UNIVERSITEIT VAN AMSTERDAM

MSc PhysicsTheoretical Physics

MASTER THESIS

Bulk Locality in AdS/CFT

Reconstruction from Bilocal Operators

by

Davide DISPENZA

10064605

September 2015

54 ECTS

Supervisor:Dr. Ben FREIVOGEL

Examiner:Prof. Dr. Erik VERLINDE

Institute for Theoretical Physics

Abstract

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 theory in d dimensions.How the correspondence works is not yet completely understood. The open question of howlocality emerges in AdS from the boundary is the focus of this project. When reconstructingquantities 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 usingquantum error correction and gauge invariance to explain how bulk information is encodedin the CFT will be analyzed. We find that this model agrees with our expectations regardinglocality and its breakdown.

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Contents

Abstract i

Introduction 1

1 Preliminaries 41.1 Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Timeslice Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Large N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Conformal Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Anti-de Sitter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 AdS/CFT Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Precursors 152.1 Bulk Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Causal Wedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.2 AdS-Rindler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 Quantum Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.2 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Results 293.1 Investigating the Timeslice Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Finite N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.2 Large N and Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Bulk Locality in AdS3/CFT2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Discussion 34

A Large N counting 36

Bibliography 39

Acknowledgements 41

ii

Introduction

One of the greatest problems in theoretical physics is the unification of quantum field theoryand 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 produceimportant quantum effects. This is why our best efforts at understanding quantum gravitystarted with the study of black holes. In the 1970s, considerations by Hawking regarding thethermodynamic 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 thedegrees of freedom of a black hole can be mapped to its surface provided the first hint at thenonlocal nature of gravity. This was already a strong indication that any local field theoreticdescription of nature should break down once gravitational effects become significant, andthat a theory of quantum gravity should also possess nonlocal properties. This was furthermotivated by the discovery of Hawking radiation, the idea that black holes radiate particlesand 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 theprocess, which contradicts unitarity, one of the fundamental properties of quantum mechan-ics. This became known as the black hole information paradox, which still remains unsolvedtoday. Various proposals have been put forward since its formulation, which all involve, insome 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 ofnature 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 holographicprinciple, 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 early1990’s when, using the thermodynamical properties of black holes, he reasoned that, givena system of volume V and surface area A, the entropy inside A is maximized if the volumecontains 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 anyclosed 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 bya topological quantum field theory, in which all physical degrees of freedom can be projectedonto the boundary”. This statement has forever changed the way we look at nature: whereasin the beginning quantum mechanics and general relativity were though of as fundamentally

1

2

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

In 1997, Maldacena conjectured a duality which would become known as the AdS/CFTcorrespondence 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 ofquantum 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 ddimensions. The higher dimensional theory is known as the bulk while the lower dimensionaltheory is known as the boundary theory. This is because, in a sense, the CFT lives on theboundary of AdS, since the mapping between the two theories is between operators whichlive on the boundary of AdS and operators which live on the CFT. We therefore have a fullcorrespondence between two radically different theories; we can translate questions regardingthe bulk theory as questions regarding the boundary, and vice versa. The key to unlocking themystery 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 hasnot yet been rigorously proved. In fact, there are still many open questions regarding how thecorrespondence works. We know that we can translate quantities in one theory to the otherby 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 ishow the theory in AdS can be local, even if it is possible to reformulate it as a theory livingonly on the boundary. This is known as the problem of the emergence of bulk locality, and it isstrictly related to the nonlocal nature of gravity. In fact, we expect locality to break down as wemove away from the semiclassical approximation of our bulk theory but, in the appropriateregime, we expect our bulk theory to be approximately local. Understanding the breakdownof bulk locality is therefore an important piece of the puzzle of quantum gravity, and it is themain focus of this project.

We can investigate the problem of bulk locality by asking how information in the CFT isencoded 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 propertiesof a quantum field theory. Bulk locality requires that fields in the bulk commute with fieldson the boundary, since they are spacelike separated. This seems to contradict the time-sliceaxiom, which states that any operator which commutes with all operators of the theory at aconstant timeslice must be proportional to the identity. This is because when we reconstructa bulk field from the boundary, we obtain an expression in terms of CFT operators, calledthe precursor operator. We can then use time evolution to express the precursor in terms ofHeisenberg picture fields on the same timeslice. If we take into account bulk locality, whichrequires the precursor operator to commute with operators on the boundary, we find thatthe timeslice axiom is violated. Related paradoxes can be found if we consider alternativeways in which we can perform bulk reconstruction. It turns out that when carrying out bulkreconstruction we can consider all of the boundary CFT, known as global reconstruction, or a

3

smaller subregion, known as the causal wedge of the bulk field. The latter proposal is knownas the subregion-subregion duality. This turns out to be very puzzling since we can havedifferent 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 isthat AdS/CFT is a quantum error correction mechanism. This would then become a questionof finding the right algorithm which correctly reproduces the AdS/CFT correspondence. Analternative explanation was put forward by Mintun, Polchinski and Rosenhaus in [5], where itis suggested that the gauge invariance of the CFT automatically employs some form of errorcorrection, 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 developedin the gauge invariance proposal. Although the gauge invariance proposal is still at an earlystage, it offers a new way to approach the problems related to bulk reconstruction. The mainclaim of this proposal is that the gauge invariance of the CFT is essentially responsible forthe correct encoding of information from the boundary to the bulk, analogous to quantum er-ror correction. It provides a simple model in which the error correcting code can be explicitlydescribed 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 ofgauge invariance in bulk reconstruction. We will do this by checking if the model is consistentwith our expectations regarding bulk locality.

This work is structured in the following way. In chapter 1, we review some properties ofquantum field theory which will be important for our discussion, followed by a brief overviewof the AdS/CFT correspondence. In chapter 2 we focus on bulk reconstruction, where we goover 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 seewhether or not our expectations regarding bulk locality are met, and conclude with a discus-sion.

Chapter 1

Preliminaries

1.1 Quantum Field Theory

1.1.1 Timeslice Axiom

Quantum field theory (QFT) is usually taught in its Lagrangian formulation, which allowsfor a more heuristic approach, without being too strict on mathematical rigor. In fact, we arestill missing a comprehensive mathematically precise formulation of quantum field theory, butthere has been some progress in this direction. Such a formulation of QFT can be useful whendefining concepts such as causality and locality. In 1964, Wightman and Streater developed anaxiomatic formalism in an attempt to define in a mathematically precise way those propertieswhich characterize a sensible QFT 1. These became known as the Wightman axioms, and theydeal with the notions of field, causality and transformations. The axiom of causality and thetimeslice axiom will have important implications in our discussion.

In any relativistic quantum field theory, causality implies that fields φ(x, t) and operatorsO(x, t) which are spacelike separated commute. In other words, a field will be able to interactonly with fields which lie inside its lightcone. Another important feature of a QFT is that anyoperator acting on the Hilbert space of the theory can be expressed as a linear combinationof products of fields. This is known as the completeness axiom and it implies that if we havean operator which commutes with all the operators of the theory, including the ones whichlie inside its lightcone, it must be proportional to the identity operator. Furthermore, usingdynamical 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 operatorsat a constant time-slice will also be proportional to the identity. The time-slice axiom has notbeen 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 gaugetheories starting from the Wightman axioms remains an unsolved problem. In fact, we will seethat the timeslice axiom is violated in a field theory with a global O(N) gauge symmetry.

1For more details see [6, 7]

4

Chapter 1. Preliminaries 5

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 moredetails, see [10, 11]. In some cases, given a theory with N fields, it is possible to greatly simplifyit 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 besolvable using perturbation theory. Let’s take the O(N) vector model as an example, where wehave 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) =

∫ddx

1

2[∂µφ(x)]2 −NV [φ2(x)/N ]

. (1.1)

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

NV =1

2m2φ2(x) +

1

8

λ

N[φ2(x)]2. (1.2)

We have, in a sense, rescaled all the second-order operators to be of order 1/N . There areseveral ways to perform the 1/N expansion of the theory, but we want a nontrivial solutionin which we will only have contributions proportional to positive powers of 1/N . It turns outthat 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 thenbe 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∑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, wecan approximate n-point functions as products of 2-point functions:

〈φ2(x1, y1)φ2(x2, y2) . . . φ2(xn, yn)〉 ∼ 〈φ2(x1, y1)〉〈φ2(x2, y2)〉 . . . 〈φ2(xn, yn)〉. (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 dealingwith a matrix field theory, the scaling of the coupling would change in the large N limit, sincevector models have N degrees of freedom while matrix models have N2 degrees of freedom.

Chapter 1. Preliminaries 6

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 multiplyingthe 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 ofconformal transformations is that they leave null geodesics invariant, meaning lightcones willbe preserved. Therefore, if two metrics differ only by a conformal factor Ω2(x), they will havethe 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 themetric such that infinity is brought at a finite proper distance. This is known as conformalcompactification, 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]. Aconformal 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 consequenceis that the theory only supports massless excitations. We will focus on 2-dimensional CFTs,where we can define complex coordinates

z = x0 + ix1, z = x0 − ix1 (1.8)

where x0 and x1 are Minkowski spacetime coordinates. The metric components are then givenby

gzz = gzz = 0, gzz = gzz =1

2(1.9)

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

Chapter 1. Preliminaries 7

transformations in two dimensions forms the global conformal group and is given by

f(z) =az + b

cz + dwith 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 givenby

S = g

∫dzdz ∂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 2comes from the change of coordinates dzdz = 2dx0dx1. The classical equation of motion for φ is∂∂φ = 0, which means that the general solution decomposes in left-moving and right-movingparts:

φ(z, z) = φ(z) + φ(z). (1.13)

The propagator for a scalar field is given by

〈φ(z, z)φ(ω, ω)〉 = − 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, ∂nφ andcombinations. An important feature of CFTs is the operator product expansion (OPE): the factthat we can approximate two local operators inserted at nearby points as a string of operatorsat one of these points. Given a time-ordered correlation function, we can approximate theinsertion of two operators Oi and Oj with their OPE:

〈Oi(z, z)Oj(ω, ω) . . .〉 =∑k

Ckij(z − ω, z − ω)〈Ok(ω, ω) . . .〉 (1.15)

The stress-energy tensor for a scalar field is given by

Tzz = 0, Tzz ≡ T (z) = −1

2∂φ∂φ, Tzz ≡ 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 isdefined as

T (z) = −1

2: ∂φ∂φ :≡ −1

2limz→ω

(∂φ(z)∂φ(ω)− 〈∂φ(z)∂φ(ω)〉) (1.17)

When taking products of normal ordered operators, Wick’s theorem holds. A primary operatorO(ω) is one whose OPE with T (z) and T (z) truncates at order (z−ω)−2 or (z−ω)−2 respectively.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)

Chapter 1. Preliminaries 8

where . . . indicates the non-singular terms. Primary operators transform in the following way:

O(z, z)→ O(z′, z′) =

(∂z′

∂z

)−h(∂z′∂z

)−hO(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. Ina CFT in d spacetime dimensions, the condition that a scalar operator O is free is equivalent tothe fact that its conformal dimension is ∆ = d−2

2 . Alternatively, a free field is characterized bythe fact that its correlation functions factorize to products of 2-point functions [13]. In a unitaryCFT, 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 ofdegrees of freedom. In fact, the free scalar field has c = c = 1; if we were to consider Dnon-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 circlescentered at the origin. This is known as radial quantization. We start by defining our theoryon an infinite space-time cylinder with the time coordinate t going from −∞ to +∞ while thespatial 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)

x and t are related to the complex plane in the following way:

z = e−iω, z = eiω (1.25)

where ω and ω are the complex coordinates on the cylinder, given by

ω = x+ iτ ω = x− iτ (1.26)

Chapter 1. Preliminaries 9

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

φ(x, t) =∑n

einxφn(t) (1.27)

φn(t) =1

2π

∫dx e−inxφ(x, t) (1.28)

In this formalism, the mode expansion of φ is given by

φ(x, t) = φ0 + 2π0t+ i∑n6=0

1

n

[αne

in(x−t) − α−nein(x+t)]

(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 relatedto the canonical creation and annihilation operators in the following way:

αn =

−i√nan for n > 0

i√−na†−n for n < 0

αn =

−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 creationoperators for n < 0 and annihilation operators for n > 0. On the cylinder, the holomorphicderivative of φ(ω, ω) is given by

∂ωφ(ω, ω) = −π0 −∑n6=0

αneinω. (1.33)

On the plane, the holomorphic derivative of φ(z, z) becomes

∂zφ(z, z) = −iπ0

z− i∑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

∑n6=0

[αne

in(x−t) + α−nein(x+t)

]. (1.35)

The commutator between two fields is given by

[φ(x, t), φ(x′, t′)] = 2i∆t+ g(∆x+ ∆t)− g(∆x−∆t) (1.36)

Chapter 1. Preliminaries 10

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

g(y) ≡∑k 6=0

1

keiky = log(1− e−iy)− log(1− eiy) (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 ±Argω > π

0 if Arg z ±Argω ≤ π1 if Arg z ±Argω ≤ −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 commutatorbetween φ and π is given by

[φ(x, t), π(x′, t′)] =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 hasits maximum possible number of Killing vectors (d + 1)(d + 2)/2. In the case of AdSd+1, theKilling vectors are SO(d,2) generators. We can embed AdSd+1 in d + 2 dimensional flat space,and express it as

−X20 −X2

d+1 +X21 + . . .+X2

d = −`2. (1.43)

Chapter 1. Preliminaries 11

If we want to cover the whole of AdSd+1 we need to use global coordinates, given by

X0 = ` coshχ cos τ (1.44)

Xd+1 = ` coshχ sin τ (1.45)

Xi = ` sinhχ xi,

d∑i=1

x2i = 1 (1.46)

and we obtain the metric

ds2 = `2(− cosh2 χdτ2 + dχ2 + sinh2 χdΩ2d−1) (1.47)

where the coordinates have the ranges

−∞ < τ <∞ χ ≥ 0 (1.48)

and dΩ2d−1 is 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 runfrom −∞ to∞, which is known as taking the “universal cover” of AdS. If we want to derivethe Penrose diagram of AdS, we need to render the radial coordinate finite, by conformalcompactification (see section 1.1.3). This can be achieved by setting

coshχ = sec ρ sinhχ = tan ρ (1.49)

which leads to the metric

ds2 =`2

cos2 ρ(−dτ2 + dρ2 + sin2 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 originwill reach spatial infinity and return in a finite amount of proper time π, given that we im-pose reflective boundary conditions 2. Massive particles, on the other hand, will not reach theboundary before returning to the origin. This means that the boundary is timelike: it has thetopology of R× Sd−1, R being the time direction. This is clearer in the diagram of AdS in threedimensions, where the boundary is given by the surface of the cylinder.

2We need to impose reflective boundary conditions in order to preserve causality, otherwise information wouldbe able to “leak” in and out of AdS. The simplest solution is then to treat AdS as a closed system.

Chapter 1. Preliminaries 12

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 inblue and timelike geodesics are drawn in red. A massless particle sent out from the origin willreach the boundary and return in a finite proper time π, while a massive particle will not beable to reach the boundary. Spacelike slices have the topology of R× Sd−1, which is visualizedon the right, where we have a the diagram of AdS3 in global coordinates. In the context ofAdS/CFT, we have a theory of quantum gravity in AdS which corresponds to a CFT living onits boundary.

1.3 AdS/CFT Correspondence

In its strongest form, the AdS/CFT correspondence states that there is a duality between acomplete (d + 1)-dimensional theory of quantum gravity in AdS space and a conformal fieldtheory in d dimensions. This mapping is holographic in the sense that a higher dimensionaltheory is completely encoded in a lower dimensional theory. It is often stated that the CFTlives on the boundary of AdS; if we think of AdS as a cylinder (see figure 1.1), we can say thatthe CFT lives on its surface. This is why the gravitational theory is often referred to as the bulkwhile CFT is often called the boundary theory.

In its original formulation (in Maldacena’s paper [2]), the correspondence is between typeIIB 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 isinstructive to see how the parameters of the two theories are related. In the bulk, we have theAdS radius in string units `/`s and the string coupling gs while on the boundary we have theYang-Mills coupling gYM and the rank N of the SU(N) gauge group, which are related in thefollowing way

`

`s= (g2

YMN)1/4 gs = g2YM . (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/`, whilequantum corrections are functions of the string coupling gs. Therefore, we need to take theAdS radius ` to be large and the string coupling gs to be small. This means that we also need

Chapter 1. Preliminaries 13

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 section 1.1.2),which is obtained by taking

g2YMN = constant N →∞ gYM → 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 anda weakly coupled gravitational theory. This makes AdS/CFT a very powerful tool for solvingproblems 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 (seesection 1.1.2), in which case we have a semi-classical theory of quantum gravity in the bulk, inthe 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 ledto a research program known as the gauge/gravity duality, where more similar conjectureshave been made relating gauge theories to gravitational theories.

The precise form of correspondence is given by:

〈e∫ddxφ(0)(x)O(x)〉CFT = 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 boundarycondition φ(0)(x) at r = 0, and ∆ is is the conformal dimension of O, which can be expressedin terms of the dimension of the CFT d, the AdS radius ` and the mass of the bulk field m. Fora scalar field, ∆ is given by

∆(∆− d) = m2`2. (1.57)

This has two solutions:

∆± =d

2±√d2

4+m2`2. (1.58)

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

−d2

4< m2 < −d

2

4+ 1. (1.59)

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

Chapter 1. Preliminaries 14

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

limr→∞

r∆Φ(r, x) = 〈O(x)〉. (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 havethe 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 correspondto the free theory, while the CFT operators with conformal dimension ∆+ will correspond toan interacting theory.

Chapter 2

Precursors

2.1 Bulk Reconstruction

Given the vast difference between the two theories, it is natural to ask exactly how the AdS/CFTcorrespondence 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 expressthe bulk field φ in terms of boundary operators. This has been studied in a series of papersby Hamilton, Kabat, Lifschytz and Lowe (HKLL) [16–20]. This is not always possible, andthe conditions under which we can successfully reconstruct the bulk from the boundary havebeen further studied in [21, 22]. From (1.60), we know that the boundary value of a bulk fieldΦ corresponds to the expectation value of a local CFT operator. The question is then: given theboundary values of Φ (which correspond to CFT one-point functions), can we fully reconstructit in the bulk?

We will now summarize the standard reconstruction procedure. We assume that the CFTpossesses a 1/N expansion, and we take N to be large. In this regime, classical gravity willbe a good bulk approximation, as explained in section 1.3. Therefore, we first solve the waveequation 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 modeexpansion of the form

Φ(r, x) =

∫dk akΦk(r, x) + h.c. (2.2)

We then take its boundary value

Φ(x) =

∫dk akϕk(x) + h.c. (2.3)

wherelimr→∞

r∆Φk(r, x) = ϕk(x). (2.4)

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

ak =

∫dx ϕ∗k(x)Φ(x). (2.5)

15

Chapter 2. Precursors 16

Figure 2.1: Global Reconstruction in AdS3. In order to reconstruct a field at the point x, weintegrate (2.7) over the whole boundary. We can express the CFT operators to be on a constanttime-slice Σ. Figure taken from [3].

Plugging this into (2.3) gives

Φ(r, x) =

∫dk

[∫dx′ ϕ∗k(x)Φ(x′)

]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 arejustified in exchanging the order of integration, this expression simplifies to

Φ(r, x) =

∫dx′ K(r, x|x′)Φ(x′) (2.7)

withK(r, x|x′) =

∫dk ϕ∗k(x

′)Φk(r, x) + h.c. (2.8)

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

Φ(r, x) =

∫dY K(r, x|Y )〈O(Y )〉+O(1/N) (2.9)

where K(r, x|Y ) is a smearing function, which obeys the bulk wave equations of motion inits r, x indices. This is not a standard boundary value problem, since the boundary data alsodepends on time. The right-hand side of (2.9) is called the precursor operator, which consistsof CFT operators which obey the bulk equations of motion, and has the boundary conditionsimposed by the extrapolate dictionary (1.60). We thus have an expression for a free, classicalbulk field in terms of appropriately smeared, CFT operators. The integration in (2.9) is carriedout over boundary CFT coordinates, which means that the precursor will be an expressioninvolving multiple CFT operators acting at different spacetime points. Therefore, the precursor

Chapter 2. Precursors 17

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

〈Φ(r1, x1)Φ(r2, x2)〉 =

∫dY dY ′ K(r1, x1|Y )K(r2, x2|Y ′)〈O(Y )O(Y ′)〉. (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, andit 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 asa function, but, as shown in [23], it does yield sensible results when regarded as a distributionintegrated against CFT expectation values. The fact that reconstruction can be carried out notonly 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 givenAdS 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 masslessscalar in AdS2, as worked out in [27]. The following is a simple example which illustrates thenon-local nature of bulk reconstruction. We will use global coordinates, which give the metricin equation (1.50). From (1.58), we know that for m = 0 we have two possible values for theconformal 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) = limρ→π/2

Φ(τ, ρ)

cos ρ, φ0,L(τ) = lim

ρ→−π/2

Φ(τ, ρ)

cos ρ(2.12)

and have the simple relationφ0,L(τ) = −φ0,R(τ + π). (2.13)

We now want to express a bulk field Φ(τ ′, ρ′) in terms of its boundary values. AdS2 is a partic-ualry simple case, since we can alternatively reconstruct Φ(τ, ρ) from either its right boundaryvalues or its left boundary values. If we choose to consider φ0,R, we will have

Φ(ρ′, τ ′) =

∫ ∞−∞

dτ K(ρ′, τ ′|τ)φ0,R(τ) (2.14)

with the smearing function

K(ρ′, τ ′|τ) =1

2θ(π

2− ρ′ − |τ − τ ′|

). (2.15)

Chapter 2. Precursors 18

Figure 2.2: On the left, the causal wedge C[A] of a bulk point x and its top-down view on theright. We have a subregionA of constant timeslice Σ and its internal bulk boundary χA. Figuretaken from [4].

We are therefore integrating over the values of φ0,R(τ) that lie within the right light-cone ofΦ(τ ′, ρ′). This means that a local bulk field near the right boundary can be expressed in termsof its right boundary values, and its precursor will therefore be localized. Alternatively, we canchoose to express the bulk field in terms of its left boundary values, in which case the precursorwill 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 subregionof a CFT Cauchy surface A, its boundary of dependence D[A] is the set of all points p such thatevery past or future-moving, timelike or null, inextendible curve (i.e. it doesn’t end at somefinite 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. Thecausal wedge of a CFT subregion in the boundary A is defined as the intersection of its causalfuture 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 boundariesof J ±[R]. AdS-Rindler coordinates are of special importance since they enable us to performthe 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 pureAdSd+1, and we takethe boundary Cauchy surface Σ to be a constant time-slice at t = 0, then the causal wedge ofone hemisphere of Σ, A is given by the AdS-Rindler wedge.

Chapter 2. Precursors 19

Figure 2.3: Rindler coordinates inAdS3, which divideAdS into two wedges, the shaded regionis 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 Minkowskispacetime, 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 = `√ρ2 − 1 sinh(τ/`)

X1 = `√ρ2 − 1 cosh(τ/`)

X2 = `ρ sinhx cos θ1

. . .

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

Xd−1 = `ρ sinhx sin θ1 . . . sin θd−2 cosφ

Xd = `ρ sinhx sin θ1 . . . sin θd−2 sinφ

Xd−1 = `ρ coshx (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 +dρ2

ρ2 − 1+ ρ2(dx2 + sinh2 xdΩ2

d−2). (2.19)

The causal surface χA can then be obtained by taking the limit ρ → 1 at fixed τ , while A itselfis given by ρ → ∞ and τ = 0. Using bulk isometries or, equivalently, boundary conformaltransformations, 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 onlyneed to access information in a CFT subregion D[A] or, if we then express the CFT operators

Chapter 2. Precursors 20

Figure 2.4: Violation of timeslice axiom in AdS-Rindler reconstruction. A top-down view ofa constant timeslice of AdS is shown. Suppose we choose to reconstruct ϕ considering theRindler wedge shaded in blue, leaving out a single boundary point Y . Bulk locality requiresϕ to commute with O(Y ). If we take the precursor representation of ϕ, we will also expectO(Y ) to commute with the CFT operators on the boundary of Rindler wedge, thus violatingthe 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 withall spacelike separated operators. Therefore, all bulk operators should commute with oper-ators on the boundary. Using the bulk reconstruction prescription explained in the previoussection, we know that we can express a bulk operator in terms of CFT operators living on itsboundary. This means that the precursor representation of a bulk field, given by (2.9), shouldalso satisfy this condition. Using the CFT Hamiltonian, we can re-write all operators on theright 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 theprecursor can be expressed in terms of CFT operators, this means that we have a nontrivialoperator which commutes with all operators in the CFT at a constant timeslice. This clearlycontradicts the time-slice axiom (see section 1.1.1). This paradox also arises in AdS-Rindler re-construction. Suppose we choose to reconstruct ϕ considering the Rindler wedge which leavesout a single boundary point Y . Bulk locality requires ϕ to commute with O(Y ). If we take theprecursor representation of ϕ, we will also expectO(Y ) to commute with the CFT operators onthe boundary of Rindler wedge, thus violating the time-slice axiom. Since locality is essentialfor any sensible quantum field theory, this suggests that bulk reconstruction should not alwaysbe possible, but that it breaks down at some point. Establishing the precise conditions underwhich this happens is then essential if we want to gain a deeper understanding of both globalreconstruction and the subregion-subregion duality.

Chapter 2. Precursors 21

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 inAdS3 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 toreconstruct it from A ∪B, B ∪ C or A ∪ C. On the right, we have illustrated a 3-site toy modelof holography, in which the boundary CFT is an O(N) gauge theory. We can define 3 distinctgauge invariant operators on the boundary which have support on any 2 points and we canreconstruct Φ using any two of these operators.

Multiple Representations AdS-Rindler reconstruction has the property that the same bulkfield 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 thennot possible to reconstruct Φ by considering any single one of these wedges. However, we canhave a situation in which we can reconstruct φ(x) in A∪B, B ∪C or A∪C (see figure 2.5). Wethus have three representations φAB , φBC , φAC for the same bulk field φ(x). Although theseoperators should all correspond to same field in the bulk, they clearly involve different CFToperators, since they have support on different CFT regions. It is then not clear in what sensethese different representations of Φ are equivalent.

2.2.1 Quantum Error Correction

The contradiction regarding the existence of multiple CFT representations of a single bulkfield 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 wewill not need the details of this proposal. We will first start with defining some concepts fromquantum information theory that we will use. A quantum system can be in a pure or mixedstate. A pure state can be written as an outer product

ρ = |ψ〉〈ψ| (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 =∑n

cn|ψn〉〈ψn|. (2.21)

Chapter 2. Precursors 22

This means that a measurement on a pure state will always yield results related to only onequantum state, whereas with a mixed state we cannot know beforehand what state we willmeasure. 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. Ifwe 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 bipartitesystem, whose Hilbert space can be written as a tensor product

H = HA ⊗HB (2.22)

then we can calculate the state of systemA from the density matrix of the whole system ρAB bytaking its partial trace with respect to B. In other words, we “trace out” B with the followingoperation:

trB(|a1〉〈a2| ⊗ |b1〉〈b2|) = |a1〉〈a2|tr(|b1〉〈b2|) (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 thefollowing properties:

(OA ⊗ OB)⊗ (O′A ⊗ O′B) = OAO′A ⊗ OBO′B (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”. Supposewe want to send a quantum state of k qutrits, but we want to ensure that the message will stillbe received in its entirety, even if some qutrits get lost. We can encode our message in a largerset of states, so that we will be able to account for partial loss of information. Let’s work out asimple example. Suppose we want to send the following state:

|ψ〉 = a|0〉+ b|1〉+ c|2〉 (2.28)

and we use three qutrits to encode one single qutrit:

|0〉 → |0〉 = |000〉+ |111〉+ |222〉

|1〉 → |1〉 = |012〉+ |120〉+ |201〉 (2.29)

|2〉 → |2〉 = |021〉+ |102〉+ |210〉

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

Chapter 2. Precursors 23

density matrix on any qutrit will give the maximally mixed state. Conversely, by looking atany 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:

|00〉 → |00〉 |01〉 → |12〉 |02〉 → |21〉

|11〉 → |01〉 |12〉 → |10〉 |20〉 → |11〉 (2.30)

|02〉 → |21〉 |10〉 → |22〉 |21〉 → |20〉.

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

(Uij ⊗ 1k)|ψ〉 = |ψ〉 ⊗ (|00〉+ |11〉+ |22〉) (2.31)

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

Oij ≡ U †ijOUij (2.32)

where O is an operator which acts on the single qutrit state (2.28). Oij will first decode themessage, act on it and encode it again. Since the message can be recovered from any twoqutrits, it is possible to construct different operators which will act in the same way on thesingle qutrit message. In a similar way, in AdS/CFT we can construct the same bulk fieldfrom different portions of boundary data, and thus have different CFT operator representationsof the same field in AdS. In this case our code subspace would be the linear span of statesgenerated by some set of precursor operators Φi(x) acting on the vacuum |Ω〉

|Ω〉, Φi(x)|Ω〉, Φi(x1)Φi(x2)|Ω〉, . . . (2.33)

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

Chapter 2. Precursors 24

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 isrelated to the gauge invariance of the boundary theory. Following the structure of [5], we willfirst 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 bulkfield at its center. This is an idealized version of a constant timeslice in AdS, where the threesites on the boundary will constitute our CFT (see figure 2.5). We take our boundary theory tobe an O(N) gauge theory consisting of N scalar fields φia, with i = 1, . . . N and a = 1, 2, 3. Thegauge 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] = 4Niφ · φ (2.34)

[φ · π + π · φ, π · π] = 4Niπ · π (2.35)

[φ · φ, π · π] = 2Ni(φ · π + π · φ). (2.36)

The matter fields are given by the vector (φia, πia) since it transforms as the fundamental rep-

resentation of this algebra. We will now address the paradoxes explained in section 2.2 usingthis simplified model. First, we will see if we can construct a gauge invariant operator in theboundary 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 findan 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 commuteswith all gauge invariants restricted to any site b 6= a, but we also need to make it commutewith those restricted to site a. We start by computing the commutators of Oa with πa · πa andφa · πa

[Oa, φa · φa] = −2Ni[(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 wenee to set them all to 0, which is obviously too strong of a requirement. We can check if wecan construct double trace operators which satisfy this requirement. Since we have chosenour theory to be O(N) invariant, we know that the generator of gauge transformations on any

Chapter 2. Precursors 25

given siteLija = φiaπ

ja − φjaπia (2.40)

will commute with all local gauge invariants. In order to make this gauge invariant, we cansimply contract two Lij ’s and we get the following bilocal gauge invariant, which still com-mutes with all local gauge invariants:

Pab = Lija Lijb . (2.41)

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

P12|Ψ〉 = −(P22 + P32)|Ψ〉 (2.46)

= −(P11 + P31)|Ψ〉. (2.47)

This means that we can express Φ(0) using any two of the three Pab’s. If we have access toany 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 errorcorrection mechanism discussed in section 2.2.1. This suggests that it is not necessary to recastholography in terms of quantum information theory, but that the gauge invariance of the CFTautomatically corrects for erasures.

Continuum Generalization

In the continuum, we consider the 2-dimensional O(N) vector model, where our theory willhave 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 section 1.3). We will consider thelarge N limit (see section 1.1.2), and we will rescale the fields such that the two-point functions

Chapter 2. Precursors 26

will be of order N0, while all higher order corrections will be powers of 1/N . The CFT actionwill be given by

S =1

8π

∫dx dt

N∑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 invariantunder 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 bilocaloperators:

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

Following the procedure outlined in section 2.1, the precursor expression for a free masslessscalar in Ads3 Φ found in [5] is given by the bilocal operator

Φ(ρ, x, t) =1√N

∫ 2π

0dx′ dx′′

[f(ρ, x|x′, x′′)φ(x′, t) · φ(x′′, t) + g(ρ, x|x′, x′′)π(x′, t) · π(x′′, t)

](2.51)

where f(ρ, x|x′, x′′) and g(ρ, x|x′, x′′) 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 ofO(1/N). We also know that for large N, higher-point functions will factorize into products oftwo-point functions (see section 1.1.2) so, in the large N limit, it will suffice to consider thetwo-point functions. This will have important consequences in evaluating if a precursor oper-ator or its commutator 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 x′, x′′, so we are carrying outglobal reconstruction. As we have seen in section 2.1, we can also consider a smaller regionof the boundary, meaning that we don’t have to consider the full range of x′, x′′. We will nowshow 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

∞∑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 thefunction hmn. Since we are in the large N limit, we are only considering the contributions fromthe two-point function, and for mn < 0 (2.52) annihilates the vacuum in both directions andmakes no contribution to the two-point function. Therefore, we are allowed to modify hmn inthe following way:

hmn(ρ, θ)→ hmn(ρ, θ) + λmn(ρ, θ) for mn < 0 (2.53)

Chapter 2. Precursors 27

where λmn is an arbitrary function. It turns out that this is related to the generator of gaugetransformations

Lij =

∞∑n=1

2

n(α

[i−nα

j]n + α

[i−nα

j]n ) + Lij0 , (2.54)

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

[Lij , αimαjn] = 0, (2.55)

for mn < 0 we can shift Φ(ρ, x, t) by

Φ→ Φ +∑m,n

1√NγmnL

ijαimαjn (2.56)

where γmn is an arbitrary function. Depending on the values of m and n, we may need totake into account normal ordering, so (2.56) may pick up a normal ordering constant. Thecommutator of Lij with a single mode αim with m < 0 is given by

[Lij , αim] = (1−N)αjm. (2.57)

We obtain a similar result for the right movers:

[Lij , αjn] = (1−N)αin for n < 0. (2.58)

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

Lijαimαjn = αimL

ijαjn + (1−N)αm · αn (2.59)

= −Nαm · αn +O(1/N) (2.60)

and similarly for m > 0, n < 0, where we would have to commute the αjn through. Therefore,we can shift Φ by

Φ→ Φ−√N∑m,n

αm · αn (2.61)

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

2.3 Motivation

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

Chapter 2. Precursors 28

framework in which we can investigate the emergence of bulk locality by studying precursoroperators, boundary operators which are related to the bulk in a nonlocal way. The study ofprecursors 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 thelanguage 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 isthat the gauge symmetry present in the boundary theory appears to play a bigger role than wethought. If this proposal is indeed correct, or at least points to the correct direction, there willbe 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 morecautious to first verify the consistency of the gauge invariance proposal before abandoning ourstandard 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 approximatelocality that we expect from holography, and if gauge invariance provides a way out of theparadoxes which seem to plague bulk reconstruction.

Chapter 3

Results

In the following, we will investigate the conditions under which the timeslice axiom is violatedin a two-dimensional CFT. As shown in section 2.2.2, we already know that in an O(N) gaugetheory on a lattice it is possible to construct a nontrivial operator which exactly commutes withall local operators of the theory at a constant timeslice. We will see if this also applies in thecontinuum. We will first focus on the question of whether or not it is possible to construct anontrivial operator which exactly commutes with all local operators of the theory at a constanttimeslice. 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 willthen consider the same AdS3/CFT2 model as in [5], and check whether or not the precursoroperator of this model respects bulk locality, to leading order in 1/N . We will attempt to findthe conditions this operator would need to satisfy.

3.1 Investigating the Timeslice Axiom

3.1.1 Finite N

The goal is to construct an operator which commutes with all local operators at a constanttimeslice. For simplicity, we will take our constant timeslice to be at t = 0. As stated in section1.1.3, 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 constructoperators of the form: φn, (∂φ)n, πn and linear combinations. Let’s first see what happens ifwe only consider combinations of φ’s and π’s. Clearly, any combination of φ(x1)’s and π(x2)’swill commute with all local operators at t = 0 except for π(x1) and φ(x2), in which case thecommutator 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 withall 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,

29

Chapter 3. Results 30

∂φ and ∂φ to the commutators with all local operators vanish simply by imposing conditionson 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 weconsider a first-order operator. Let’s take the operator

O1 ≡∫ b

adx f(x)φ(x) (3.2)

and commute it with π(x′) with a < x′ < b

[O1(x), π(x′)] = if(x′). (3.3)

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

O2 ≡∫ b

adx1

∫ d

cdx2 f(x1, x2) φ(x1)φ(x2). (3.4)

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

[O2(x1, x2), π(x3)] = i

∫ b

adx1 f(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: weconsider 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 anoperator which commutes with all the gauge invariants of the theory at a given time-slice. It ispossible to construct an operator which commutes with all the local gauge invariants by usingthe 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 ofthis model, given by (2.40). Since the gauge invariant quantities are the same as those in thediscrete model, we already know that Lij(x) commutes with all gauge invariants. To leading

Chapter 3. Results 31

order in 1/N (see appendix A), the mode expansion of Lij(x) is given by

Lij(x) =i

2π

∑m,n 6=0

1

m

(α[imα

j]n − α

[i−mα

j]−n

)ei(m+n)x +

1

πφ

[i0π

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, andwe are looking for a nontrival operator which commutes with all local gauge invariants, sowe will consider the O(N) generator at a specific point x. This is not a trivial operator, itscontribution to the two-point function is can be found using (A.14) and (A.4). This operatoris not gauge invariant, but we can contract two Lij ’s to construct a gauge invariant bilocaloperator 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 modelpresented in 2.2.2 procedure can be generalized and that, using gauge invariance, we can stillconstruct an operator which violates the timeslice axiom.

3.2 Bulk Locality in AdS3/CFT2

The precursor operator found in [5] is of the form

O′2 =1√N

∫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 thecylinder. We will see if it is possible to make O′2 with the local gauge invariants of the the-ory by only considering leading order terms in 1/N and by imposing appropriate conditionson the smearing functions f(x, y) and g(x, y). The commutators of O′2 with the local gaugeinvariants are given by

[O′2(x, y),1√Nφ(z) · φ(z)] = − i

N

∫ 2π

0dx [g(z, x) + g(x, z)][φ(z) · π(x) + π(x) · φ(z)] (3.10)

[O′2(x, y),1√Nφ(z) · π(z)] = [O′2(x, y), π(z) · φ(z)]

=i

N

∫ 2π

0dx [f(x, z) + f(z, x)][φ(z) · φ(x)]− [g(x, z) + g(z, x)][π(x) · π(z)] (3.11)

[O′2(x, y),1√Nπ(z) · π(z)] =

2i

N

∫ 2π

0dx [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, aftera straightforward calculation we find that these conditions are too strong, as they would also

Chapter 3. Results 32

make O′2 itself vanish. The question is then if we can impose some weaker conditions onf(x, y) and g(x, y) which make the commutator approximately vanish (to leading order in1/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 theresults from appendix A, we find that the leading order contributions of the terms that appearin the commutators (3.10)-(3.12) are given by

1

Nφ(z) · π(x) =

1

N

1

2πφ0 · π0 +

i

2π

∑n>0

cos[n(z − x)] +O(1/N) (3.13)

1

Nπ(x) · φ(z) =

1

N

1

2ππ0 · φ0 −

i

2π

∑n>0

cos[n(z − x)] +O(1/N) (3.14)

1

Nφ(z) · φ(x) =

1

Nφ2

0 +∑n>0

2

ncos[n(z − x)] +O(1/N) (3.15)

1

Nπ(z) · π(x) =

1

2

∑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 toset ∫ 2π

0dx [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 thefollowing conditions on f(x, z):

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

∫ 2π

0dx 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 thecommutator (3.11) vanish

∫ 2π

0dx f(x, z)

[1

Nφ2

0 +∑n>0

2

ncos[n(z − x)]

]= 0 (3.20)

∫ 2π

0dx [g(x, z) + g(z, x)]

[∑n>0

n

2cos[n(z − x)]

]= 0 (3.21)

Evaluating the infinite sum in (3.20) yields

∑n>0

2

ncos[n(z − x)] = − log

[2 sin2

(x− z

2

)]− 2πim+ (3.22)

Chapter 3. Results 33

where we have used the properties of the complex logarithm (1.39). If we evaluate the infinitesum in (3.21), we get ∑

n>0

n

2cos[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)

∫ 2π

0dx f(x, z) = 0∫ 2π

0dx f(x, z) log

[sin2

(x− z

2

)]= 0∫ 2π

0dx [g(z, x) + g(x, z)] = 0∫ 2π

0dx

[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/CFT2

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

Discussion

In an attempt to gain insight into the inconsistencies which arise in bulk reconstruction, wehave focused on the validity of the timeslice axiom in a two-dimensional CFT and the issue ofbulk locality in the case of theAdS3/CFT2 O(N) vector model. We first considered a single freescalar field in a two-dimensional CFT, and checked whether or not it was possible to constructa nontrivial operator which commuted with all local operators. As expected, we found thatit was not possible, as stated by the timeslice axiom. Once we imposed a global O(N) gaugesymmetry, we only needed to consider the gauge invariant operators of the theory. In this case,we found that we could combine gauge generators at different spatial points and construct anontrivial operator which commuted with all local gauge invariants. We therefore found thatgauge invariance causes a violation of the timeslice axiom. This can be traced back to ourcurrent limitations, since a complete axiomatic formulation of gauge theories is still a work inprogress in mathematical physics. It should be noted that we have only considered a globalgauge symmetry, whereas in a more realistic model we would need to have a local gaugesymmetry, which would then require one to fix a gauge, and our candidate operator wouldthen also need to commute with the gauge fields of the theory. Performing a similar analysisin the case of a local gauge symmetry could shed more light on this issue. In any case, thisalready suggests that, in the context of AdS/CFT, gauge invariance does play a role in theparadoxes that arise in bulk reconstruction.

We then considered the AdS3/CFT2 O(N) vector model precursor and verified that bulklocality was respected. In this case, we had two factors at play: large N and gauge invariance.For finite N, the precursor could not exactly commute with all gauge invariants of the the-ory. For large N, we found that it was possible to impose a set of conditions on the smearingfunctions that enabled the precursor to commute with all local gauge invariants, to leadingorder in 1/N . Our results agree with the standard picture of holography, where bulk localityis expected to hold only approximately, after taking the large N limit of the boundary theory,thus suppressing quantum and stringy corrections. At finite N, our bulk theory should ceaseto be local. It seems, however, that taking the large N limit of the boundary theory is not theonly factor that plays a role in preserving bulk locality, but that gauge invariance also simpli-fies things by restricting the set of operators of the theory that generate physical states. In thecontext of the subregion-subregion duality, our results cannot provide a clearer indication ofwhether gauge invariance plays a fundamental role. The question of whether or not gaugeinvariance is sufficient in the explanation of the freedom of choosing a CFT subregion in bulkreconstruction remains open. More insight could be gained by expanding our analysis of theAdS3/CFT2 O(N) vector model. In this work, we have only considered one of the two choicesof precursor operator which are available from the AdS/CFT dictionary. We have taken φ2 as

34

Chapter 3. Results 35

our precursor operator, which has conformal dimension zero, but the precursor with confor-mal dimension two ∂φ∂φ is also admissible. In this case we would need to consider a precursorof the form ∫

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

and we would expect to find a similar set of constraints on f(x, y) and g(x, y) which satisfy therequirement of bulk locality. Furthermore, in our analysis we only modeled global reconstruc-tion, by taking the region of integration of the precursor to cover the whole boundary. Onecould also attempt to model AdS-Rindler reconstruction, by appropriately restricting the re-gion of integration, check if bulk locality still holds, and if so under what conditions. It shouldalso be noted that we have not explicitly calculated the smearing functions present in the pre-cursor operator; it would therefore be interesting to see if these actually satisfy the conditionsthat we found.

In the more general context of quantum gravity, our results are in agreement with the non-local nature of gravity. We know that in the classical limit, gravity should be approximatelylocal, which is also reflected in our calculation, where we show that holography enables us todefine precursor operators which respect bulk locality. The paradox regarding the timesliceaxiom remains unsolved, but an objection can be made to requiring this property to alwayshold in AdS/CFT. Given that we do not possess a rigorous formulation of AdS/CFT, it is quitean ambitious task to attempt to reconcile holography with axiomatic QFT and the fact thatwe quickly run into problems reflects this. Since bulk locality is expected to break down, weshould not expect all properties of local field theory to hold in AdS/CFT.

The proposals that we have seen attempt to make statements about bulk locality and bulkreconstruction more precise, by using quantum error correction and gauge invariance. Theproposal that AdS/CFT is actually a quantum error correcting mechanism is the most radical,while the gauge invariance proposal attempts to reconcile this with the symmetries alreadypresent in the CFT. Which of these proposals is correct remains an open question, but they areboth providing fertile ground for new ideas on the emergence of bulk locality. It could turn outthat neither of these proposals is entirely correct, but it seems that they are pointing in the rightdirection, since they are able to address the problem of locality, one of the most fundamentalissues of AdS/CFT, and quantum gravity in general. We are still far from achieving a completeunderstanding of holography. The inconsistencies which we are faced with when carrying outbulk reconstruction, even in the more simple cases strongly suggest that something is missingin the current approach. If gauge invariance and quantum error correction do play an essentialrole in bulk reconstruction, they could also be important ingredients in the formulation of atheory of quantum gravity.

Appendix A

Large N counting

In the following calculation, we will set

π0|0〉 = 0, (A.1)

meaning that we will take our scalar field to be completely delocalized. Setting one of the zeromodes to annihilate the vacuum will not change the physics of our theory, so we could havealso set a similar condition for φ0. In this calculation, we will come across second order termsinvolving α, α, π0 and φ0. Let’s consider each term and see how it contributes to the two-pointfunction, to leading order in 1/N . Since π0 · α and π0 · α always annihilate the vacuum in bothdirections, they will not contribute to the 2-point function. The other terms give the followingcontributions:

φ0 · π0, (φ0 · π0)† = π0 · φ0 (A.2)

〈(φ0 · π0)(π0 · φ0)〉 = 〈(i+ π0 · φ0)(π0 · φ0)〉 = 0 (A.3)

〈(φ0 · π0)φ20〉 = 〈(i+ π0 · φ0)φ2

0〉 = iN〈φ20〉 (A.4)

〈φ20(π0 · φ0)〉 = 〈φ2

0(φ0 · π0 − i)〉 = −iN〈φ20〉 (A.5)

This term will give an O(1) contribution.

αn · αm, (αn · αm)† = α−m · α−n (A.6)

〈(αn · αm)(. . .)〉 = 0 for n < 0 (A.7)

〈(. . .)(α−m · α−n)〉 = 0 for n < 0 (A.8)

〈(α−m · α−n)(. . .)〉 = 0 form > 0 (A.9)

〈(. . .)(αn · αm)〉 = 0 form > 0 (A.10)

36

Appendix A. Large N counting 37

We have that αn · αm and αn · αm will not contribute to the 2-point function for n < 0,m > 0,so we get

∑m,n 6=0

1

n(αn · αm − α−n · α−m)ei(nz+mx) ∼

∑m<0n>0

1

n

[αn · αmei(nz+mx) + αn · αme−i(nz+mx)

](A.11)

=∑m<0n>0

1

n

[(αm · αn +Nnδm+n)ei(nz+mx) + (αn · αm +Nnδm+n)e−i(nz+mx)

](A.12)

∼ N∑n>0

[ein(z−x) + e−in(z−x)

](A.13)

= 2N∑n>0

cos[n(z − x)] = −∑m,n 6=0

1

n(αm · αn − α−m · α−n)ei(nz+mx) (A.14)

where in the second line we have used the commutation relations (1.31). This will also give anO(1) contribution. The rest of the terms will not contribute to the two-point functions for thefollowing values of m,n:

αn · φ0, (αn · φ0)† = α−n · φ0 (A.15)

〈(αn · φ0)(. . .)〉 = 0 for n < 0 (A.16)

〈(. . .)(φ0 · α−n)〉 = 0 for n < 0 (A.17)

〈(α−n · φ0)(. . .)〉 = 0 for n > 0 (A.18)

〈(. . .)(φ0 · αn)〉 = 0 for n > 0 (A.19)

αn · αm, (αn · αm)† = α−m · α−n (A.20)

〈(αn · αm)(. . .)〉 = 0 for n < 0 orm < 0

〈(. . .)(α−m · α−n)〉 = 0 for n < 0 orm < 0

〈(α−m · α−n)(. . .)〉 = 0 for n > 0 orm > 0

〈(. . .)(αn · αm)〉 = 0 for n > 0 orm > 0

mn < 0 (A.21)

This means that these terms can still contribute to the two-point function for any given valueof m,n; however, it turns out that their contributions to the 2-point function are O(1/N). Let’sconsider αn · φ0. This annihilates the vacuum for n < 0 (see (A.15)), so we will take n > 0:

1

N2〈αinφi0φ

j0α

jm〉 =

1

N2〈φi0φ

j0(nδijδm+n + αjmα

in)〉 (A.22)

∼ 1

N2〈φ2

0〉 ∼ O(1/N). (A.23)

Appendix A. Large N counting 38

Similarly, we will consider αn · αm with n > 0,m > 0:

1

N2〈αinαimα

jn′α

jm′〉 =

1

N2〈αinα

jn′(mδm+m′δij + αjm′α

im)〉 (A.24)

∼ 1

N2〈αn · αn′〉 ∼ O(1/N). (A.25)

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Acknowledgements

I would like to thank my supervisor, Ben Freivogel, for his constant guidance, patience andavailability.

I would also like to thank my second supervisor, Erik Verlinde, for providing helpful feed-back and bringing up interesting discussion points during my presentation.

Special thanks also go to Laurens Kabir and Robert Jefferson, who were always happy to helpme out whenever I got stuck on a problem.

Finally, I would like to thank my fellow students (Zan, Giulio, Lefkos, António and Jorrit justto name a few) and all my friends and family for their support.

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