# Britto, Cachazo, Feng ## New Recursion Relations for Tree Amplitudes of Gluons \[Links: [arXiv](https://arxiv.org/abs/hep-th/0412308), [PDF](https://arxiv.org/pdf/hep-th/0412308.pdf)\] \[Abstract: We present new recursion relations for tree amplitudes in gauge theory that give very compact formulas. Our relations give any tree amplitude as a sum over terms constructed from products of two amplitudes of fewer particles multiplied by a Feynman propagator. The two amplitudes in each term are physical, in the sense that all particles are on-shell and momentum conservation is preserved. This is striking, since it is just like adding certain factorization limits of the original amplitude to build up the full answer. As examples, we recompute all known tree-level amplitudes of up to seven gluons and show that our recursion relations naturally give their most compact forms. We give a new result for an eight-gluon amplitude, $A(1^+,2^-,3^+,4^-,5^+,6^-,7^+,8^-)$. We show how to build any amplitude in terms of three-gluon amplitudes only.\] ## Refs - this is the original paper for [[0058 BCFW|BCFW]] - later paper [[2005#Britto, Cachazo, Feng, Witten]] # Buchel, Liu, Starinets ## Coupling constant dependence of the shear viscosity in N=4 supersymmetric Yang-Mills theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/0406264), [PDF](https://arxiv.org/pdf/hep-th/0406264.pdf)\] \[Abstract: Gauge theory - gravity duality predicts that the [[0430 Holographic shear viscosity|shear viscosity]] of N=4 supersymmetric $SU(N_c)$ Yang-Mills plasma at temperature $T$ in the limit of large $N_c$ and large 't Hooft coupling $g^2_{YM} N_c$ is independent of the coupling and equals to $\pi N_c^2 T^3/8$. In this paper, we compute the leading correction to the shear viscosity in inverse powers of 't Hooft coupling using the $\alpha'$-corrected low-energy effective action of type IIB string theory. We also find the correction to the ratio of shear viscosity to the volume entropy density (equal to $1/4\pi$ in the limit of infinite coupling). The correction to 1/4\pi scales as $(g^2_{YM} N_c)^{-3/2}$ with a positive coefficient.\] # Burrington, Liu, Sabra ## AdS$_5$ Black Holes with Fermionic Hair \[Links: [arXiv](https://arxiv.org/abs/hep-th/0412155), [PDF](https://arxiv.org/pdf/hep-th/0412155.pdf)\] \[Abstract: The study of new [[0178 BPS|hep-th/0412155]] objects in AdS$_5$ has led to a deeper understanding of [[0001 AdS-CFT|AdS/CFT]]. To help complete this picture, and to fully explore the consequences of the supersymmetry algebra, it is also important to obtain new solutions with bulk fermions turned on. In this paper we construct superpartners of the 1/2 BPS black hole in AdS$_5$ using a natural set of fermion zero modes. We demonstrate that these superpartners, carrying fermionic hair, have conserved charges differing from the original bosonic counterpart. To do so, we find the R-charge and dipole moment of the new system, as well as the mass and angular momentum, defined through the boundary stress tensor. The complete set of superpartners fits nicely into a chiral representation of AdS$_5$ supersymmetry, and the spinning solutions have the expected gyromagnetic ratio, $g=1$.\] # Calabrese, Cardy ## Entanglement Entropy and Quantum Field Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/0405152), [PDF](https://arxiv.org/pdf/hep-th/0405152.pdf)\] \[Abstract: We carry out a systematic study of [[0301 Entanglement entropy|entanglement entropy]] in relativistic quantum field theory. This is defined as the von Neumann entropy $S_A=-\operatorname{Tr} \rho_A \log \rho_A$ corresponding to the reduced density matrix $\rho_A$ of a subsystem $A$. For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge $c$, we re-derive the result $S_A\sim(c/3) \log(l)$ of [[HolzheyLarsenWilczek1994|Holzhey et al.]] when A is a finite interval of length $l$ in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when $A$ consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length $\xi$ is large but finite, we show that $S_A\sim{\cal A}(c/6)\log\xi$, where $\cal A$ is the number of boundary points of $A$. These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite-size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free-field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.\] ## Refs - introduces the replica trick to calculate the [[0301 Entanglement entropy|entanglement entropy]] - [[0293 Renyi entropy]] # Dixon, Glover, Khoze ## MHV Rules for Higgs Plus Multi-Gluon Amplitudes \[Links: [arXiv](https://arxiv.org/abs/hep-th/0411092), [PDF](https://arxiv.org/pdf/hep-th/0411092.pdf)\] \[Abstract: We use tree-level perturbation theory to show how non-supersymmetric one-loop scattering amplitudes for a Higgs boson plus an arbitrary number of partons can be constructed, in the limit of a heavy top quark, from a generalization of the scalar graph approach of [[0352 CSW relations|Cachazo, Svrcek and Witten]]. The Higgs boson couples to gluons through a top quark loop which generates, for large top mass, a dimension-5 operator $H \operatorname{tr} G^2$. This effective interaction leads to amplitudes which cannot be described by the standard [[0061 Maximally helicity violating amplitudes|MHV]] rules; for example, amplitudes where all of the gluons have positive helicity. We split the effective interaction into the sum of two terms, one holomorphic (selfdual) and one anti-holomorphic (anti-selfdual). The holomorphic interactions give a new set of MHV vertices -- identical in form to those of pure gauge theory, except for momentum conservation -- that can be combined with pure gauge theory MHV vertices to produce a tower of amplitudes with more than two negative helicities. Similarly, the anti-holomorphic interactions give anti-MHV vertices that can be combined with pure gauge theory anti-MHV vertices to produce a tower of amplitudes with more than two positive helicities. A Higgs boson amplitude is the sum of one MHV-tower amplitude and one anti-MHV-tower amplitude. We present all MHV-tower amplitudes with up to four negative-helicity gluons and any number of positive-helicity gluons (NNMHV). These rules reproduce all of the available analytic formulae for Higgs + $n$-gluon scattering ($n<=5$) at tree level, in some cases yielding considerably shorter expressions.\] # Eynard ## All genus correlation functions for the hermitian 1-matrix model \[Links: [arXiv](https://arxiv.org/abs/hep-th/0407261), [PDF](https://arxiv.org/pdf/hep-th/0407261.pdf)\] \[Abstract: We rewrite the loop equations of the hermitian [[0197 Matrix model|matrix model]], in a way which allows to compute all the correlation functions, to all orders in the topological $1/N^2$ expansion, as residues on an hyperelliptical curve. Those residues, can be represented diagrammaticaly as Feynmann graphs of a cubic interaction field theory on the curve.\] # Fuchs, Runkel, Schweigert (Mar) ## TFT construction of RCFT correlators III: Simple currents \[Links: [arXiv](https://arxiv.org/abs/hep-th/0403157), [PDF](https://arxiv.org/pdf/hep-th/0403157)\] \[Abstract: We use simple currents to construct symmetric special Frobenius algebras in modular tensor categories. We classify such simple current type algebras with the help of abelian group cohomology. We show that they lead to the modular invariant torus partition functions that have been studied by Kreuzer and Schellekens. We also classify boundary conditions in the associated conformal field theories and show that the boundary states are given by the formula proposed in [hep-th/0007174](https://arxiv.org/abs/hep-th/0007174). Finally, we investigate conformal defects in these theories.\] # Fuchs, Runkel, Schweigert (Dec) ## TFT construction of RCFT correlators IV: Structure constants and correlation functions \[Links: [arXiv](https://arxiv.org/abs/hep-th/0412290), [PDF](https://arxiv.org/pdf/hep-th/0412290)\] \[Abstract: We compute the fundamental correlation functions in two-dimensional rational conformal field theory, from which all other correlators can be obtained by sewing: the correlators of three bulk fields on the sphere, one bulk and one boundary field on the disk, three boundary fields on the disk, and one bulk field on the cross cap. We also consider conformal defects and calculate the correlators of three defect fields on the sphere and of one defect field on the cross cap. Each of these correlators is presented as the product of a structure constant and the appropriate conformal two- or three-point block. The structure constants are expressed as invariants of ribbon graphs in three-manifolds.\] # Gukov, Martinec, Moore, Strominger (Mar, b) ## Chern-Simons Gauge Theory and the AdS${}_3$/CFT${}_2$ Correspondence \[Links: [arXiv](https://arxiv.org/abs/hep-th/0403225), [PDF](https://arxiv.org/pdf/hep-th/0403225)\] \[Abstract: The bulk partition function of pure [[0089 Chern-Simons theory|Chern-Simons theory]] on a three-manifold is a state in the space of [[0031 Conformal block|conformal blocks]] of the dual boundary RCFT, and therefore transforms non-trivially under the boundary modular group. In contrast the bulk partition function of AdS${}_3$ string theory is the modular-invariant partition function of the dual CFT on the boundary. This is a puzzle because AdS${}_3$ string theory formally reduces to pure Chern-Simons theory at long distances. We study this puzzle in the context of massive Chern-Simons theory. We show that the puzzle is resolved in this context by the appearance of a chiral "spectator boson" in the boundary CFT which restores modular invariance. It couples to the conformal metric but not to the gauge field on the boundary. Consequently, we find a generalization of the standard Chern-Simons/RCFT correspondence involving "nonholomorphic conformal blocks" and nonrational boundary CFTs. These generalizations appear in the long-distance limit of AdS${}_3$ string theory, where the role of the spectator boson is played by other degrees of freedom in the theory.\] # Huang ## Vertex operator algebras, the Verlinde conjecture and modular tensor categories \[Links: [arXiv](https://arxiv.org/abs/math/0412261), [PDF](https://arxiv.org/pdf/math/0412261)\] \[Abstract: Let $V$ be a simple vertex operator algebra satisfying the following conditions: (i) The homogeneous subspaces of $V$ of weights less than 0 are 0, the homogeneous subspace of $V$ of weight 0 is spanned by the vacuum and $V'$ is isomorphic to $V$ as a $V$-module. (ii) Every weak $V$-module gradable by nonnegative integers is completely reducible. (iii) $V$ is $C_2$-cofinite. We announce a proof of the Verlinde conjecture for $V$, that is, of the statement that the matrices formed by the fusion rules among irreducible $V$-modules are diagonalized by the matrix given by the action of the modular transformation $\tau\mapsto -1/\tau$ on the space of characters of irreducible $V$-modules. We discuss some consequences of the Verlinde conjecture, including the [[0643 Verlinde formula|Verlinde formula]] for the fusion rules, a formula for the matrix given by the action of $\tau\mapsto -1/\tau$ and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of $V$-modules when $V$ satisfies in addition the condition that irreducible $V$-modules not equivalent to $V$ has no nonzero elements of weight 0. In particular, the category of $V$-modules has a natural structure of [[0644 Modular tensor category]].\] # Kovtun, Son, Starinets ## Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics \[Links: [arXiv](https://arxiv.org/abs/hep-th/0405231), [PDF](https://arxiv.org/pdf/hep-th/0405231.pdf)\] \[Abstract: The ratio of shear viscosity to volume density of entropy can be used to characterize how close a given fluid is to being perfect. Using string theory methods, we show that this ratio is equal to a universal value of $\hbar/4\pi k_B$ for a large class of strongly interacting quantum field theories whose dual description involves black holes in anti--de Sitter space. We provide evidence that this value may serve as a lower bound for a wide class of systems, thus suggesting that black hole horizons are dual to the most ideal fluids.\] ## Refs - OG for [[0430 Holographic shear viscosity|KSS]] bound # Maldacena, Maoz ## Wormholes in AdS3 \[Links: [arXiv](https://arxiv.org/abs/hep-th/0401024), [PDF](https://arxiv.org/pdf/hep-th/0401024.pdf)\] \[Abstract: We construct a few Euclidean supergravity solutions with multiple boundaries. We consider examples where the corresponding boundary field theory is well defined on each boundary. We point out that these configurations are [[0249 Factorisation problem|puzzling]] from the AdS/CFT point of view. A proper understanding of the AdS/CFT dictionary for these cases might yield some information about the physics of closed universes.\] ## Summary - *constructs* some ==Euclidean== solutions with multiple boundaries - *demonstrates* [[0249 Factorisation problem]] ## Obstacle - cannot have AdS with positive boundary curvature which solve Einstein’s equations - [[WittenYau1999]] - [[CaiGalloway2000]] - solution: turn on YM fields ## Genus-2 wormhole - used in [[2020#Belin, de Boer]] # Marolf ## States and boundary terms: subtleties of Lorentzian AdS/CFT \[Links: [arXiv](https://arxiv.org/abs/hep-th/0412032), [PDF](https://arxiv.org/pdf/hep-th/0412032.pdf)\] \[Abstract: \] # Nakayama (Review) ## Liouville Field Theory -- A decade after the revolution \[Links: [arXiv](https://arxiv.org/abs/hep-th/0402009), [PDF](https://arxiv.org/pdf/hep-th/0402009.pdf)\] \[Abstract: We review recent developments (up to January 2004) of the [[0562 Liouville theory|Liouville field theory]] and its [[0197 Matrix model|matrix model]] dual. This review consists of three parts. In part I, we review the bosonic Liouville theory. After briefly reviewing the necessary background, we discuss the bulk structure constants (the [[0598 DOZZ formula|DOZZ formula]]) and the boundary states (the FZZT brane and the ZZ brane). Various applications are also presented. In part II, we review the supersymmetric extension of the Liouville theory. We first discuss the bulk structure constants and the branes as in the bosonic Liouville theory, and then we present the matrix dual descriptions with some applications. In part III, the Liouville theory on unoriented surfaces is reviewed. After introducing the crosscap state, we discuss the matrix model dual description and the tadpole cancellation condition. This review also includes some original material such as the derivation of the conjectured dual action for the $N = 2$ Liouville theory from other known dualities and the comparison of the Liouville crosscap state with the $c = 0$ unoriented matrix model. This is based on my master's thesis submitted to Department of Physics, Faculty of Science, University of Tokyo on January 2004.\] # Papadimitriou, Skenderis ## AdS/CFT correspondence and Geometry \[Links: [arXiv](https://arxiv.org/abs/hep-th/0404176), [PDF](https://arxiv.org/pdf/hep-th/0404176.pdf)\] \[Abstract: \] ## Summary - [[0209 Holographic renormalisation|counter term in holographic renormalisation]] - combines optimally elements from all previous methods and supersedes them in efficiency # Seiberg, Shih ## Minimal String Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/0409306), [PDF](https://arxiv.org/pdf/hep-th/0409306.pdf)\] \[Abstract: We summarize recent progress in the understanding of [[0583 Minimal string theory|minimal string theory]], focusing on the worldsheet description of physical operators and [[0156 D-brane|D-branes]]. We review how a geometric interpretation of minimal string theory emerges naturally from the study of the D-branes. This simple geometric picture ties together many otherwise unrelated features of minimal string theory, and it leads directly to a worldsheet derivation of the dual matrix model.\] # Szabados (Review) ## Quasi-Local Energy-Momentum and Angular Momentum in General Relativity \[Links: [Inspire](https://inspirehep.net/literature/1252892), [DOI](https://doi.org/10.12942/lrr-2009-4)\] \[Abstract: The present status of the [[0595 Quasi-local energy|quasi-local mass]], energy-momentum and angular-momentum constructions in general relativity is reviewed. First, the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local quantities, are recalled. Then, the various specific constructions and their properties (both successes and deficiencies are discussed. Finally, some of the (actual and potential) applications of the quasi-local concepts and specific constructions are briefly mentioned.\]