We derive a simple relation between the Mellin amplitude for AdS/CFT correlation functions and the bulk S-Matrix in the flat spacetime limit, proving a conjecture of Penedones. As a consequence of the Operator Product Expansion, the Mellin amplitude for any unitary CFT must be a meromorphic function with simple poles on the real axis. This provides a powerful and suggestive handle on the locality vis-a-vis analyticity properties of the S-Matrix. We begin to explore analyticity by showing how the familiar poles and branch cuts of scattering amplitudes arise from the holographic description. For this purpose we compute examples of Mellin amplitudes corresponding to 1-loop and 2-loop Witten diagrams in AdS. We also examine the flat spacetime limit of conformal blocks, implicitly relating the S-Matrix program to the Bootstrap program for CFTs. We use this connection to show how the existence of small black holes in AdS leads to a universal prediction for the conformal block decomposition of the dual CFT.
We explore the momentum analyticity of the static transverse polarization tensor of a 2+1 dimensional holographic superconductor in its normal phase with a nonzero chemical potential, aiming at finding the holographic counterpart of the singularities underlying the Friedel-like oscillations of an ordinary field theory. We prove that the polarization tensor is a meromorphic function with an infinite number of poles located on the complex momentum plane off real axis. With the aid of the WKB approximation these poles are found to lies asymptotically along two straight lines parallel to the imagi…
The static electric polarization of a holographic field theory dual to the Einstein-Maxwell theory in the background of $AdS_4$ with a Reissner-Nordstöm (AdS-RN) black hole is investigated. We prove that the holographic polarization is a meromorphic functions in complex momentum plane and locate analytically the asymptotic distribution of the poles along two straight lines parallel to the imaginary axis for a large momentum magnitude. The results are compared with the numerical result on Friedel-like poles of the same holographic model reported in the literature and with the momentum singulari…
We compute holographic Rényi entropies for spherical entangling surfaces on the boundary while considering third order Lovelock gravity with negative cosmological constant in the bulk. Our study shows that third order Lovelock black holes with hyperbolic event horizon are unstable, and at low temperatures those with smaller mass are favoured, giving rise to first order phase transitions in the bulk. We determine regions in the Lovelock parameter space in arbitrary dimensions, where bulk phase transitions happen and where boundary causality constraints are met. We show that each of these points…
The region near a critical point is studied using holographic models of second-order phase transitions. In a previous paper, we argued that the quantum circuit complexity of the vacuum ($C_0$) is the largest at the critical point. When deforming away from the critical point by a term $\int d^d x \, τ\, O_Δ$ the complexity $C(τ)$ has a piece non-analytic in $τ$, namely $C_0 -C(τ) \sim |τ-τ_c|^{ν(d-1)} + \mathrm{analytic} $. Here, as usual, $ν=\frac{1}{d-Δ}$ and $ξ$ is the correlation length $ξ\sim |τ-τ_c|^{-ν}$ and there are possible logarithmic corrections to this expression. That was derived …
The constraints implied by analyticity in two-dimensional factorised S-matrix theories are reviewed. Whenever the theory is not time-reversal invariant, it is argued that the familiar condition of `Real analyticity' for the S-matrix amplitudes has to be superseded by a different one known as `Hermitian analyticity'. Examples are provided of integrable quantum field theories whose (diagonal) two-particle S-matrix amplitudes are Hermitian analytic but not Real analytic. It is also shown that Hermitian analyticity is consistent with the bootstrap equations and that it ensures the equivalence betw…
In this work, we present a holographic renormalization scheme for asymptotically anti-de Sitter spacetimes in which the dual renormalization scheme of the boundary field theory is dimensional regularization. This constitutes a new level of precision in the holographic dictionary and paves the way for the exact matching of scheme dependent quantities, such as holographic beta functions, with field theory computations. Furthermore, the renormalization procedure identifies a local source field which satisfies the equations of motion along renormalization group flows, resolving a long-standing puz…
The flow of the action induced by changing $N$ is computed in large $N$ matrix models. It is shown that the change in the action is non-analytic. This non-analyticity appears at the origin of the space of matrices if the action is even.
The mapping between bulk supergravity fields in anti-de Sitter space and operators in the dual boundary conformal field theory usually relies heavily on the available global symmetries. In the present work, we study a generalization of this mapping to time dependent situations, for the simple case of collapsing shock waves in two spacetime dimensions. The construction makes use of analyticity of the conformal field theory and the properties of the asymptotic bulk geometry to reconstruct the non-analytic bulk observables. Many of the features of this construction are expected to apply to higher…
We consider weakly-coupled QFT in AdS at finite temperature. We compute the holographic thermal two-point function of scalar operators in the boundary theory. We present analytic expressions for leading corrections due to local quartic interactions in the bulk, with an arbitrary number of derivatives and for any number of spacetime dimensions. The solutions are fixed by judiciously picking an ansatz and imposing consistency conditions. The conditions include analyticity properties, consistency with the operator product expansion, and the Kubo-Martin-Schwinger condition. For the case without an…
Holographic bounds have been derived using explicitly gravitational arguments. Motivated by explicit constructions of bulk wavepackets from observables in the boundary CFT, we derive a holographic bound in the context of the gauge/gravity correspondence within the dual field theory. We verify the consistency of the bound with the program of determining the Black Hole S-Matrix from the AdS/CFT correspondence.
We show an explicit connection between the solution to the equations of motion in the Gaussian functional approximation and the minimum of the (Gaussian) effective potential/action of the linear $Σ$ model, as well as with the N/D method in dispersion theory. The resulting equations contain analytic functions with branch cuts in the complex mass squared plane. Therefore the minimum of the effective action may lie in the complex mass squared plane. Many solutions to these equations can be found on the second, third, etc. Riemann sheets of the equation, though their physical interpretation is not…
Analyticity and crossing properties of four point function are investigated in conformal field theories in the frameworks of Wightman axioms. A Hermitian scalar conformal field, satisfying the Wightman axioms, is considered. The crucial role of microcausality in deriving analyticity domains is discussed and domains of analyticity are presented. A pair of permuted Wightman functions are envisaged. The crossing property is derived by appealing to the technique of analytic completion for the pair of permuted Wightman functions. The operator product expansion of a pair of scalar fields is studied …
Imprints of flat space analyticity in de Sitter S-matrix
Kyoto University（京都大学）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；The University of Tokyo（东京大学）；Research Center for the Early Universe (RESCEU), Graduate School of Science, The University of Tokyo（东京大学理学研究科早期宇宙研究中心）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心）；National Institute of Technology (KOSEN), Gunma College（国立冈山工业高等专门学校）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Yokohama National University（横滨国立大学）；Yamaguchi University（山口大学）；Kavli Institute for the Physics and Mathematics of the Universe, Todai Institute for Advanced Study, The University of Tokyo（东京大学前沿研究院 Kavli 宇宙物理学与数学研究所）；Okayama University of Science（冈山理科大学）
The recent proposal by Polchinski and Susskind for the holographic flat space S-matrix is discussed. By using Feynman diagrams we argue that in principle all the information about the S-matrix in the interacting field theory in the bulk of the anti-de Sitter space is encoded into the data on the timelike boundary. The problem of locality of interpolating field is discussed and it is suggested that the interpolating field lives in a quantum Boltzmannian Hilbert space.
We investigate the consequence of the energy-momentum conservation law for the holographic S-matrix from AdS/CFT correspondence. It is shown that the conservation law is not a natural consequence of conformal invariance in the large N limit. We predict a new singularity for the four point correlation function of a marginal operator. Only the two point scattering amplitude is explicitly calculated, and the result agrees with what is expected.
Bros, Epstein and Glaser proved crossing symmetry of the S-matrix of a theory without massless fields by using certain analyticity properties of the off-shell momentum space Green's function in the complex momentum plane. The latter properties follow from representing the momentum space Green's function as Fourier transform of the position space Green's function, satisfying certain properties implied by the underlying local quantum field theory. We prove the same analyticity properties of the momentum space Green's functions in superstring field theory by directly working with the momentum spa…
The generating function of the cumulants in random matrix models, as well as the cumulants themselves, can be expanded as asymptotic (divergent) series indexed by maps. While at fixed genus the sums over maps converge, the sums over genera do not. In this paper we obtain alternative expansions both for the generating function and for the cumulants that cure this problem. We provide explicit and convergent expansions for the cumulants, for the remainders of their perturbative expansion (in the size of the maps) and for the remainders of their topological expansion (in the genus of the maps). We…
We consider a massive, neutral, scalar field theory of mass $m_0$ in a five dimensional flat spacetime. Subsequently, one spatial dimension is compactified on a circle, $S^1$, ofradius $R$. The resulting theory is defined in the manifold, $R^{3,1}\otimes S^1$. The mass spectrum is a state of lowest mass, $m_0$, and a tower of massive Kaluza-Klein states. The analyticity property of the elastic scattering amplitude is investigated in the Lehmann-Symanzik-Zimmermann (LSZ) formulation of this theory. In the context of nonrelativistic potential scattering, for the $R^3\otimes S^1$ spatial geometry…
The Real EFThedron: A Moment Geometry for the Nonperturbative S-Matrix
Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学与应用科学研究所有）；School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院基础物理与数学学院，UCAS）；Department of Physics, Blackett Laboratory, Imperial College London（帝国理工学院布莱克特物理系）；International Centre for Theoretical Physics Asia-Pacific（理论物理学亚太中心）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）
We consider the four-point function of the stress tensor multiplet superprimary in $\mathcal{N}=4$ super-Yang-Mills (SYM) with gauge group $SU(N)$ in the large $N$ and large 't Hooft coupling $λ\equiv g_\text{YM}^2N$ limit, which is holographically dual to the genus expansion of IIB string theory on $AdS_5\times S^5$. In \cite{Binder:2019jwn} it was shown that the integral of this correlator is related to derivatives of the mass deformed $\mathcal{N}=2^*$ sphere free energy, which was computed using supersymmetric localization to leading order in $1/N^2$ for finite $λ$. We generalize this comp…
The Gaussian Hermitian matrix model was recently proposed to have a dual string description with worldsheets mapping to a sphere target space. The correlators were written as sums over holomorphic (Belyi) maps from worldsheets to the two-dimensional sphere, branched over three points. We express the matrix model correlators by using the fuzzy sphere construction of matrix algebras, which can be interpreted as a string field theory description of the Belyi strings. This gives the correlators in terms of trivalent ribbon graphs that represent the couplings of irreducible representations of su(2)…
We investigate a deformed matrix model proposed by Kazakov et.al. in relation to Witten's two-dimensional black hole. The existing conjectures assert the equivalence of the two by mapping each to a deformed c=1 theory called the sine-Liouville theory. We point out that the matrix theory in question may be naturally interpreted as a gauged quantum mechanics deformed by insertion of an exponentiated Wilson loop operator, which gives us more direct and holographic map between the two sides. The matrix model in the usual scaling limit must correspond to the bosonic SL(2,R)/U(1) theory in genus exp…
The scaling limit used recently to derive matrix models, and a certain analyticity assumption, are invoked to argue that the agreement between some matrix model calculations and supergravity is a consequence of string world sheet duality.
Using a numerical simulation of the classical dynamics of the plane-wave and flat space matrix models of M-theory, we study the thermalization, equilibrium thermodynamics and fluctuations of these models as we vary the temperature and the size of the matrices, N. We present our numerical implementation in detail and several checks of its precision and consistency. We show evidence for thermalization by matching the time-averaged distributions of the matrix eigenvalues to the distributions of the appropriate Traceless Gaussian Unitary Ensemble of random matrices. We study the autocorrelations a…
We use the formalisms of Holographic Space-time (HST) and Matrix Theory[11] to investigate the claim of [1] that old black holes contain a firewall, i.e. an in-falling detector encounters highly excited states at a time much shorter than the light crossing time of the Schwarzschild radius. In both formalisms there is no dramatic change in particle physics inside the horizon until a time of order the Schwarzschild radius. The Matrix Theory formalism has been shown to give rise to an S-matrix, which coincides with effective supergravity for an infinite number of low energy amplitudes. We conclud…
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；Department of Physics and Astronomy, University of Waterloo（滑铁卢大学物理与天文学系）；Instituto de Física Teórica, UNESP, ICTP South American Institute for Fundamental Research（圣保罗州立大学理论物理研究所，ICTP南美基础研究所）
We review unitarity and crossing constraints on scattering amplitudes for particles with spin in four dimensional quantum field theories. As an application we study two to two scattering of neutral spin 1/2 fermions in detail. Assuming Mandelstam analyticity of its scattering amplitude, we use the numerical S-matrix bootstrap method to estimate various non-perturbative bounds on quartic and cubic (Yukawa) couplings.
We use analyticity arguments to conjecture a one-loop gravity scattering amplitude with an arbitrary number of external legs possessing the same helicity. This result also gives the complete perturbative S-matrix of self-dual gravity.
Contents: 1. Introduction. 2. The Early Years. 2.1 Living Without Field Theory. 2.2 Initial Postulates. 2.3 Further Postulates. 2.4 QCD and a Final Postulate. 2.5 Analyticity in Field Theory. 3. Axiomatic S-Matrix Theory. 3.1 Unitarity, Bubble Diagrams and Landau Diagrams. 3.2 The Landau Equations and the ``$+~ α$'' Condition. 3.3 Macrocausality and Essential Support. 3.4 The Structure Theorem. 3.5 Local Discontinuity Formulae. 3.6 Good and Bad Functions and the Steinmann Relations. 3.7 Global Discontinuity Formulae. 3.8 CPT, Hermitian Analyticity, etc. 3.9 Holonomy. 4. Asymptotic S-Matrix The…
We set up the formalism of holographic renormalization for the matter-coupled two-dimensional maximal supergravity that captures the low-lying fluctuations around the non-conformal D0-brane near-horizon geometry. As an application we compute holographically one- and two-point functions of the BFSS matrix quantum mechanics and its supersymmetric $SO(3)\times SO(6)$ deformation.
We study the two point correlation function of a local operator on an $n$-sheeted replica manifold corresponding to the half-space in the vacuum state of a conformal field theory. In analogy with the inverse Laplace transform, we define the Renyi transform of this correlation function, which is a function of one complex variable $w$, dual to the Renyi parameter $n$. Inspired by the inversion formula of Caron-Huot, we argue that if the Renyi transform $f(w)$ has bounded behavior at infinity in the complex $w$ plane, the discontinuity of the Renyi transform disc $f(w)$ provides the unique analyt…
We show that the Aharonov-Bohm effect in the nuclear matrix model derives the statistical nature of nucleons in holographic QCD. For N_c= odd (even), the nucleon is shown to be a fermion (boson).
We revisit the perturbative S-matrix of c=1 string theory from the worldsheet perspective. We clarify the origin of the leg pole factors, the non-analyticity of the string amplitudes, and the validity as well as limitations of earlier computations based on resonance momenta. We compute the tree level 4-point amplitude and the genus one 2-point reflection amplitude by numerically integrating Virasoro conformal blocks with DOZZ structure constants on the sphere and on the torus, with sufficiently generic complex Liouville momenta, and find agreement with known answers from the c=1 matrix model.
Matrix theory and the AdS/CFT correspondence provide nonperturbative holographic formulations of string theory. In both cases the finite N theories can be thought of as infrared regulated versions of flat space string theory in which removing the cutoff is equivalent to letting N go to infinity. In this paper we consider the nature of this limit. In both cases the holographic mapping becomes completely nonlocal. In matrix theory this corresponds to the growth of D0-brane bound states with N. For the AdS/CFT correspondence there is a similar delocalization of the holographic image of a system a…
Using Chebyshev polynomials combined with some mild combinatorics, we provide a new formula for the analytical planar limit of a random matrix model with a one-cut potential $V$. For potentials $V(x)=x^{2}/2-\sum_{n\ge1}a_{n}x^{n}/n$, as a power series in all $a_{n}$, the formal Taylor expansion of the analytic planar limit is exactly the formal planar limit. In the case $V$ is analytic in infinitely many variables $\{a_{n}\}_{n\ge1}$ (on the appropriate spaces), the planar limit is also an analytic function in infinitely many variables and we give quantitative versions of where this is define…
We provide, for non-experts, a brief overview of holographic QCD and a review of a recent proposal of matrix-description of multi-baryon systems in holographic QCD. Based on the matrix model, we derive the baryon interaction at short distances in multi-flavor holographic QCD. We show that there is a very universal repulsive core of inter-baryon forces for generic number of flavors. This is consistent with a recent lattice QCD analysis for N_f = 2, 3 where repulsive core looks universal. We also provide a comparison of our results with the lattice QCD and the operator product expansion (OPE) an…
Using the AdS/CFT correspondence, we probe the scale-dependence of thermalization in strongly coupled field theories following a quench, via calculations of two-point functions, Wilson loops and entanglement entropy in d=2,3,4. In the saddlepoint approximation these probes are computed in AdS space in terms of invariant geometric objects - geodesics, minimal surfaces and minimal volumes. Our calculations for two-dimensional field theories are analytical. In our strongly coupled setting, all probes in all dimensions share certain universal features in their thermalization: (1) a slight delay in…
Matrix quantum mechanics offers an attractive environment for discussing gravitational holography, in which both sides of the holographic duality are well-defined. Similarly to higher-dimensional implementations of holography, collapsing shell solutions in the gravitational bulk correspond in this setting to thermalization processes in the dual quantum mechanical theory. We construct an explicit, fully nonlinear supergravity solution describing a generic collapsing dilaton shell, specify the holographic renormalization prescriptions necessary for computing the relevant boundary observables, an…
We consider constraints on the S-matrix of any gapped, Lorentz invariant quantum field theory in 3+1 dimensions due to crossing symmetry, analyticity and unitarity. We extremize cubic couplings, quartic couplings and scattering lengths relevant for the elastic scattering amplitude of two identical scalar particles. In the cases where our results can be compared with the older S-matrix literature they are in excellent agreement. We also extremize a cubic coupling in 2+1 dimensions which we can directly compare to a universal bound for a QFT in AdS. This paper generalizes our previous 1+1 dimens…
We argue that nonperturbative CFT correlation functions admit a Mellin amplitude representation. Perturbative Mellin representation readily follows. We discuss the main properties of nonperturbative CFT Mellin amplitudes: subtractions, analyticity, unitarity, Polyakov conditions and polynomial boundedness at infinity. Mellin amplitudes are particularly simple for large N CFTs and 2D rational CFTs. We discuss these examples to illustrate our general discussion. We consider subtracted dispersion relations for Mellin amplitudes and use them to derive bootstrap bounds on CFTs. We combine crossing,…
We calculate the Ohmic and Hall conductivities from linear response theory in a system consisting of the intersection of Nc Dp-branes and Nf << N_c Dq-branes. Agreement is found between previous results found in a macroscopic approach comparing induced currents with external electric fields. The issue of how to deal with mixed operators is raised. The retarded Green's function is given by a matrix that can be computed from the boundary action similarly to the Green's function for uncoupled operators.
We study a precise and computationally tractable notion of operator complexity in holographic quantum theories, including the ensemble dual of Jackiw-Teitelboim gravity and two-dimensional holographic conformal field theories. This is a refined, "microcanonical" version of K-complexity that applies to theories with infinite or continuous spectra (including quantum field theories), and in the holographic theories we study exhibits exponential growth for a scrambling time, followed by linear growth until saturation at a time exponential in the entropy $\unicode{x2014}$a behavior that is characte…
Using Matrix Theory as a concrete example of a fundamental holographic theory, we show that the emergent macroscopic spacetime displays a new macroscopic quantum structure, holographic geometry, and a new observable phenomenon, holographic noise, with phenomenology similar to that previously derived on the basis of a quasi-monochromatic wave theory. Traces of matrix operators on a light sheet with a compact dimension of size $R$ are interpreted as transverse position operators for macroscopic bodies. An effective quantum wave equation for spacetime is derived from the Matrix Hamiltonian. Its s…
We revisit analytical methods for constraining the nonperturbative $S$-matrix of unitary, relativistic, gapped theories in $d \geq 3$ spacetime dimensions. We assume extended analyticity of the two-to-two scattering amplitude and use it together with elastic unitarity to develop two natural expansions of the amplitude. One is the threshold (non-relativistic) expansion and the other is the large spin expansion. The two are related by the Froissart-Gribov inversion formula. When combined with crossing and a local bound on the discontinuity of the amplitude, this allows us to constrain scattering…
The bulk S-Matrix can be given a non-perturbative definition in terms of the flat space limit of AdS/CFT. We show that the unitarity of the S-Matrix, ie the optical theorem, can be derived by studying the behavior of the OPE and the conformal block decomposition in the flat space limit. When applied to perturbation theory in AdS, this gives a holographic derivation of the cutting rules for Feynman diagrams. To demonstrate these facts we introduce some new techniques for the analysis of conformal field theories. Chief among these is a method for conglomerating local primary operators to extract…
We propose a local renormalization group procedure where length scale is changed in spacetime dependent way. Combining this scheme with an earlier observation that high energy modes in renormalization group play the role of dynamical sources for low energy modes at each scale, we provide a prescription to derive background independent holographic duals for field theories. From a first principle construction, it is shown that the holographic theory dual to a D-dimensional matrix field theory is a (D+1)-dimensional quantum theory of gravity coupled with matter fields of various spins. The gravit…
We consider CFT states defined by adding nonlocal multi-trace sources to the Euclidean path integral defining the vacuum state. For holographic theories, we argue that these states correspond to states in the gravitational theory with a good semiclassical description but with a more general structure of bulk entanglement than states defined from single-trace sources. We show that at leading order in large N, the entanglement entropies for any such state are precisely the same as those of another state defined by appropriate single-trace effective sources; thus, if the leading order entanglemen…
