Bulk Locality in AdS/CFT Reconstruction from Bilocal Operators

MSc Physics

Theoretical Physics

M

T

Bulk Locality in AdS/CFT

Reconstruction from Bilocal Operators

by

Davide D

10064605

The AdS/CFT correspondence is a holographic mapping between a (d+1)-dimensional the-ory of quantum gravity in anti-de Sitter space and a conformal field thethe-ory in d dimensions. How the correspondence works is not yet completely understood. The open question of how locality emerges in AdS from the boundary is the focus of this project. When reconstructing quantities in AdS from the boundary theory, we run into inconsistencies, which have been re-cently formulated as precise paradoxes. A proposal which addresses these paradoxes using quantum error correction and gauge invariance to explain how bulk information is encoded in the CFT will be analyzed. We find that this model agrees with our expectations regarding locality and its breakdown.

Abstract i

Introduction 1

1 Preliminaries 4

1.1 Quantum Field Theory . . . 4

1.1.1 Timeslice Axiom . . . 4

1.1.2 Large N . . . 5

1.1.3 Conformal Invariance . . . 6

1.2 Anti-de Sitter Space . . . 10

1.3 AdS/CFT Correspondence. . . 12 2 Precursors 15 2.1 Bulk Reconstruction . . . 15 2.1.1 Causal Wedges . . . 18 2.1.2 AdS-Rindler . . . 19 2.2 Paradoxes . . . 20

2.2.1 Quantum Error Correction . . . 21

2.2.2 Gauge Invariance . . . 24

2.3 Motivation . . . 27

3 Results 29 3.1 Investigating the Timeslice Axiom . . . 29

3.1.1 Finite N . . . 29

3.1.2 Large N and Gauge Invariance . . . 30

3.2 Bulk Locality in AdS3/CF T2 . . . 31

Discussion 34

A Large N counting 36

Bibliography 39

Acknowledgements 41

One of the greatest problems in theoretical physics is the unification of quantum field theory and general relativity. In most cases, gravitational effects are negligible at the quantum scale. However, in the proximity of a black hole, the spacetime curvature is large enough to produce important quantum effects. This is why our best efforts at understanding quantum gravity started with the study of black holes. In the 1970s, considerations by Hawking regarding the thermodynamic properties of black holes led Bekenstein to the formulation of the area law

SBH =

A 4

which states that the entropy of a black hole is proportional to its area. The fact that all the degrees of freedom of a black hole can be mapped to its surface provided the first hint at the nonlocal nature of gravity. This was already a strong indication that any local field theoretic description of nature should break down once gravitational effects become significant, and that a theory of quantum gravity should also possess nonlocal properties. This was further motivated by the discovery of Hawking radiation, the idea that black holes radiate particles and may eventually evaporate completely. Hawking’s calculation of black hole radiation pro-duced a puzzling result: it seems that, as a black hole evaporates, information is lost in the process, which contradicts unitarity, one of the fundamental properties of quantum mechan-ics. This became known as the black hole information paradox, which still remains unsolved today. Various proposals have been put forward since its formulation, which all involve, in some way or another, a modification of our current physical theories or new physics alto-gether. The black hole information paradox strongly indicates that our accepted description of nature breaks down when quantum gravitational effects come into play. These important dis-coveries about the quantum properties of black holes led to the formulation of the holographic principle, a profound statement about quantum gravity with far-reaching implications.

The holographic principle is an idea which was first introduced by ’t Hooft in the early 1990’s when, using the thermodynamical properties of black holes, he reasoned that, given a system of volume V and surface area A, the entropy inside A is maximized if the volume contains a black hole. Therefore, the maximum number of states in any spatial volume is de-termined only by the size of its surface area. “It means that”, ’t Hooft writes in [1] , “given any closed surface, we can represent all that happens inside it by degrees of freedom on this sur-face itself. This, one may argue, suggests that quantum gravity should be described entirely by a topological quantum field theory, in which all physical degrees of freedom can be projected onto the boundary”. This statement has forever changed the way we look at nature: whereas in the beginning quantum mechanics and general relativity were though of as fundamentally

incompatible with each other, we are now beginning to realize that they may be two sides of the same coin. This paved the way for what is now considered one of the greatest discoveries in theoretical physics of the past 20 years, the AdS/CFT correspondence.

In 1997, Maldacena conjectured a duality which would become known as the AdS/CFT correspondence in his groundbreaking paper [2] , providing the first implementation of holog-raphy. The AdS/CFT correspondence is a mapping between a (d+1)-dimensional theory of quantum gravity in Anti-de Sitter space (a maximally symmetric spacetime with constant neg-ative curvature) and a conformal field theory (a scale invariant quantum field theory) in d dimensions. The higher dimensional theory is known as the bulk while the lower dimensional theory is known as the boundary theory. This is because, in a sense, the CFT lives on the boundary of AdS, since the mapping between the two theories is between operators which live on the boundary of AdS and operators which live on the CFT. We therefore have a full correspondence between two radically different theories; we can translate questions regarding the bulk theory as questions regarding the boundary, and vice versa. The key to unlocking the mystery of quantum gravity now lies in the understanding of how this correspondence works. Since its formulation, the correspondence has passed many nontrivial checks, but it has not yet been rigorously proved. In fact, there are still many open questions regarding how the correspondence works. We know that we can translate quantities in one theory to the other by using the AdS/CFT dictionary, but various inconsistencies arise when we attempt to recon-struct the bulk theory from the boundary. One of the most puzzling aspects of holography is how the theory in AdS can be local, even if it is possible to reformulate it as a theory living only on the boundary. This is known as the problem of the emergence of bulk locality, and it is strictly related to the nonlocal nature of gravity. In fact, we expect locality to break down as we move away from the semiclassical approximation of our bulk theory but, in the appropriate regime, we expect our bulk theory to be approximately local. Understanding the breakdown of bulk locality is therefore an important piece of the puzzle of quantum gravity, and it is the main focus of this project.

We can investigate the problem of bulk locality by asking how information in the CFT is encoded in the bulk, and under what conditions bulk reconstruction breaks down. In fact, bulk reconstruction turns out to give rise to paradoxes regarding the fundamental properties of a quantum field theory. Bulk locality requires that fields in the bulk commute with fields on the boundary, since they are spacelike separated. This seems to contradict the time-slice axiom, which states that any operator which commutes with all operators of the theory at a constant timeslice must be proportional to the identity. This is because when we reconstruct a bulk field from the boundary, we obtain an expression in terms of CFT operators, called the precursor operator. We can then use time evolution to express the precursor in terms of Heisenberg picture fields on the same timeslice. If we take into account bulk locality, which requires the precursor operator to commute with operators on the boundary, we find that the timeslice axiom is violated. Related paradoxes can be found if we consider alternative ways in which we can perform bulk reconstruction. It turns out that when carrying out bulk reconstruction we can consider all of the boundary CFT, known as global reconstruction, or a

smaller subregion, known as the causal wedge of the bulk field. The latter proposal is known as the subregion-subregion duality. This turns out to be very puzzling since we can have different representations for the same bulk operator which should be, in principle, equivalent. These paradoxes were addressed in [3,4] where, in an attempt to resolve them, an infor-mation theoretic explanation of bulk reconstruction was proposed, where the main claim is that AdS/CFT is a quantum error correction mechanism. This would then become a question of finding the right algorithm which correctly reproduces the AdS/CFT correspondence. An alternative explanation was put forward by Mintun, Polchinski and Rosenhaus in [5], where it is suggested that the gauge invariance of the CFT automatically employs some form of error correction, and that it is an essential ingredient in ensuring the consistency of bulk reconstruc-tion. This project will focus on verifying the consistency of the model of holography developed in the gauge invariance proposal. Although the gauge invariance proposal is still at an early stage, it offers a new way to approach the problems related to bulk reconstruction. The main claim of this proposal is that the gauge invariance of the CFT is essentially responsible for the correct encoding of information from the boundary to the bulk, analogous to quantum er-ror correction. It provides a simple model in which the erer-ror correcting code can be explicitly described in the CFT, using only gauge invariance. This is why we wish to investigate this pro-posal further, focusing on the issue of bulk locality, in an attempt to shed light on the role of gauge invariance in bulk reconstruction. We will do this by checking if the model is consistent with our expectations regarding bulk locality.

This work is structured in the following way. In chapter 1, we review some properties of quantum field theory which will be important for our discussion, followed by a brief overview of the AdS/CFT correspondence. In chapter 2 we focus on bulk reconstruction, where we go over the paradoxes that arise in this framework and review some of the proposed solutions. Finally, in chapter 3 we analyze the model used in the gauge invariance proposal and see whether or not our expectations regarding bulk locality are met, and conclude with a discus-sion.

Preliminaries

1.1

Quantum Field Theory

Quantum field theory (QFT) is usually taught in its Lagrangian formulation, which allows for a more heuristic approach, without being too strict on mathematical rigor. In fact, we are still missing a comprehensive mathematically precise formulation of quantum field theory, but there has been some progress in this direction. Such a formulation of QFT can be useful when defining concepts such as causality and locality. In 1964, Wightman and Streater developed an axiomatic formalism in an attempt to define in a mathematically precise way those properties which characterize a sensible QFT1_{. These became known as the Wightman axioms, and they} deal with the notions of field, causality and transformations. The axiom of causality and the timeslice axiom will have important implications in our discussion.

In any relativistic quantum field theory, causality implies that fields φ(x, t) and operators O(x, t) which are spacelike separated commute. In other words, a field will be able to interact only with fields which lie inside its lightcone. Another important feature of a QFT is that any operator acting on the Hilbert space of the theory can be expressed as a linear combination of products of fields. This is known as the completeness axiom and it implies that if we have an operator which commutes with all the operators of the theory, including the ones which lie inside its lightcone, it must be proportional to the identity operator. Furthermore, using dynamical laws, it should be possible to evolve operators backwards and forwards in time. Therefore we can express any operator at an arbitrary time in terms of fields at a constant (arbitrarily small) time slice. This means that, any operator which commutes with all operators at a constant time-slice will also be proportional to the identity. The time-slice axiom has not been proven in general, but it is known to hold in the case of interacting scalar field theories (see [8]). Although these axioms are generally believed to hold for any QFT, constructing gauge theories starting from the Wightman axioms remains an unsolved problem. In fact, we will see that the timeslice axiom is violated in a field theory with a global O(N) gauge symmetry.

1_{For more details see [6,7]}

1.1.2 Large N

An important ingredient for our discussion will be the large N limit of quantum field theories. We will only give a brief overview of the topic; for a general introduction see [9] and for more details, see [10,11]. In some cases, given a theory with N fields, it is possible to greatly simplify it by expressing the solutions of the theory in terms of a 1/N expansion, given that N is large. It is a very powerful tool, since it enables us to solve theories which would not otherwise be solvable using perturbation theory. Let’s take the O(N) vector model as an example, where we have N scalar fields φ. The trick is to express the action as a function of N in a convenient way. With hindsight, we can write the action for this theory in the following way:

S(φ) = Z ddx 1 2[∂µφ(x)] 2_{− N V [φ}2_{(x)/N ]}  . (1.1)

In the case of φ4 theory, V will be given by

N V = 1 2m 2_φ2_{(x) +} 1 8 λ N[φ 2_(x)]2_{. (1.2)}

We have, in a sense, rescaled all the second-order operators to be of order 1/N . There are several ways to perform the 1/N expansion of the theory, but we want a nontrivial solution in which we will only have contributions proportional to positive powers of 1/N . It turns out that this can be achieved by defining a new coupling g

g = λ/N (1.3)

and by taking the large N limit while keeping g fixed. The leading order contribution will then be given by the two-point function. In fact,by the central limit theorem, for large N, the O(N) invariants of the theory, given by

φ2(x, y) =

N

X

i=1

φi(x)φ(y)i (1.4)

will have a normal distribution. This means that, using the same logic as in mean field theory, where we assume that the expectation values of these operators have small fluctuations, we can approximate n-point functions as products of 2-point functions:

hφ2(x1, y1)φ2(x2, y2) . . . φ2(xn, yn)i ∼ hφ2(x1, y1)ihφ2(x2, y2)i . . . hφ2(xn, yn)i. (1.5)

This is known as large N factorization. With the 1/N expansion, we then have a way of calcu-lating corrections order by order. So far we have considered a vector model; if we were dealing with a matrix field theory, the scaling of the coupling would change in the large N limit, since vector models have N degrees of freedom while matrix models have N2 _{degrees of freedom.}

We would then define our new coupling as

g2 = λ/N (1.6)

and take the large N limit while taking g fixed. This is known as the ’t Hooft limit. 1.1.3 Conformal Invariance

A conformal transformation is a change of coordinates which is implemented by multiplying the metric by a spacetime-dependent function:

gµν(x) → Ω2(x)gµν(x). (1.7)

Effectively, a conformal transformation is a local change of scale. An important property of conformal transformations is that they leave null geodesics invariant, meaning lightcones will be preserved. Therefore, if two metrics differ only by a conformal factor Ω2_{(x), they will have}

the same causal structure. This is particularly useful if we want to visualize the causal struc-ture of spacetimes at large distances, since we can choose a function Ω(x) that rescales the metric such that infinity is brought at a finite proper distance. This is known as conformal compactification, and its visual representation is called a Penrose diagram.

Conformal Field Theories in Two Dimensions

The following is a brief overview of the main properties of CFTs, for more details see [12]. A conformal field theory (CFT) is a field theory which is invariant under conformal transforma-tions. This means that the theory looks the same at all length scales. A direct consequence is that the theory only supports massless excitations. We will focus on 2-dimensional CFTs, where we can define complex coordinates

z = x0+ ix1, z = x¯ 0− ix1 (1.8)

where x0_{and x}1_{are Minkowski spacetime coordinates. The metric components are then given}

by

gzz = g ¯z ¯z = 0, gz ¯z= g¯zz =

1

2 (1.9)

Functions which depend on z are called holomorphic (or left-moving) functions while func-tions which depend on ¯zare called anti-holomorphic (or right-moving) functions. The holo-morphic and anti-holoholo-morphic derivatives are given by

∂z ≡ ∂ =

1

2(∂0− i∂1), ∂z¯≡ ¯∂ = 1

2(∂0+ i∂1). (1.10) Any holomorphic function f (z) is a conformal transformation. It’s a global conformal trans-formation if it’s invertible, otherwise it’s only local. The complete set of global conformal

transformations in two dimensions forms the global conformal group and is given by f (z) = az + b

cz + d with ad − bc = 1 and a, b, c, d ∈ C. (1.11) We will illustrate the main features of 2d CFTs with the free scalar field. Its action is given by

S = g Z

dzd¯z ∂φ(z, ¯z) ¯∂φ(z, ¯z) (1.12) where g is the normalization, which from now on we will take to be 1/4π. The extra factor of 2 comes from the change of coordinates dzd¯z = 2dx0dx1. The classical equation of motion for φ is ∂ ¯∂φ = 0, which means that the general solution decomposes in left-moving and right-moving parts:

φ(z, ¯z) = φ(z) + φ(¯z). (1.13) The propagator for a scalar field is given by

hφ(z, ¯z)φ(ω, ¯ω)i = − ln(z − ω) − ln(¯z − ¯ω). (1.14) A local operator, or field, is any local expression that can be written down, such as φn_{, ∂}n_φand

combinations. An important feature of CFTs is the operator product expansion (OPE): the fact that we can approximate two local operators inserted at nearby points as a string of operators at one of these points. Given a time-ordered correlation function, we can approximate the insertion of two operators Oiand Ojwith their OPE:

hOi(z, ¯z)Oj(ω, ¯ω) . . .i =

X

k

C_ijk(z − ω, ¯z − ¯ω)hOk(ω, ¯ω) . . .i (1.15)

The stress-energy tensor for a scalar field is given by Tz ¯z= 0, Tzz ≡ T (z) = − 1 2∂φ∂φ, Tz ¯¯z≡ ¯T (¯z) = − 1 2 ¯ ∂φ ¯∂φ. (1.16) In order to compute OPEs, we need to take normal ordering into account, which in a CFT is defined as

T (z) = −1

2 : ∂φ∂φ :≡ − 1

2z→ωlim(∂φ(z)∂φ(ω) − h∂φ(z)∂φ(ω)i) (1.17)

When taking products of normal ordered operators, Wick’s theorem holds. A primary operator O(ω) is one whose OPE with T (z) and ¯T (¯z)truncates at order (z−ω)−2or (¯z− ¯ω)−2respectively. The OPE of a primary operator has the form:

T (z)O(ω, ¯ω) = hO(ω, ¯ω) (z − ω)2 + ∂O(ω, ¯ω) z − ω + . . . (1.18) ¯ T (¯z)O(ω, ¯ω) = ˜hO(ω, ¯ω) (¯z − ¯ω)2 + ¯ ∂O(ω, ¯ω) ¯ z − ¯ω + . . . (1.19)

where . . . indicates the non-singular terms. Primary operators transform in the following way: O(z, ¯z) → O(z0, ¯z0) = ∂z 0 ∂z −h  ∂ ¯z0 ∂ ¯z −˜h O(z, ¯z). (1.20) The conformal dimension of an operator is the quantity

∆ = h + ˜h (1.21)

and it is analogous to the dimension we associate to fields in QFT by dimensional analysis. In a CFT in d spacetime dimensions, the condition that a scalar operator O is free is equivalent to the fact that its conformal dimension is ∆ = d−2₂ . Alternatively, a free field is characterized by the fact that its correlation functions factorize to products of 2-point functions [13]. In a unitary CFT, the left-moving TT OPE has the form

T (z)T (ω) = c/2 (z − ω)4 + 2T (ω) (z − ω)2 + ∂T (ω) z − ω + . . . (1.22) ¯ T (¯z) ¯T (¯ω) = ˜c/2 (¯z − ¯ω)4 + 2T (¯ω) (¯z − ¯ω)2 + ∂T (¯ω) ¯ z − ¯ω + . . . (1.23) where c and ˜c are called the central charges of a theory, which are related to the number of degrees of freedom. In fact, the free scalar field has c = ˜c = 1; if we were to consider D non-interacting free scalars, we would have c = ˜c = D.

Radial Quantization

We will quantize the free scalar field by defining the spatial direction along concentric circles centered at the origin. This is known as radial quantization. We start by defining our theory on an infinite space-time cylinder with the time coordinate t going from −∞ to +∞ while the spatial coordinate x is periodically defined, its period being the circumference of the cylinder, which we will set to 2π. We will therefore have the following boundary conditions for φ:

φ(x, t) = φ(x + 2π, t). (1.24)

xand t are related to the complex plane in the following way:

z = e−iω, z = e¯ i¯ω (1.25)

where ω and ¯ωare the complex coordinates on the cylinder, given by

where we have replaced t with the Euclidean time coordinate t → −iτ . The Fourier expansion of a free scalar field φ(x, t) is given by:

φ(x, t) =X n einxφn(t) (1.27) φn(t) = 1 2π Z dx e−inxφ(x, t) (1.28)

In this formalism, the mode expansion of φ is given by φ(x, t) = φ0+ 2π0t + i X n6=0 1 n h αnein(x−t)− ˜α−nein(x+t) i (1.29) with [φ0, π0] = i (1.30) [αm, αn] = [ ˜αm, ˜αn] = mδm+n (1.31)

The αn’s are the left-moving modes, while the ˜αn’s are the right movers, and they are related

to the canonical creation and annihilation operators in the following way:

αn= ( −i√nan for n > 0 i√−na†_−n for n < 0 α˜n= ( −i√na−n for n > 0 i√−na†n for n < 0 (1.32)

They decouple because of the periodicity condition on φ given by (1.24). They are creation operators for n < 0 and annihilation operators for n > 0. On the cylinder, the holomorphic derivative of φ(ω, ¯ω)is given by

∂ωφ(ω, ¯ω) = −π0−

X

n6=0

αneinω. (1.33)

On the plane, the holomorphic derivative of φ(z, ¯z)becomes ∂zφ(z, ¯z) = −i π0 z − i X n6=0 αnz−n−1 (1.34)

where we have used (1.20). The conjugate momentum is given by: π(x, t) = ∂L ∂ ˙φ = 1 4π∂tφ(x, t) = π0 2π+ 1 4π X n6=0 h αnein(x−t)+ ˜α−nein(x+t) i . (1.35)

The commutator between two fields is given by

where g(y) is given by the following infinite sum: g(y) ≡X

k6=0

1 ke

iky_{= log(1 − e}−iy_{) − log(1 − e}iy_{) (1.37)}

Using the properties of the complex logarithm, this simplifies to g(y) = log(−e−iy) − 2πim−

= πi − iy − 2iπm− (1.38) where m±=      −1 if Arg z ± Arg ω > π 0 if Arg z ± Arg ω ≤ π 1 if Arg z ± Arg ω ≤ −π (1.39)

so the commutator (1.36) vanishes. Note that this also holds for the limiting cases where ∆x = ±∆t and ∆x = ∆t = 0; this can be checked by taking the limit of g(y). The commutator between φ and π is given by

[φ(x, t), π(x0, t0)] = i

2[δ(∆x + ∆t) + δ(∆x − ∆t)], (1.40) while commutator between two conjugate momenta is 0.

1.2

Anti-de Sitter Space

Anti-de Sitter space is a maximally symmetric solution to Einstein’s equation in the vacuum, with a negative cosmological constant:

Rµν = Λgµν (1.41)

where, in d + 1 dimensions, the cosmological constant Λ is related to the AdS radius ` by Λ = −d(d − 1)

2`2 (1.42)

and the curvature is given by κ = −`2. A (d + 1)-dimensional maximally symmetric space has its maximum possible number of Killing vectors (d + 1)(d + 2)/2. In the case of AdSd+1, the

Killing vectors are SO(d,2) generators. We can embed AdSd+1in d + 2 dimensional flat space,

and express it as

If we want to cover the whole of AdSd+1we need to use global coordinates, given by X0= ` cosh χ cos τ (1.44) Xd+1= ` cosh χ sin τ (1.45) Xi= ` sinh χ ˆxi, d X i=1 ˆ x2_i = 1 (1.46)

and we obtain the metric

ds2= `2(− cosh2χdτ2+ dχ2+ sinh2χdΩ2_d−1) (1.47)

where the coordinates have the ranges

−∞ < τ < ∞ χ ≥ 0 (1.48)

and dΩ2_d−1is the metric on a unit d − 1 sphere. Note that the time coordinate is actually defined periodically: it already covers all of AdS once in the interval [0, 2π), but we can allow it to run from −∞ to ∞, which is known as taking the “universal cover” of AdS. If we want to derive the Penrose diagram of AdS, we need to render the radial coordinate finite, by conformal compactification (see section1.1.3). This can be achieved by setting

cosh χ = sec ρ sinh χ = tan ρ (1.49) which leads to the metric

ds2 = ` 2 cos2_ρ(−dτ 2_{+ dρ}2_{+ sin}2_ρdΩ2 d−1) (1.50) with −∞ < τ < ∞ 0 ≤ ρ ≤ π/2. (1.51) In two dimensions, the range of ρ extends to

−π/2 ≤ ρ ≤ π/2. (1.52)

As we can see from the Penrose diagram in figure 1.1, light signals sent out from the origin will reach spatial infinity and return in a finite amount of proper time π, given that we im-pose reflective boundary conditions2_{. Massive particles, on the other hand, will not reach the} boundary before returning to the origin. This means that the boundary is timelike: it has the topology of R × Sd−1, R being the time direction. This is clearer in the diagram of AdS in three dimensions, where the boundary is given by the surface of the cylinder.

2_{We need to impose reflective boundary conditions in order to preserve causality, otherwise information would}

τ = 0 τ = π

ρ = 0 ρ = π/2

ρ _τ

Figure 1.1: On the left, we have the Penrose diagram of AdS. Lightlike geodesics are drawn in blue and timelike geodesics are drawn in red. A massless particle sent out from the origin will reach the boundary and return in a finite proper time π, while a massive particle will not be able to reach the boundary. Spacelike slices have the topology of R × Sd−1_{, which is visualized}

on the right, where we have a the diagram of AdS3 in global coordinates. In the context of

AdS/CFT, we have a theory of quantum gravity in AdS which corresponds to a CFT living on its boundary.

1.3

AdS/CFT Correspondence

In its strongest form, the AdS/CFT correspondence states that there is a duality between a complete (d + 1)-dimensional theory of quantum gravity in AdS space and a conformal field theory in d dimensions. This mapping is holographic in the sense that a higher dimensional theory is completely encoded in a lower dimensional theory. It is often stated that the CFT lives on the boundary of AdS; if we think of AdS as a cylinder (see figure1.1), we can say that the CFT lives on its surface. This is why the gravitational theory is often referred to as the bulk while CFT is often called the boundary theory.

In its original formulation (in Maldacena’s paper [2]), the correspondence is between type IIB string theory on AdS5 × S5 and N = 4 supersymmetric Yang-Mills theory in four

di-mensions. Although for our purposes we will not need to know all the technical details, it is instructive to see how the parameters of the two theories are related. In the bulk, we have the AdS radius in string units `/`sand the string coupling gs while on the boundary we have the

Yang-Mills coupling gY M and the rank N of the SU(N) gauge group, which are related in the

following way

` `s

= (g_{Y M}2 N )1/4 gs = g2Y M. (1.53)

In order to have a good semiclassical approximation in the bulk, we need to suppress both “stringy” corrections and quantum corrections. Stringy corrections are functions of `s/`, while

quantum corrections are functions of the string coupling gs. Therefore, we need to take the

N to be large, so we need to take the large N limit of the boundary theory in a sensible way. In the case of N = 4 supersymmetric Yang-Mills, we take the ’t Hooft limit (see section1.1.2), which is obtained by taking

g_{Y M}2 N =constant N → ∞ gY M → 0 (1.54)

which in the bulk corresponds to taking gs→ 0

` `s

=constant. (1.55)

We therefore have a duality between a gauge theory in the strongly coupled ’t Hooft limit and a weakly coupled gravitational theory. This makes AdS/CFT a very powerful tool for solving problems in strongly coupled theories by reformulating them in their dual (weakly-coupled) description.

The correspondence has been widely studied for the large N limit of the boundary CFT (see section1.1.2), in which case we have a semi-classical theory of quantum gravity in the bulk, in the sense that we quantize the matter fields while treating the gravitational field as classical. The strongest form of the conjecture states that the correspondence holds for all values of N. The scope of the duality has been expanded to include a larger class of theories, which has led to a research program known as the gauge/gravity duality, where more similar conjectures have been made relating gauge theories to gravitational theories.

The precise form of correspondence is given by: heR ddxφ(0)(x)O(x)_i CF T = ZAdS  r∆−dφ(x, r)|r=0= φ(0)(x)  (1.56) where O is an operator inserted in CFT partition function with a coupling parameter φ(0)(x).

The right hand side is the AdS partition function of the scalar field φ, given some boundary condition φ(0)(x)at r = 0, and ∆ is is the conformal dimension of O, which can be expressed

in terms of the dimension of the CFT d, the AdS radius ` and the mass of the bulk field m. For a scalar field, ∆ is given by

∆(∆ − d) = m2`2. (1.57)

This has two solutions:

∆±= d 2± r d2 4 + m 2_`2_{. (1.58)}

∆+is always an allowed solution but, as shown in [14], ∆−is also admissible for

−d

2

4 < m

2_{< −}d2

4 + 1. (1.59)

This is because, within this range of values of m2, there are two different AdS-invariant ways of quantizing the AdS theory. In the “extrapolate” version of the correspondence, we have the

following relation between a bulk field Φ and a CFT operator O: lim

r→∞r

∆_{Φ(r, x) = hO(x)i. (1.60)}

Therefore, for (1.59) there will be two different dual CFT’s, given by the two choices of bound-ary conditions we can impose on Φ at r → ∞. It has been conjectured in [15] that if we have the O(N) vector model (see section 1.1.2) on the boundary, this will correspond to a higher-spin gauge theory in AdS. The CFT operators with conformal dimension ∆−will correspond

to the free theory, while the CFT operators with conformal dimension ∆+will correspond to

Precursors

2.1

Bulk Reconstruction

Given the vast difference between the two theories, it is natural to ask exactly how the AdS/CFT correspondence works, and how information in one theory is “encoded” in the other. If the re-lation (1.60) is true, an interesting question to ask is if it is possible to invert it and express the bulk field φ in terms of boundary operators. This has been studied in a series of papers by Hamilton, Kabat, Lifschytz and Lowe (HKLL) [16–20]. This is not always possible, and the conditions under which we can successfully reconstruct the bulk from the boundary have been further studied in [21,22]. From (1.60), we know that the boundary value of a bulk field Φcorresponds to the expectation value of a local CFT operator. The question is then: given the boundary values of Φ (which correspond to CFT one-point functions), can we fully reconstruct it in the bulk?

We will now summarize the standard reconstruction procedure. We assume that the CFT possesses a 1/N expansion, and we take N to be large. In this regime, classical gravity will be a good bulk approximation, as explained in section1.3. Therefore, we first solve the wave equation in AdS which, in the case of a scalar field, is given by

(2 − m2)Φ(r, x) = 0 (2.1)

where we can express Φ(r, x) in terms of creation and annihilation ak operators in a mode

expansion of the form

Φ(r, x) = Z

dk akΦk(r, x) +h.c. (2.2)

We then take its boundary value

Φ(x) = Z dk akϕk(x) +h.c. (2.3) where lim r→∞r ∆_Φ k(r, x) = ϕk(x). (2.4)

If the mode functions ϕk(x)are orthogonal, we can invert (2.3) and get

ak =

Z

dx ϕ∗_k(x)Φ(x). (2.5)

Figure 2.1: Global Reconstruction in AdS3. In order to reconstruct a field at the point x, we

integrate (2.7) over the whole boundary. We can express the CFT operators to be on a constant time-slice Σ. Figure taken from [3].

Plugging this into (2.3) gives Φ(r, x) = Z dk Z dx0ϕ∗_k(x)Φ(x0)  Φk(r, x) +h.c. (2.6)

This already gives us a way to express a bulk field in terms of its boundary values. If we are justified in exchanging the order of integration, this expression simplifies to

Φ(r, x) = Z dx0K(r, x|x0)Φ(x0) (2.7) with K(r, x|x0) = Z dk ϕ∗_k(x0)Φk(r, x) +h.c. (2.8)

We therefore have the following expression for a bulk field in terms of the expectation values of the corresponding CFT operator O:

Φ(r, x) = Z

dY K(r, x|Y )hO(Y )i + O(1/N ) (2.9) where K(r, x|Y ) is a smearing function, which obeys the bulk wave equations of motion in its r, x indices. This is not a standard boundary value problem, since the boundary data also depends on time. The right-hand side of (2.9) is called the precursor operator, which consists of CFT operators which obey the bulk equations of motion, and has the boundary conditions imposed by the extrapolate dictionary (1.60). We thus have an expression for a free, classical bulk field in terms of appropriately smeared, CFT operators. The integration in (2.9) is carried out over boundary CFT coordinates, which means that the precursor will be an expression involving multiple CFT operators acting at different spacetime points. Therefore, the precursor

operator will be highly non-local. To leading order in 1/N , bulk-to-bulk correlation functions can be expressed in terms of correlation functions between non-local operators in the CFT:

hΦ(r1, x1)Φ(r2, x2)i =

Z

dY dY0K(r1, x1|Y )K(r2, x2|Y0)hO(Y )O(Y0)i. (2.10)

As mentioned earlier, this procedure does not always work, which means that the smear-ing function K(r, x|Y ) does not always exist. It can be explicitly calculated in global AdS, and it has been done in [16]. There are some subtleties regarding Poincare and AdS-Rindler re-construction, while in AdS-Schwarzschild K(r, x|y) does not exist; this has been discussed in [21–23]. What is important for our discussion is that in AdS-Rindler, K(r, x|y) does not exist as a function, but, as shown in [23], it does yield sensible results when regarded as a distribution integrated against CFT expectation values. The fact that reconstruction can be carried out not only in global AdS, but also in smaller regions such as Poincare or AdS-Rindler, led to the so-called subregion-subregion duality proposal, which states that bulk reconstruction of a given AdS field should be possible, given that the AdS region we consider includes its causal wedge [24–26].

AdS2 Example We will now go over a simple case of global reconstruction for a massless

scalar in AdS2, as worked out in [27]. The following is a simple example which illustrates the

non-local nature of bulk reconstruction. We will use global coordinates, which give the metric in equation (1.50). From (1.58), we know that for m = 0 we have two possible values for the conformal dimension of the corresponding CFT operator:

∆+= 1, ∆−= 0. (2.11)

For simplicity, we will consider ∆+, in which case the two boundary values of a bulk field Φ

will be defined by φ0,R(τ ) = lim ρ→π/2 Φ(τ, ρ) cos ρ , φ0,L(τ ) =ρ→−π/2lim Φ(τ, ρ) cos ρ (2.12)

and have the simple relation

φ0,L(τ ) = −φ0,R(τ + π). (2.13)

We now want to express a bulk field Φ(τ0, ρ0)in terms of its boundary values. AdS2is a

partic-ualry simple case, since we can alternatively reconstruct Φ(τ, ρ) from either its right boundary values or its left boundary values. If we choose to consider φ0,R, we will have

Φ(ρ0, τ0) = Z ∞

−∞

dτ K(ρ0, τ0|τ )φ_0,R(τ ) (2.14)

with the smearing function

K(ρ0, τ0|τ ) = 1 2θ π 2 − ρ 0_{− |τ − τ}0_|_. (2.15)

Figure 2.2: On the left, the causal wedge C[A] of a bulk point x and its top-down view on the right. We have a subregion A of constant timeslice Σ and its internal bulk boundary χA. Figure

taken from [4].

We are therefore integrating over the values of φ0,R(τ ) that lie within the right light-cone of

Φ(τ0, ρ0). This means that a local bulk field near the right boundary can be expressed in terms of its right boundary values, and its precursor will therefore be localized. Alternatively, we can choose to express the bulk field in terms of its left boundary values, in which case the precursor will be highly non-local.

2.1.1 Causal Wedges

A Cauchy surface Σ in the CFT is a line containing all points at a single time. Given a subregion of a CFT Cauchy surface A, its boundary of dependence D[A] is the set of all points p such that every past or future-moving, timelike or null, inextendible curve (i.e. it doesn’t end at some finite point) through p must intersect A. The bulk causal future/past of a boundary region R, indicated by J±[R], is given by the set of bulk points which are causally connected to it. The causal wedge of a CFT subregion in the boundary A is defined as the intersection of its causal future and past:

C[A] ≡ J+[D[A]] ∩ J−[D[A]]. (2.16) The causal surface χA, the “rim” of the wedge, is defined as the intersection of the boundaries

of J±[R]. AdS-Rindler coordinates are of special importance since they enable us to perform the reconstruction of a bulk field only using the information in D[A], since they naturally di-vide the geometry into causal wedges. If we take the geometry to be pure AdSd+1, and we take

the boundary Cauchy surface Σ to be a constant time-slice at t = 0, then the causal wedge of one hemisphere of Σ, A is given by the AdS-Rindler wedge.

Figure 2.3: Rindler coordinates in AdS3, which divide AdS into two wedges, the shaded region

is the right Rindler wedge. Figure taken from [3]. 2.1.2 AdS-Rindler

Rindler coordinates describe an observer moving along a constant-acceleration path. In Minkowski spacetime, they describe the near-horizon limit of a black hole. In AdS, Rindler coordinates di-vide AdS in two parts, known as Rindler wedges. Starting from the embedding equation (1.43), we can define AdS-Rindler coordinates in the following way:

X0 = `pρ2<sub>− 1 sinh(τ /`)</sub>

X1 = `pρ2<sub>− 1 cosh(τ /`)</sub>

X2 = `ρ sinh x cos θ1

. . .

Xd−2= `ρ sinh x sin θ1. . . sin θd−3cos θd−2

Xd−1= `ρ sinh x sin θ1. . . sin θd−2cos φ

Xd= `ρ sinh x sin θ1. . . sin θd−2sin φ

Xd−1= `ρ cosh x (2.17)

with the following ranges:

x ≥ 0 ρ > 1 − ∞ < τ < ∞ 0 ≤ θi ≤ π 0 ≤ φ < 2π. (2.18)

The Rindler-AdS metric is then given by: ds2 = −(ρ2− 1)2dτ2+ dρ

2

ρ2_{− 1}+ ρ

2_(dx2_{+ sinh}2_xdΩ2

d−2). (2.19)

The causal surface χAcan then be obtained by taking the limit ρ → 1 at fixed τ , while A itself

is given by ρ → ∞ and τ = 0. Using bulk isometries or, equivalently, boundary conformal transformations, we can reproduce the causal wedge for any circular region in Σ. Therefore, in order to reconstruct a bulk field in AdS-Rindler which is close to the boundary, we will only need to access information in a CFT subregion D[A] or, if we then express the CFT operators

Figure 2.4: Violation of timeslice axiom in AdS-Rindler reconstruction. A top-down view of a constant timeslice of AdS is shown. Suppose we choose to reconstruct ϕ considering the Rindler wedge shaded in blue, leaving out a single boundary point Y . Bulk locality requires ϕto commute with O(Y ). If we take the precursor representation of ϕ, we will also expect O(Y )to commute with the CFT operators on the boundary of Rindler wedge, thus violating the time-slice axiom. Image taken from [3]

on the same timeslice, A.

2.2

Paradoxes

Bulk Locality The aforementioned framework for bulk reconstruction can lead to inconsis-tencies regarding bulk locality. Bulk locality requires an operator in the bulk to commute with all spacelike separated operators. Therefore, all bulk operators should commute with oper-ators on the boundary. Using the bulk reconstruction prescription explained in the previous section, we know that we can express a bulk operator in terms of CFT operators living on its boundary. This means that the precursor representation of a bulk field, given by (2.9), should also satisfy this condition. Using the CFT Hamiltonian, we can re-write all operators on the right hand side of (2.9) in terms of Heisenberg picture fields on a single time-slice in the CFT. In the case of global reconstruction, since we are considering the whole boundary, this im-plies that the precursor should commute with all CFT operators at a single timeslice. Since the precursor can be expressed in terms of CFT operators, this means that we have a nontrivial operator which commutes with all operators in the CFT at a constant timeslice. This clearly contradicts the time-slice axiom (see section1.1.1). This paradox also arises in AdS-Rindler re-construction. Suppose we choose to reconstruct ϕ considering the Rindler wedge which leaves out a single boundary point Y . Bulk locality requires ϕ to commute with O(Y ). If we take the precursor representation of ϕ, we will also expect O(Y ) to commute with the CFT operators on the boundary of Rindler wedge, thus violating the time-slice axiom. Since locality is essential for any sensible quantum field theory, this suggests that bulk reconstruction should not always be possible, but that it breaks down at some point. Establishing the precise conditions under which this happens is then essential if we want to gain a deeper understanding of both global reconstruction and the subregion-subregion duality.

A B C Φ x1 x2 x3 Φ

Figure 2.5: Bulk reconstruction. On the left, we have a top-down view of a constant timeslice in AdS3 with three non-overlapping causal wedges A, B and C. Since Φ lies outside of A, B, C,

we can’t recover it from the CFT by looking at any of the given causal wedges. However, the union of any two of these subregions will include Φ, and we will therefore be able to reconstruct it from A ∪ B, B ∪ C or A ∪ C. On the right, we have illustrated a 3-site toy model of holography, in which the boundary CFT is an O(N) gauge theory. We can define 3 distinct gauge invariant operators on the boundary which have support on any 2 points and we can reconstruct Φ using any two of these operators.

Multiple Representations AdS-Rindler reconstruction has the property that the same bulk field φ(x) can be reconstructed from more than one CFT region. Suppose we have three non-overlapping causal wedges A, B, C, and a bulk field Φ which lies outside all of them. It is then not possible to reconstruct Φ by considering any single one of these wedges. However, we can have a situation in which we can reconstruct φ(x) in A ∪ B, B ∪ C or A ∪ C (see figure2.5). We thus have three representations φAB, φBC, φAC for the same bulk field φ(x). Although these

operators should all correspond to same field in the bulk, they clearly involve different CFT operators, since they have support on different CFT regions. It is then not clear in what sense these different representations of Φ are equivalent.

2.2.1 Quantum Error Correction

The contradiction regarding the existence of multiple CFT representations of a single bulk field can be reconciled by using the language of quantum error correction, as claimed in [3]. We will only provide an intuitive explanation of their argument, since for our purposes we will not need the details of this proposal. We will first start with defining some concepts from quantum information theory that we will use. A quantum system can be in a pure or mixed state. A pure state can be written as an outer product

ρ = |ψihψ| (2.20)

while a mixed state is a statistical distribution of multiple states, each with an assigned proba-bility, and it has the form

ρ =X

n

This means that a measurement on a pure state will always yield results related to only one quantum state, whereas with a mixed state we cannot know beforehand what state we will measure. The components of ρ in some basis form the density matrix of the system. A max-imally mixed state is one whose density matrix is proportional to the identity operator 1. If we are dealing with a composition of independent physical systems, we can still extract infor-mation about one of two systems, while ignoring the other ones. Suppose we have a bipartite system, whose Hilbert space can be written as a tensor product

H = H_A⊗ H_B (2.22)

then we can calculate the state of system A from the density matrix of the whole system ρABby

taking its partial trace with respect to B. In other words, we “trace out” B with the following operation:

trB(|a1iha2| ⊗ |b1ihb2|) = |a1iha2|tr(|b1ihb2|) (2.23)

and obtain the reduced density matrix

ρA=trB(ρAB). (2.24)

Moreover, we can define products of operators acting on systems A and B which have the following properties: (OA⊗ ˜OB) ⊗ (O0A⊗ ˜O 0 B) = OAOA0 ⊗ ˜OBO˜0B (2.25) tr(OA⊗ ˜OB) =tr(OA)tr( ˜OB) (2.26) (OA⊗ ˜OB)†= O † A⊗ ˜O † B. (2.27)

Generally, an error correcting algorithm consists in encoding a message of k bits in n > k bits. The simplest quantum error correction protocol deals with three-state “qutrits”. Suppose we want to send a quantum state of k qutrits, but we want to ensure that the message will still be received in its entirety, even if some qutrits get lost. We can encode our message in a larger set of states, so that we will be able to account for partial loss of information. Let’s work out a simple example. Suppose we want to send the following state:

|ψi = a|0i + b|1i + c|2i (2.28)

and we use three qutrits to encode one single qutrit:

|0i → |˜0i = |000i + |111i + |222i

|1i → |˜1i = |012i + |120i + |201i (2.29) |2i → |˜2i = |021i + |102i + |210i

which span what is known as the code subspace | ˜ψi. Note that if we only look at one of the qutrits in the code subspace, we will not be able to recover any information, since the reduced

density matrix on any qutrit will give the maximally mixed state. Conversely, by looking at any two qutrits from the code subspace we will be able to recover the entire message (2.28). We can use the following unitary transformation Uij, which only acts on two states:

|00i → |00i |01i → |12i |02i → |21i

|11i → |01i |12i → |10i |20i → |11i (2.30) |02i → |21i |10i → |22i |21i → |20i.

If we act with this on the code subspace, we will always be able to recover the original message |ii, regardless of the two states we act on

(Uij⊗ 1k)| ˜ψi = |ψi ⊗ (|00i + |11i + |22i) (2.31)

The idea is that, using this protocol, it’s possible to construct operators which are defined in terms of two of the three qutrit states of (2.29), but which act in the same way on the original message (2.28). This can be done by constructing operators with the following form

˜

O_ij ≡ U_ij†OU_ij (2.32)

where O is an operator which acts on the single qutrit state (2.28). Oij will first decode the

message, act on it and encode it again. Since the message can be recovered from any two qutrits, it is possible to construct different operators which will act in the same way on the single qutrit message. In a similar way, in AdS/CFT we can construct the same bulk field from different portions of boundary data, and thus have different CFT operator representations of the same field in AdS. In this case our code subspace would be the linear span of states generated by some set of precursor operators Φi(x)acting on the vacuum |Ωi

|Ωi, Φ_i(x)|Ωi, Φi(x1)Φi(x2)|Ωi, . . . (2.33)

The freedom of choosing a causal wedge in AdS-Rindler bulk reconstruction corresponds to the different code subspaces that we can consider. This is the essence of this proposal: the claim in [3] is that holography is essentially a quantum error correcting algorithm and that, using the AdS-Rindler reconstruction, statements regarding the possibility of recovering a bulk field from a given a CFT subregion translate into statements involving the possibility of correcting for erasures in the code subspace. Recasting holography in this framework enables us to make the state more precisely under which conditions bulk reconstruction breaks down, thus also addressing the paradox regarding bulk locality. In fact, if we consider AdS/CFT as an error correcting mechanism, the requirement that a bulk field at the center of AdS must commute with all boundary operators can be relaxed. The commutator of the bulk field precursor with the boundary operators must vanish only when acting on the states in the code subspace. What is missing from this proposal is an explicit realization of the quantum error correcting code in the CFT. This led to a new insight regarding the role of gauge invariance in bulk reconstruction,

which resulted in an alternative solution to these paradoxes. 2.2.2 Gauge Invariance

In [5], it is argued that the freedom of choice of causal wedge to reconstruct a bulk field is related to the gauge invariance of the boundary theory. Following the structure of [5], we will first consider a toy model of holography on a discrete lattice, which directly relates to the mul-tiple representations paradox, and and we will then generalize this model in the continuum.

Discrete Model

The simplified version of AdS/CFT we will analyze consists of a three-site lattice with a bulk field at its center. This is an idealized version of a constant timeslice in AdS, where the three sites on the boundary will constitute our CFT (see figure2.5). We take our boundary theory to be an O(N) gauge theory consisting of N scalar fields φia, with i = 1, . . . N and a = 1, 2, 3. The

gauge invariant quantities on a single site are φ · φ, π · π, φ · π and they generate an SL(2,R) algebra:

[φ · φ, φ · π + π · φ] = 4N iφ · φ (2.34) [φ · π + π · φ, π · π] = 4N iπ · π (2.35) [φ · φ, π · π] = 2N i(φ · π + π · φ). (2.36) The matter fields are given by the vector (φi

a, πai) since it transforms as the fundamental

rep-resentation of this algebra. We will now address the paradoxes explained in section2.2using this simplified model. First, we will see if we can construct a gauge invariant operator in the boundary which commutes with all local operators. With the requirement of gauge invari-ance, we are only interested in local gauge invariant operators, which makes it easier to find an operator which commutes with all of them. One possible candidate for a local precursor is

Oa= Aφa· φa+ Bφa· πa+ Cπa· φa+ Dπa· πa. (2.37)

This is a single trace operator, since it contains one sum over the gauge index. It commutes with all gauge invariants restricted to any site b 6= a, but we also need to make it commute with those restricted to site a. We start by computing the commutators of Oawith πa· πaand

φa· πa

[Oa, φa· φa] = −2N i[(B + C)φa· φa+ D(φa· πa+ πa· φa)] (2.38)

[Oa, φa· πa] = N [2Aiφa· φa+ iC(φa· πa− πa· φa) − 2iDπa· πa]. (2.39)

If we impose constraints on A, B, C, D such that the commutators vanish, we find that we nee to set them all to 0, which is obviously too strong of a requirement. We can check if we can construct double trace operators which satisfy this requirement. Since we have chosen our theory to be O(N) invariant, we know that the generator of gauge transformations on any

given site

Lij_a = φi_aπj_a− φj_aπ_ai (2.40) will commute with all local gauge invariants. In order to make this gauge invariant, we can simply contract two Lij_{’s and we get the following bilocal gauge invariant, which still}

com-mutes with all local gauge invariants:

Pab = LijaL ij

b . (2.41)

We therefore find that imposing a global gauge invariance enables us to violate the timeslice axiom. However, it should be noted that a more realistic model would have a local gauge invariance, meaning that we would need to fix a gauge and our candidate operator would also need to commute with the gauge fields. If we want to reconstruct a bulk field, we can express it in terms of the Pab’s. If we want the precursor expression for a bulk field at the origin, we

will have

Φ(0) = P12+ P23+ P31. (2.42)

We can now express the precursor in terms of the total O(N) generator

Lij = Lij₁ + Lij₂ + Lij₃ (2.43)

in the following way:

P12= LijLij2 − P22− P32 (2.44)

= LijLij₁ − P11− P31. (2.45)

Since the total O(N) generator annihilates physical states |Ψi, we will have

P12|Ψi = −(P22+ P32)|Ψi (2.46)

= −(P11+ P31)|Ψi. (2.47)

This means that we can express Φ(0) using any two of the three Pab’s. If we have access to

any two sites, we can fully reconstruct Φ(0) and correct for the erasure of the missing site. If we only have access to one site, however, we lose all information, much like in the error correction mechanism discussed in section2.2.1. This suggests that it is not necessary to recast holography in terms of quantum information theory, but that the gauge invariance of the CFT automatically corrects for erasures.

Continuum Generalization

In the continuum, we consider the 2-dimensional O(N) vector model, where our theory will have N free massless scalars. According to the AdS/CFT correspondence, the operator φ2, with ∆ = 0, will be dual to a free massless scalar in AdS3 (see section1.3). We will consider the

will be of order N0_{, while all higher order corrections will be powers of 1/N . The CFT action} will be given by S = 1 8π Z dx dt N X i=1 ∂µφi(x, t)∂µφi(x, t) (2.48)

The theory is O(N) invariant, which means that it is invariant under rotations in N dimensions. Note that so far we are considering a global gauge symmetry. More precisely, (2.48) is invariant under the following transformations:

φ(x, t) → Aφ(x, t), A−1= AT (2.49)

where A is an N × N matrix. The gauge invariants of this theory are given by the bilocal operators:

φ(x, t) · φ(y, t0), φ(x, t) · π(y, t0), π(x, t) · φ(y, t0), π(x, t) · π(y, t0). (2.50)

Following the procedure outlined in section2.1, the precursor expression for a free massless scalar in Ads3Φfound in [5] is given by the bilocal operator

Φ(ρ, x, t) = √1 N Z 2π 0 dx0dx00f (ρ, x|x0_{, x}00_)φ(x0_{, t) · φ(x}00_{, t) + g(ρ, x|x}0_{, x}00_)π(x0_{, t) · π(x}00_{, t)} (2.51) where f (ρ, x|x0, x00)and g(ρ, x|x0, x00)are the appropriate smearing functions. The 1/√N fac-tor ensures that the two-point function will be O(1), while the three-point function will be of O(1/N). We also know that for large N, higher-point functions will factorize into products of two-point functions (see section 1.1.2) so, in the large N limit, it will suffice to consider the two-point functions. This will have important consequences in evaluating if a precursor oper-ator or its commutoper-ator actually contributes to the two-point function at leading order in 1/N . In the expression (2.51), we are integrating over the full range of x0, x00, so we are carrying out global reconstruction. As we have seen in section2.1, we can also consider a smaller region of the boundary, meaning that we don’t have to consider the full range of x0, x00. We will now show how this is related to the O(N) gauge symmetry of the theory.

The precursor (2.51) can be expressed in terms of left and right-moving modes α, ˜α:

Φ(ρ, x, t) = √1 N ∞ X m,n=−∞ hmn(ρ, x) αm(t) · ˜αn(t) (2.52)

and the freedom of choosing a causal wedge corresponds to the freedom in the choice of the function hmn. Since we are in the large N limit, we are only considering the contributions from

the two-point function, and for mn < 0 (2.52) annihilates the vacuum in both directions and makes no contribution to the two-point function. Therefore, we are allowed to modify hmnin

the following way:

where λmnis an arbitrary function. It turns out that this is related to the generator of gauge transformations Lij = ∞ X n=1 2 n(α [i −nαj]n + ˜α [i −nα˜j]n) + L ij 0, (2.54)

where Lij₀ acts on the zero modes. Since Lij _{annihilates all physical states, and}

[Lij, αi_mα˜j_n] = 0, (2.55)

for mn < 0 we can shift Φ(ρ, x, t) by

Φ → Φ +X m,n 1 √ NγmnL ij_αi mα˜jn (2.56)

where γmn is an arbitrary function. Depending on the values of m and n, we may need to

take into account normal ordering, so (2.56) may pick up a normal ordering constant. The commutator of Lij with a single mode αimwith m < 0 is given by

[Lij, αi_m] = (1 − N )αj_m. (2.57)

We obtain a similar result for the right movers:

[Lij, ˜αj_n] = (1 − N ) ˜αi_n for n < 0. (2.58)

So for m < 0, n > 0, we have

Lijαi_mα˜j_n= αi_mLijα˜j_n+ (1 − N )αm· ˜αn (2.59)

= −N αm· ˜αn+ O(1/N ) (2.60)

and similarly for m > 0, n < 0, where we would have to commute the ˜αjnthrough. Therefore,

we can shift Φ by Φ → Φ − √ N X m,n αm· ˜αn (2.61)

which is a specific case of the transformation (2.53). There seems to be a relation between the freedom of choice of the smearing function of the precursor operator and the gauge symmetry of the CFT. It is not yet clear whether or not gauge invariance plays a fundamental role in reconstruction, and still remains an open question.

2.3

Motivation

The only way of deepening our understanding of how holography works is by analyzing how bulk information is stored on the boundary. This is a notoriously difficult problem, especially if we try to make quantitative statements, since it is inextricably connected to the issue of approximate locality and its breakdown. The approaches that we have explained provide a

framework in which we can investigate the emergence of bulk locality by studying precursor operators, boundary operators which are related to the bulk in a nonlocal way. The study of precursors is therefore essential in the understanding of holography.

The two proposed solutions to the paradoxes presented in this chapter are closely related, but fundamentally different. On the one hand, we have the very radical proposal that holog-raphy is in fact a highly complex quantum computer, and that we therefore need to change the language we use to study AdS/CFT if we wish to gain any further insight into its workings. On the other hand, we have a more conservative viewpoint which attempts to incorporate ele-ments of quantum error correction in the standard framework of AdS/CFT. The novelty here is that the gauge symmetry present in the boundary theory appears to play a bigger role than we thought. If this proposal is indeed correct, or at least points to the correct direction, there will be no need to adopt the language of information theory to study the more fundamental ques-tions regarding holography, such as the emergence of bulk locality. It would therefore be more cautious to first verify the consistency of the gauge invariance proposal before abandoning our standard picture of holography and adopting a new paradigm.

With these considerations in mind, our choice is to investigate the gauge invariance pro-posal further, by checking if this model incorporates the properties related to approximate locality that we expect from holography, and if gauge invariance provides a way out of the paradoxes which seem to plague bulk reconstruction.

Results

In the following, we will investigate the conditions under which the timeslice axiom is violated in a two-dimensional CFT. As shown in section2.2.2, we already know that in an O(N) gauge theory on a lattice it is possible to construct a nontrivial operator which exactly commutes with all local operators of the theory at a constant timeslice. We will see if this also applies in the continuum. We will first focus on the question of whether or not it is possible to construct a nontrivial operator which exactly commutes with all local operators of the theory at a constant timeslice. We will first see what happens at finite N, by considering a single free scalar. Then, we will impose a global O(N) gauge invariance and take the large N limit of the theory. We will then consider the same AdS3/CF T2 model as in [5], and check whether or not the precursor

operator of this model respects bulk locality, to leading order in 1/N . We will attempt to find the conditions this operator would need to satisfy.

3.1

Investigating the Timeslice Axiom

The goal is to construct an operator which commutes with all local operators at a constant timeslice. For simplicity, we will take our constant timeslice to be at t = 0. As stated in section

1.1.3, since we are dealing with a free field, n-point functions factorize to products of two-point functions, so we will restrict our analysis to second order operators. We can construct operators of the form: φn_{, (∂φ)}n_{, π}n_{and linear combinations. Let’s first see what happens if}

we only consider combinations of φ’s and π’s. Clearly, any combination of φ(x1)’s and π(x2)’s

will commute with all local operators at t = 0 except for π(x1) and φ(x2), in which case the

commutator will diverge. If we extend this construction to an operator involving derivatives, we can consider an operator of the following form:

O(z, ¯z) = a φ(z, ¯z) + b ∂φ(z) + c ¯∂φ(¯z) (3.1) and see if we can impose any conditions on the coefficients a, b, c that make O commute with all local operators. However, taking a derivative with respect to z or ¯z will change the k-dependence of the infinite sum (1.37), so it will not be possible to make the contributions of φ,

∂φand ¯∂φto the commutators with all local operators vanish simply by imposing conditions on a, b, c, since they cannot depend on k.

Smearing

Another way in which we can try to construct an operator that commutes with all local op-erators is by smearing over one of the spatial coordinates. Let’s first see what happens if we consider a first-order operator. Let’s take the operator

O₁ ≡ Z b

a

dx f (x)φ(x) (3.2)

and commute it with π(x0)with a < x0 < b

[O1(x), π(x0)] = if (x0). (3.3)

This can only be set to 0 if f (x) vanishes in the region of integration. Let’s now consider a bilocal operator φ(x1)φ(x2)and integrate it against a function f (x1, x2):

O2 ≡ Z b a dx1 Z d c dx2f (x1, x2) φ(x1)φ(x2). (3.4)

When we commute this operator with π(x3), where x3 ∈ [a, b], x/ 3∈ [c, d], we get

[O2(x1, x2), π(x3)] = i

Z b

a

dx1f (x1, x3)φ(x1). (3.5)

Again, this will only commute if f (x1, x2)vanishes in the boundary of integration. Therefore,

in the case of a single free scalar field, it is not possible to construct an operator which com-mutes with all local operators. We will now see what happens if we add two ingredients: we consider N free scalar fields and we take our theory to be gauge invariant.

3.1.2 Large N and Gauge Invariance

If we impose a global O(N) gauge invariance, we will only need to consider the gauge invari-ants of our theory, since they generate the physical states. Our goal is then to construct an operator which commutes with all the gauge invariants of the theory at a given time-slice. It is possible to construct an operator which commutes with all the local gauge invariants by using the generator of gauge transformations at an arbitrary point, given by

Lij(x) = φi(x)πj(x) − φj(x)πi(x), (3.6)

which is the continuum version of the operator that was found in the lattice simplification of this model, given by (2.40). Since the gauge invariant quantities are the same as those in the discrete model, we already know that Li_{j(x)commutes with all gauge invariants. To leading}

order in 1/N (see appendix A), the mode expansion of Lij_{(x)is given by} Lij(x) = i 2π X m,n6=0 1 m  α_m[iαj]_n − ˜α[i−mα˜ j] −n  ei(m+n)x+ 1 πφ [i 0π j] 0 (3.7)

Integrating this expression over all possible values of x will yield the total O(N) generator, given by (2.54). However, the total gauge generator would annihilate all physical states, and we are looking for a nontrival operator which commutes with all local gauge invariants, so we will consider the O(N) generator at a specific point x. This is not a trivial operator, its contribution to the two-point function is can be found using (A.14) and (A.4). This operator is not gauge invariant, but we can contract two Lij’s to construct a gauge invariant bilocal operator which still commutes with all local gauge invariants

P(x, y) = L(x) · L(y), (3.8)

which is the continuum version of (2.41). We can therefore conclude that the lattice model presented in2.2.2procedure can be generalized and that, using gauge invariance, we can still construct an operator which violates the timeslice axiom.

3.2

Bulk Locality in AdS

/CFT

The precursor operator found in [5] is of the form O0₂ = √1

N Z

dx dy [f (x, y) φ(x) · φ(y) + g(x, y) π(x) · π(y)] (3.9)

which does not commute with all local gauge invariants. We will consider global reconstruc-tion, so the region of integration for both x and y will be [0, 2π], given that we are on the cylinder. We will see if it is possible to make O₂0 with the local gauge invariants of the the-ory by only considering leading order terms in 1/N and by imposing appropriate conditions on the smearing functions f (x, y) and g(x, y). The commutators of O₂0 with the local gauge invariants are given by

[O0₂(x, y),√1 Nφ(z) · φ(z)] = − i N Z 2π 0 dx [g(z, x) + g(x, z)][φ(z) · π(x) + π(x) · φ(z)] (3.10) [O0₂(x, y),√1 Nφ(z) · π(z)] = [O 0 2(x, y), π(z) · φ(z)] = i N Z 2π 0 dx {[f (x, z) + f (z, x)][φ(z) · φ(x)] − [g(x, z) + g(z, x)][π(x) · π(z)]} (3.11) [O0₂(x, y),√1 Nπ(z) · π(z)] = 2i N Z 2π 0 dx [f (x, z)φ(x) · π(z) + f (z, x)π(z) · φ(x)]. (3.12) Apart from simply setting f (x, y) = g(x, y) = 0, we see that in order to make (3.10) and (3.11) vanish, we can also impose f (x, y) = −f (y, x) and g(x, y) = −g(y, x). However, after a straightforward calculation we find that these conditions are too strong, as they would also

make O0₂ itself vanish. The question is then if we can impose some weaker conditions on f (x, y) and g(x, y) which make the commutator approximately vanish (to leading order in 1/N). Now, using the mode expansions of φ(x) and π(x), we will see how the commutators (3.10)-(3.12) contribute to the two-point function. Using the results from appendix Using the results from appendix A, we find that the leading order contributions of the terms that appear in the commutators (3.10)-(3.12) are given by

1 Nφ(z) · π(x) = 1 N 1 2πφ0· π0+ i 2π X n>0 cos[n(z − x)] + O(1/N ) (3.13) 1 Nπ(x) · φ(z) = 1 N 1 2ππ0· φ0− i 2π X n>0 cos[n(z − x)] + O(1/N ) (3.14) 1 Nφ(z) · φ(x) = 1 Nφ 2 0+ X n>0 2 ncos[n(z − x)] + O(1/N ) (3.15) 1 Nπ(z) · π(x) = 1 2 X n>0 n cos[n(z − x)] + O(1/N ). (3.16)

Using (3.13) and (3.14), we find that the commutator (3.10) becomes of the form 1 N 1 2π[φ(z) · π(x) + π(x) · φ(z)] ∼ 1 N(φ0· π0+ π0· φ0) (3.17) which is independent of x, z. Therefore, to make this contribution vanish it would suffice to set

Z 2π

0

dx [g(z, x) + g(x, z)] = 0. (3.18) From (3.13) and (3.14), we also see that in order to make (3.12) vanish, we need to impose the following conditions on f (x, z):

f (x, z) = f (z, x),

Z 2π

0

dx f (x, z) = 0 (3.19)

Finally, (3.15) and (3.16) lead to the following constraints on f (x, z) and g(x, z) which make the commutator (3.11) vanish Z 2π 0 dx f (x, z) " 1 Nφ 2 0+ X n>0 2 ncos[n(z − x)] # = 0 (3.20) Z 2π 0 dx [g(x, z) + g(z, x)] " X n>0 n 2 cos[n(z − x)] # = 0 (3.21)

Evaluating the infinite sum in (3.20) yields X n>0 2 ncos[n(z − x)] = − log  2 sin2 x − z 2  − 2πim+ (3.22)

where we have used the properties of the complex logarithm (1.39). If we evaluate the infinite sum in (3.21), we get X n>0 n 2 cos[n(z − x)] = 1 4[cos(x − z) − 1] (3.23)

The conditions we need to impose on f (x, z) and g(x, z) simplify to

f (x, z) = f (z, x) Z 2π 0 dx f (x, z) = 0 Z 2π 0 dx f (x, z) log  sin2 x − z 2  = 0 Z 2π 0 dx [g(z, x) + g(x, z)] = 0 Z 2π 0 dx[g(x, z) + g(z, x)] sin2[(x − z)/2] = 0 (3.24) (3.25) (3.26) (3.27)

A class of solutions is given by

f (x, y) = cos(x − y) − 2 cos[2(x − y)] (3.28) g(x, z) = sin2[(x − y)/2][α cos(x − y) + β cos[2(x − y)]. (3.29)

where α and β are arbitrary coefficients. We can therefore conclude that, in an AdS3/CF T2

O(N) vector model, it is possible for a precursor of the form (3.9) to commute with all local gauge invariants, to leading order in 1/N , given that the smearing functions satisfy the condi-tions given by (3.24)-(3.29).