electric-type coefficient $G\mu_\ell=[length]^{2\ell+1}$ measuring the $\ell^{th}$-order mass multipolar moment $GM_{a_1... a_\ell}$ induced in a star by an external $\ell^{th}$-order gravito-electric tidal field $G_{a_1... a_\ell}$; (ii) a gravito-magnetic-type coefficient $G\sigma_\ell=[length]^{2\ell+1}$ measuring the $\ell^{th}$ spin multipole moment $G S_{a_1... a_\ell}$ induced in a star by an external $\ell^{th}$-order gravito-magnetic tidal field $H_{a_1... a_\ell}$; and (iii) a dimensionless ''shape'' [[0581 Tidal Love numbers|Love number]] $h_\ell$ measuring the distortion of the shape of the surface of a star by an external $\ell^{th}$-order gravito-electric tidal field. All the dimensionless tidal coefficients $G\mu_\ell/R^{2\ell+1}$, $G\sigma_ł/R^{2\ell+1}$ and $h_\ell$ (where $R$ is the radius of the star) are found to have a strong sensitivity to the value of the star's ''compactness'' $c\equiv GM/(c_0^2 R)$ (where we indicate by $c_0$ the speed of light). In particular, $G\mu_ł/R^{2ł+1}\sim k_\ell$ is found to strongly decrease, as $c$ increases, down to a zero value as $c$ is formally extended to the ''black-hole (BH) limit'' $c^{BH}=1/2$. The shape Love number $h_\ell$ is also found to significantly decrease as $c$ increases, though it does not vanish in the formal limit $c\to c^{BH}$. The formal vanishing of $\mu_\ell$ and $\sigma_\ell$ as $c\to c^{BH}$ is a consequence of the [[0455 Black hole uniqueness theorems|no-hair]] properties of black holes; this suggests, but in no way proves, that the effective action describing the gravitational interactions of black holes may not need to be augmented by nonminimal worldline couplings.\] # Denef, Hartnoll, Sachdev ## Black hole determinants and quasinormal modes \[Links: [arXiv](https://arxiv.org/abs/0908.2657), [PDF](https://arxiv.org/pdf/0908.2657.pdf)\] \[Abstract: We derive an expression for functional determinants in thermal spacetimes as a product over the corresponding [[0325 Quasi-normal modes|quasinormal modes]]. As simple applications we give efficient computations of scalar determinants in thermal AdS, [[0086 Banados-Teitelboim-Zanelli black hole|BTZ black hole]] and de Sitter spacetimes. We emphasize the conceptual utility of our formula for discussing '$1/N corrections to strongly coupled field theories via the [[0001 AdS-CFT|holographic correspondence]].\] ## Expressions - partition function $Z=\sum_{g_{\star}} \operatorname{det}\left(-\nabla_{g_{\star}}^2\right)^{ \pm 1} e^{-S_E\left[g_{\star}\right]}$ - (complex) bosonic fields$\frac{1}{\operatorname{det}\left(-\nabla_{g_{\star}}^2\right)}=e^{\mathrm{Pol}} \prod_{z_{\star}} \frac{\left|z_{\star}\right|}{4 \pi^2 T}\left|\Gamma\left(\frac{i z_{\star}}{2 \pi T}\right)\right|^2$ - fermionic fields$Z_F=e^{\mathrm{Pol}(\Delta)} \prod_{z_{\star}} \frac{2 \pi}{\Gamma\left(\frac{1}{2}+\frac{i z_{\star}}{2 \pi T}\right) \Gamma\left(\frac{1}{2}-\frac{i \bar{z}_{\star}}{2 \pi T}\right)}$ # Deruelle, Sasaki, Sendouda, Yamauchi ## Hamiltonian formulation of f(Riemann) theories of gravity \[Links: [arXiv](https://arxiv.org/abs/0908.0679), [PDF](https://arxiv.org/pdf/0908.0679.pdf)\] \[Abstract: We present a canonical formulation of gravity theories whose Lagrangian is an arbitrary function of the Riemann tensor. Our approach allows a unified treatment of various subcases and an easy identification of the degrees of freedom of the theory.\] ## Notations - $\varphi^{\mu \nu \rho \sigma}=\frac{\partial f}{\partial \varrho_{\mu \nu \rho \sigma}}$ - $\varrho_{\mu \nu \rho \sigma}=\mathcal{R}_{\mu \nu \rho \sigma}$ ## Conversion to 1st order DEs - why: - needed to introduce non-minimal coupling to matter - important for studying well-posedness of initial value problem, number of independent Cauchy data, global charges, stability, positivity of energy, and junction conditions ## Boundary action - see [[2018#Jiang, Zhang]] for better notations - GHY $\bar{S}=-\oint_{\partial \mathcal{M}} \mathrm{d} \Sigma_{\mu} n^{\mu} \Psi \cdot K$ (2.19) - $\Psi^{i j} \equiv-2 h^{i k} h^{j l} n^{\mu} n^{\nu} \varphi_{k \mu l \nu}$ ## For F(R) - $\Psi^{i j}=\Phi h^{i j}$ - $f^{\prime}(\varrho)=\Phi$ (scalar degree of freedom) - => $\bar{S}=-\oint_{\partial \mathcal{M}} \mathrm{d} \Sigma_{\mu} n^{\mu} f^{\prime}(\varrho)h^{i j} K_{ij}$ ## Subtlety - this does not work for Lovelock gravity ## Refs - [[0006 Higher-derivative gravity]] # Edalati, Jottar, Leigh ## Transport Coefficients at Zero Temperature from Extremal Black Holes \[Links: [arXiv](https://arxiv.org/abs/0910.0645), [PDF](https://arxiv.org/pdf/0910.0645.pdf)\] \[Abstract: \] ## Refs - [[0472 CMT for extremal BH]] - generalises [[2009#Faulkner, Liu, McGreevy, Vegh]] ## Summary - shows that [[0430 Holographic shear viscosity]] still satisfies $\eta=s/4\pi$ at zero temperature - the conductivity has a power law behaviour near $\omega\to0$, along with a Drude $\delta$-function peak # Faulkner, Liu, McGreevy, Vegh ## Emergent quantum criticality, Fermi surfaces, and AdS$_2$ \[Links: [arXiv](https://arxiv.org/abs/0907.2694), [PDF](https://arxiv.org/pdf/0907.2694.pdf)\] \[Abstract: Gravity solutions dual to $d$-dimensional field theories at finite charge density have a near-horizon region which is AdS$_2 \times R^{d-1}$. The scale invariance of the AdS$_2$ region implies that at low energies the dual field theory exhibits emergent quantum critical behavior controlled by a ($0+1$)-dimensional CFT. This interpretation sheds light on recently-discovered [[0535 Holographic Fermi surface|holographic descriptions of Fermi surfaces]], allowing an analytic understanding of their low-energy excitations. For example, the scaling behavior near the Fermi surfaces is determined by conformal dimensions in the emergent IR CFT. In particular, when the operator is marginal in the IR CFT, the corresponding spectral function is precisely of the "Marginal Fermi Liquid" form, postulated to describe the optimally doped cuprates.\] ## Three forms of quantum critical behaviour (discussed in this paper) - scaling behaviour of spectral density - periodic behaviour in $\log \omega$ at small momentum - Fermi surfaces ## Scaling limit - define $r-r_*=\lambda \frac{R_2^2}{\zeta}, \quad t=\lambda^{-1} \tau, \quad \lambda \rightarrow 0$ with $\zeta, \tau$ finite - finite temperature - $d s^2=\frac{R_2^2}{\zeta^2}\left(-\left(1-\frac{\zeta^2}{\zeta_0^2}\right) d \tau^2+\frac{d \zeta^2}{1-\frac{\zeta^2}{\zeta_0^2}}\right)+\frac{r_*^2}{R^2} d \vec{x}^2$ - AdS$_2$ BH cross $\mathbb{R}^{d-1}$ - zero temperature - $d s^2=\frac{R_2^2}{\zeta^2}\left(-d \tau^2+d \zeta^2\right)+\frac{r_*^2}{R^2} d \vec{x}^2$ - empty AdS$_2$ cross $\mathbb{R}^{d-1}$ ## Low-frequency limit - cannot naively expand around $\omega=0$ as the EOM is singular as $\omega\to0$ - divide the radial direction into two regions - solve in each region and then match - the result is - $G_R(\omega, k)=K \frac{b_{+}^{(0)}+\omega b_{+}^{(1)}+O\left(\omega^2\right)+\mathcal{G}_k(\omega)\left(b_{-}^{(0)}+\omega b_{-}^{(1)}+O\left(\omega^2\right)\right)}{a_{+}^{(0)}+\omega a_{+}^{(1)}+O\left(\omega^2\right)+\mathcal{G}_k(\omega)\left(a_{-}^{(0)}+\omega a_{-}^{(1)}+O\left(\omega^2\right)\right)}$. - $\mathcal{G}_k$ is Green's function for the IR CFT - RG interpretation - the exact zero-frequency retarded Green's function $G_R(\omega=0, \vec{k})$ is governed by UV physics: asymptotic coefficients $a_+$ and $b_+$ - the small $\omega$ expansion has two parts - analytic part governed by UV physics - non-analytical part proportional to the retarded Green's function of $\mathcal{O}_{k}$ in the IR CFT ## Bosons versus fermions - the discussion works for any spin - due to Bose statistics of the boundary operator, the behaviour corresponds to the instability of the ground state # Giombi, Yin ## Higher Spin Gauge Theory and Holography: The Three-Point Functions \[Links: [arXiv](https://arxiv.org/abs/0912.3462), [PDF](https://arxiv.org/pdf/0912.3462)\] \[Abstract: In this paper we calculate the tree level three-point functions of [[0013 Vasiliev theory|Vasiliev]]'s [[0421 Higher-spin gravity|higher spin]] gauge theory in AdS4 and find agreement with the correlators of the free field theory of $N$ massless scalars in three dimensions in the $O(N)$ singlet sector. This provides substantial evidence that Vasiliev theory is dual to the free field theory, thus verifying a conjecture of Klebanov and Polyakov. We also find agreement with the critical $O(N)$ vector model, when the bulk scalar field is subject to the alternative boundary condition such that its dual operator has classical dimension 2.\] # Hartman, Song, Strominger ## The Kerr-Fermi Sea \[Links: [arXiv](https://arxiv.org/abs/0912.4265), [PDF](https://arxiv.org/pdf/0912.4265.pdf)\] \[Abstract: The presence of a massive scalar field near a Kerr black hole is known to produce instabilities associated with bound [[0616 Superradiance|superradiant]] modes. In this paper we show that for massive fermions, rather than inducing an instability, the bound superradiant modes condense and form a Fermi sea which extends well outside the ergosphere. The shape of this Fermi sea in phase space and various other properties are analytically computed in the semiclassical WKB approximation. The low energy effective theory near the black hole is described by ripples in the Fermi surface. Expressions are derived for their dispersion relation and the effective force on particles which venture into the sea.\] # Heemskerk, Penedones, Polchinski, Sully ## Holography from CFT \[Links: [arXiv](https://arxiv.org/abs/0907.0151), [PDF](https://arxiv.org/pdf/0907.0151.pdf), [talk](https://youtu.be/qgpvY1_Y398)\] \[Abstract: The locality of bulk physics at distances below the AdS length scale is one of the remarkable aspects of [[0001 AdS-CFT|AdS/CFT duality]], and one of the least tested. It requires that the AdS radius be large compared to the Planck length and the string length. In the CFT this implies a large-$N$ expansion and a gap in the spectrum of anomalous dimensions. We conjecture that the implication also runs in the other direction, so that any CFT with a large-$N$ expansion and a large gap has a local bulk dual. For an abstract CFT we formulate the consistency conditions, most notably crossing symmetry, and show that the conjecture is true in a broad range of CFT’s, to first nontrivial order in $1/N^2$ : in any CFT with a gap and a large-$N$ expansion, the four-point correlator is generated via the AdS/CFT dictionary from a local bulk interaction. We establish this result by a counting argument on each side, and also investigate various properties of some explicit solutions.\] ## Refs - relation to [[0128 Bulk point singularity|bulk point singularity]] - using bulk point singularity (assuming their existence) to extract flat space scattering amplitudes - relation to [[0122 Holographic CFT|holographic CFT]] - this paper conjectures that large-$N$ + large gap to higher-spin single-trace operators = weakly coupled, local gravity dual - [[ChowdhuryGhosh2021]][](https://arxiv.org/pdf/2107.06266.pdf) - supports that any CFT with a large charge expansion and a gap in the spectrum has an AdS bulk dual ## Summary - precision holography (the bulk has locality at string scale and not smeared over AdS scale) is not only sufficient not also necessary for the boundary CFT to have most of its operators having large conformal dimension - "most" here means all single-trace operators of spin greater than two - (sec.5) calculate four-point amplitudes arising from various local bulk interactions, resolve them into partial waves, and show that the results agree with those found from the abstract CFT conditions - (sec.6) identify the Lorentizian CFT singularity associated with bulk locality, and show that it arises from the sum over partial waves # Hofman ## Higher derivative gravity, causality, and positivity of energy in a UV complete QFT ## Refs - relation between causality in the bulk and CFT [[0119 Positivity bounds|positivity bounds]] # Horowitz, Roberts ## Zero Temperature Limit of Holographic Superconductors \[Links: [arXiv](https://arxiv.org/abs/0908.3677), [PDF](https://arxiv.org/pdf/0908.3677.pdf)\] \[Abstract: We consider [[0431 Holographic superconductor|holographic superconductors]] whose bulk description consists of gravity minimally coupled to a Maxwell field and charged scalar field with general potential. We give an analytic argument that there is no "hard gap": the real part of the conductivity at low frequency remains nonzero (although typically exponentially small) even at zero temperature. We also numerically construct the gravitational dual of the ground state of some holographic superconductors. Depending on the charge and dimension of the condensate, the infrared theory can have emergent conformal or just Poincare symmetry. In all cases studied, the area of the horizon of the dual black hole goes to zero in the extremal limit, consistent with a nondegenerate ground state.\] ## Refs - [[0472 CMT for extremal BH]] # Iqbal, Liu ## Real-time response in AdS/CFT with application to spinors \[Links: [arXiv](https://arxiv.org/abs/0903.2596), [PDF](https://arxiv.org/pdf/0903.2596.pdf)\] \[Abstract: We discuss a simple derivation of the real-time AdS/CFT prescription as an analytic continuation of the corresponding problem in Euclidean signature. We then extend the formalism to spinor operators and apply it to the examples of real-time fermionic correlators in CFTs dual to pure AdS and the BTZ black hole.\] ## Refs - a similar investigation was given in [[GubserPufuRocha2008]][](http://arxiv.org/abs/0808.0407) - proposal of [[0473 Retarded Green's function|retarded Green's function]] calculation in [[2002#Son, Starinets]] - an earlier work [[2008#Iqbal, Liu]] ## Summary - justify the prescription for computing [[0473 Retarded Green's function|retarded Green's function]] proposed in [[2002#Son, Starinets]] - by analytically continuing the Euclidean dictionary - extends to spinors (which is a first order system) # Liu, McGreevy, Vegh ## Non-Fermi liquids from holography \[Links: [arXiv](https://arxiv.org/abs/0903.2477), [PDF](https://arxiv.org/pdf/0903.2477.pdf)\] \[Abstract: We report on a potentially new class of non-Fermi liquids in (2+1)-dimensions. They are identified via the [[0103 Two-point functions|response functions]] of composite fermionic operators in a class of strongly interacting quantum field theories at finite density, computed using the [[0001 AdS-CFT|AdS/CFT]] correspondence. We find strong evidence of [[0535 Holographic Fermi surface|Fermi surfaces]]: gapless fermionic excitations at discrete shells in momentum space. The spectral weight exhibits novel phenomena, including particle-hole asymmetry, discrete scale invariance, and scaling behavior consistent with that of a critical Fermi surface postulated by Senthil.\] ## Refs - [[0461 Fermions in AdS-CFT]] - [[0535 Holographic Fermi surface]] # Mars (Review) ## Present status of the Penrose inequality \[Links: [arXiv](https://arxiv.org/abs/0906.5566), [PDF](https://arxiv.org/pdf/0906.5566.pdf)\] \[Abstract: The [[0476 Penrose inequality|Penrose inequality]] gives a lower bound for the total mass of a spacetime in terms of the area of suitable surfaces that represent black holes. Its validity is supported by the cosmic censorship conjecture and therefore its proof (or disproof) is an important problem in relation with gravitational collapse. The Penrose inequality is a very challenging problem in mathematical relativity and it has received continuous attention since its formulation by Penrose in the early seventies. Important breakthroughs have been made in the last decade or so, with the complete resolution of the so-called Riemannian Penrose inequality and a very interesting proposal to address the general case by Bray and Khuri. In this paper, the most important results on this field will be discussed and the main ideas behind their proofs will be summarized, with the aim of presenting what is the status of our present knowledge in this topic.\] # McGreevy (Notes) ## Holographic duality with a view toward many-body physics \[Links: [arXiv](https://arxiv.org/abs/0909.0518), [PDF](https://arxiv.org/pdf/0909.0518.pdf)\] \[Abstract: These are notes based on a series of lectures given at the KITP workshop "Quantum Criticality and the AdS/CFT Correspondence" in July, 2009. The goal of the lectures was to introduce condensed matter physicists to the AdS/CFT correspondence. Discussion of string theory and of supersymmetry is avoided to the extent possible.\] # Nguyen, Spradlin, Volovich, Wen ## The Tree Formula for MHV Graviton Amplitudes \[Links: [arXiv](https://arxiv.org/abs/0907.2276), [PDF](https://arxiv.org/pdf/0907.2276.pdf)\] \[Abstract: We present and prove a formula for the [[0061 Maximally helicity violating amplitudes|MHV]] scattering amplitude of $n$ gravitons at tree level. Some of the more interesting features of the formula, which set it apart as being significantly different from many more familiar formulas, include the absence of any vestigial reference to a cyclic ordering of the gravitons--making it in a sense a truly gravitational formula, rather than a recycled Yang-Mills result, and the fact that it simultaneously manifests both $S_{n-2}$ symmetry as well as large-$z$ behavior that is $O(1/z^2)$ term-by-term, without relying on delicate cancellations. The formula is seemingly related to others by an enormous simplification provided by $O(n^n)$ iterated Schouten identities, but our proof relies on a complex analysis argument rather than such a brute force manipulation. We find that the formula has a very simple link representation in [[0330 Twistor theory|twistor space]], where cancellations that are non-obvious in physical space become manifest.\] ## Refs - [[0061 Maximally helicity violating amplitudes]] - better expression later: [[2012#Hodges]] ## Summary - *presents and proves* a [[0061 Maximally helicity violating amplitudes|MHV]] formula for gravity - simultaneously manifests both $S_{n-2}$ symmetry and large $r$ fall-off ($O(1/z^2)$) term by term # Skenderis, van Rees ## Holography and wormholes in 2+1 dimensions \[Links: [arXiv](https://arxiv.org/abs/0912.2090), [PDF](https://arxiv.org/pdf/0912.2090.pdf)\] \[Abstract: \] ## Refs - talks about [[0002 3D gravity]] - apply real-time dictionary [[2008#Skenderis, van Rees (May)]] [[2008#Skenderis, van Rees (Dec)]] to solutions found in [[AminneborgBengtssonBrillHolstPeldan1997]] and [[1999#Brill]] ## Summary - dual states of a class of [[0002 3D gravity]] - multi-boundary, each asymptotic region isometric to [[0086 Banados-Teitelboim-Zanelli black hole]] - dual states capture the non-trivial topology behind horizons ## A counting puzzle - The puzzle - spacetimes are uniquely determined given a Riemann surface of genus $g$ and $m$ boundaries -> $6g-6+3m$ parameters (see [[1999#Brill]]) - but each outer region is characterised by just the mass -> $m$ parameters - NOT in Euclidean - holographic one-point function captures the non-trivial topology, and do contain enough parameters # Wall (Jan) ## Ten proofs of GSL \[Links: [arXiv](https://arxiv.org/abs/0901.3865), [PDF](https://arxiv.org/pdf/0901.3865.pdf)\] \[Abstract: Ten attempts to prove the [[0082 Generalised second law|GSL]] of Thermodyanmics are described and critiqued. Each proof provides valuable insights which should be useful for constructing future, more complete proofs. Rather than merely summarizing previous research, this review offers new perspectives, and strategies for overcoming limitations of the existing proofs. A long introductory section addresses some choices that must be made in any formulation the GSL: Should one use the Gibbs or the Boltzmann entropy? Should one use the global or the apparent horizon? Is it necessary to assume any entropy bounds? If the area has quantum fluctuations, should the GSL apply to the average area? The definition and implications of the classical, hydrodynamic, semiclassical and full quantum gravity regimes are also discussed. A lack of agreement regarding how to define the "quasi-stationary" regime is addressed by distinguishing it from the "quasi-steady" regime.\] # Wall (Oct) ## Proving the Achronal Averaged Null Energy Condition from the Generalized Second Law \[Links: [arXiv](https://arxiv.org/abs/0910.5751), [PDF](https://arxiv.org/pdf/0910.5751)\] \[Abstract: A null line is a complete achronal null geodesic. It is proven that for any quantum fields minimally coupled to semiclassical Einstein gravity, the [[0417 Averaged null energy condition|averaged null energy condition]] (ANEC) on null lines is a consequence of the [[0082 Generalised second law|generalized second law]] of thermodynamics for causal horizons. Auxiliary assumptions include CPT and the existence of a suitable renormalization scheme for the generalized entropy. Although the ANEC can be violated on general geodesics in curved spacetimes, as long as the ANEC holds on null lines there exist theorems showing that semiclassical gravity should satisfy positivity of energy, topological censorship, and should not admit closed timelike curves. It is pointed out that these theorems fail once the linearized graviton field is quantized, because then the renormalized shear squared term in the Raychaudhuri equation can be negative. A "shear-inclusive" generalization of the ANEC is proposed to remedy this, and is proven under an additional assumption about perturbations to horizons in classical general relativity.\]