We give a procedure for deriving certain bulk S-matrix elements from corresponding boundary correlators. These are computed in the plane wave limit, via an explicit construction of certain boundary sources that give bulk wavepackets. A critical role is played by a specific singular behavior of the lorentzian boundary correlators. It is shown in examples how correlators derived from the bulk supergravity exhibit the appropriate singular structure, and reproduce the corresponding S-matrix elements. This construction thus provides a nontrivial test for whether a given boundary conformal field theory can reproduce bulk physics, and where it does, supplies a prescription to extract bulk S-matrix elements in the plane wave limit.
We compute thermal 2-point correlation functions in the black brane $AdS_5$ background dual to 4d CFT's at finite temperature for operators of large scaling dimension. We find a formula that matches the expected structure of the OPE. It exhibits an exponentiation property, whose origin we explain. We also compute the first correction to the two-point function due to graviton emission, which encodes the proper time from the event horizon to the black hole singularity.
In this paper, we study the bulk local states in AdS/CFT correspondence in the large $N$ limit using the formula explicitly relating the bulk local operators and the CFT local operators. We identify the bulk local state in terms of CFT local states and find that the bulk local state corresponds to a CFT state supported in the whole space, which means a version of the subregion duality is not valid. On the other hand, CFT states supported in a space region are expressed in terms of the bulk states supported in a certain region. We find that the quantum error correction proposal is not realized …
We study non-Gaussian bulk 2d CFTs in AdS$_2$ using boundary CFT techniques and recent results in JT/Schwarzian gravity. We highlight the constraints on the operator content of a theory imposed by the boundary conditions by examining the relation between correlator coefficients and OPE coefficients in the presence of a boundary. We then calculate bulk and boundary correlators for various boundary conditions. Schwarzian techniques are used to calculate gravitational correlators perturbatively in $1/c$.
We develop the representation of free spinor fields in the bulk of Lorentzian anti-de Sitter space in terms of smeared operators in the dual conformal field theory. To do this we expand the bulk field in a complete set of normalizable modes, work out the extrapolate dictionary for spinor fields, and show that the bulk field can be reconstructed from its near-boundary behavior. In some cases chirality and reality conditions can be imposed in the bulk. We study the action of the CFT modular Hamiltonian on bulk fermions to show that they transform with the expected spinor Lie derivative, and we c…
These lecture notes are based on a series of lectures given at the XIII Modave summer school in mathematical physics. We review the construction due to Hamilton, Kabat, Lifschytz and Lowe for reconstructing local bulk operators from CFT operators in the context of AdS/CFT and show how to recover bulk correlation functions from this definition. Building on the work of these authors, it has been noted that the bulk displays quantum error correcting properties. We will discuss tensor network toy models to exemplify these remarkable features. We will discuss the role of gauge invariance and of dif…
We explain our proposal for an alternative bulk reconstruction of AdS/CFT correspondences from a scalar field by the flow method. By smearing and then normalizing a primary field in a $d$ dimensional CFT, we construct a bulk field, through which a $d+1$ dimensional AdS space emerges.
In this paper, we show that the bulk reconstruction in the AdS/CFT correspondence is rather simple and has an intuitive picture, by showing that the HKLL bulk reconstruction formula can be simplified. We also reconstruct the wave packets in the bulk theory from the CFT primary operators. With these wave packets, we discuss the causality and duality constraints and find our picture is only the consistent one. Our picture of the bulk reconstruction can be applied to the asymptotic AdS spacetime.
AdS/CFT correspondence gives us a bulk-boundary dictionary between CFT operators and local fields in the bulk. Can a bulk-boundary dictionary exist for a non local bulk field? In this paper we consider a particular non local theory and show that a bulk boundary dictionary relating it to CFT operators in the bulk can indeed be constructed. In this dictionary the scale of non locality of the bulk theory is related to the conformal dimension of the dual CFT operator.
In the framework of bulk reconstruction, we elucidate the relationship between the action of CFT modular Hamiltonians on bulk operators, the possible equation of motion for the bulk operators, and the charge distribution at infinity corresponding to such bulk fields. In particular for scalar fields interacting with gravity or with gauge fields, we show how CFT considerations of the action of the modular Hamiltonian constrain the possible bulk equation of motion to be consistent with bulk gauge invariance and diffeomorphism invariance. In fact we show that requiring that the action of the modul…
We study the action of the CFT total modular Hamiltonian on the CFT representation of bulk fields with spin. In the vacuum of the CFT the total modular Hamiltonian acts as a bulk Lie derivative, reducing on the RT surface to a boost perpendicular to the RT surface. This enables us to reconstruct bulk fields with spin from the CFT. On fields with gauge redundancies the total modular Hamiltonian acts as a bulk Lie derivative together with a compensating bulk gauge (or diffeomorphism) transformation to restore the original gauge. We consider the Lie algebra generated by the total modular Hamilton…
It is a long-standing conjecture that any CFT with a large central charge and a large gap $Δ_{\text{gap}}$ in the spectrum of higher-spin single-trace operators must be dual to a local effective field theory in AdS. We prove a sharp form of this conjecture by deriving numerical bounds on bulk Wilson coefficients in terms of $Δ_{\text{gap}}$ using the conformal bootstrap. Our bounds exhibit the scaling in $Δ_{\text{gap}}$ expected from dimensional analysis in the bulk. Our main tools are dispersive sum rules that provide a dictionary between CFT dispersion relations and S-matrix dispersion rela…
AdS_4/CFT_3 duals from M2-branes at hypersurface singularities and their deformations
我们构造了与M理论Freund-Rubin解AdS_4 x V_{5,2} /Z_k（有或无挠G流）全息对偶的N=2的三维Chern-Simons-Einstein理论，其中V_{5,2}是齐次Sasaki-Einstein七维流形.这些理论的整体对称群一般是SU（2）× U（1）× U（1）_R，因此它们是非环面的。场论可以被认为是模型族中的n=2个成员，用正整数n标记，产生于多个M2-膜在某些超曲面奇点处。我们描述了这些模型可以通过广义Hanany-Witten膜结构工程。AdS_4 x V_{5,2} /Z_k解可以变形为翘曲几何R^{1，2} x T^* S^4/Z_k，具有通过四球的自对偶G通量.我们表明，这种解决方案是对偶的超对称质量变形，这精确地修改了经典的模空间的场论变形的几何。
A common method to prepare states in AdS/CFT is to perform the Euclidean path integral with sources turned on for single-trace operators. These states can be interpreted as coherent states of the bulk quantum theory associated to Lorentzian initial data on a Cauchy slice. In this paper, we discuss the extent to which arbitrary initial data can be obtained in this way. We show that the initial data must be analytic and define the subset of it that can be prepared by imposing bulk regularity. Turning this around, we show that for generic analytic initial data the corresponding Euclidean section …
We establish the Lorentzian AdS$_2$/CFT$_1$ correspondence from a reconstruction of all bulk points through the kinematic-space approach. The OPE block is exactly a bulk local operator. We formulate the correspondence between the bulk propagator in the non-interacting scalar field theory and the conformal block in CFT$_1$. When we consider the stress tensor, the variation probes the variation of AdS$_2$ metric. The reparameterization provides the asymptotic boundary of the bulk spacetime as in the derivation of the Schwarzian theory from two-dimensional dilaton gravity theory. Finally, we find…
We advance a construction for the covariant holographic entanglement negativity for time dependent mixed states of disjoint intervals in $(1+1)$ dimensional conformal field theories ($CFT_{1+1}$) dual to bulk non static $AdS_3$ geometries. Application of our proposal to such mixed states in a $CFT_{1+1}$ dual to bulk non extremal and extremal rotating BTZ black holes exactly reproduces the replica technique results in the large central charge limit. We also investigate the time dependent holographic entanglement negativity for such mixed states in a $CFT_{1+1}$ dual to a bulk Vaidya-$AdS_3$ ge…
We present a Lorentzian inversion formula valid for any defect CFT that extracts the bulk channel CFT data as an analytic function of the spin variable. This result complements the already obtained inversion formula for the corresponding defect channel, and makes it now possible to implement the analytic bootstrap program for defect CFT, by going back and forth between both channels. A crucial role in our derivation is played by the Calogero-Sutherland description of defect blocks which we review. As first applications we obtain the large-spin limit of bulk CFT data necessary to reproduce the …
We count the number of independent solutions to crossing constraints of four point functions involving charged scalars and charged fermions in a CFT with large gap in the spectrum. To find the CFT data we employ recently developed analytical functionals to charged fields. We compute the corresponding higher dimensional flat space S matrices in an independent group theoretic manner and obtain agreement with our CFT counting of ambiguities. We also write down the local lagrangians explicitly. Our work lends further evidence to \cite{Heemskerk:2009pn} that any CFT with a large charge expansion an…
The $p$-adic AdS/CFT is a holographic duality based on the $p$-adic number field $\mathbb{Q}_p$. For a $p$-adic CFT living on $\mathbb{Q}_p$ and with complex-valued fields, the bulk theory is defined on the Bruhat-Tits tree, which can be viewed as the bulk dual of $\mathbb{Q}_p$. We propose that bulk theory can be formulated as a lattice gauge theory of PGL$(2,\mathbb{Q}_p)$ on the Bruhat-Tits tree, and show that the Wilson line networks in this lattice gauge theory can reproduce all the correlation functions of the boundary $p$-adic CFT.
We study a set of CFT operators suitable for reconstructing a charged bulk scalar field $ϕ$ in AdS${}_3$ (dual to an operator ${\cal O}$ of dimension $Δ$ in the CFT) in the presence of a conserved spin-$n$ current in the CFT. One has to sum a tower of smeared non-primary scalars $\partial_{+}^{m} J^{(m)}$, where $J^{(m)}$ are primaries with twist $Δ$ and spin $m$ built from ${\cal O}$ and the current. The coefficients of these operators can be fixed by demanding that bulk correlators are well-defined: with a simple ansatz this requirement allows us to calculate bulk correlators directly from t…
