# Bakalov, Kirillov ## On the Lego-Teichmuller game \[Links: [arXiv](https://arxiv.org/abs/math/9809057), [PDF](https://arxiv.org/pdf/math/9809057.pdf)\] \[Abstract: For a smooth oriented surface $S$, denote by $M(S)$ the set of all ways to represent $S$ as a result of gluing together standard spheres with holes (''the Lego game''). In this paper we give a full set of simple moves and relations which turn $M(S)$ into a connected and simply-connected 2-complex. Results of this kind were first obtained by Moore and Seiberg, but their paper contains serious gaps. Our proof is based on a different approach and is much more rigorous.\] # Banks, Douglas, Horowitz, Martinec ## AdS Dynamics from Conformal Field Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/9808016), [PDF](https://arxiv.org/pdf/hep-th/9808016)\] \[Abstract: We explore the extent to which a local string theory dynamics in anti-de Sitter space can be determined from its proposed Conformal Field Theory (CFT) description. Free fields in the bulk are constructed from the CFT operators, but difficulties are encountered when one attempts to incorporate interactions. We also discuss general features of black hole dynamics as seen from the CFT perspective. In particular, we argue that the singularity of AdS$_3$ black holes is resolved in the CFT description.\] # Bern, Del Duca, Schmidt ## The Infrared Behavior of One-Loop Gluon Amplitudes at Next-to-Next-to-Leading Order \[Links: [arXiv](https://arxiv.org/abs/hep-ph/9810409), [PDF](https://arxiv.org/pdf/hep-ph/9810409.pdf)\] \[Abstract: \] ## Summary - *gives* the one-loop gluon splitting and soft functions to all orders in $\epsilon$ - leaving the calculational details and a complete listing of the one-loop splitting and soft functions, including fermions, to a forthcoming paper (which?) ## Splitting functions - eq.5 ## Soft functions - $\operatorname{Soft}^{1-\mathrm{loop}}\left(a, k^{\pm}, b\right)$=-\operatorname{Soft}^{\mathrm{tree}}\left(a, k^{\pm}, b\right) c_{\Gamma}\left(\frac{\mu^{2}\left(-s_{a b}\right)}{\left(-s_{a k}\right)\left(-s_{k b}\right)}\right)^{\epsilon}\left(\frac{1}{\epsilon^{2}}+\frac{\pi^{2}}{6}+\frac{7 \pi^{4}}{360} \epsilon^{2}\right)+\mathcal{O}\left(\epsilon^{3}\right)$ (eq.e14) - to order $O(\epsilon^0)$ this agrees with what one can extract from 4 and 5 point amplitudes (at order $O(\epsilon^0)$) # Bern, Dixon, Perelstein, Rozowsky (Sep) ## One-loop n-point helicity amplitudes in (self-dual) gravity \[Links: [arXiv](https://arxiv.org/abs/hep-th/9809160), [PDF](https://arxiv.org/pdf/hep-th/9809160.pdf)\] \[Abstract: \] # Bern, Dixon, Perelstein, Rozowsky (Nov) ## Multi-leg one-loop gravity amplitudes from gauge theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/9811140), [PDF](https://arxiv.org/pdf/hep-th/9811140.pdf)\] \[Abstract: \] ## Refs - this is important for [[0067 Double copy]] ## What I need from it - split function for gravitons ## Notation - $s_{i j} \equiv\left(k_{i}+k_{j}\right)^{2}$ - $\langle i j\rangle[j i]=s_{i j}=2 k_{i} \cdot k_{j}$ ## Graviton splitting function - at *any* loop order (including tree) $M_{n}^{\operatorname{loop}}\left(\ldots, a^{\lambda_{a}}, b^{\lambda_{b}}, \ldots\right) \stackrel{a \| b}{\longrightarrow} \sum_{\lambda} \operatorname{Split}_{-\lambda}^{\mathrm{gravity}}\left(z, a^{\lambda_{a}}, b^{\lambda_{b}}\right) \times M_{n-1}^{\operatorname{loop}}\left(\ldots, P^{\lambda}, \ldots\right)$ - from gauge split function - $\operatorname{Split}_{-(\lambda+\tilde{\lambda})}^{\operatorname{gravity}}\left(z, 1^{\lambda_{1}+\tilde{\lambda}_{1}}, 2^{\lambda_{2}+\tilde{\lambda}_{2}}\right)$=-s_{12} \times \operatorname{Split}_{-\lambda}^{\operatorname{tree}}\left(z, 1^{\lambda_{1}}, 2^{\lambda_{2}}\right) \times \operatorname{Split}_{-\tilde{\lambda}}^{\operatorname{tree}}\left(z, 2^{\tilde{\lambda}_{2}}, 1^{\tilde{\lambda}_{1}}\right)$ - $\operatorname{Split}_{-\tilde{\lambda}}^{\operatorname{tree}}$ are gauge splitting amplitudes - results - $\text{Split}_{+}^{\text {gravity }}\left(z, a^{+}, b^{+}\right)=0$ - $\text{Split}_-^\text{gravity} \left(z, a^{+}, b^{+}\right)=-\frac{1}{z(1-z)} \frac{[a b]}{\langle a b\rangle}$ - $\text{Split}_{+}^{\text {gravity }}\left(z, a^{-}, b^{+}\right)=-\frac{z^{3}}{1-z} \frac{[a b]}{\langle a b\rangle}$ - translation to [[2019#Pate, Raclariu, Strominger, Yuan]] - $z=k_i/k_P=\frac{\omega_i}{\omega_i+\omega_j}$, $1-z=k_j/k_P=\frac{\omega_j}{\omega_i+\omega_j}$ - $\langle ij\rangle=z_i^\alpha z_{j\alpha}=z_i-z_j$ - $[i j] \equiv\left[\bar{z}_{i} \bar{z}_{j}\right]:=\bar{z}_{i}^{\dot{\alpha}} \bar{z}_{j \dot{\alpha}}=\bar z_i-\bar z_j$ # Bohm ## *Inhomogeneous* Einstein metrics on low-dimensional spheres and other low-dimensional spaces \[Links: [PDF](https://link.springer.com/content/pdf/10.1007/s002220050261.pdf)\] \[Abstract: \] ## Refs - [[2003#Hartnoll]] AdS Einstein metrics ## Summary - OG for [[0427 Bohm metrics]] # Freedman, Mathur, Matusis, Rastelli ## Correlation functions in AdS/CFT \[Links: [arXiv](https://arxiv.org/abs/hep-th/9804058), [PDF](https://arxiv.org/pdf/hep-th/9804058.pdf)\] \[Abstract: \] ## Refs - later on 4-pt function [[1999#D'Hoker, Freedman, Mathur, Matusis, Rastelli]] - this is one of the earliest papers on [[0001 AdS-CFT]] - good for [[0109 Witten diagrams]] computations # Gopakumar, Vafa ## On the Gauge Theory/Geometry Correspondence \[Links: [arXiv](https://arxiv.org/abs/hep-th/9811131), [PDF](https://arxiv.org/pdf/hep-th/9811131)\] \[Abstract: The 't Hooft expansion of $SU(N)$ [[0089 Chern-Simons theory|Chern-Simons theory]] on $S^3$ is proposed to be exactly dual to the topological closed string theory on the $S^2$ blow up of the conifold geometry. The $B$-field on the $S^2$ has magnitude $Ng_s=\lambda$, the 't Hooft coupling. We are able to make a number of checks, such as finding exact agreement at the level of the partition function computed on *both* sides for arbitrary $\lambda$ and to all orders in $1/N$. Moreover, it seems possible to derive this correspondence from a linear sigma model description of the conifold. We propose a picture whereby a perturbative D-brane description, in terms of holes in the closed string worldsheet, arises automatically from the coexistence of two phases in the underlying $U(1)$ gauge theory. This approach holds promise for a derivation of the [[0001 AdS-CFT|AdS/CFT]] correspondence.\] # Gubser, Klebanov, Polyakov ## Gauge Theory Correlators from Non-Critical String Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/9802109), [PDF](https://arxiv.org/pdf/hep-th/9802109.pdf)\] \[Abstract: We suggest a means of obtaining certain Green's functions in 3+1-dimensional ${\cal N} = 4$ supersymmetric Yang-Mills theory with a large number of colors via non-critical string theory. The non-critical string theory is related to critical string theory in anti-deSitter background. We introduce a boundary of the anti-deSitter space analogous to a cut-off on the Liouville coordinate of the two-dimensional string theory. Correlation functions of operators in the gauge theory are related to the dependence of the supergravity action on the boundary conditions. From the quadratic terms in supergravity we read off the anomalous dimensions. For operators that couple to massless string states it has been established through absorption calculations that the anomalous dimensions vanish, and we rederive this result. The operators that couple to massive string states at level $n$ acquire anomalous dimensions that grow as $2 (n g_{YM} \sqrt {2 N} )^{1/2}$ for large 't Hooft coupling. This is a new prediction about the strong coupling behavior of large $N$ SYM theory.\] # Henningson, Skenderis (Jun) ## The Holographic Weyl anomaly \[Links: [arXiv](https://arxiv.org/abs/hep-th/9806087), [PDF](https://arxiv.org/pdf/hep-th/9806087.pdf)\] \[Abstract: \] ## Refs - a later view of this in the same year [[HenningsonSkenderis199812]] - holographic calculation of [[0306 Weyl anomaly]] ## Conventions - Riemann - see footnote on p.4 - Riemann differs from [[0283 Lagrangian variation toolkit|modern convention]] by a sign - coordinate - used $\rho=z^2$ (so $\epsilon$ in this paper is the square of $\epsilon$ in papers using $z$) # Horowitz, Myers ## The AdS/CFT Correspondence and a New Positive Energy Conjecture for General Relativity \[Links: [arXiv](https://arxiv.org/abs/hep-th/9808079), [PDF](https://arxiv.org/pdf/hep-th/9808079.pdf)\] \[Abstract: We examine the [[0001 AdS-CFT|AdS/CFT correspondence]] when the gauge theory is considered on a compactified space with supersymmetry breaking boundary conditions. We find that the corresponding supergravity solution has a negative energy, in agreement with the expected negative Casimir energy in the field theory. Stability of the gauge theory would imply that this supergravity solution has minimum energy among all solutions with the same boundary conditions. Hence we are lead to conjecture a new [[0116 Positive energy theorem|positive energy theorem]] for asymptotically locally Anti-de Sitter spacetimes. We show that the candidate minimum energy solution is stable against all quadratic fluctuations of the metric.\] ## Refs - [[0231 Bulk solutions for CFTs on non-trivial geometries]] - origin of [[0407 Horowitz-Myers conjecture|Horowitz-Myers conjecture]] ## Summary - constructs AdS soliton solutions - conjectures that these are minimum energy solutions -> new [[0116 Positive energy theorem|positive energy theorem]] - shows that they are stable against all ==quadratic== metric fluctuations ## Energy in [[0001 AdS-CFT|AdS/CFT]] - the energy of spacetime should equal to that in CFT - in asymptotically global AdS - AdS: there is an established [[0116 Positive energy theorem|positive energy theorem]] - CFT: stability of gauge theory vacuum - boundary topology $S^1\times S^{p-1}\times R$ - AdS: a ground state lower than pure AdS is proposed in this paper - (for $p=3$) $\rho_{\mathrm{SUGRA}}=-\frac{\pi^{2}}{8} \frac{N^{2}}{\beta^{4}} F\left(\beta^{2} / l^{2}\right)$ - $F(0)=1$: to leading order it matches the result for $S^1\times R^{p}$ (expected because at leading order $S^2$ is effectively $R^2$) - CFT: Casimir - boundary topology $S^1\times R^{p}$ - **AdS soliton** $d s^{2}=\frac{r^{2}}{l^{2}}\left[\left(1-\frac{r_{0}^{p+1}}{r^{p+1}}\right) d \tau^{2}+\left(d x^{i}\right)^{2}-d t^{2}\right]+\left(1-\frac{r_{0}^{p+1}}{r^{p+1}}\right)^{-1} \frac{l^{2}}{r^{2}} d r^{2}$ - $i=1,...,p-1$ - (for $p=3$) $\rho_{\text {SUGRA }}=\frac{E}{V_{2} \beta}=-\frac{\pi^{2}}{8} \frac{N^{2}}{\beta^{4}}$ - c.f. $\rho_{\text {gauge }}=-\frac{\pi^{2}}{6} \frac{N^{2}}{\beta^{4}}$ (a factor of 3/4) # Louko, Marolf ## Single-exterior black holes and the AdS-CFT conjecture \[Links: [arXiv](https://arxiv.org/abs/hep-th/9808081), [PDF](https://arxiv.org/pdf/hep-th/9808081.pdf)\] \[Abstract: In the context of the conjectured [[0001 AdS-CFT|AdS-CFT]] correspondence of string theory, we consider a class of asymptotically Anti-de Sitter black holes whose conformal boundary consists of a single connected component, identical to the conformal boundary of Anti-de Sitter space. In a simplified model of the boundary theory, we find that the boundary state to which the black hole corresponds is pure, but this state involves correlations that produce thermal expectation values at the usual Hawking temperature for suitably restricted classes of operators. The energy of the state is finite and agrees in the semiclassical limit with the black hole mass. We discuss the relationship between the black hole topology and the correlations in the boundary state, and speculate on generalizations of the results beyond the simplified model theory.\] ## Summary - overall state is pure, but it looks thermal if we focus on a constraint class of operators - tracing over left or right moving sector gives a thermal state ## Comments - single side -> boundary is pure ## Refs - [[0253 RT for pure BHs]] # Maldacena, Michelson, Strominger ## Anti-de Sitter Fragmentation \[Links: [arXiv](https://arxiv.org/abs/hep-th/9812073), [PDF](https://arxiv.org/pdf/hep-th/9812073.pdf)\] \[Abstract: Low-energy, near-horizon scaling limits of black holes which lead to string theory on AdS$_2 \times S^2$ are described. Unlike the higher-dimensional cases, in the simplest approach all finite-energy excitations of AdS$_2 \times S^2$ are suppressed. Surviving zero-energy configurations are described. These can include tree-like structures in which the AdS$_2 \times S^2$ throat branches as the horizon is approached, as well as disconnected AdS$_2 \times S^2$ universes. In principle, the [[0004 Black hole entropy|black hole entropy]] counts the quantum ground states on the moduli space of such configurations. In a nonsupersymmetric context AdS$_D$ for general $D$ can be unstable against instanton-mediated fragmentation into disconnected universes. Several examples are given.\] # Olum ## Superluminal travel requires negative energies \[Links: [arXiv](https://arxiv.org/abs/gr-qc/9805003), [PDF](https://arxiv.org/pdf/gr-qc/9805003.pdf)\] \[Abstract: I investigate the relationship between faster-than-light travel and weak-energy-condition violation, i.e., negative energy densities. In a general spacetime it is difficult to define faster-than-light travel, and I give an example of a metric which appears to allow superluminal travel, but in fact is just flat space. To avoid such difficulties, I propose a definition of superluminal travel which requires that the path to be traveled reach a destination surface at an earlier time than any neighboring path. With this definition (and assuming the generic condition) I prove that superluminal travel requires weak-energy-condition violation.\] # O'Raifeartaigh, Pawlowski, Sreedhar ## Duality in Quantum Liouville Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/9811090), [PDF](https://arxiv.org/pdf/hep-th/9811090.pdf)\] \[Abstract: The quantisation of the two-dimensional [[0562 Liouville theory|Liouville field theory]] is investigated using the path integral, on the sphere, in the large radius limit. The general form of the $N$-point functions of vertex operators is found and the three-point function is derived explicitly. In previous work it was inferred that the three-point function should possess a two-dimensional lattice of poles in the parameter space (as opposed to a one-dimensional lattice one would expect from the standard Liouville potential). Here we argue that the two-dimensionality of the lattice has its origin in the duality of the quantum mechanical Liouville states and we incorporate this duality into the path integral by using a two-exponential potential. Contrary to what one might expect, this does not violate conformal invariance; and has the great advantage of producing the two-dimensional lattice in a natural way.\] # Runkel ## Boundary structure constants for the A-series Virasoro minimal models \[Links: [arXiv](https://arxiv.org/abs/hep-th/9811178), [PDF](https://arxiv.org/pdf/hep-th/9811178.pdf)\] \[Abstract: We consider A-series [[0612 Modular invariance|modular invariant]] Virasoro minimal models on the upper half plane. From Lewellen's sewing constraints a necessary form of the bulk and boundary structure constants is derived. Necessary means that any solution can be brought to the given form by rescaling of the fields. All constants are expressed essentially in terms of fusing ($F$-)matrix elements and the normalisations are chosen such that they are real and no square roots appear. It is not shown in this paper that the given structure constants solve the sewing constraints, however random numerical tests show no contradiction and agreement of the bulk structure constants with Dotsenko and Fateev. In order to facilitate numerical calculations a recursion relation for the $F$-matrices is given.\] # Visser, Bassett, Liberati ## Superluminal censorship \[Links: [arXiv](https://arxiv.org/abs/gr-qc/9810026), [PDF](https://arxiv.org/pdf/gr-qc/9810026.pdf)\] \[Abstract: We argue that "effective'' [[0115 Superluminality|superluminal]] travel, potentially caused by the tipping over of light cones in Einstein gravity, is always associated with violations of the [[0480 Null energy condition|null energy condition (NEC)]]. This is most easily seen by working perturbatively around Minkowski spacetime, where we use linearized Einstein gravity to show that the [[0480 Null energy condition|NEC]] forces the light cones to contract (narrow). Given the [[0480 Null energy condition|NEC]], the Shapiro time delay in any weak gravitational field is always a delay relative to the Minkowski background, and never an advance. Furthermore, any object travelling within the lightcones of the weak gravitational field is similarly delayed with respect to the minimum traversal time possible in the background Minkowski geometry.\] ## Related - [[0115 Superluminality]] - [[0091 Boundary causality]] - [[0247 Energy conditions]] ## Summary - near flat space, violation of [[0480 Null energy condition|NEC]] <=> effective [[0115 Superluminality|superluminality]] # Witten (Feb) ## AdS space and holography \[Links: [arXiv](https://arxiv.org/abs/hep-th/9802150), [PDF](https://arxiv.org/pdf/hep-th/9802150.pdf)\] \[Abstract: Recently, it has been proposed by Maldacena that large $N$ limits of certain conformal field theories in $d$ dimensions can be described in terms of supergravity (and string theory) on the product of d+1-dimensional AdS space with a compact manifold. Here we elaborate on this idea and propose a precise correspondence between conformal field theory observables and those of supergravity: correlation functions in conformal field theory are given by the dependence of the supergravity action on the asymptotic behaviour at infinity. In particular, dimensions of operators in conformal field theory are given by masses of particles in supergravity. As quantitative confirmation of this correspondence, we note that the Kaluza-Klein modes of Type IIB supergravity on AdS$_5$\times S^5$ match with the chiral operators of $\mathcal{N}=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the $\mathcal{N}=4$ theory has a large $N$ phase transition related to the thermodynamics of AdS black holes.\] ## Summary - precise correspondence between CFT and supergravity observables: [[0001 AdS-CFT|AdS/CFT]] - a Hamiltonian version of the correspondence (with some assumptions) - show the $\mathcal{N}=4$ theory has a large $N$ phase transition related to AdS black holes - example: KK modes of Type IIB SUGRA <-> chiral operators in [[0155 N=4 SYM]] ## The structure - (1) Introduction - some review and proposes a direct dictionary - defines AdS and show that global symmetry of AdS = conformal symmetry of its boundary - (2) Boundary behaviour - 2.1 Euclidean AdS: define Euclidean metric and its compactification - 2.2 Massless field equations - 2.3 Ansatz for the effective action - 2.4 calculations for massless scalar ## KK modes <-> chiral operators - KK modes for AdS${}_5\times S^5$ calculated in [[KimRomansVanNieuwenhuizen1985]] and [[GunaydinMarcus1985]] - massive, massless, tachyonic (within BF bound) <-> irrelevant, marginal, relevant - stringy <-> divergent dimensions (large $N$ and large t'Hooft) ## The dictionary - scalar field $\left\langle\exp \int_{\mathbf{S}^{d}} \phi_{0} \mathcal{O}\right\rangle_{C F T}=Z_{S}\left(\phi_{0}\right)$ - metric field $Z_{C F T}(h)=Z_{S}(h)$ - gauge field (in AdS) $\left\langle\exp \left(\int_{\mathbf{S}^{d}} J_{a} A_{0}^{a}\right)\right\rangle_{C F T}=Z_{S}\left(A_{0}\right)$ - n.b. in all cases, the AdS partition function is computed as a function of the boundary values of the bulk fields, and then this function is interpreted as a generating functional of the CFT correlators, where the sources are the boundary values # Witten (Mar) ## Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories \[Links: [arXiv](https://arxiv.org/abs/hep-th/9803131), [PDF](https://arxiv.org/pdf/hep-th/9803131.pdf)\] \[Abstract: The [[0001 AdS-CFT|correspondence]] between supergravity (and string theory) on AdS space and boundary conformal field theory relates the thermodynamics of ${\cal N}=4$ super Yang-Mills theory in four dimensions to the thermodynamics of Schwarzschild black holes in Anti-de Sitter space. In this description, quantum phenomena such as the spontaneous breaking of the center of the gauge group, magnetic confinement, and the mass gap are coded in classical geometry. The correspondence makes it manifest that the entropy of a very large AdS Schwarzschild black hole must scale "holographically" with the volume of its horizon. By similar methods, one can also make a speculative proposal for the description of large $N$ gauge theories in four dimensions without [[0359 Supersymmetry|supersymmetry]].\]