# Amsel, Marolf ## Supersymmetric Multi-trace Boundary Conditions in AdS \[Links: [arXiv](https://arxiv.org/abs/0808.2184), [PDF](https://arxiv.org/pdf/0808.2184.pdf)\] \[Abstract: Boundary conditions for massive fermions are investigated in AdS$_d$ for $d \ge 2$. For fermion masses in the range $0 \le |m| < 1/2\ell$ with $\ell$ the AdS length, the standard notion of normalizeability allows a choice of boundary conditions. As in the case of scalars at or slightly above the [[0528 Breitenlohner-Freedman bound|Breitenlohner-Freedman (BF) bound]], such boundary conditions correspond to multi-trace deformations of any CFT dual. By constructing appropriate boundary superfields, for $d=3,4,5$ we identify joint scalar/fermion boundary conditions which preserve either ${\cal N}=1$ supersymmetry or ${\cal N}=1$ superconformal symmetry on the boundary. In particular, we identify boundary conditions corresponding via AdS/CFT (at large $N$) to a 595-parameter family of double-trace marginal deformations of the low-energy theory of $N$ M2-branes which preserve ${\cal N} =1$ superconformal symmetry. We also establish that (at large $N$ and large 't Hooft coupling \lambda) there are no marginal or relevant multi-trace deformations of 3+1 ${\cal N} =4$ super Yang-Mills which preserve even ${\cal N}=1$ supersymmetry.\] # Arkani-Hamed, Kaplan ## On tree amplitudes in gauge theory and gravity \[Links: [arXiv](https://arxiv.org/abs/0801.2385), [PDF](https://arxiv.org/pdf/0801.2385.pdf)\] \[Abstract: The [[0058 BCFW|BCFW recursion]] relations provide a powerful way to compute tree amplitudes in gauge theories and gravity, but only hold if some amplitudes vanish when two of the momenta are taken to infinity in a particular complex direction. This is a very surprising property, since individual Feynman diagrams all diverge at infinite momentum. In this paper we give a simple physical understanding of amplitudes in this limit, which corresponds to a hard particle with (complex) light-like momentum moving in a soft background, and can be conveniently studied using the background field method exploiting background light-cone gauge. An important role is played by enhanced spin symmetries at infinite momentum--a single copy of a "Lorentz" group for gauge theory and two copies for gravity--which together with Ward identities give a systematic expansion for amplitudes at large momentum. We use this to study tree amplitudes in a wide variety of theories, and in particular demonstrate that certain pure gauge and gravity amplitudes do vanish at infinity. Thus the BCFW recursion relations can be used to compute completely general gluon and graviton tree amplitudes in any number of dimensions. We briefly comment on the implications of these results for computing massive 4D amplitudes by KK reduction, as well understanding the unexpected cancelations that have recently been found in loop-level gravity amplitudes.\] ## Summary - *proves* [[0058 BCFW|BCFW]] using background field method # Brigante, Liu, Myers, Shenker, Yaida ## Viscosity Bound and Causality Violation \[Links: [arXiv](https://arxiv.org/abs/0802.3318), [PDF](https://arxiv.org/pdf/0802.3318.pdf)\] \[Abstract: In recent work we showed that, for a class of conformal field theories (CFT) with [[0425 Gauss-Bonnet gravity|Gauss-Bonnet gravity]] dual, the shear viscosity to entropy density ratio, $\eta/s$, could violate the conjectured Kovtun-Starinets-Son viscosity bound, $\eta/s\geq1/4\pi$. In this paper we argue, in the context of the same model, that tuning $\eta/s$ below $(16/25)(1/4\pi)$ induces microcausality violation in the CFT, rendering the theory inconsistent. This is a concrete example in which inconsistency of a theory and a lower bound on viscosity are correlated, supporting the idea of a possible universal lower bound on $\eta/s$ for all consistent theories.\] # Casini ## Relative entropy and the Bekenstein bound \[Links: [arXiv](https://arxiv.org/abs/0804.2182), [PDF](https://arxiv.org/pdf/0804.2182.pdf)\] \[Abstract: \] ## Proving Bekenstein bound - reformulates [[0418 Bekenstein bound]] as $\Delta S \leq \Delta K$ - but since $S_{\mathrm{rel}}(\rho \mid \sigma)=\Delta K-\Delta S$ and relative entropy cannot be negative, the Bekenstein bound is immediately derived # Compere, Marolf ## Setting the boundary free in AdS/CFT \[Links: [arXiv](https://arxiv.org/abs/0805.1902), [PDF](https://arxiv.org/pdf/0805.1902.pdf)\] \[Abstract: \] ## Comments from Don > The main difference between what Compère and I did and the Neumann end-of-the-world brane is that the former is at an asymptotic boundary while the latter lies at finite distance.  As a result, the two somewhat different notions of “Neumann BCs.”  In the latter case we have K_ij =0, while in the former it is just a subleading component that vanishes. > Still, one might ask of the end of the world brane is something like a TT-bar deformation of a CFT coupled to gravity.  It’s not clear to me that this works, and I am not sure what it would mean, but it is an interesting question. > In fact, you might do better to consider and e d of the world brane with finite tension.  This is essentially a Randall-Sundrum brane, which is generally thought of as being dual to a CFT+gravity with a cut off.  The one issue here is that I think that story applies to RS branes with negative tension, while one generally takes the EOW tension to be positive.  But perhaps the other sign works as well. > So, yes, there is probably some further story here that could be better understood. ## Summary - choose Neumann or mixed BC rather than Dirichlet -> gives dynamics to the boundary metric -> the CFT couples to gravity - main task: show that fluctuations of the boundary metric are ==normalisable== - key point: CT also contributes to the norm ## Refs - related to [[0181 AdS-BCFT]] (my thought) ## Logic - promoting boundary metric to dynamic field: $Z_{\text {induced }}:=\int \mathcal{D} g_{0} Z_{C F T}\left[g_{(0)}\right]$ - doing this in the bulk: (eq.1.5) $Z_{\text {induced }}=\int \mathcal{D} g_{0} Z_{\text {Dir }}\left[g_{(0)}\right]=\int(\mathcal{D} g) e^{-S_{D i r}}=: Z_{N e u}$ (typo in v2) - where $\mathcal{D} g=\mathcal{D} g_{(0)}(\mathcal{D} g)_{g_{(0)}}$ - stationarity of the $S_{Dir}$ requires $T^{ij}$ because $\delta S_{D i r}=E O M+\frac{1}{2} \int_{\partial M} d^{d} x \sqrt{-g_{(0)}} T^{i j} \delta g_{(0) i j}$ # Dixon, Magnea, Sterman ## Universal structure of subleading infrared poles in gauge theory amplitudes \[Links: [arXiv](https://arxiv.org/abs/0805.3515), [PDF](https://arxiv.org/pdf/0805.3515.pdf)\] \[Abstract: \] ## Refs - later by author [[2021#Magnea]] - root [[0010 Celestial holography]] - [[0009 Soft theorems]] - [[0078 Collinear limit]] ## Summary - *uses* ==eikonal approximation== - *obtains* loop corrections to soft factorisation # Dyer, Hinterbichler ## Boundary terms, variational principles and higher derivative modified gravity \[Links: [arXiv](https://arxiv.org/abs/0809.4033), [PDF](https://arxiv.org/pdf/0809.4033.pdf)\] \[Abstract: We discuss the criteria that must be satisfied by a well-posed variational principle. We clarify the role of [[0138 Variational principle|Gibbons-Hawking-York type boundary terms]] in the actions of [[0006 Higher-derivative gravity|higher derivative models of gravity]], such as $F(R)$ gravity, and argue that the correct boundary terms are the naive ones obtained though the correspondence with [[0140 Scalar-tensor theory|scalar-tensor theory]], despite the fact that variations of normal derivatives of the metric must be fixed on the boundary. We show in the case of $F(R)$ gravity that these boundary terms reproduce the correct [[0487 ADM mass|ADM energy]] in the hamiltonian formalism, and the correct [[0004 Black hole entropy|entropy for black holes]] in the semi-classical approximation.\] ## Summary - discuss **criteria** for [[0138 Variational principle|variational principle in higher derivative gravity]] - correct boundary terms are naive ones obtained through [[0140 Scalar-tensor theory|scalar-tensor theory]] - for ==f(R)==, the boundary terms reproduce: - correct ADM in Hamiltonian method - correct BH [[0004 Black hole entropy|entropy]] semi-classically ## The criteria - the *unique* solutions to EL equations given boundary and initial data extremises the action - the action naturally wants *end point data*, but usually these are just as good as initial data (both the field and its first time derivative) except for unfortunate choices (footnote 2) ## Unfortunate choices - e.g. see section III.A - the equations of motion want initial data, but action principle wants end point data - **a set of measure zero**: finding degeneracies in the initial value problem is equal to finding all eigenvalues -> they are a discrete set ## Constraints - when $\text{det}\frac{\partial^2L}{\partial \dot{q}^{i}\partial\dot{q}^{j}}\ne0$ - almost always there in high energy physics, except scalar fields on fixed backgrounds - d.o.f. - separate variables into **constrained** and **unconstrained** ones - each choice of end point data for unconstrained variables determine the data for the constrained variables - the action (after looking at its variation) must be such that only the unconstrained variables need to be fixed at the boundary, even thought all variables can appear in the boundary term ## Gauge - a sign: when there is arbitrary function in the solution -> gauge freedom - in the bulk: need to identify equivalent classes (gauge fixing) - **if a gauge transformation does not vanish at boundaries** - need to identify equivalent classes at the boundaries too - two types: fix end-point data, or fix something else - e.g. in GR: - time diffeomorphism (generated by Hamiltonian constraint) => equivalent classes of boundary data - spatial diffeomorphism (generated by momentum constraint) => identify end point data (past and future) # Giombi, Maloney, Yin ## One-loop Partition Functions of 3D Gravity \[Links: [arXiv](https://arxiv.org/abs/0804.1773), [PDF](https://arxiv.org/pdf/0804.1773.pdf)\] \[Abstract: We consider the one-loop partition function of free quantum field theory in locally Anti-de Sitter space-times. In three dimensions, the one loop determinants for scalar, gauge and graviton excitations are computed explicitly using heat kernel techniques. We obtain precisely the result anticipated by Brown and Henneaux: the partition function includes a sum over "boundary excitations" of [[0002 3D gravity|AdS3]], which are the Virasoro descendants of empty Anti-de Sitter space. This result also allows us to compute the one-loop corrections to the Euclidean action of the [[0086 Banados-Teitelboim-Zanelli black hole|BTZ black hole]] as well its higher genus generalizations.\] # Giombi, Yin ## ZZ boundary states and fragmented AdS2 spaces \[Links: [arXiv](https://arxiv.org/abs/0808.0923), [PDF](https://arxiv.org/pdf/0808.0923)\] \[Abstract: In this paper we show that [[0642 Boundary Liouville CFT|Liouville gravity on the strip]] with Zamolodchikov-Zamolodchikov (ZZ) boundary conditions has a semi-classical interpretation in terms of fragmented AdS2 spacetime geometries. Further, we study the three-point functions of the ZZ boundary states, and show that they are dominated by multi-AdS2 instantons in the classical limit.\] # Guica, Hartman, Song, Strominger ## The Kerr/CFT Correspondence \[Links: [arXiv](https://arxiv.org/abs/0809.4266), [PDF](https://arxiv.org/pdf/0809.4266.pdf)\] \[Abstract: Quantum gravity in the region very near the horizon of an extreme Kerr black hole (whose angular momentum and mass are related by $J=GM^2$) is considered. It is shown that consistent boundary conditions exist, for which the [[0060 Asymptotic symmetry|asymptotic symmetry]] generators form one copy of the [[0032 Virasoro algebra|Virasoro algebra]] with [[0033 Central charge|central charge]] $c_L=12J / \hbar$. This implies that the near-horizon quantum states can be identified with those of (a chiral half of) a two-dimensional conformal field theory (CFT). Moreover, in the extreme limit, the Frolov-Thorne vacuum state reduces to a thermal density matrix with dimensionless temperature $T_L=1/2\pi$ and conjugate energy given by the zero mode generator, $L_0$, of the Virasoro algebra. Assuming unitarity, the [[0406 Cardy formula|Cardy formula]] then gives a microscopic entropy $S_{micro}=2\pi J / \hbar$ for the CFT, which reproduces the macroscopic Bekenstein-Hawking entropy $S_{macro}=\text{Area} / 4\hbar G$. The results apply to any consistent unitary quantum theory of gravity with a Kerr solution. We accordingly conjecture that extreme Kerr black holes are holographically dual to a chiral two-dimensional conformal field theory with central charge $c_L=12J / \hbar$, and in particular that the near-extreme black hole GRS 1915+105 is approximately dual to a CFT with $c_L \sim 2 \times 10^{79}$.\] ## Refs - OG for the extremal [[0520 Kerr-CFT correspondence|Kerr/CFT correspondence]] # Gukov, Witten ## Branes and Quantization \[Links: [arXiv](https://arxiv.org/abs/0809.0305), [PDF](https://arxiv.org/pdf/0809.0305)\] \[Abstract: The problem of quantizing a symplectic manifold $(M,\omega)$ can be formulated in terms of the $A$-model of a complexification of $M$. This leads to an interesting new perspective on quantization. From this point of view, the Hilbert space obtained by quantization of $(M,\omega)$ is the space of $(\mathcal{B}_{cc},\mathcal{B}')$ strings, where $\mathcal{B}_{cc}$ and $\mathcal{B}'$ are two $A$-branes; $\mathcal{B}'$ is an ordinary Lagrangian $A$-brane, and $\mathcal{B}_{cc}$ is a space-filling coisotropic $A$-brane. $\mathcal{B}'$ is supported on $M$, and the choice of $\omega$ is encoded in the choice of $\mathcal{B}_{cc}$. As an example, we describe from this point of view the representations of the group $SL(2,\mathbb{R})$. Another application is to [[0089 Chern-Simons theory|Chern-Simons gauge theory]].\] ## Quantisation They provide a new approach to quantisation using 2d sigma-models. The goal is to get closer to a systematic theory of quantisation. ## $A$-model They need a good $A$-model, which has complex-valued correlation functions etc rather than just functions of a formal deformation parameter. For example, the most familiar $A$-model observables are obtained from sums over worldsheet instantons of different degrees. Having a good $A$-model means that such sums are not just formal power series but converge to complex-valued functions. # Hartnoll, Herzog, Horowitz (Oct) ## Holographic Superconductors \[Links: [arXiv](https://arxiv.org/abs/0810.1563), [PDF](https://arxiv.org/pdf/0810.1563.pdf)\] \[Abstract: It has been shown that a [[0431 Holographic superconductor|gravitational dual to a superconductor]] can be obtained by coupling anti-de Sitter gravity to a Maxwell field and charged scalar. We review our earlier analysis of this theory and extend it in two directions. First, we consider all values for the charge of the scalar field. Away from the large charge limit, backreaction on the spacetime metric is important. While the qualitative behaviour of the dual superconductor is found to be similar for all charges, in the limit of arbitrarily small charge a new type of black hole instability is found. We go on to add a perpendicular magnetic field $B$ and obtain the London equation and magnetic penetration depth. We show that these holographic superconductors are Type II, i.e., starting in a normal phase at large $B$ and low temperatures, they develop superconducting droplets as $B$ is reduced.\] ## Related topics - [[0495 Hairy black holes]] - [[0455 Black hole uniqueness theorems]] - [[0431 Holographic superconductor]] ## Summary - relaxes an earlier limitation where the charge of the operator was large and the backreaction of the metric was therefore legally neglected - adds magnetic field ## Setup - Lagrangian: $\mathcal{L}=R+\frac{6}{L^2}-\frac{1}{4} F^{a b} F_{a b}-V(|\psi|)-|\nabla \psi-i q A \psi|^2$ - choice of potential: $V(\psi)=-\frac{2}{L^2}\psi^2$ - ansatz: - $d s^2=-g(r) e^{-\chi(r)} d t^2+\frac{d r^2}{g(r)}+r^2\left(d x^2+d y^2\right)$ - $A=\phi(r) d t, \quad \psi=\psi(r)$ - no impurities => translation invariance - at the horizon: - four independent parameters: - $r_+$ - $\psi_+=\psi(r_+)$ - $E_+=\phi'(r_+)$ - $\chi_+=\chi(r_+)$ - BC: $\phi(r_+)=0$ for the gauge connection to be regular - at infinity: - $\chi=0$, using scaling of boundary time - $\phi=\mu-\frac{\rho}{r}+\cdots$ - $\psi=\frac{\psi^{(1)}}{r}+ \frac{\psi^{(2)}}{r^2}+\cdots$ - extra scaling symmetries set - $L=1$: the usual rescaling of AdS scale - $r_+=1$: involves scaling of the planar directions - horizon to infinity map (of physical parameters): - $\left(\psi_{+}, E_{+}\right) \mapsto\left(\mu, \rho, \psi^{(1)}, \psi^{(2)}, \epsilon\right)$ ## Order parameter near critical temperature - $\left\langle\mathcal{O}_1\right\rangle \sim\left\langle\mathcal{O}_2\right\rangle \sim\left(T_c-T\right)^{1 / 2}$ - which is the classic 1/2 mean field exponent of Landau-Ginzburg theory # He, Zhang ## Consistency Conditions on S-Matrix of Spin 1 Massless Particles \[Links: [arXiv](https://arxiv.org/abs/0811.3210), [PDF](https://arxiv.org/pdf/0811.3210.pdf)\] \[Abstract: Motivated by new techniques in the computation of scattering amplitudes of massless particles in four dimensions, like [[0058 BCFW|BCFW recursion relations]], the question of how much structure of the S-matrix can be determined from purely S-matrix arguments has received new attention. The BCFW recursion relations for massless particles of spin 1 and 2 imply that the whole tree-level S-matrix can be determined in terms of three-particle amplitudes (evaluated at complex momenta). However, the known proofs of the validity of the relations rely on the Lagrangian of the theory, either by using Feynman diagrams explicitly or by studying the effective theory at large complex momenta. This means that a purely S-matrix theoretic proof of the relations is still missing. The aim of this paper is to provide such a proof for spin 1 particles by extending the four-particle test introduced by [[2007#Benincasa, Cachazo|P. Benincasa and F. Cachazo]] in [arXiv:0705.4305](https://arxiv.org/abs/0705.4305) to all particles. We show how $n$-particle tests imply that the rational function built from the BCFW recursion relations possesses all the correct factorization channels including holomorphic and anti-holomorphic [[0078 Collinear limit|collinear limits]]. This in turn implies that they give the correct S-matrix of the theory.\] # Hofman, Maldacena ## Conformal collider physics: Energy and charge correlations \[Links: [arXiv](https://arxiv.org/abs/0803.1467), [PDF](https://arxiv.org/pdf/0803.1467.pdf)\] \[Abstract: We study observables in a conformal field theory which are very closely related to the ones used to describe hadronic events at colliders. We focus on the correlation functions of the energies deposited on calorimeters placed at a large distance from the collision. We consider initial states produced by an operator insertion and we study some general properties of the energy correlation functions for conformal field theories. We argue that the small angle singularities of energy correlation functions are controlled by the twist of non-local [[0450 Light-ray operators|light-ray operators]] with a definite spin. We relate the charge two point function to a particular moment of the parton distribution functions appearing in deep inelastic scattering. The one point energy correlation functions are characterized by a few numbers. For ${\cal N}=1$ superconformal theories the one point function for states created by the R-current or the stress tensor are determined by the two parameters $a$ and $c$ characterizing the [[0306 Weyl anomaly|conformal anomaly]]. Demanding that the measured energies are positive we get bounds on $a/c$. We also give a prescription for computing the energy and charge correlation functions in theories that have a gravity dual. The prescription amounts to probing the falling string state as it crosses the AdS horizon with gravitational [[0117 Shockwave|shock waves]]. In the leading, two derivative, gravity approximation the energy is uniformly distributed on the sphere at infinity, with no fluctuations. We compute the stringy corrections and we show that they lead to small, non-gaussian, fluctuations in the energy distribution. Corrections to the one point functions or antenna patterns are related to [[0006 Higher-derivative gravity|higher derivative corrections]] in the bulk.\] ## Related - [[0493 Conformal collider bounds]] # Iqbal, Liu ## Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm \[Links: [arXiv](https://arxiv.org/abs/0809.3808), [PDF](https://arxiv.org/pdf/0809.3808.pdf)\] \[Abstract: We show that at the level of linear response the low frequency limit of a strongly coupled field theory at finite temperature is determined by the horizon geometry of its gravity dual, i.e. by the "[[0229 Membrane paradigm|membrane paradigm]]" fluid of classical black hole mechanics. Thus generic boundary theory transport coefficients can be expressed in terms of geometric quantities evaluated at the horizon. When applied to the stress tensor this gives a simple, general proof of the universality of the [[0430 Holographic shear viscosity|shear viscosity]] in terms of the universality of gravitational couplings, and when applied to a conserved current it gives a new general formula for the conductivity. Away from the low frequency limit the behavior of the boundary theory fluid is no longer fully captured by the horizon fluid even within the derivative expansion; instead we find a nontrivial evolution from the horizon to the boundary. We derive flow equations governing this evolution and apply them to the simple examples of charge and momentum diffusion.\] ## Summary - OG for understanding [[0430 Holographic shear viscosity|holographic shear viscosity]] as universal behaviour of the Green's function near the horizon ## Green function - the one-point function due to a source $\phi_0$ is given by the thermal retarded correlator of $\mathcal{O}$: - $\langle\mathcal{O}(\omega, \vec{k})\rangle_{\mathrm{QFT}}=-G^{R}(\omega, \vec{k}) \phi_{0}(\omega, \vec{k})$ - the low-frequency limit of this defines a transport coefficient: - $\chi=-\lim _{\omega \rightarrow 0} \lim _{\vec{k} \rightarrow 0} \frac{1}{\omega} \operatorname{Im} G^{R}(\omega, \vec{k}) .$ - this means the low frequency limit response is just: - $\langle\mathcal{O}\rangle_{\mathrm{QFT}}=-\chi \partial_{t} \phi_{0}(t)$ - examples - [[0430 Holographic shear viscosity|shear viscosity]], $\eta$: $\mathcal{O}=T_{xy}$, off-diagonal elements of the stress tensor - DC conductivity, $\sigma$: take $\mathcal{O}=J_z$, a component of the electric current ## Away from horizon - idea: take the [[0229 Membrane paradigm|membrane]] to the boundary - consider a fictitious membrane at *each* constant-radius hypersurface and introduce a linear response function for each of them - one can then derive a flow equation for the radius-dependent response function; at generic momenta this evolves nontrivially from the horizon to the boundary, where it determines the response of the dual field theory # Lee ## A Non-Fermi Liquid from a Charged Black Hole; A Critical Fermi Ball \[Links: [arXiv](https://arxiv.org/abs/0809.3402), [PDF](https://arxiv.org/pdf/0809.3402.pdf)\] \[Abstract: Using the [[0001 AdS-CFT|AdS/CFT]] correspondence, we calculate a fermionic spectral function in a 2+1 dimensional non-relativistic quantum field theory which is dual to a gravitational theory in the AdS$_4$ background with a charged black hole. The spectral function shows no quasiparticle peak but the [[0535 Holographic Fermi surface|Fermi surface]] is still well defined. Interestingly, all momentum points inside the Fermi surface are critical and the gapless modes are defined in a *critical Fermi ball* in the momentum space.\] ## Refs - [[0535 Holographic Fermi surface]] # Schuster, Toro ## Constructing the Tree-Level Yang-Mills S-Matrix Using Complex Factorization \[Links: [arXiv](https://arxiv.org/abs/0811.3207), [PDF](https://arxiv.org/pdf/0811.3207.pdf)\] \[Abstract: A remarkable connection between [[0058 BCFW|BCFW recursion relations]] and constraints on the S-matrix was made by [[2007#Benincasa, Cachazo|Benincasa and Cachazo]] in [0705.4305](https://arxiv.org/abs/0705.4305), who noted that mutual consistency of different BCFW constructions of four-particle amplitudes generates non-trivial (but familiar) constraints on three-particle coupling constants --- these include gauge invariance, the equivalence principle, and the lack of non-trivial couplings for spins gt;2$. These constraints can also be derived with weaker assumptions, by demanding the existence of four-point amplitudes that factorize properly in all unitarity limits with complex momenta. From this starting point, we show that the BCFW prescription can be interpreted as an algorithm for fully constructing a tree-level S-matrix, and that complex factorization of general BCFW amplitudes follows from the factorization of four-particle amplitudes. The allowed set of BCFW deformations is identified, formulated entirely as a statement on the three-particle sector, and using only complex factorization as a guide. Consequently, our analysis based on the physical consistency of the S-matrix is entirely independent of field theory. We analyze the case of pure [[0071 Yang-Mills|Yang-Mills]], and outline a proof for gravity. For Yang-Mills, we also show that the well-known scaling behavior of BCFW-deformed amplitudes at large $z$ is a simple consequence of factorization. For gravity, factorization in certain channels requires asymptotic behavior $~1/z^2$.\] # Sekino, Susskind ## Fast scramblers \[Links: [arXiv](https://arxiv.org/abs/0808.2096), [PDF](https://arxiv.org/pdf/0808.2096.pdf)\] \[Abstract: \] ## Conjecture: 1) The most rapid scramblers take a time logarithmic in the number of degrees of freedom. 2) Matrix quantum mechanics (systems whose degrees of freedom are n by n matrices) saturate the bound. 3) Black holes are the fastest scramblers in nature. # Sharipov (Notes) ## A note on Kosmann-Lie derivatives of Weyl spinors \[Links: [arXiv](https://arxiv.org/abs/0801.0622), [PDF](https://arxiv.org/pdf/0801.0622.pdf)\] \[Abstract: Kosmann-Lie derivatives in the bundle of Weyl spinors are considered. It is shown that the basic spin-tensorial fields of this bundle are constants with respect to these derivatives.\] # Skenderis, van Rees (May) ## Real-time gauge/gravity duality \[Links: [arXiv](https://arxiv.org/abs/0805.0150), [PDF](https://arxiv.org/pdf/0805.0150.pdf)\] \[Abstract: We present a general prescription for the holographic computation of real-time $n$-point functions in non-trivial states. In QFT such real-time computations involve a choice of a time contour in the complex time plane. The holographic prescription amounts to ''filling in'' this contour with bulk solutions: real segments of the contour are filled in with Lorentzian solutions while imaginary segments are filled in with Riemannian solutions and appropriate matching conditions are imposed at the corners of the contour. We illustrate the general discussion by computing the 2-point function of a scalar operator using this prescription and by showing that this leads to an unambiguous answer with the correct $i\epsilon$ insertions.\] ## Refs - closely related paper [[2008#Skenderis, van Rees (Dec)]] - develops a real-time [[0001 AdS-CFT|dictionary]] - bulk dual of [[0042 Schwinger-Keldysh techniques|Schwinger-Keldysh]] - used in [[2016#Dong, Lewkowycz, Rangamani]] ## Summary - prescription for relating in and out states in QFT to initial & final data for bulk fields - examples: ==free scalar fields in pure AdS==; but works for other fields (including metric) ## In and out states - initial and/or final boundary needed for bulk fields. they should correspond to in and out states for the field theory - but this had not been worked out previously ## Strategy - first specify contour in field theory, depending on what we want to compute - then fill in the bulk piece-wise - at the junctions, appropriate **matching conditions** are used to join them - matching condition will be described in [[2008#Skenderis, van Rees (Dec)]] - require *stationarity* of combined on-shell action w.r.t. variation of $\phi_\pm$ ## Preparing bulk state - Euclidean PI in the bulk is just like the HH state - see also [[2004#Marolf]] and [[2001#Maldacena]] ## Key result: vacuum to vacuum correlator - $\left\langle 0\left|T \exp \left(-i \int_{\delta M_{L}} d^{d} x \sqrt{-g} \phi_{(0)} \mathcal{O}\right)\right| 0\right\rangle=\exp \left(i I_{L}\left[\phi_{(0)}, \phi_{-}, \phi_{+}\right]-I_{E}\left[0, \phi_{-}\right]-I_{E}\left[0, \phi_{+}\right]\right)$ - we have set sources to zero because we are looking at vac-to-vac correlator ## Example: free scalar field in pure AdS - first find Lorentzian solution with spatial boundary condition (in order to obtain $n$-pt function) - the coefficients of the normalisable modes represent the state, which is unfixed a priori - then one finds Euclidean solution on the half-ball - on the whole ball, boundary condition completely fix the bulk solution, but on the half ball, boundary condition on the hemisphere does not specify the solution - requiring continuity and stationarity at the junctions gives unique solution - this fixes the Lorentzian coefficients uniquely (also the Euclidean but that is not needed) - -> obtain $n$-point function using the Lorentzian solution - in summary, contour is completely fixed by the matching to the Euclidean caps, and this contour reproduces the $i\epsilon$ prescription on the boundary # Skenderis, van Rees (Dec) ## Real-time gauge/gravity duality: prescription, renormalization and examples \[Links: [arXiv](https://arxiv.org/abs/0812.2909), [PDF](https://arxiv.org/pdf/0812.2909.pdf)\] \[Abstract: We present a comprehensive analysis of the prescription we recently put forward for the computation of real-time correlation functions using gauge/gravity duality. The prescription is valid for any holographic supergravity background and it naturally maps initial and final data in the bulk to initial and final states or density matrices in the field theory. We show in detail how the technique of holographic renormalization can be applied in this setting and we provide numerous illustrative examples, including the computation of time-ordered, Wightman and retarded 2-point functions in Poincare and global coordinates, thermal correlators and higher-point functions.\] ## Refs - [[0042 Schwinger-Keldysh techniques]] - earlier work [[2008#Skenderis, van Rees (May)]] ## Summary - shows how to apply holographic renormalisation in the Lorentzian setting - shows continuity of stress tensor (across corners of time contour: Euclidean to Lorentzian, or Lorentzian timefold) for bulk gravity (harder than for scalar field) - computation of ==Wightman, time-ordered, retarded 2-pt. functions== ## Gravity - variational principle - need boundary counterterms and corner terms - finding matching condition requires varying the interface data with respect to the action, which works only if there is good variational principle for each segment - labelling the interface - using [[0011 Fefferman-Graham expansion|FG expansion]] makes it no longer possible to have the interface at $t=0$, but now it is given by a function ==$t=f_\pm(r,x^a)$== - turns out that this does not enter the expression for the stress tensor, which is good, because boundary CFT does not know about $f$ - this is needed in [[2016#Dong, Lewkowycz, Rangamani]] - imposing the matching conditions - matching conditions: continuation and stationarity - if $N^2_{(0)}$ is continuous (only when two segments have same signature) - either $f$ has to be continuous up to and including $e^{-dr}$ (no corner at all) - or opposite (again up to and including $e^{-dr}$) (timefold) - if discontinuous: all terms up to and including $e^{-dr}$ in $f$ vanish - one-point function of **stress tensor** is continuous across the corner (intersection of infinity and the connecting boundary between two pieces of the metric) - need to show that variation does not give a localised term at the corner - continuity of stress tensor follows from continuity of all locally determined terms in FG expansion *and* $g_{(d)ij}$ - continuity of $g_{(d)ij}$ can also be checked using double ADM expansion # Wang, Yau ## Isometric embeddings into the Minkowski space and new quasi-local mass \[Links: [arXiv](https://arxiv.org/abs/0805.1370), [PDF](https://arxiv.org/pdf/0805.1370.pdf)\] \[Abstract: The definition of [[0595 Quasi-local energy|quasi-local mass]] for a bounded space-like region in space-time is essential in several major unsettled problems in general relativity. The quasi-local mass is expected to be a type of flux integral on the boundary two-surface and should be independent of whichever space-like region it bounds. An important idea which is related to the Hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi-local mass from the difference of the extrinsic geometries. This approach has been taken by Brown-York and Liu-Yau (see also related works) to define such notions using the isometric embedding theorem into the Euclidean three-space. However, there exist surfaces in the Minkowski space whose quasilocal mass is strictly positive. It appears that the momentum information needs to be accounted for to reconcile the difference. In order to fully capture this information, we use isometric embeddings into the Minkowski space as references. In this article, we first prove an existence and uniqueness theorem for such isometric embeddings. We then solve the boundary value problem for Jang's equation as a procedure to recognize such a surface in the Minkowski space. In doing so, we discover new expression of quasi-local mass. The new mass is positive when the ambient space-time satisfies the dominant energy condition and vanishes on surfaces in the Minkowski space. It also has the nice asymptotic behavior at spatial and null infinity. Some of these results were announced in [29].\] ## Summary - origin of [[0595 Quasi-local energy#Wang-Yau energy|Wang-Yau energy]]