# Bakalov, Kirillov (Lectures) ## Lectures on Tensor Categories and Modular Functors \[Links: [AMS](https://bookstore.ams.org/view?ProductCode=ULECT/21)\] # Birmingham, Sachs, Solodukhin ## Conformal Field Theory Interpretation of Black Hole Quasi-normal Modes \[Links: [arXiv](https://arxiv.org/abs/hep-th/0112055), [PDF](https://arxiv.org/pdf/hep-th/0112055.pdf)\] \[Abstract: We obtain exact expressions for the [[0325 Quasi-normal modes|quasi-normal modes]] of various spin for the [[0086 Banados-Teitelboim-Zanelli black hole|BTZ]] black hole. These modes determine the relaxation time of black hole perturbations. Exact agreement is found between the quasi-normal frequencies and the location of the poles of the [[0473 Retarded Green's function|retarded correlation function]] of the corresponding perturbations in the dual conformal field theory. This then provides a new quantitative test of the [[0001 AdS-CFT|AdS/CFT]] correspondence.\] # Chrusciel, Herzlich ## The mass of asymptotically hyperbolic Riemannian manifolds \[Links: [arXiv](https://arxiv.org/abs/math/0110035), [PDF](https://arxiv.org/pdf/math/0110035)\] \[Abstract: We present a set of global invariants, called "mass integrals", which can be defined for a large class of asymptotically hyperbolic Riemannian manifolds. When the "boundary at infinity" has spherical topology one single invariant is obtained, called the [[0592 Gravitational energy|mass]]; we show [[0116 Positive energy theorem|positivity]] thereof. We apply the definition to conformally compactifiable manifolds, and show that the mass is completion-independent. We also prove the result, closely related to the problem at hand, that conformal completions of conformally compactifiable manifolds are unique.\] # Fukuma, Matsuura ## Holographic Renormalization Group Structure in Higher-Derivative Gravity \[Links: [arXiv](https://arxiv.org/abs/hep-th/0112037), [PDF](https://arxiv.org/pdf/hep-th/0112037.pdf)\] \[Abstract: \] <!-- - low citation ---> # Fukuma, Matsuura, Sakai ## Higher-derivative gravity and the AdS/CFT correspondence \[Links: [arXiv](https://arxiv.org/abs/hep-th/0103187), [PDF](https://arxiv.org/pdf/hep-th/0103187.pdf)\] \[Abstract: \] ## Refs - project [[hayward_pert]] - [[2002#Fukuma, Matsuura, Sakai]] ## Summary - talks about perturbative higher-derivative gravity - talks about boundary conditions ## Boundary conditions - transform $\mathcal{L}\left(g, \dot{g}, \cdots, g^{(N+1)}\right)$ into a Hamilton system with $(N=1)$ pairs of canonical variables, $\left(g, Q^{a}\right),\left(p, P_{a}\right)(a=1, \cdots, N)$, and choose Dirichlet for $g$ and Neumann for $Q^a$. # Gao, Wald ## The "physical process" version of the first law and the generalized second law for charged and rotating black holes \[Links: [arXiv](https://arxiv.org/abs/gr-qc/0106071), [PDF](https://arxiv.org/pdf/gr-qc/0106071.pdf)\] \[Abstract: \] # Harrington, Preskill ## Achievable rates for the Gaussian quantum channel \[Links: [arXiv](https://arxiv.org/abs/quant-ph/0105058), [PDF](https://arxiv.org/pdf/quant-ph/0105058.pdf)\] \[Abstract: We study the properties of quantum stabilizer codes that embed a finite-dimensional protected code space in an infinite-dimensional Hilbert space. The stabilizer group of such a code is associated with a symplectically integral lattice in the phase space of $2N$ canonical variables. From the existence of symplectically integral lattices with suitable properties, we infer a lower bound on the quantum capacity of the Gaussian quantum channel that matches the one-shot coherent information optimized over Gaussian input states.\] # Hosomichi ## Bulk-Boundary Propagator in Liouville Theory on a Disc \[Links: [arXiv](https://arxiv.org/abs/hep-th/0108093), [PDF](https://arxiv.org/pdf/hep-th/0108093.pdf)\] \[Abstract: We study [[0562 Liouville theory|Liouville theory]] on worldsheets [[0548 Boundary CFT|with boundary]] using the solutions of Knizhnik-Zamolodchikov equation involving a degenerate representation of the [[0032 Virasoro algebra|Virasoro algebra]]. The expression for bulk-boundary propagator on a disc is proposed.\] # Huisken, Ilmanen ## The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality \[Links: [DOI](https://doi.org/10.4310/jdg/1090349447)\] \[Abstract: Let $M$ be an asymptotically flat 3-manifold of nonnegative scalar curvature. The Riemannian Penrose Inequality states that the area of an outermost minimal surface $N$ in $M$ is bounded by the [[0487 ADM mass|ADM mass]] $m$ according to the formula $|N| \le 16πm^2$. We develop a theory of weak solutions of the inverse mean curvature flow, and employ it to prove this inequality for each connected component of $N$ using Geroch's monotonicity formula for the ADM mass. Our method also proves positivity of Bartnik's gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry.\] # Maldacena ## Eternal black holes in AdS \[Links: [arXiv](https://arxiv.org/abs/hep-th/0106112), [PDF](https://arxiv.org/pdf/hep-th/0106112.pdf)\] \[Abstract: We propose a dual non-perturbative description for maximally extended Schwarzschild Anti-de-Sitter spacetimes. The description involves two copies of the conformal field theory associated to the AdS spacetime and an initial entangled state. In this context we also discuss a version of the information loss paradox and its resolution.\] ## Summary - *proposes* that two-sided BH = TFD - *raises* a information paradox - and proposes a resolution (which is not enough/correct) ## Remarks - the famous BH = TFD paper - mentioned in [[2008#Skenderis, van Rees (May)]] ## Information paradox - also (better) explained in [[Rsc0035 Harlow Jerusalem lectures]] - benefit of AdS: BH live forever so we can wait an arbitrarily long time for objects thrown in at early time to completely become radiation - the two point function - both on one side: $\left\langle\Psi\left|T\left(\mathcal{O}_{1}(t, \phi) \mathcal{O}_{1}(0,0)\right)\right| \Psi\right\rangle \sim \sum_{n=-\infty}^{\infty} \frac{1}{\left.\left[\cosh \left(\frac{2 \pi t}{\beta}\right)-\cosh \left(\frac{2 \pi(\phi+2 \pi n)}{\beta}\right)-i \epsilon\right)\right]^{2 \Delta}}$ - on two sides: $\left\langle\Psi\left|\mathcal{O}_{1}\left(t_{1}, \phi_{1}\right) \mathcal{O}_{2}\left(t_{2}, \phi_{2}\right)\right| \Psi\right\rangle \sim \sum_{n=-\infty}^{\infty} \frac{1}{\left(\cosh \left(\frac{2 \pi\left(t_{1}+t_{2}\right)}{\beta}\right)+\cosh \left(\frac{2 \pi\left(\phi_{1}-\phi_{2}+2 \pi n\right)}{\beta}\right)\right)^{2 \Delta}}$ - decays as a function of $t$ (or $t_1+t_2$ which is the same thing as the only sum matters) - paradox: in a unitary theory it cannot decay forever - reason: there will be fluctuations at late times (see [[Rsc0035 Harlow Jerusalem lectures]]) - **resolutions** - Maldacena: there will be other Euclidean saddles - problem: not enough to just know that they exist; also different Euclidean saddles prepare different states - "Harlow's resolution" (mentioned in [[Rsc0035 Harlow Jerusalem lectures]]): studying large time separation uses the infinite collection of modes near the horizon which has infinite entropy; but since BH has finite entropy these modes cannot be trusted # Page ## Thermodynamics of Near-Extreme Black Holes \[Links: [arXiv](https://arxiv.org/abs/hep-th/0012020), [PDF](https://arxiv.org/pdf/hep-th/0012020.pdf)\] \[Abstract: The [[0127 Black hole thermodynamics|thermodynamics]] of nearly-extreme charged black holes depends upon the number of ground states at fixed large charge and upon the distribution of excited energy states. Here three possibilities are examined: (1) Ground state highly degenerate (as suggested by the large semiclassical Hawking entropy of an extreme Reissner-Nordstrom black hole), excited states not. (2) All energy levels highly degenerate, with macroscopic energy gaps between them. (3) All states nondegenerate (or with low degeneracy), separated by exponentially tiny energy gaps. I suggest that in our world with broken supersymmetry, this last possibility seems most plausible. An experiment is proposed to distinguish between these possibilities, but it would take a time that is here calculated to be more than about 10^837 years.\] # Policastro, Son, Starinets ## Shear viscosity of strongly coupled $N=4$ supersymmetric Yang-Mills plasma \[Links: [arXiv](https://arxiv.org/abs/hep-th/0104066), [PDF](https://arxiv.org/pdf/hep-th/0104066.pdf)\] \[Abstract: Using the anti-de Sitter/conformal field theory correspondence, we relate the shear viscosity $\eta$ of the finite-temperature [[0155 N=4 SYM|N=4 supersymmetric Yang-Mills theory]] in the large $N$, strong-coupling regime with the absorption cross section of low-energy gravitons by a near-extremal black three-brane. We show that in the limit of zero frequency this cross section coincides with the area of the horizon. From this result we find $\eta=\pi/8 N^2T^3$. We conjecture that for finite 't Hooft coupling $(g_YM)^2N$ the shear viscosity is $\eta=f((g_YM)^2N) N^2T^3$, where $f(x)$ is a monotonic function that decreases from $O(x^{-2}\ln^{-1}(1/x))$ at small $x$ to $\pi/8$ when $x\to\infty$.\] # Ponsot, Teschner ## Boundary Liouville Field Theory: Boundary Three Point Function \[Links: [arXiv](https://arxiv.org/abs/hep-th/0110244), [PDF](https://arxiv.org/pdf/hep-th/0110244.pdf)\] \[Abstract: [[0562 Liouville theory|Liouville field theory]] is considered on domains with conformally invariant boundary conditions. We present an explicit expression for the three point function of boundary fields in terms of the fusion coefficients which determine the monodromy properties of the [[0031 Conformal block|conformal blocks]].\] # Reall ## Classical and Thermodynamic Stability of Black Branes \[Links: [arXiv](https://arxiv.org/abs/hep-th/0104071), [PDF](https://arxiv.org/pdf/hep-th/0104071.pdf)\] \[Abstract: \] ## Summary - *shows* that the [[0442 Gregory-Laflamme instability]] occurs for many more solutions than just the ones considered by them - *proves* [[0443 Gubser-Mitra conjecture]] using Euclidean path integral ## Euclidean path integral v.s. stability - it appears that the classical Lorentzian threshold unstable mode corresponds precisely to a Euclidean negative mode - Local thermodynamic stability of a magnetically charged black brane is equivalent to positivity of the specific heat of the black hole obtained by dimensional reduction. # Stanev (Lectures) ## Two Dimensional Conformal Field Theory on Open and Unoriented Surfaces \[Links: [arXiv](https://arxiv.org/abs/hep-th/0112222), [PDF](https://arxiv.org/pdf/hep-th/0112222)\] \[Abstract: Introduction to [[0003 2D CFT|two dimensional conformal field theory]] on [[0548 Boundary CFT|open]] and [[0620 Non-orientable CFT|unoriented]] surfaces. The construction is illustrated in detail on the example of SU(2) WZW models.\] # Strominger ## The dS/CFT Correspondence \[Links: [arXiv](https://arxiv.org/abs/hep-th/0106113), [PDF](https://arxiv.org/pdf/hep-th/0106113)\] \[Abstract: A holographic duality is proposed relating quantum gravity on dS$_D$ ($D$-dimensional de Sitter space) to conformal field theory on a single $S^{D-1}$ (($D-1$)-sphere), in which bulk de Sitter correlators with points on the boundary are related to CFT correlators on the sphere, and points on $I^+$ (the future boundary of $dS_D$) are mapped to the antipodal points on $S^{D-1}$ relative to those on $I^-$. For the case of $dS_3$, which is analyzed in some detail, the central charge of the CFT$_2$ is computed in an analysis of the asymptotic symmetry group at $I^\pm$. This [[0545 de Sitter quantum gravity|dS/CFT]] proposal is supported by the computation of correlation functions of a massive scalar field. In general the dual CFT may be non-unitary and (if for example there are sufficently massive stable scalars) contain complex conformal weights. We also consider the physical region $O^-$ of dS$_3$ corresponding to the causal past of a timelike observer, whose holographic dual lives on a plane rather than a sphere. $O^-$ can be foliated by asymptotically flat spacelike slices. Time evolution along these slices is generated by $L_0+\bar L_0$, and is dual to scale transformations in the boundary CFT$_2$.\] # Zamolodchikov, Zamolodchikov ## Liouville field theory on a pseudosphere \[Links: [arXiv](https://arxiv.org/abs/hep-th/0101152), [PDF](https://arxiv.org/pdf/hep-th/0101152)\] \[Abstract: [[0562 Liouville theory|Liouville field theory]] is considered with boundary conditions corresponding to a quantization of the classical Lobachevskiy plane (i.e. euclidean version of AdS$_2$). We solve the bootstrap equations for the out-vacuum wave function and find an infinite set of solutions. This solutions are in one to one correspondence with the degenerate representations of the Virasoro algebra. Consistency of these solutions is verified by both boundary and modular bootstrap techniques. Perturbative calculations lead to the conclusion that only the ''basic'' solution corresponding to the identity operator provides a ''natural'' quantization of the Lobachevskiy plane.\]