If the AdS/CFT conjecture holds, every question about bulk physics can be answered by the boundary CFT. But we still don't know how to translate all the questions about bulk physics to questions about the boundary CFT. Completing this bulk-boundary dictionary is the aim of the bulk reconstruction program, which we review in these lectures. We cover the HKLL construction, bulk reconstruction in AdS/Rindler, mirror operator construction of Papadodimas and Raju and the Marolf-Wall paradox. These notes are based on lectures given at ST4 2018.
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…
These lectures review recent developments in our understanding of the emergence of local bulk physics in AdS/CFT. The primary topics are sufficient conditions for a conformal field theory to have a semiclassical dual, bulk reconstruction, the quantum error correction interpretation of the correspondence, tensor network models of holography, and the quantum Ryu-Takayanagi formula.
We introduce the AdS/CFT correspondence as a natural extension of QFT in a fixed AdS background. We start by reviewing some general concepts of CFT, including the embedding space formalism. We then consider QFT in a fixed AdS background and show that one can define boundary operators that enjoy very similar properties as in a CFT, except for the lack of a stress tensor. Including a dynamical metric in AdS generates a boundary stress tensor and completes the CFT axioms. We also discuss some applications of the bulk geometric intuition to strongly coupled QFT. Finally, we end with a review of th…
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.
The bulk reconstructions in AdS/CFT and its cousins are essential to understand the holographic nature of quantum gravity. In this work, we try to study the bulk reconstruction in the AdS$_3$/WCFT$_2$ correspondence. After deriving the bulk-boundary propagator, which is different from the usual one in AdS/CFT, we define the bulk proto-scalar field by using the Virasoro-Kac-Moody symmetry in two different ways. One is to impose the bulk primary conditions on the field and construct the field algebraically. The other is to use the bulk-boundary vacuum OPE block, which can be read by applying the…
We discuss the extension of the HKLL (Hamilton, Kabat, Lifschytz, and Lowe) bulk reconstruction for non-interacting scalar fields corresponding to conformal weights $Δ$ smaller than the original condition $Δ> d-1$. We give explicit formulas for the cases $d-2<Δ\leq d-1$ and $Δ=d-s$ with integer $s$. In the latter case we show that smearing CFT fields over a region of the boundary consisting of points light-like separated from the bulk point is sufficient for bulk reconstruction, whereas in general smearing over all light-like and space-like separated points is required.
The bulk reconstruction formula for a Euclidean anti-de Sitter space is directly related to the inverse of the Gel'fand-Graev-Radon transform. Correlation functions of a conformal scalar field theory in the boundary are thereby related to correlation functions of a self-interacting scalar field theory in the bulk at different loop orders.
The bulk reconstruction program aims to obtain representations of bulk fields as operators in the boundary CFT. In this paper we extend the program by obtaining the boundary representation for a scalar field in a rotating BTZ black hole. We find that the representation of the field near the inner horizon shows novel features. We also obtain a representation for fields inside the horizon as operators in a single boundary CFT using mirror operator construction.
We study the structure of entanglement wedges in the Kasner-AdS geometry, which provides an example of AdS/CFT engineered cosmological singularity. We investigate the specific limitations of causal reconstruction methods, imposed by the presence of the cosmological singularities, and we show the supremacy of modular reconstruction. This model provides an example where modular reconstruction based on a proper operator subalgebra is more powerful than the strongest possible causal reconstruction, based on the complete operator algebra.
In a recent paper, a novel surface growth approach for reconstructing bulk geometry and matter fields was proposed, it was shown that this picture can be explicitly realized by the one-shot entanglement distillation tensor network and the surface state correspondence. In the present paper, we give direct analysis for the growth of the bulk minimal surfaces in asymptotically AdS 3 spacetime and show that bulk geometry can be efficiently reproduced in this way, which provides further support for the surface growth approach in entanglement wedge reconstruction.
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.
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…
Bulk Reconstruction of Scalar Excitations in Flat$_3$/CCFT$_2$ and the Flat Limit from (A)dS$_3$/CFT$_2$
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Department of Physics, Kyoto University（京都大学物理学系）；Inamori Research Institute for Science（稻盛理工研究院）
We propose a reconstruction of general bulk surfaces in any dimension in terms of the differential entropy in the boundary field theory. In particular, we extend the proof of Headrick et al. to calculate the area of a general class of surfaces, which have a 1-parameter foliation over a closed manifold. The area can be written in terms of extremal surfaces whose boundaries lie on ring-like regions in the field theory. We discuss when this construction has a description in terms of spatial entanglement entropy and suggest lessons for a more complete and covariant approach.
In this paper, the two reconstruction methods, light-cone cuts method and hole-ography, are combined to provide complete bulk metrics of locally AdS$_3$ static spacetimes. As examples, our method is applied to the geometries of pure AdS$_3$, AdS$_3$ soliton, and BTZ black hole, and we see them successfully reconstructed. The light-cone cuts method is known to have difficulty in obtaining conformal factors, while the hole-ography in describing temporal components. Combining the two methods, we overcome the disadvantages and give complete metrics for a class of holographic theories such that ent…
We propose a surface growth approach to reconstruct the bulk spacetime geometry, motivated by Huygens'principle of wave propagation. We first construct a tensor network corresponding to a special surface growth picture with spherical symmetry and fractal feature using the one-shot entanglement distillation (OSED)method and show that the resulting tensor network is equivalent to the MERA-like tensor network, which gives a proof that the MERA-like tensor network is indeed a discretized version of the time slice of AdS spacetime, rather than just an analogy. Furthermore, we generalize the origina…
The problem of how the boundary encodes the bulk in AdS/CFT is still a subject of study today. One of the major issues that needs more elucidation is the problem of subregion duality; what information of the bulk a given boundary subregion encodes. Although the proof given by Dong, Harlow, and Wall states that the entanglement wedge of the bulk should be encoded in boundary subregions, no explicit procedure for reconstructing the entanglement wedge was given so far. In this paper, mode sum approach to obtaining smearing functions for a single bulk scalar is generalised to include bulk reconstr…
We introduce new tools for studying modular flow in AdS/CFT. These tools allow us to efficiently extract bulk information related to causality and locality. For example, we discuss the relation between analyticity in modular time and entanglement wedge nesting which can then be used to extract the location of the Ryu-Takayanagi (RT) surface directly from the boundary theory. Probing the RT surface close to the boundary our results reduce to the recent proof of the Quantum Null Energy Condition. We focus on heavy probe operators whose correlation functions are determined by spacelike geodesics.…
We study the portion of an asymptotically Anti de Sitter geometry's bulk where the metric can be reconstructed, given the areas of minimal 2-surfaces anchored to a fixed boundary subregion. We exhibit situations in which this region can reach parametrically far outside of the entanglement wedge. If the setting is furthermore holographic, so that the bulk geometry is dual to a state in a conformal field theory (CFT), these minimal 2-surface areas can be deduced from the expectation values of operators localized within the boundary subregion. This presents us with an alternative: Either the redu…
This is a paper compiled by students of the 2008 Summer School on Particles, Fields, and Strings held at the University of British Columbia on lectures given by Veronika Hubeny as understood and interpreted by the authors. We start with an introduction to the AdS/CFT duality. More specifically, we discuss the correspondence between relativistic, conformal hydrodynamics and Einstein's theory of gravity. Within our framework the Einstein equations are an effective description for the string theory in the bulk of AdS_5 spacetime and the hydrodynamic fluid equations represent the conformal field t…
We continue the studies of our earlier proposal for an AdS/CFT correspondence for time-dependent supergravity backgrounds. We note that by performing a suitable change of variables, the dual super Yang-Mills theory lives on a flat base space, and the time-dependence of the supergravity background is entirely encoded in the time-dependent couplings (gauge and axionic) and their supersymmetric completion. This form of the SYM allows a detailed perturbative analysis to be performed. In particular the one-loop Wilsonian effective action of the boundary SYM theory is computed. By using the holograp…
We provide a formula to reconstruct bulk spacetime metrics inside black holes by the time dependence of complexity in the dual quantum field theory, based on the complexity=volume (CV) conjecture in the holographic duality.
How do subregion boundary representations transform under conformal transformations? In this paper we conjecture a transformation rule and provide evidence for it. We also show how this transformation rule helps mitigate the subregion reconstruction paradox presented by Alhmeiri et al.
We discuss the holographic reconstruction of static thin bubble walls in BTZ black hole geometries. We consider two reconstruction prescriptions suggested in recent years: hole-ography and light-cone cuts, in the context of thin bubble walls, and comment on their applicability in the presence of non-trivial matter in the bulk. We find that while the light-cone cuts prescription goes through within its own limitations, the current hole-ographic approaches are inadequate to describe bubble spacetimes completely. Much like entanglement shadows found around BTZ black holes and conical defects in t…
In the bulk reconstruction program, one constructs boundary representations of bulk fields. We investigate the relation between the global/Poincare and AdS-Rindler representations for AdS$_2$. We obtain the AdS-Rindler smearing function for massive and massless fields and show that the global and AdS-Rindler boundary representations are related by conformal transformations. We also use the boundary representations of creation and annihilation operators to compute the Bogoliubov transformation relating global modes to AdS-Rindler modes for both massive and massless particles.
Three introductory lectures: on Yangians and their representations; on Yangian symmetry in 1+1D integrable (bulk) field theory; and on the effect of a boundary upon this symmetry.
In the light of the duality between physics in the bulk of anti-de Sitter space and a conformal field theory on the boundary, we review the M2, D3 and M5 branes and how their near-horizon geometry yields the compactification of D=11 supergravity on S^{7}, Type IIB supergravity on S^{5} and D=11 supergravity on S^{4}, respectively. We discuss the ``Membrane at the End of the Universe'' idea and its relation to the corresponding superconformal singleton theories that live on the boundary of the AdS_{4}, AdS_{5} and AdS_{7} vacua. The massless sectors of these compactifications are described by t…
In this work, a relation is found between state dependence of bulk observables in the gauge/gravity correspondence and nonperturbative diffeomorphism invariance. Certain bulk constraints, such as the black hole information paradox, appear to obstruct the existence of a linear map from bulk operators to exact CFT operators that is valid over the entire expected range of validity of the bulk effective theory. By formulating the bulk gravitational physics in the Hartle-Hawking framework to address these nonperturbative IR questions, I will demonstrate, in the context of eternal AdS-Schwarzschild,…
These notes introduce a handful of core ideas from quantum information science that figure prominently in modern research on quantum gravity. The central concept that forms the base of these notes is that of a quantum channel; that is, the most general physically-reasonable map between quantum states and between operators on Hilbert space. After reviewing some fundamentals, we will study channels and their properties, and then go on to formulate quantum error correction in terms of quantum channels. Along the way, we will see how a handful of problems in high energy physics, such as the black …
We construct asymptotically AdS black hole solutions, with a self-interacting bulk scalar field, in the context of 5D general relativity. As the observable universe is characterized by spatial flatness, we focus to solutions where the horizon of the black hole, and subsequently all 3D hypersurfaces for fixed radial coordinate,have zero spatial curvature. We examine two cases for the black hole scalar hair: a) an exponential decaying scalar field profile and b) an inverse power scalar field profile. The scalar black hole solutions we present in this paper, are characterized by four functions f(…
The Ryu-Takayanagi and Hubeny-Rangamani-Takayanagi formulae suggest that bulk geometry emerges from the entanglement structure of the boundary theory. Using these formulae, we build on a result of Alexakis, Balehowsky, and Nachman to show that in four bulk dimensions, the entanglement entropies of boundary regions of disk topology uniquely fix the bulk metric in any region foliated by the corresponding HRT surfaces. More generally, for a bulk of any dimension $d \geq 4$, knowledge of the (variations of the) areas of two-dimensional boundary-anchored extremal surfaces of disk topology uniquely …
These lecture notes are intended to serve as an introduction to applied AdS/CFT with numerics for an audience of graduate students and others with little background in the subject. The presentation begins with a poor man's review of current status of quantum gravity, where AdS/CFT correspondence is believed to be the well formulated quantum gravity in the anti-de Sitter space. Then we present the basic ingredients in applied AdS/CFT and introduce the relevant numerics for solving differential equations into which the bulk dynamics collapses. To demonstrate how to apply AdS/CFT with numerics, w…
In this paper, we study the AdS-Rindler reconstruction. The CFT operators naively given by the holographic dictionary for the AdS-Rindler reconstruction contain tachyonic modes, which are inconsistent with the causality and unitarity of the CFT. Therefore, the subregion duality and the entanglement wedge reconstruction do not hold. We also find that the tachyonic modes in the AdS-Rindler patch lead to arbitrary high-energy or trans-Planckian modes in the global AdS. It means that the mode expansion of the Rindler patch is sensitive to the UV limit of the theory, that is, quantum gravity. In ad…
In this work we present a "modest" holographic reconstruction of the bulk geometry in asymptotically flat spacetime using the two-point correlators of boundary quantum field theory (QFT) in asymptotically flat spacetime. The boundary QFT lives on the null boundary of the spacetime, namely null infinity and/or the Killing horizons. The bulk reconstruction relies on two unrelated results: (i) there is a bulk-to-boundary type correspondence between free quantum fields living in the bulk manifold and free quantum fields living on its null boundary, and (ii) one can construct the metric by making u…
Light-cone cuts have recently been proposed as a method to reconstruct the conformal metric of a holographic spacetime. We explore how additional information about the bulk geometry gets encoded in the structure of these light-cone cuts. In particular, we study how the hyperbolic angle related to a cusp in the light-cone cut encodes information about the matter content of the spacetime. We provide an explicit numerical example reconstructing the metric for a 4- dimensional spacetime composed by the superposition of a boson star and a gas of radiation in AdS.
We consider the reconstruction of a Lifshitz spacetime from three perspectives: differential entropy (or "hole-ography"), causal wedges and entanglement wedges. We find that not all time-varying bulk curves in vacuum Lifshitz can be reconstructed via the differential entropy approach, adding a caveat to the general analysis of \cite{Headrick:2014eia}. We show that the causal wedge for Lifshitz spacetimes degenerates, while the entanglement wedge requires the additional consideration of a set of boundary-emanating light-sheets. The need to include bulk surfaces with no clear field theory interp…
Perturbative bulk reconstruction in AdS/CFT starts by representing a free bulk field $ϕ^{(0)}$ as a smeared operator in the CFT. A series of $1/N$ corrections must be added to $ϕ^{(0)}$ to represent an interacting bulk field $ϕ$. These corrections have been determined in the literature from several points of view. Here we develop a new perspective. We show that correlation functions involving $ϕ^{(0)}$ suffer from ambiguities due to analytic continuation. As a result $ϕ^{(0)}$ fails to be a well-defined linear operator in the CFT. This means bulk reconstruction can be understood as a procedure…
Holographic duality implies that the geometric properties of the gravitational bulk theory should be encoded in the dual field theory. These naturally include the metric on dimensions that become compact near the conformal boundary, as is the case for any asymptotically locally $\text{AdS}_n\times\mathbb{S}^k$ spacetime. Almost all previous work on metric reconstruction ignores these dimensions and would thus at most apply to dimensionally-reduced metrics. In this work, we generalize the approach to bulk reconstruction using light-cone cuts and propose a prescription to obtain the full higher-…
We prove that the metric of a general holographic spacetime can be reconstructed (up to an overall conformal factor) from distinguished spatial slices - "light-cone cuts" - of the conformal boundary. Our prescription is covariant and applies to bulk points in causal contact with the boundary. Furthermore, we describe a procedure for determining the light-cone cuts corresponding to bulk points in the causal wedge of the boundary in terms of the divergences of correlators in the dual field theory. Possible extensions for determining the conformal factor and including the cuts of points outside o…
We outline a holographic recipe to reconstruct $α'$ corrections to AdS (quantum) gravity from an underlying CFT in the strictly planar limit ($N\rightarrow\infty$). Assuming that the boundary CFT can be solved in principle to all orders of the 't Hooft coupling $λ$, for scalar primary operators, the $λ^{-1}$ expansion of the conformal dimensions can be mapped to higher curvature corrections of the dual bulk scalar field action. Furthermore, for the metric pertubations in the bulk, the AdS/CFT operator-field isomorphism forces these corrections to be of the Lovelock type. We demonstrate this by…
