# Aharony, Gubser, Maldacena, Ooguri, Oz ## Large N field theories, string theory and gravity \[Links: [arXiv](https://arxiv.org/abs/hep-th/9905111), [PDF](https://arxiv.org/pdf/hep-th/9905111.pdf)\] \[Abstract: We review the [[0001 AdS-CFT|holographic correspondence]] between field theories and string/M theory, focusing on the relation between compactifications of string/M theory on Anti-de Sitter spaces and conformal field theories. We review the background for this correspondence and discuss its motivations and the evidence for its correctness. We describe the main results that have been derived from the correspondence in the regime that the field theory is approximated by classical or semiclassical gravity. We focus on the case of the $\mathcal{N}=4$ supersymmetric gauge theory in four dimensions, but we discuss also field theories in other dimensions, conformal and non-conformal, with or without supersymmetry, and in particular the relation to QCD. We also discuss some implications for black hole physics.\] ## Refs - contains review of [[1997#Maldacena]] - [[0001 AdS-CFT]] # Balasubramanian, Kraus ## A Stress Tensor for Anti-de Sitter Gravity \[Links: [arXiv](https://arxiv.org/abs/hep-th/9902121), [PDF](https://arxiv.org/pdf/hep-th/9902121.pdf)\] \[Abstract: We propose a procedure for computing the [[0592 Gravitational energy|boundary stress tensor]] associated with a gravitating system in asymptotically anti-de Sitter space. Our definition is free of ambiguities encountered by previous attempts, and correctly reproduces the masses and angular momenta of various spacetimes. Via the [[0001 AdS-CFT|AdS/CFT]] correspondence, our classical result is interpretable as the expectation value of the stress tensor in a quantum conformal field theory. We demonstrate that the conformal anomalies in two and four dimensions are recovered. The two dimensional stress tensor transforms with a Schwarzian derivative and the expected central charge. We also find a nonzero ground state energy for global AdS$_5$, and show that it exactly matches the Casimir energy of the dual $N=4$ super Yang-Mills theory on $S^3 x R$.\] # Balasubramanian, Ross ## Holographic particle detection \[Links: [arXiv](https://arxiv.org/abs/hep-th/9906226), [PDF](https://arxiv.org/pdf/hep-th/9906226.pdf)\] \[Abstract: In anti-de Sitter (AdS) space, classical supergravity solutions are represented "holographically" by conformal field theory (CFT) states in which operators have expectation values. These 1-point functions are directly related to the asymptotic behaviour of bulk fields. In some cases, distinct supergravity solutions have identical asymptotic behaviour; so dual expectation values are insufficient to distinguish them. We argue that non-local objects in the gauge theory can resolve the ambiguity, and explicitly show that collections of point particles in AdS$_3$ can be detected by studying kinks in dual CFT Green functions. Three dimensional black holes can be formed by collision of such particles. We show how black hole formation can be detected in the holographic dual, and calculate CFT quantities that are sensitive to the distribution of matter inside the event horizon.\] ## Caveat - the CFT propagator in this paper is in fact the restriction of bulk propagator to the boundary - -> raise concerns about causality - -> addressed in [[2000#Louko, Marolf, Ross]] - answer: it is in fact causal, but stationary phase approximation is only valid in appropriately analytic spacetimes, and NOT in the actual spacetime considered in this paper ## Refs - later paper [[2000#Louko, Marolf, Ross]] ## Summary - some supergravity solutions have identical asymptotic behaviour so dual expectation values are insufficient to distinguish them - -> need non-local objects - collections of point particles can be obtained by studying kinks in the dual CFT Green functions - CFT also sensitive to distribution of matter*inside* black hole horizon - uses a ==stationary phase approximation== to obtain the propagator in CFT from geodesics in supergravity solutions # Behrend, Pearce, Petkova, Zuber ## Boundary Conditions in Rational Conformal Field Theories \[Links: [arXiv](https://arxiv.org/abs/hep-th/9908036), [PDF](https://arxiv.org/pdf/hep-th/9908036.pdf)\] \[Abstract: We develop further the theory of [[0096 Rational CFT|Rational Conformal Field Theories]] (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy's equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph $G$ to each RCFT such that the conformal boundary conditions are labelled by the nodes of $G$. This approach is carried to completion for $sl(2)$ theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the A-D-E classification. We also review the current status for [[0601 Weiss-Zumino-Witten models|WZW]] $sl(3)$ theories. Finally, a systematic generalization of the formalism of Cardy-Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk-boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints.\] # Bousso (May) ## A Covariant Entropy Conjecture \[Links: [arXiv](https://arxiv.org/abs/hep-th/9905177), [PDF](https://arxiv.org/pdf/hep-th/9905177.pdf)\] \[Abstract: \] ## Refs - [[0082 Generalised second law]] ## Summary - an upper bound on the entropy defined on the lightcone ## Statement of the conjecture - Let $L$ be a hypersurface generated by surface-orthogonal null geodesics with non-positive expansion from a codimension-2 surface with area $A$ - stop when the light-ray reaches a boundary or singularity OR when the expansion becomes positive (this happens only at caustics) - then the entropy on $L$ is bounded by $S\le A/4$ ## Conditions - dominant energy condition - the spacetime is inextendable and contains no naked singularities - prevent an arbitrary amount of entropy entering or leaving such boundaries ## Some examples - a normal sphere vs a very entropic sphere - entropic spheres have more entropy and one may think it is hard to have a bound on it because the area can decrease indefinitely; but lightrays in an entropic region also requires more affine time to cover the interior and will hit the singularity sooner; so the lightsheet is not as extended and has less entropy # Bray ## Proof of the Riemannian Penrose Conjecture Using the Positive Mass Theorem \[Links: [arXiv](https://arxiv.org/abs/math/9911173), [PDF](https://arxiv.org/pdf/math/9911173)\] \[Abstract: We prove the Riemannian [[0476 Penrose inequality|Penrose conjecture]], an important case of a conjecture made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we consider a Riemannian 3-manifold as a totally geodesic submanifold of a space-time in the context of general relativity, then outermost minimal spheres with total area $A$ correspond to apparent horizons of black holes contributing a mass $\sqrt{A/16\pi}$, scalar curvature corresponds to local energy density at each point, and the rate at which the metric becomes flat at infinity corresponds to total mass. The Riemannian Penrose conjecture then states that the total mass of an asymptotically flat 3-manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes. The flow of metrics we define continuously evolves the original 3-metric to a Schwarzschild 3-metric, which represents a spherically symmetric black hole in vacuum. We define the flow such that the area of the minimal spheres (which flow outward) and hence the mass contributed by the black holes in each of the metrics in the flow is constant, and then use the positive mass theorem to show that the total mass of the metrics is nonincreasing. Then since the total mass equals the mass of the black holes in a Schwarzschild metric, the Riemannian Penrose conjecture follows. This result improves upon the beautiful work of Huisken and Ilmanen, who used inverse mean curvature flows of surfaces to show that the total mass is at least the mass contributed by the largest black hole.\] # Brill ## Black holes and wormholes in 2+1 dimensions \[Links: [arXiv](https://arxiv.org/abs/gr-qc/9904083), [PDF](https://arxiv.org/pdf/gr-qc/9904083.pdf)\] \[Abstract: \] ## Refs - in [[0002 3D gravity]] - extends earlier work [[AminneborgBengtssonBrillHolstPeldan1997]] ## Sum - focus on ==sourceless== Einstein equations -> no particles - ==arbitrary number of asymptotic regions== and ==different genus== - (sec.2) time-symmetric ones - (sec.3) time evolution of time-symmetric ones - (sec.4) non-time-symmetric: have angular momentum ## Time-symmetric geometries - consider isometries on the time-symmetric surface, and later extend to full spacetime by extending the isometry to the whole AdS (like rotating in $SO(4)$ but for $SO(2,2)$) - simple solutions ![[Brill1999_embedded.png]] - obtain by the [[0099 Quotient method in AdS3]] - fundamental domain - doubling trick - other topologies - use pairs of pants - counting - $6g+3k-6$ dimensional Teichmüller space - $g$ genus, $k$ asymptotic AdS regions # Chamblin, Emparan, Johnson, Myers ## Charged AdS Black Holes and Catastrophic Holography \[Links: [arXiv](https://arxiv.org/abs/hep-th/9902170), [PDF](https://arxiv.org/pdf/hep-th/9902170.pdf)\] \[Abstract: We compute the properties of a class of charged black holes in anti-de Sitter space-time, in diverse dimensions. These black holes are solutions of consistent Einstein-Maxwell truncations of gauged supergravities, which are shown to arise from the inclusion of rotation in the transverse space. We uncover rich thermodynamic phase structures for these systems, which display classic critical phenomena, including structures isomorphic to the van der Waals-Maxwell liquid-gas system. In that case, the phases are controlled by the universal 'cusp' and 'swallowtail' shapes familiar from catastrophe theory. All of the thermodynamics is consistent with field theory interpretations via [[0001 AdS-CFT|holography]], where the dual field theories can sometimes be found on the world volumes of coincident rotating branes.\] # Chekhov, Fock ## Quantum Teichmüller space \[Links: [arXiv](https://arxiv.org/abs/math/9908165), [PDF](https://arxiv.org/pdf/math/9908165)\] \[Abstract: We describe explicitly a noncommutative deformation of the $*$-algebra of functions on the Teichmüller space of Riemann surfaces with holes equivariant w.r.t. the mapping class group action.\] # D'Hoker, Freedman, Mathur, Matusis, Rastelli ## Graviton exchange and complete 4-pt functions in AdS/CFT \[Links: [arXiv](https://arxiv.org/abs/hep-th/9903196), [PDF](https://arxiv.org/pdf/hep-th/9903196.pdf)\] \[Abstract: \] ## Refs - causality issues of [[finiteN_bcau]] - earlier work on 2-pt function [[]] ## Summary - graviton exchange diagrams for correlation functions of arbitrary ==scalar== operators - computes 4-pt amplitude of ==dilaton and axion field in IIB supergravity== - leading power singularity agrees with expected stress tensor contribution in a double OPE expansion - subleading orders: logs # Del Duca, Dixon, Maltoni ## New Color Decompositions for Gauge Amplitudes at Tree and Loop Level \[Links: [arXiv](https://arxiv.org/abs/hep-ph/9910563), [PDF](https://arxiv.org/pdf/hep-ph/9910563.pdf)\] \[Abstract: Recently, a color decomposition using structure constants was introduced for purely gluonic tree amplitudes, in a compact form involving only the linearly independent subamplitudes. We give two proofs that this decomposition holds for an arbitrary number of gluons. We also present and prove similar decompositions at one loop, both for pure gluon amplitudes and for amplitudes with an external quark-antiquark pair.\] ## Summary - proves [[0245 Kleiss-Kuijf relations|Kleiss-Kuijf relations]], reducing the number of independent terms in the colour decomposition from $(n-1)!$ to $(n-2)!$ # Dunne (Lectures) ## Aspects of Chern-Simons Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/9902115), [PDF](https://arxiv.org/pdf/hep-th/9902115.pdf)\] \[Abstract: Lectures at the 1998 Les Houches Summer School: Topological Aspects of Low Dimensional Systems. These lectures contain an introduction to various aspects of [[0089 Chern-Simons theory|Chern-Simons gauge theory]]: (i) basics of planar field theory, (ii) canonical quantization of Chern-Simons theory, (iii) Chern-Simons vortices, and (iv) radiatively induced Chern-Simons terms.\] # Felder, Frohlich, Fuchs, Schweigert (Sep) ## Conformal boundary conditions and three-dimensional topological field theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/9909140), [PDF](https://arxiv.org/pdf/hep-th/9909140)\] \[Abstract: We present a general construction of all correlation functions of a two-dimensional rational conformal field theory, for an arbitrary number of bulk and boundary fields and arbitrary topologies. The correlators are expressed in terms of Wilson graphs in a certain three-manifold, the connecting manifold. The amplitudes constructed this way can be shown to be modular invariant and to obey the correct factorization rules.\] # Felder, Frohlich, Fuchs, Schweigert (Dec) ## Correlation functions and boundary conditions in RCFT and three-dimensional topology \[Links: [arXiv](https://arxiv.org/abs/hep-th/9912239), [PDF](https://arxiv.org/pdf/hep-th/9912239)\] \[Abstract: We give a general construction of correlation functions in rational conformal field theory on a possibly non-orientable surface with boundary in terms of 3-dimensional topological quantum field theory. The construction applies to any modular category. It is proved that these correlation functions obey modular and factorization rules. Structure constants are calculated and expressed in terms of the data of the modular category.\] # Figueroa-O'Farrill (Notes) ## On the supersymmetries of anti de Sitter vacua \[Links: [arXiv](https://arxiv.org/abs/hep-th/9902066), [PDF](https://arxiv.org/pdf/hep-th/9902066.pdf)\] \[Abstract: We present details of a geometric method to associate a Lie superalgebra with a large class of bosonic [[0332 Supergravity|supergravity]] vacua of the type AdS $\times X$, corresponding to elementary branes in [[0517 M-theory|M-theory]] and type II string theory.\] # Giddings (a) ## The Boundary S matrix and the AdS to CFT dictionary - shows equivalence of two of the [[0001 AdS-CFT]] # Horowitz, Hubeny ## Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium \[Links: [arXiv](https://arxiv.org/abs/hep-th/9909056), [PDF](https://arxiv.org/pdf/hep-th/9909056.pdf)\] \[Abstract: We investigate the decay of a scalar field outside a Schwarzschild anti de Sitter black hole. This is determined by computing the complex frequencies associated with [[0325 Quasi-normal modes|quasinormal modes]]. There are qualitative differences from the asymptotically flat case, even in the limit of small black holes. In particular, for a given angular dependence, the decay is always exponential - there are no power law tails at late times. In terms of the [[0001 AdS-CFT|AdS/CFT correspondence]], a large black hole corresponds to an approximately thermal state in the field theory, and the decay of the scalar field corresponds to the decay of a perturbation of this state. Thus one obtains the timescale for the approach to thermal equilibrium. We compute these timescales for the strongly coupled field theories in three, four, and six dimensions which are dual to string theory in asymptotically AdS spacetimes.\] ## Refs - [[0325 Quasi-normal modes]] - holographic understanding on the CFT: [[2005#Kovtun, Starinets]] ## Summary - [[0325 Quasi-normal modes|QNM]] for AdS black holes is different from flat space - the timescale of the (exponential) decaying to equilibrium is given by the imaginary part of the lowest quasinormal frequency $\tau=1 / \omega_{I}$ ## Definitions - for a QNM: - near infinity, purely outgoing $\Phi \sim e^{-i \omega\left(t-r_{*}\right)}$ - near horizon, purely ingoing $\Phi \sim e^{-i \omega\left(t+r_{*}\right)}$ - complex frequency: $\omega=\omega_{R}-i \omega_{I}$ ## Large v.s. small BH - for large BH, due to additional symmetry, the frequencies can only depend on the BH temperature $T \sim r_{+} / R^{2}$ - for small BH, they can depend on both of the dimensional parameters - they do NOT behave like BHs in flat spacetime - reason: the late time behaviour is affected by waves bouncing off the potential at large radius # Horowitz, Itzhaki ## Black holes, shock waves, and causality in the AdS /CFT correspondence \[Links: [arXiv](https://arxiv.org/abs/hep-th/9901012), [PDF](https://arxiv.org/pdf/hep-th/9901012.pdf)\] \[Abstract: We find the expectation value of the energy-momentum tensor in the CFT corresponding to a moving black hole in AdS. Boosting the black hole to the speed of light, keeping the total energy fixed, yields a [[0117 Shockwave|gravitational shock wave]] in AdS. The analogous procedure on the field theory side leads to "light cone'' states, i.e., states with energy-momentum tensor localized on the light cone. The correspondence between the gravitational shock wave and these light cone states provides a useful tool for testing causality. We show, in several examples, how the CFT reproduces the causal relations in AdS.\] ## Summary - obtains the AdS [[0117 Shockwave|shockwave]] from boosted black hole - the [[0129 Dual of shockwaves|dual of shockwave]] is a light-cone state in CFT: stress tensor localised to a light cone - the [[0117 Shockwave|shockwave solution]] does not have $\alpha^\prime$ corrections (both in flat space and in AdS) ## Comments - some overlapping results with [[DanielssonKeski-VakkuriKruczenski1998]][](https://arxiv.org/abs/hep-th/9812007) # Kummer, Vassilevich ## Hawking radiation from dilaton gravity in 1 + 1 dimensions: a pedagogical review \[Links: [arXiv](https://arxiv.org/abs/gr-qc/9907041), [PDF](https://arxiv.org/pdf/gr-qc/9907041.pdf)\] \[Abstract: \] ## Summary - review the [[0240 Anti-evaporation]] problem and its resolution - parent [[0241 2D models of 4D physics]] # Nojiri, Odintsov ## On the conformal anomaly from higher derivative gravity in AdS/CFT correspondence \[Links: [arXiv](https://arxiv.org/abs/hep-th/9903033), [PDF](https://arxiv.org/pdf/hep-th/9903033.pdf)\] \[Abstract: \] ## Summary - finds [[0306 Weyl anomaly]] a la [[0209 Holographic renormalisation]] # Polchinski ## S-Matrices from AdS Spacetime \[Links: [arXiv](https://arxiv.org/abs/hep-th/9901076), [PDF](https://arxiv.org/pdf/hep-th/9901076.pdf)\] \[Abstract: In the large-$N$ limit of $d=4$, $\mathcal{N}=4$ gauge theory, the dual AdS spacetime becomes flat. We identify a gauge theory correlator whose large-$N$ limit is the flat spacetime S-matrix.\] ## Summary - early proposal of getting S-matrix [[0454 Flat holography from AdS-CFT|from AdS/CFT]] # Ponsot, Teschner ## Liouville bootstrap via harmonic analysis on a noncompact quantum group \[Links: [arXiv](https://arxiv.org/abs/hep-th/9911110), [PDF](https://arxiv.org/pdf/hep-th/9911110.pdf)\] \[Abstract: The purpose of this short note is to announce results that amount to a verification of the [[0036 Conformal bootstrap|bootstrap]] for [[0562 Liouville theory|Liouville theory]] in the generic case under certain assumptions concerning existence and properties of fusion transformations. Under these assumptions one may characterize the fusion and braiding coefficients as solutions of a system of functional equations that follows from the combination of consistency requirements and known results. This system of equations has a unique solution for irrational [[0033 Central charge|central charge]] $c>25$. The solution is constructed by solving the Clebsch-Gordan problem for a certain continuous series of quantum group representations and constructing the associated Racah-coefficients. This gives an explicit expression for the fusion coefficients. Moreover, the expressions can be continued into the strong coupling region $1<c<25$, providing a solution of the bootstrap also for this region.\] ## Summary - shows that there is a system of functional equations for the [[0573 Crossing kernel|fusion]] coefficients, which for $c>25$ has at most one solution # Runkel ## Structure constants for the D-series Virasoro minimal models \[Links: [arXiv](https://arxiv.org/abs/hep-th/9908046), [PDF](https://arxiv.org/pdf/hep-th/9908046.pdf)\] \[Abstract: In this paper expressions are given for the bulk and [[0548 Boundary CFT|boundary]] structure constants of D-series Virasoro minimal models on the upper half plane. It is the continuation of an earlier work on the A-series. The solution for the boundary theory is found first and then extended to the bulk. The [[0612 Modular invariance|modular invariant]] bulk field content is recovered as the maximal set of bulk fields consistent with the boundary theory. It is found that the structure constants are unique up to redefinition of the fields and in the chosen normalisation exhibit a manifest $Z_2$-symmetry associated to the D-diagram. The solution has been subjected to random numerical tests against the constraints it has to fulfill.\] ## Caveat The normalisation of the fields contains extra factors of $(-1)^s$ where $s$ is the spin. See eq.(56) for bulk fields and eq.(53) for the bulk-boundary OPE. # Segal (Lectures) ## Topological Field Theory \[Links: [ITP](https://web.math.ucsb.edu/~drm/conferences/ITP99/segal/)\] # Townsend (Review) ## Killing spinors, supersymmetries and rotating intersecting branes \[Links: [arXiv](https://arxiv.org/abs/hep-th/9901102), [PDF](https://arxiv.org/pdf/hep-th/9901102.pdf)\] \[Abstract: I review a recently proposed method for determining the symmetry superalgebra of a supergravity configuration from its [[0533 Killing spinor|Killing spinors]], and its application to the ‘near-horizon' limits of various rotating and intersecting branes.\] ## Refs - [[0533 Killing spinor]]