The Emergence of Space and Gravity from Entanglement in AdS3/CFT2

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University of Amsterdam

MSc Physics

Track: Theoretical Physics

Master Thesis

The Emergence of Space and Gravity from

Entanglement in AdS

3

/CFT

2 by Manus Visser, MA BSc 5794722 August 2014 60 ECTS Supervisor:

Prof. dr. Erik Verlinde dr. Diego HofmanExaminer:

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“I would not call [entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.”

— E. Schrödinger, Discussion of probability relations between separated sys-tems, Proc. Cam. Philos. Soc., 31 (1935), pp. 555-563.

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Abstract

The Ryu-Takayanagi formula S = A/(4G) relates entanglement entropy in a CFT to the area of a minimal surface in AdS space. This formula tells us that the connectedness of AdS space emerges from entanglement in the CFT. In this thesis we check the validity of this formula in a particular example of the AdS3/CFT2 correspondence. We show

that the geodesic length in AdS3 with a conical defect agrees asymptotically with the

entanglement entropy of a low-energy excited state in a 2D CFT. Furthermore, recently the Ryu-Takayanagi formula has been used to understand the emergence of gravity from CFT physics. It has been argued that the linearised vacuum Einstein equations are dual to a first law of entanglement in a holographic CFT. We apply this argument to AdS3/CFT2, and show that the metric perturbation giving rise to a conical defect in

pure AdS can be derived from a first law of entanglement. Lastly, possible extensions of the argument to the non-linear level and to Einstein’s equations with stress-energy tensor are discussed.

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Populairwetenschappelijke samenvatting

Over de oorsprong van ruimte en zwaartekracht

Heb je je wel eens afgevraagd waarom water bij kamertemperatuur een vloeistof is, en geen gas? Het is welbekend dat water uit H2O-moleculen bestaat. Bij

kamertemper-atuur zijn deze watermoleculen met elkaar verbonden door zogeheten waterstofbruggen. Deze verbinding is zo sterk dat de watermoleculen samen een vloeistof vormen. Bij een temperatuur hoger dan 100 graden, daarentegen, breken de waterstofbruggen, en zwer-ven de moleculen los in de ruimte. In dat geval vormen de moleculen een gas, namelijk waterdamp.

We zien dus dat op kleine schaal “water” er anders uitziet dan op grote schaal. Op kleine schaal bestaat water uit moleculen met lege ruimte ertussen, terwijl op grote schaal water vloeibaar, of beter gezegd, “continu verbonden” is. Hiermee bedoelen we dat er in principe geen lege ruimtes aanwezig zijn in een bad met water – op een paar luchtbellen na dan. Dit lijkt wellicht een triviale observatie – we kunnen immers zwemmen in water zonder grote obstakels tegen te komen waar “geen water” is – maar als je je bedenkt waar water uit bestaat (moleculen met veel lege ruimte ertussen), dan is dit juist heel wonderlijk.

Deze continue verbondenheid van water is een voorbeeld van emergentie. Emergentie is simpel gezegd het fenomeen dat op grote schaal er nieuwe eigenschappen optreden die op kleine schaal niet aanwezig zijn. Een ander voorbeeld van emergentie is temperatuur. Temperatuur is een maat voor de gemiddelde snelheid van een heleboel moleculen. We kunnen dus niet zeggen dat één molecuul een temperatuur heeft. We kunnen pas een temperatuur meten als we heel veel moleculen samennemen – grofweg 1023 _{moleculen of}

meer. Kortom, temperatuur emergeert uit de gezamenlijke interactie van moleculen, die op zichzelf genomen geen temperatuur hebben. De natuurkundige en Nobelprijs winnaar Phil Anderson heeft het fenomeen emergentie daarom samengevat met de slogan “More is diﬀerent”. Een heleboel moleculen samen geven aanleiding tot andere eigenschappen, zoals temperatuur – en zelfs andere natuurwetten, zoals we straks zullen zien.

In deze scriptie stel ik een soortgelijke vraag over ruimte: waarom is de ruimte om ons heen continu verbonden? De analogie tussen ruimte en water is zeer treﬀend: net als je in water kunt zwemmen, kun je door de ruimte heenlopen, of zweven, zonder plekken tegen te komen waar “geen ruimte” is. Bij water kun je je nog iets voorstellen bij een plek “waar geen water is”, omdat er ook luchtbellen in water voorkomen. Maar ruimte is zoiets vanzelfsprekends dat we ons niet kunnen voorstellen dat het er niet is.

Recentelijk zijn er in de natuurkunde aanwijzingen gevonden dat de verbondenheid van ruimte, net als de verbondenheid van water, emergent is. Dat wil zeggen dat op een fundamenteel niveau ruimte niet verbonden is. En een ruimte die niet verbonden is, kun je haast geen ruimte noemen. Maar als er op kleine schaal geen ruimte is, wat is dan de oorsprong van ruimte? Net zoals water uit H2O-moleculen bestaat, verwachten we dat

ruimte ook uit kleinere bouwstenen is opgebouwd.

In mijn scriptie heb ik een bepaald voorstel onderzocht voor deze bouwstenen van de ruimte. Om dit voorstel te begrijpen moet ik eerst een ander begrip introduceren:

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A

B

_A

_B

Figure 1: Gedachte-experiment.

verstrengeling. Verstrengeling is een belangrijk fenomeen uit de kwantummechanica – de theorie van de fundamentele deeltjes – waarbij twee of meer deeltjes zodanig zijn verbon-den, dat het ene deeltje niet meer volledige beschreven kan worden zonder het andere deeltje te noemen. Twee deeltjes kunnen in de kwantumwereld (d.w.z. op een schaal van ongeveer 10 10_{m of kleiner) zo sterk met elkaar verstrengeld zijn dat een meting aan het}

ene deeltje invloed heeft op de toestand van het andere deeltje. Daarbij hoeven zich deelt-jes niet per se naast elkaar te bevinden; ze kunnen ook ruimtelijk van elkaar gescheiden zijn. Verstrengeling wordt daarom ook wel een “niet-lokale kwantumcorrelatie” genoemd. De Canadees Mark van Raamsdonk was de eerste natuurkundige die de verbonden-heid van ruimte aan het begrip verstrengeling koppelde. Zijn hypothese is dat de verbon-denheid van ruimte emergeert uit de verstrengeling van microscopische toestanden. Je kunt je de verstrengeling als een soort lijm voorstellen die alle microtoestanden bij elkaar houdt. De lijm (vergelijkbaar met de waterstofbruggen tussen H2O-moleculen) zorgt er

uiteindelijk voor dat ruimte op macroniveau verbonden is.

Van Raamsdonks hypothese is een zeer kwalitatieve uitspraak, maar we kunnen haar ook kwantitatief maken. Van Raamsdonk baseerde zijn hypothese oorspronkelijk op een eenvoudige formule, die van toepassing is op de volgende situatie. Stel je voor dat we de ruimte (we nemen voor het gemak een vierkant) in twee gebieden opdelen A en B, zoals weergegeven in het linkerdeel van figuur 1. Dan geldt de volgende formule

S = A 4GN

, (0.1)

waarbij S een maat is voor de hoeveelheid verstrengeling tussen de twee gebieden A en B (de zogeheten “verstrengelingsentropie”) , A is de oppervlakte (“area”) die de twee gebieden scheidt, en GN is de zwaartekrachtsconstante van Newton. Deze formule stelt

dus een maat voor verstrengeling (de verstrengelingsentropie’) gelijk aan een geometrische grootheid (de oppervlakte). In mijn scriptie heb ik deze formule gecheckt voor een specifiek soort ruimte.

Laten we nu het volgende gedachte-experiment uitvoeren. Stel dat we de verstren-gelingsentropie S naar nul sturen, dan zegt deze formule dat de oppervlakte A ook nul wordt. Oftewel het grensgebied dat A en B scheidt wordt een punt (aangezien een punt geen oppervlakte heeft). Deze situatie hebben we weergegeven in het rechterdeel van

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figuur 1. Het is duidelijk te zien dat de gebieden A en B van elkaar gescheiden zijn. Met andere woorden, als er geen verstrengeling is tussen de ruimtes A en B, dan zijn ze niet met elkaar verbonden. We kunnen dit ook omdraaien: als A en B wel verstrengeld zijn (en de verstrengelingsentropie S niet nul is), dan resulteert dat in de verbondenheid van de gebieden A en B. We concluderen dus dat de formule ons vertelt dat de verbonden-heid van ruimte emergeert uit verstrengeling! Dit komt overeen met Van Raamsdonks hypothese.

To nu toe hebben we het alleen over ruimte gehad. Einsteins algemene relativiteitstheorie vertelt ons echter dat ruimte en zwaartekracht nauw met elkaar verbonden zijn. Kort gezegd, geeft de kromming van ruimte(tijd) aanleiding tot zwaartekracht. We hoeven dit hier niet precies te begrijpen. Het gaat er vooral om dat niet alleen ruimte, maar ook zwaartekracht emergent blijkt te zijn. Hiermee bedoelen we dat zwaartekracht, net als temperatuur, op fundamenteel niveau niet bestaat, en alleen verschijnt als een “thermo-dynamisch fenomeen” op grotere schaal. De belangrijkste aanwijzing voor de emergentie van zwaartekracht is dat de natuurwetten die zwaartekracht beschrijven (genaamd de “Einstein vergelijkingen”) equivalent zijn aan de wetten van de thermodynamica (die temperatuur, druk, warmte, et cetera beschrijven). Aangezien temperatuur en druk emergente grootheden zijn, wordt aangenomen dat zwaartekracht ook emergent is. In het tweede deel van mijn scriptie bestudeer ik waaruit zwaartekracht emergeert, oftewel wat de oorsprong van zwaartekracht is.

In een aantal zeer recente artikelen (van onder andere Van Raamsdonk) wordt de oorsprong van zwaartekracht gerelateerd aan de fysica van verstrengeling. Het blijkt dat Einsteins vergelijkingen ook equivalent zijn aan een bepaalde wet die verstrengeling beschrijft. Deze wet van verstrengeling stelt de verandering in verstrengelingsentropie gelijk aan de verandering in energie van een toestand. Het is interessant dat niet alleen deze verstrengelingswet nodig is om de Einstein vergelijkingen af te leiden, maar ook formule (0.1) die verstrengelingsentropie aan oppervlakte relateert. Formule (0.1) en de verstrengelingswet vormen samen de belangrijkste aannames voor de afleiding van de Einstein vergelijkingen. In hoofdstuk 5 van deze scriptie kan deze afleiding worden teruggevonden.

We concluderen dat de fysica van verstrengeling essentieel is voor de emergentie van ruimte en zwaartekracht. Ruimte, of beter gezegd oppervlakte, wordt gedefinieerd door de verstrengelingsentropie (formule 0.1). En zwaartekracht is grofweg het resultaat van het feit dat formule (0.1) altijd moet blijven gelden. Dat wil zeggen: als er een massa wordt geplaatst in de ruimte, dan wordt de ruimte zodanig gekromd dat formule (0.1) weer geldt. Deze afgemeten kromming van de ruimte resulteert in zwaartekracht. Kortom, de onveranderlijkheid van formule (0.1) geeft aanleiding tot zwaartekracht.

Daarmee neemt ons begrip van ruimte en zwaartekracht een andere wending dan Einstein ooit had kunnen vermoeden. Einstein noemde verstrengeling een spukhafte Fernwirkung (“spooky action at a distance"), omdat het de snelheidslimiet van infor-matieoverdracht leek te schenden. Het is ironisch te noemen dat nu juist dit fenomeen zo’n belangrijke rol speelt in de oorsprong van Einsteins theorie van zwaartekracht.

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Introduction

In the study of gravity, the ideas of holography and emergence have played a crucial role. On the one hand, holography posits that theories with gravity are equivalent to lower dimensional quantum theories without gravity [67, 64]. On the other hand, gravity is believed to be emergent, since the basic equations that govern gravity closely resemble the laws of thermodynamics [10, 47, 68]. Emergence here means that on a fundamental level gravity does not exist; it rather arises as a thermodynamic phenomenon in a course-grained description. By combining the idea of emergence with that of holography, it seems natural to assume that the microscopic theory from which gravity emerges is precisely given by the holographic dual of the (classical) gravitational theory. This means that the degrees of freedom in the theory without gravity can be viewed as more fundamental than those in the gravity theory.

Furthermore, since gravity is intimately connected with spacetime geometry, it is believed that space is also emergent. In holographic settings the gravity theory is defined on a higher dimensional space than its dual quantum theory, so the holographic (radial) direction can be interpreted as an emergent space direction. It is not (yet) clear whether time is also emergent in holographic settings. Thus, gravity and space are supposed to be emergent in a holographic scenario.

A central question in the study of gravity is to understand how it emerges from mi-croscopic physics. In other words, we would like to know what the origin of gravity and of space is. Recently, it has been suggested that the physics of quantum entanglement plays a crucial role in the emergence of space [60, 69, 65]. The universal quantity of interest here is the entanglement entropy, which measures the quantum correlations between two subsets of degrees of freedom in general quantum systems. The relationship between entanglement entropy and the dual spacetime geometry has been made precise in the AdS/CFT correspondence, which is a holographic duality between certain d-dimensional conformal field theories (CFT) and (d + 1)-dimensional gravity theories in anti-de Sitter space (AdS) [53]. Ryu and Takayanagi (RT) have proposed that entanglement entropy in the CFT is proportional to the area of extremal surfaces in AdS [60, 59, 56]

SA= kBc 3 ~ Area 4GN .

In this thesis we use natural units c = ~ = kB = 1, in which case the area law reduces to

SA= A/4GN. It has been argued that the RT formula implies that (the connectedness

of) AdS space is emergent from quantum entanglement in the CFT [69, 70]. 5

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In the present thesis we study the Ryu-Takayanagi formula in AdS3/CFT2, which is a

well understood example of holography. We specialise to three diﬀerent (asymptotically) AdS3 spacetimes: pure AdS, AdS with a conical defect, and the BTZ black hole. The

latter two are quotients of the isometry group of pure AdS. We will confirm that the RT formula holds for these three cases by comparing holographic calculations with known 2D CFT results. Special attention should be paid to the check of the RT formula for the conical defect geometry, since these are new results (see section 4.2).

In recent work [50, 39], the holographic formula for entanglement entropy has also been used to understand the emergence of gravity from CFT physics. It has been argued that linearised gravity emerges from a ‘first law’ of entanglement and from the holo-graphic formula for entanglement entropy. The first law of entanglement SA= hHAi

is an exact quantum generalisation of the ordinary first law of thermodynamics. In a holographic context, this first law can be interpreted as a constraint on the metric per-turbation around pure AdS, that is dual to a small variation of the CFT vacuum state. It will be shown that that an infinite set of such constraints is exactly equivalent to the requirement that the metric perturbation satisfies the gravitational equations of motion, linearised about pure AdS. For theories in which the entanglement entropy is computed by the Ryu-Takayanagi formula, the linearised Einstein equations are derived from the first law. In this thesis, we consider again the conical defect geometry as a toy model, and show that the metric perturbation that establishes a conical defect in pure AdS follows from the first law of entanglement together with the RT formula (see section 5.3). This thesis is organised as follows. Chapters 2 4 cover the emergence of AdS space in three bulk dimensions; and chapter 5 is about the emergence of linearised gravity in general dimensions. To begin with, in chapter 1 we review the basics of the AdS3/CFT2

correspondence. In chapter 2 we define entanglement entropy and explain the Ryu-Takayanagi formula. In chapter 3 we will calculate the entanglement entropy in two-dimensional conformal field theory for diﬀerent configurations. And in chapter 4 we verify that these 2D CFT results are consistent with the spatial geodesic lengths in the three (asymptotically) AdS3 spacetimes under consideration. Moreover, we discuss

the relation between entanglement entropy and black hole entropy in this chapter. The last chapter 5 reviews [39], in which the authors argue that the linearised gravitational equations of motion in pure AdS are equivalent to a first law of entanglement in the CFT. Finally, in section 5.5 we discuss possible extensions of this argument to the non-linear level and to Einstein’s equations with stress-energy tensor.

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Chapter 1

The AdS

₃

/

CFT

₂

correspondence

AdS/CFT is the best known setting in which entanglement entropy can be directly related to geometry. In this chapter we review some basics of the AdS/CFT correspondence in 2+1 dimensions. We will address the following questions: What is AdS/CFT (section 1.1)? Why is 2+1-dimensional gravity interesting (section 1.2)? What are the metrics and global symmetries of 3D AdS (section 1.3)? What are the dual states and symmetries in 2D CFT (section 1.4)? How can thermodynamic entropy be computed in 2D CFT (section 1.5)?

1.1 The holographic principle

The holographic principle states that a (quantum) theory of gravity in d + 1 dimensions is equivalent to a quantum theory without gravity in one space dimension less. The idea of holography was first developed by ’t Hooft and Susskind [67, 64], and was inspired by the study of black holes. An important result about black holes is namely that their entropy is proportional to the area of the event horizon. This result is known as the Bekenstein-Hawking formula [10, 42]

SBH = Area

4GN

. (1.1)

The area law contrasts with the usual entropy of thermodynamic systems, which is proportional to the volume. Since entropy measures the number of possible microscopic degrees of freedom associated to a given macroscopic state, the area law entails that the number of microscopic degrees of freedom in a black hole system is the same as that of a quantum theory on the horizon. More generally, it is expected that in a theory of gravity the degrees of freedom contained in any spatial volume can be described by a quantum theory on the boundary surface of this volume.

The first concrete example of the holographic principle was the AdS/CFT correspon-dence. This correspondence relates a gravitational theory in (d + 1)-dimensional anti-de Sitter spacetime (AdS) to a conformally invariant field theory (CFT) without gravity in ddimensions. The AdS spacetime is referred to as the ‘bulk’, an the degrees of freedom

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1.2. GRAVITY IN 2+1 DIMENSIONS 8 of the CFT live on the conformal boundary of AdS. A prototype example of AdS/CFT relates string theory on AdS5⇥S5to four-dimensional supersymmetric Yang-Mills theory.

Maldacena conjectured in [53] that these two theories are exactly equivalent, meaning they are two distinct descriptions of the same physics.1

The exactness of the AdS/CFT duality has not been proven (yet), but has profound consequences for the supposed emergence of AdS spacetime and gravity. If the AdS/CFT duality is indeed exact, then it is implausible that gravitation in AdS is emergent from the boundary degrees of freedom. This is because the bulk theory describes exactly the same physics as the CFT, and consequently does not contain any new, emergent phenomena. However, if the duality is inexact, then the boundary degrees of freedom can be viewed as more fundamental than the bulk degrees of freedom. In this case, the radial direction in AdS emerges from the microscopic degrees of freedom on the boundary.2

In the present thesis, we take the latter ‘emergence’ point of view, mainly because of the recently established relationship between quantum entanglement in the CFT and geometry in AdS [60, 65, 70]. Entanglement is a quantum mechanical property, whereas the geometry to which it is related is classical. On the basis of this relationship, Van Raamsdonk has argued that quantum entanglement and geometry stand to each to other in an emergent relationship. Therefore, we speculate that the connection between en-tanglement in the CFT and geometry in AdS – which has been quantified by Ryu and Takayanagi, see chapter 2 – is also an example of emergence. In section 2.5 we will further discuss Van Raamsdonk’s argument for the emergence of (the connectedness of) space from entanglement.

Below we will study the AdS/CFT correspondence in 2 + 1 bulk dimensions. This example of AdS/CFT can be viewed as a toy model for higher dimensional cases. In sections 1.2 and 1.3 we will discuss the gravity side of this correspondence, and in section 1.4 we will treat the CFT side.

1.2 Gravity in 2+1 dimensions

Classical pure 2+1 dimensional gravity is described by the Einstein-Hilbert action IEH =

1 16⇡GN

Z

d3xp g (R 2⇤) , (1.2)

where ⇤ is the cosmological constant, and GN is the 3D Newton constant. For

anti-de Sitter spacetime the cosmological constant ⇤ is related to the AdS radius L by: ⇤ = 1/L2_.

The resulting Euler-Lagrange equations are the standard vacuum Einstein equations Rµ⌫

1

2gµ⌫R + ⇤gµ⌫ = 0. (1.3)

1_{It should be mentioned that this thesis does not provide an introduction to the AdS/CFT dictionary.}

For a review of AdS/CFT, we refer the reader to [33, 3].

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1.2. GRAVITY IN 2+1 DIMENSIONS 9 The first essential feature of 2+1 dimensional vacuum gravity is that it has no local degrees of freedom. This follows from a simple counting argument [25, 26]. The phase space of three dimensional Einstein gravity consists of a spatial metric on a constant-time hypersurface (which has three independent components), and its conjugate momentum (another three degrees of freedom per point). However, the theory also has three con-straints on initial conditions (following from Einstein’s equations) and three arbitrary coordinate choices (following from diﬀeomorphism invariance). Hence, these constraints completely fix the physical degrees of freedom at any point in spacetime. This means that there are no propagating degrees of freedom like gravitational waves in three dimensions. A second (related) feature is that three dimensional spacetimes which satisfy the vacuum Einstein equations (1.3) are maximally symmetric. Taking the trace of Einstein’s equation (1.3) yields that the Ricci scalar is proportional to the cosmological constant: R = 6⇤, and thus Rµ⌫ = 2⇤gµ⌫. This already tells us that solutions of the vacuum

Einstein equations have constant curvature, and can be classified locally by the sign of their cosmological constant. The modifier ‘locally’ is necessary to account for possible global diﬀerences, as will be explained below. From these facts and from the vanishing of the Weyl tensor Cµ⌫⇢ in three dimensions, it follows that the Riemann tensor is

maximally symmetric:3

Rµ⌫⇢ = ⇤(gµ⇢g⌫ gµ g⌫⇢). (1.4)

Note that the Riemann tensor does not contain any derivatives of the metric and hence is completely locally defined. In other words, curvature is concentrated at the location of matter, and there is no dynamics that governs changes in the curvature. This is essentially the same as the first feature: there are no propagating degrees of freedom in three dimensions.

Thirdly, although three-dimensional gravity is locally trivial, globally it is not. We are still free to vary the global properties of a three dimensional spacetime, such as its topology. A topologically nontrivial 3D spacetime has a finite number of global degrees of freedom. The AdS solutions which we will discuss in the next sections are examples of such topologically diﬀerent spacetimes: locally they are completely identical, but glob-ally they can be distinguished from each other. These solutions can be generated by taking quotients of (the isometry group of) the vacuum AdS spacetime, as we will show in appendix B. This is analogous to how a torus T2 _{can be obtained from the complex}

plane : T2 _{= /( + ⌧ )}_{, which means that the torus is locally identical to the complex}

plane but globally the former is the quotient of the latter.

Why are we interested in AdS3/CFT2? The topological nature of 3D gravity makes

AdS3/CFT2 the cleanest and best understood example of holography. It is a clean

example since we only have to examine the global properties of asymptotically AdS spacetimes to distinguish them from each other. Furthermore, holography in three (bulk)

3_{In n dimensions the Riemann tensor can be written in terms of the metric tensor g}

µ⌫, the Ricci

tensor Rµ⌫ (and its trace), and the Weyl tensor Cµ⌫⇢ : Rµ⌫⇢ = Cµ⌫⇢ +_{n 2}2 (gµ[⇢R ]⌫ g⌫[⇢R ]µ) 2

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1.3. METRICS AND GLOBAL SYMMETRIES OF 3D ADS 10 dimensions is well understood, because 3D Einstein gravity can be described by Chern-Simons theory. Chern-Chern-Simons theory is a topological field theory that reduces to a 1 + 1 dimensional conformal field theory on the (conformal) boundary of AdS3 [1, 73].

Given certain boundary conditions, Witten demonstrated that the Chern-Simons action is equivalent to a Wess-Zumino-Witten action on the boundary [74]. Moreover, even before the connection between Chern-Simons theory and 3D gravity was known, Brown and Henneaux already showed that the asymptotic symmetry group of AdS3is equivalent

to the (pseudo)-conformal group in CFT2[16]. Although Chern-Simons formalism is very

useful in describing 3D gravity, we will use the metric formalism instead in this thesis. For the purpose of this thesis, AdS3/CFT2is useful because holographic entanglement

entropy has been fully worked out in this setting. Cardy and Calabrese [20] have calcu-lated the entanglement entropy for many systems in 2D CFT, and Ryu and Takayanagi have checked these results holographically [60]. Moreover, due to the topological nature of 3D gravity, the emergence of space from entanglement is straightforward in AdS3/CFT2.

Changes in the entanglement in the CFT only give rise to global eﬀects in AdS. Later on, we will look at a particular example (i.e. AdS with a conical defect) where this global eﬀect can be described by only one parameter. It is believed that the lessons we learn from this example about the emergence of space can be generalised to higher dimensions.

1.3 Metrics and global symmetries of 3D AdS

Let us now focus on (maximally symmetric) 3D spacetimes with a negative cosmological constant, called anti-de Sitter spacetime (denoted by AdS3). First we will make clear

how AdS3 can be embedded in 2,2. Then we will introduce the coordinate frames of

three (asymptotically) AdS3 spacetimes: empty AdS, AdS with a conical defect, and the

BTZ black hole. These are the three main examples in this thesis.

The latter two solutions (AdS with a conical defect and the BTZ black hole) are so-called ‘asymptotically’ AdS3 spacetimes. This means that they behave similarly to

pure AdS3 in the limit r ! 1, with certain falloﬀ conditions. The falloﬀ conditions at

infinity are called ‘boundary conditions’. In 2+1 dimensions these asymptotic spacetime can actually be obtained as quotient spaces of pure AdS3. We will show this explicitly

in appendix B.

1.3.1 Embedding of AdS3

AdS3 can be embedded in flat 2,2, with has coordinates (T1, T2, X1, X2) and metric

[23, 31]

ds2 = (dT1)2 (dT2)2+ (dX1)2+ (dX2)2. (1.5) Anti-de Sitter spacetime is a hyperboloid in the embedding space

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1.3. METRICS AND GLOBAL SYMMETRIES OF 3D ADS 11 where L is the AdS radius. We haven chosen a Lorentzian signature for the embedding metric. Note that the isometry group of AdS3is by construction SO(2, 2). The isometries

become even more evident when we combine the embedding coordinates (T1_{, T}2_{, X}1_{, X}2₎

into a 2 ⇥ 2 matrix X = 1 L ✓ T1+ X1 T2+ X2 T2+ X2 T1 X1 ◆ 2 SL(2, ). (1.7)

AdS3 is often called the group manifold of SL(2, ), since the embedding constraint

(1.6) is equivalent to requiring that the matrix X has unit determinant: det X = 1. Furthermore, the metric of the embedding space (1.5) is the same as the Killing-Cartan metric on the group manifold of SL(2, )

ds2= 1 2Tr(X

1_dXX 1_dX). _(1.8)

This metric is invariant under left X ! YLX and right X ! XYR group actions,

where YL, YR2 SL(2, ). The isometries are, hence, elements of two copies of SL(2, ),

but in addition we need to mod out by 2, since the group actions X ! YLXYR and

X ! ( YL)X( YR) are identical. Therefore, we conclude that Lorentzian AdS3 has an

SL(2, )_{⇥ SL(2, )/} 2⇠= SO(2, 2)group of isometries.

The generators of SO(2, 2) are the six linearly independent Killing vectors H = T1@_T2 T2@_T1 B11= T1@_X1+ X1@_T1 B12= T1@_X2 + X2@_T1

J12= X1@_X2 X2@_X1 B21= T2@_X1+ X1@_T2 B22= T2@_X2 + X2@_T2. (1.9)

They form a basis for the Lie algebra so(2, 2) ⇠= sl(2, ) sl(2, ). The Killing vector H = @⌧ generates time translations in the embedding space, J12generates rotations, and

the other four B vectors generate boosts. 1.3.2 Vacuum AdS3

Global coordinates. Now let us construct some intrinsic coordinate systems, which do not require us to think of AdS3 as embedded in 2,2. The most common coordinate

system covering the entire AdS3 manifold is given by the following parametrisation [31]

T1 = L cosh ⇢ sin ⌧ T2 = L cosh ⇢ cos ⌧ X1 = L sinh ⇢ cos X2 = L sinh ⇢ sin

(1.10)

where the coordinates range over ⌧ 2 [0, 2⇡), 2 [0, 2⇡) and ⇢ 2 [0, 1). With this parametrisation the induced metric on the hyperboloid (1.6) becomes

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1.3. METRICS AND GLOBAL SYMMETRIES OF 3D ADS 12

ds2 = L2( cosh2⇢d⌧2+ d⇢2+ sinh2⇢d 2). (1.11) Note that the metric has a Lorentzian signature. This is the AdS3 metric in so-called

global coordinates. These coordinates have a strange feature, namely that ⌧ is periodic. In order to avoid closed timelike curves, it is common to forget that the metric (1.11) came from a particular embedding and declare that ⌧ ranges from 1 to 1. The space thus obtained is actually the universal cover of anti-de Sitter space, but in the following we will just refer to it as AdS.

One of the advantages of working with global coordinates is that we can draw pictures. That is, global AdS3can be represented as an infinite solid cylinder. To see this, we define

an new radial coordinate via sinh ⇢ = tan , so that the metric becomes ds2 = L

2

cos2 d⌧

2_{+ d} 2_{+ sin}2 _d 2 _, _(1.12)

where 2 [0, ⇡/2]. The terms within parentheses define the metric of an infinite solid cylinder, with = ⇡/2 on the boundary and = 0 in the center. Hence, the metric of global AdS3 is conformal to the metric on the cylinder. On the cylinder = ⇡/2 we have

an induced conformal structure ⇠ d⌧2_{+ d} 2_{, so we conclude that the cylinder is the}

conformal boundary of global AdS3. The CFT lives on this conformal boundary of AdS.

Lastly, we point out another useful set of global coordinates. By introducing a new radial r = L sinh ⇢ and time coordinate t = L⌧, the metric takes the form

ds2 = ✓ 1 + r 2 L2 ◆ dt2+ ✓ 1 + r 2 L2 ◆ 1 dr2+ r2d 2. (1.13) These global coordinates range over t 2 ( 1, 1), 2 [0, 2⇡) and r 2 [0, 1).

Poincaré coordinates. The Poincaré coordinates (t, x, z) are given by the following parametrisation of the embedding constraint

T1= Lt/z X1= Lx/z T2+ X2= L2/z

T2 X2= t2+ x2+ z2 /z.

(1.14)

This leads to the Poincaré metric ds2 = L

2

z2( dt

2_{+ dx}2_{+ dz}2_), _(1.15)

with coordinates t and x ranging from 1 to 1. The coordinate z behaves as a radial coordinate and divides the hyperboloid into two charts, as can be seen from the third

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1.3. METRICS AND GLOBAL SYMMETRIES OF 3D ADS 13 equation in (1.14). The first chart is the region z > 0, that is T2 _> _X2_{, and corresponds}

to one half of the hyperboloid. The other half of the hyperboloid T2_< _X2 _corresponds

to the second chart, i.e. the region z < 0. Usually the so-called ‘Poincaré patch of AdS’ is chosen to be the chart corresponding to z > 0. It is obvious, but important to mention that both Poincaré charts are not valid on AdS3 entirely, and cover only part of

the spacetime.

Furthermore, we consider the region separating the two Poincaré charts: z = 0. This region is conformally flat, as follows from the metric (1.15). Moreover, it corresponds to the conformal boundary of AdS, which can be seen by taking the near-boundary limit of global coordinates (1.13), i.e. r ! 1,

ds2 = r 2 L2dt 2₊L2 r2dr 2_{+ r}2_d 2_. _(1.16)

By substituting z = L2_/r _{and x = L we arrive again at the Poincaré metric (1.15).}

The only diﬀerence between the near-boundary limit of global coordinates and proper Poincaré coordinates is that is not periodically identified in the latter. We conclude from this limit that Poincaré AdS has a planar conformal boundary, and hence the CFT lives on a plane in this case. As we have seen above, the cost of having this simple boundary is that the Poincaré coordinates do not cover the entire AdS spacetime.

Let us also mention the map between global coordinates (1.11) and Poincaré coordi-nates (1.15) [55]

1

z = cosh ⇢ cos ⌧ + sinh ⇢ cos t = z cosh ⇢ sin ⌧

x = z sinh ⇢ sin

(1.17) This map will turn out to be useful in section 4.1 to relate the geodesic distance in global coordinates to the one in Poincaré coordinates.

Lastly, the Killing vector fields of the Poincaré patch are

L@x (=B21 J12) L@t (=B12 H) t@t+ x@x+ z@z (= B22) x@t+ t@x (=B11) (1.18) 1 L[(2xt@t+ (x 2_{+ t}2 _z2_)@ x+ 2xz@z] (= B21 J12) 1 L[(x 2_{+ t}2_{+ z}2_)@ t+ 2xt@x+ 2tz@z] (= B12 H)

As indicated on right hand side, they are linear combination of the Killing vectors (1.9) of the embedding space. Equation (1.18) will be used in section 5.2 to determine the Killing vector of the AdS-Rindler patch.

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1.3. METRICS AND GLOBAL SYMMETRIES OF 3D ADS 14

r

t 2⌅(1-)

Figure 1.1: The conical defect metric can be constructed by taking out a wedge of pure AdS. By identifying the end lines of the wedge, we obtain AdS3 with a conical defect.

The deficit angle of the resulting spacetime is 2⇡(1 ). 1.3.3 AdS3 with a conical defect

Let us introduce another set of embedding coordinates T1= ircos( )

T2=pr2_/ 2_{+ L}2_{sin( t/L)}

X1= r sin( )

X2= ipr2_/ 2_{+ L}2_{cos( t/L)}

(1.19)

In these coordinates the induced metric on the hyperboloid is ds2= ✓ 2₊ r2 L2 ◆ dt2+ ✓ 2₊ r2 L2 ◆ 1 dr2+ r2d 2, (1.20) where the coordinates range over t 2 [0, 1), r 2 [0, 1) and 2 [0, 2⇡). The parameter ranges from 1 (empty AdS) to 0 (Poincaré AdS). Between these two values the metric has a conical singularity at r = 0. The deficit angle corresponding to the case 0 < < 1 can be obtained by rescaling the coordinates

t0 = t , r0 = r/ and 0 = , (1.21)

leading to the metric ds2 = ✓ 1 +r02 L2 ◆ dt02+ ✓ 1 +r02 L2 ◆ 1 dr02+ r02d 02, (1.22)

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1.3. METRICS AND GLOBAL SYMMETRIES OF 3D ADS 15 where the angular coordinate is restricted to the range 0  0 _{ 2⇡ . Hence, the deficit}

angle of this geometry is

0 _{= 2⇡(1} _). _(1.23)

Thus we observe that the spacetime has a conical defect in the -direction. Actually the conical defect is the only diﬀerence between this geometry and global AdS. This follows immediately from the metric (1.22), which looks exactly like that of global AdS (1.13), except for the diﬀerent range of the angular coordinate. Later on, in chapter 5, we will view this conical defect geometry as a metric perturbation around a pure AdS background. From the discussion above, it should be clear that this perturbation is completely determined by one parameter .

Usually, this solution to the vacuum Einstein’s equations is excluded from the physical spectrum, since the conical singularity that it possesses is naked, i.e. it is not shielded by a horizon. Nevertheless, the metric has an interesting interpretation: the conical defect could be caused by a point particle at the origin r = 0. Deser and Jackiw found that the mass m of the point particle is related to the deficit angle by [35, 36]

m = 1 4G =

0

8⇡G, (1.24)

so m ranges from 0 (empty AdS) to 1/4G (Poincaré AdS). Note that this mass is not equal to the ADM mass, M = 2_/8G_{, which is commonly used to describe the mass of}

the BTZ black hole. Due to this point mass interpretation, it could be argued that the conical defect metric is still a meaningful solution.

1.3.4 BTZ black holes

Our next example of a non-trivial 3D anti-de Sitter spacetime is the BTZ black hole, named after the authors of [7]. This black hole solution is quite special, since it is asymptotically anti-de Sitter and since it has no curvature singularity. Nevertheless, it shares all generic features with other black holes, such as Schwarzschild or Kerr black holes. To wit, the BTZ black hole has an event horizon, it has thermodynamic properties, and it appears as the final state of collapsing matter [23]. In fact, we will see below that the non-rotating BTZ black hole is an AdS-Schwarzschild black hole in three spacetime dimensions.

Metrics for BTZ black hole. The metric for a BTZ black hole with mass M and angular momentum J is [7]

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1.3. METRICS AND GLOBAL SYMMETRIES OF 3D ADS 16 where the lapse N? _{and shift functions N are}

N?(r) = f (r) = ✓ 8GM + r 2 L2 + 16G2J2 r2 ◆1/2 , N (r) = 4GJ r2 . (1.26) The metric is singular when the lapse function N vanishes, which happens at two real values of r given by r2_±= 4GM L2 0 @1 ± s 1 ✓ J M L ◆21 A , or (1.27) r++ r p 8GL = p LM J and rp+ r 8GL = p LM + J, (1.28) where r+is the event horizon or black hole horizon, and r is the inner (Cauchy) horizon.

These are merely coordinate singularities, very similar to the singularity at r = 2GM of the ordinary Schwarzschild solution. The only true singularity is located at r = 0, but this is a singularity on the causal structure and not in the curvature, since the curvature is everywhere finite and constant.

Furthermore, the (tt)-component of the metric vanishes at

r2_erg = 8GM L2= r₊2 + r2, (1.29) which is the surface of infinite redshift. The three special radii are related by: r  r+ rerg. The region between r+ and rerg determines an ergosphere, just like for a 3+1

dimensional Kerr black hole. This means that timelike curves in this region necessarily have d /dt > 0 (when J > 0), so all observers are forced to move in the direction of rotation of the black hole [23].

It follows from (1.27), that the mass and angular momentum can be written in terms of the horizons as M = r 2 ++ r2 8GL2 , and J = 2r+r 8GL . (1.30)

The values of M and J are restricted by the condition that the horizons are real-valued: M > 0 and |J|  ML. In the extreme case, |J| = ML, the inner and outer horizon coincide: r+= r .

Using (1.30) we can also write the BTZ metric in terms of the horizons as ds2= (r 2 _r2 +)(r2 r2) r2_L2 dt 2₊ L2r2 (r2 _r2 +)(r2 r2) dr2+ r2⇣d r+r Lr2 dt ⌘2 . (1.31)

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1.3. METRICS AND GLOBAL SYMMETRIES OF 3D ADS 17 Below we will be mainly interested in the non-rotating case, i.e. when J = 0. The non-rotating black hole has no inner horizon, r = 0, and no ergosphere, rerg = 0. The

metric thus simplifies to

ds2= r 2 _r2 + L2 dt 2₊ L2 r2 _r2 + dr2+ r2d 2, = ✓ 8GM + r 2 L2 ◆ dt2+ ✓ 8GM + r 2 L2 ◆ 1 dr2+ r2d 2. (1.32a) (1.32b) Notice that this metric is not singular any more at r = 0. We still call it a black hole though, because it has an event horizon r+.

Let us consider three interesting limits or continuations of the non-rotating BTZ black hole:

(i) M = 0: This is the ground state of the BTZ black hole, also called the massless BTZ black hole. Eﬀectively the black hole disappears in this limit, since there is no event horizon and no singularity. Therefore it should be regarded as empty space. Note that the metric of the massless BTZ geometry is identical to the Poincaré metric, except for the fact that is not periodically identified in the latter.

(ii) 1/8G < M < 0: This corresponds to solutions with a conical defect in the -direction. The conical defect metric can be obtained from (1.32a) by the analytic continuation r+! i L. Note that again the event horizon disappears (because r+

is complex), but a naked singularity becomes apparent. The ADM mass of this solution can be expressed as: M = 2_/8G_{, with 0 < < 1.}

(iii) M = 1/8G: This can be recognised as global AdS3.

To sum up, we see that the physical spectrum of the BTZ geometry can be analytically continued to the conical defect solution and global AdS. Usually, solutions containing a naked singularity are excluded from the spectrum. From that point of view, global AdS emerges as a ‘bound state’, separated from the continuous black hole spectrum by a mass gap of 1/8G [6]. We argued in the previous section, however, that the conical defect solutions are meaningful as solutions containing a point mass.

BTZ black hole thermodynamics. Let us summarise the thermodynamic properties of BTZ black holes [25]. The Hawking temperature is

TH =  2⇡ = r2 + r2 2⇡r+L2 , (1.33)

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1.4. SATES AND LOCAL CONFORMAL SYMMETRIES IN 2D CFT 18

⌦H =

r r+L

. (1.34)

Both quantities, TH and ⌦H, follow naturally from the demand that there are no conical

singularities in the Euclidean black hole metric, as we will see below (this derivation of the Hawking temperature is known as the Euclidean path integral method [27]). Fur-thermore, the first law of thermodynamics for BTZ black holes is

dM = THdSBH+ ⌦HdJ. (1.35)

The first term is analogous to the heat flow dQ in thermodynamics, whereas the second term represents the work dW done on the black hole by adding a bit of matter to it. The only quantity which we have not mentioned so far is the famous Bekenstein-Hawking entropy SBH (1.1). By inserting the expressions for M, J, TH,and ⌦H into the first law,

it is straightforward to check that the entropy of the black hole is proportional to its area: SBH = 2⇡r+ 4GN = 2⇡ r L 8G(M L + J) + 2⇡ r L 8G(M L J), (1.36) where in the last step we used expression (1.28) to rewrite r+. In section 1.5 we will review

Strominger’s proof that the entropy of the BTZ black hole matches the thermodynamic entropy in the CFT [63].

1.4 Sates and local conformal symmetries in 2D CFT

Let us now move to the dual theory of gravity in AdS3, i.e. a two dimensional conformal

field theory. A 2D CFT is defined as a local quantum field theory with local conformal invariance in two dimensions. In this section we will first motivate why a gravitational theory in AdS3 is dual to a 2D CFT, by looking at the symmetries on both sides of the

duality. It turns out that, given certain boundary conditions, the asymptotic symmetry algebra is isomorphic to the conformal algebra in two dimensions. An important result in AdS3/CFT2 is that the asymptotic symmetry algebra can be centrally-extended to a

Virasoro algebra, for which the central charge has been computed by Brown and Hen-neaux [16]. After stating this result, we will furthermore describe which states in the CFT are dual to the asymptotically AdS3 spacetimes of the previous section.

Virasoro algebra and central charge. In section 1.3 we have seen that the isometry group of global AdS3is SL(2, )⇥SL(2, ). The isometry group is the set of symmetries

that leave the pure AdS3 metric invariant. It is well-known that SL(2, ) ⇥ SL(2, ) is

the global conformal group in two dimensions on the plane or on the cylinder. However, the full symmetry group of a CFT is the local conformal group. These local conformal symmetries can actually be derived from the ‘asymptotic symmetries’ of AdS space.

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1.4. SATES AND LOCAL CONFORMAL SYMMETRIES IN 2D CFT 19 Asymptotic symmetries are defined as deformations of spacetime that act non-trivially at infinity and that respect certain boundary conditions. By ‘deformation of spacetime’ we mean a variation of the metric under a diﬀeomorphism generated by a vector field ⇣µ_.

The variation of the metric is then given by the Lie derivative along ⇣µ_:

⇣gµ⌫ =L⇣gµ⌫.

Such a variation is generated by a conserved charge Q⇣, which is defined by the Poisson

bracket ⇣gµ⌫ = {gµ⌫, Q⇣}. Now a deformation of spacetime ‘acts non-trivially’ at the

boundary if the corresponding charge is non-zero. All asymptotic symmetries which diﬀer by terms with vanishing charges are taken to be identical. Furthermore, the boundary conditions are basically restrictions on the allowed finite deformations (or excitations) of the background geometry.4 _{By definition the boundary conditions are preserved by the}

asymptotic symmetries.

The asymptotic symmetries form an algebra, since they inherit a Lie algebra structure from the Lie commutator of diﬀeomorphisms.5 _{The asymptotic symmetry algebra should}

contain the isometry algebra as a subalgebra. In fact, Brown and Henneaux [16] have shown that the asymptotic symmetry algebra of AdS3is an infinite-dimensional extension

of the isometry algebra sl(2, ) sl(2, ). In two dimensions this infinite-dimensional extension is the conformal algebra, whose elements generate local conformal symmetries. Hence, given the infinitesimal generators ln, ¯lnwith n 2 – which are the Fourier modes

corresponding to the asymptotic Killing vectors ⇣µ _{– the asymptotic symmetry algebra}

of AdS3 is

[lm, ln] = (m n)lm+n,

[¯lm, ¯ln] = (m n)¯lm+n,

[lm, ¯ln] = 0.

These are two independent copies of the Witt algebra. Each of the two copies contains a finite subalgebra generated by l 1, l0, l1, which together form the isometry algebra of

AdS3. Hence the isometry algebra is indeed a subalgebra of the asymptotic symmetry

algebra.

Noether’s theorem now tells us that the local conformal symmetry is associated to a conserved charge. Let us denote the conserved charges associated to the vector fields ln(¯ln)by Ln( ¯Ln), i.e.

Ln⌘ Qln, L¯n⌘ Q¯ln.

This means that the charges Lnand ¯Lnare generators of local conformal transformations.

They are known as conformal (Noether) charges. It was shown in [17] that the algebra

4_{In general, the set of boundary conditions is not unique, but it should meet certain consistency}

requirements [43]: (1) they are invariant under the isometry group; (2) they allow for deformations of physical interest (for example, in the case of AdS3, the BTZ black hole); and (3) they yield finite and

well-defined charges. These requirements restrict the window of consistent and interesting boundary conditions.

5_{The algebra of asymptotic symmetries is often loosely speaking called the ‘asymptotic symmetry}

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1.4. SATES AND LOCAL CONFORMAL SYMMETRIES IN 2D CFT 20 of conformal charges is isomorphic to the central extension of the asymptotic symmetry algebra. Hence, the conformal charge algebra is not given by the Witt algebra, but by its central extension: the Virasoro algebra. Thus, we conclude that the generators Ln

and ¯Ln form two copies of the Virasoro algebra [40]

[Lm, Ln] = (m n)Lm+n+ cL 12m(m 2 ₁₎ m+n,0, [ ¯Lm, ¯Ln] = (m n) ¯Lm+n+ cR 12m(m 2 ₁₎ m+n,0, [Lm, ¯Ln] = 0. (1.37) where L 1, L0, L1 span again an sl(2, ) subalgebra. The real numbers cL and cR are

the left and right-moving central charges of the conformal field theory. In order to avoid local gravitational anomalies, we will often assume that the left- and right-moving central charges are equal: cL = cR = c. Further, the term which is proportional to cL,R/12

corresponds to a shift of the zero eigenmodes L0, ¯L0 (which can be interpreted as the

Casimir energy on the cylinder).

Lastly, Brown and Henneaux have computed the central charge belonging to the central extension of the asymptotic symmetry algebra [16]

c = 3L 2GN

. (1.38)

This formula is important in relating CFT observables to AdS quantities. For example, in chapter 4 we will make extensive use of this formula to relate entanglement entropy in the CFT to a geodesic length in AdS space.

To conclude, we have explained that the Virasoro algebra with corresponding cen-tral charge is equivalent to the (cencen-tral extension of the) asymptotic symmetry algebra of AdS3. It is well-known that the representations of the Virasoro algebra classify the states

of a two dimensional conformal field theory. However, below we will not review how the representations classifies the full spectrum of the theory. Instead we will focus on the states in the CFT that are dual to the asymptotically AdS spacetimes of the previous section.

Dual states in the CFT. Recall that the asymptotically AdS spacetimes correspond to diﬀerent ADM masses. In order to find the dual states in the CFT to these diﬀerent spacetimes, we need the relations for the mass and angular momentum in terms of the Virasoro generators M L = L0 cL 24 + ¯L0 cR 24, (1.39) J = L0 L¯0, (1.40)

where L0 and ¯L0 are the lowest Virasoro generators. From a CFT perspective these

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1.4. SATES AND LOCAL CONFORMAL SYMMETRIES IN 2D CFT 21 scale transformations on the plane, which correspond to time translations on the cylinder (in the so-called ‘radial quantization’). Since the Hamiltonian is the generator of time translations, it is natural to associate the right hand side of (1.39) with mass (or energy) in AdS. On the other hand, the linear combination L0 L¯0 generates rotations, and is

hence associated to angular momentum in AdS.

Now remember from our discussion on p. 17 that in AdS3 there is a mass gap in the

black hole spectrum between global AdS and the massless BTZ black hole. We pointed out that global AdS3 has a mass M = 1/(8G), whereas the massless BTZ geometry

obviously has zero mass M = 0. Coussaert and Henneaux [30] have explained this mass gap in a supersymmetric two dimensional conformal field theory as the Casimir energy. In a theory with superconformal symmetry it is known that the Ramond ground state and the Neveu-Schwarz ground state have diﬀerent L0 and ¯L0 eigenvalues. The

Neveu-Schwarz (NS) ground state has vanishing L0and ¯L0 eigenvalues. Hence, according (1.39)

the NS vacuum corresponds to a spacetime with mass M = c/(12L) = 1/(8G), i.e. global AdS3. In contrast, the Ramond (R) ground state has an eigenvalue L0 = c/12,

which corresponds to a zero mass M = 0, i.e. massless BTZ or Poincaré AdS3.

Furthermore, we have seen that spacetimes which are inside the mass gap 1

8G < M < 0

correspond to AdS with a conical defect. These conical geometries are dual to an excited state in the CFT. More precisely, the excited state is a primary state, which is created by a twist operator. In the next chapter we will further explain what a twist operator is. But for now it will suﬃce to state that a twist operator creates ‘twisted’ boundary conditions, which translate exactly to the diﬀerent periodicity of the angular coordinate in AdS (with corresponding deficit angle, eq. 1.23).

Lastly, the AdS/CFT dictionary states that the BTZ black hole is dual to a thermal state in the CFT. There is, however, also another AdS3 solution dual to a thermal state

in the CFT: AdS3 filled with thermal radiation. The BTZ black hole hole is only the

dominant classical solution above a certain critical temperature, Tc = (2⇡L) 1. Below

this temperature, thermal AdS is more probable than the BTZ solution. The change in the most dominant classical solution at temperature Tc is referred to as the

Hawking-Page phase transition.6 _{It is important to realise that both classical solutions (BTZ and}

thermal AdS) are possible for all temperatures, but one is more probable than the other depending on the temperature.

To sum up, we have the following duality between states in the CFT and geometries in AdS

M = 1

8G : global AdS3 $ NS vacuum state

1

8G < M < 0 : AdS3 with a conical defect $ twist operator M = 0 : Poincaré AdS3/massless BTZ $ R vacuum state

M > 0 : BTZ black hole & thermal AdS $ thermal state

6_{In appendix C we will show explicitly that there exists a sharp first-order phase transition between}

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1.5. THERMAL ENTROPY: A DERIVATION OF THE CARDY FORMULA 22

1.5 Thermal entropy: a derivation of the Cardy formula

In this section we will derive a formula for the thermodynamic entropy associated to a thermal state in the CFT. This formula was originally found by Cardy [22], but here we will closely follow Carlip’s presentation of the derivation [24]. The Cardy formula is of great importance for the AdS/CFT correspondence, since it matches precisely with the horizon entropy of BTZ black holes, as shown by Strominger [63]. At the end of this section we will consider the implications of this match for the interpretation of black hole entropy. But before we move to the actual derivation of the Cardy formula, we will first review the modular invariance of a conformal field theory. Modular invariance plays a crucial role in the derivation of the Cardy formula.

Modular invariance. Recall that the conformal boundary of asymptotically AdS3

spacetimes is a cylinder. We can change the conformal boundary by making the imaginary time direction periodic, i.e. tE ⇠ tE + , with inverse temperature = 1/T. Then the

conformal boundary becomes a torus. Hence the CFT lives in this case on the torus, and AdS spacetime (which has a finite temperature now) corresponds to the interior of the torus.

A two-dimensional torus can be obtained from the complex plane (with coordinate !) by modding out a lattice generated by

! _{7! ! + 1} ! _{7! ! + ⌧}

with Im ⌧ > 0. This means that the opposite edges of the parallelogram (unit cell) are identified to generate a torus (see Figure 1.2). The complex number ⌧ is called the modular parameter (or modulus) of the torus. For global AdS3 filled with thermal

radiation it can be shown that the modular parameter of the boundary torus is ⌧AdS =

✓ 2⇡+

i

2⇡L, (1.42)

where is the inverse temperature, and ✓ is the so-called “angular potential”. The angular potential ✓ is the canonical conjugate of the angular momentum J, just like the inverse temperature is the canonical conjugate of the Hamiltonian H. In appendix C we will show explicitly that the modular parameter of thermal AdS is given by (1.42).

Now there exist certain transformations that change the modular parameter but leave the torus invariant. These are called modular transformations, and they form the modular group that is isomorphic to SL(2, )/ 2 ⇠= P SL(2, ). Elements of the modular group

can be written as [40]

⌧ _7! a⌧ + b

c⌧ + d, with

a, b, c, d₂

ad bc = 1. (1.43)

Although the modular parameter changes under such transformations – and hence de-scribes a diﬀerent unit cell on the upper half-plane – the lattice itself is preserved up

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1.5. THERMAL ENTROPY: A DERIVATION OF THE CARDY FORMULA 23 Im Re 1 0 1+ -1/ A-cycle B -cyc le

(a) Complex plane

A-cycle B-cycle

(b) Torus

Figure 1.2: The two-dimensional torus can be obtained by modding out the complex plane by the lattice generated by the complex numbers 1 and ⌧, i.e. 2 _{= /( + ⌧ )}_.

The shape of the torus is invariant under modular transformations. In particular, we have illustrated the modular transformations T (green unit cell) and S (orange unit cell) on the complex plane.

to an overall scaling and rotation. That is, under transformation (1.43) the area of the unit cell changes as Im ⌧ 7! Im ⌧

|c⌧+d|2. But if one normalises the lattice basis {1, ⌧} to

n 1 p Im ⌧, ⌧ p Im ⌧ o

, then the area of the unit cell is equal to 1 and it stays the same under the modular transformation (1.43). Given this normalisation, a modular transformation only rotates the lattice by an irrelevant phase. Hence, the shape of the torus is left invari-ant by the action of SL(2, ). Moreover, we expect that the conformal field theory living on the torus is itself invariant under the action of the modular group. Therefore, we will assume below that the CFT partition function Z(⌧) on the torus is modular invariant.

Two important transformations that generate the whole modular group are

T : ⌧ 7! ⌧ + 1, (1.44)

S : ⌧ 7! 1

⌧. (1.45)

They are generators in the sense that each modular transformation can be written as a composition of powers of S and T . On the lattice S represents an inversion in the unit circle followed by a reflection with respect to the imaginary axis, whereas T represents a unit translation to the right. This has been illustrated in Figure 1.2. On the torus the S transformation has a simpler interpretation: it eﬀectively interchanges the roles of the A- and B-cycle of the torus. This is because it transforms the normalised basis as follows

S : ⇢ 1 p Im ⌧, ⌧ p Im ⌧ 7 ! ⇢ |⌧| p Im ⌧, e i p Im ⌧ , (1.46)

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1.5. THERMAL ENTROPY: A DERIVATION OF THE CARDY FORMULA 24 where the exponent represents a rotation of the unit cell. The invariance under S is crucial in deriving Cardy’s formula.

CFT partition function. The partition function is defined canonically as a trace over the CFT Hilbert space

Z = Trhe H+i✓Ji. (1.47)

On the torus the partition function becomes a function of the modular parameter ⌧, and its complex conjugate ¯⌧. By using the relations for the Hamiltonian H (1.39), the angular momentum J (1.40), and the modular parameter ⌧ (1.42) the partition function can be rewritten as

Z(⌧, ¯⌧ ) = ei✓(cL cR)24 Tr

h

e2⇡i⌧ (L0 cL₂₄)_e 2⇡i¯⌧ ( ¯L0 cR₂₄)i_. (1.48)

Note that if the left- and right-moving central charges are equal, i.e. cL = cR = c,

then the prefactor vanishes. Furthermore, the shifts cL,R/24are caused by the Casimir

eﬀect. For notational simplicity, we suppress the ¯⌧-dependence from now on, and restore it at the end of the computation. The trace can then be expressed as

Z(⌧ ) = X N 0 ⇢(N )e2⇡i⌧ (N 24c)= Z ₁ 0 dN ⇢(N ) e2⇡i⌧ (N 24c), (1.49)

where ⇢(N) is the number of states (or entropy density) with energy eigenvalue L0= N.

In the microcanonical ensemble, the entropy is essentially the logarithm of the number of states, i.e. S = log ⇢(N). At zero temperature there exists only one state (the vacuum state) so in that case the entropy is zero, but at finite temperature we expect a microscopic degeneracy for a fixed energy N.

Below we will compute the entropy for large values of N and for high temperatures. We will take the modular parameter to be purely imaginary, i.e. ⌧ = i /(2⇡L), so the high-temperature limit T ! 1 corresponds to ⌧ ! 0. Furthermore, as indicated above, we will assume that Z(⌧) is modular invariant. This means in particular that Z(⌧ ) = Z( 1/⌧ ). We want to use this fact to determine the density of states ⇢(N) for large N and for ⌧ ! 0 (we will see below that these limits are related to each other). From the density of states one can easily deduce a formula for the entropy, called the “Cardy formula”.

Cardy formula. One can derive an expression for ⇢(N) by taking the inverse Laplace transform of (1.49). The density of states is then given by the following contour integral

⇢(N ) = Z

C

d⌧ Z(⌧ ) e 2⇡i⌧ (N 24c), (1.50)

where the contour C is close to the real line. Note that in the high-temperature limit ⌧ ! 0 the contour integral is dominated by the behaviour of Z(⌧). It seems, however,

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1.5. THERMAL ENTROPY: A DERIVATION OF THE CARDY FORMULA 25 that we cannot determine the behaviour of Z(⌧), because Z(⌧) itself is computed by (1.49) which is defined in terms of ⇢(N).

Now Cardy’s key insight was to use the modular invariance of Z(⌧) to relate the high-temperature behaviour to the low-temperature behaviour. In particular, the partition function is invariant under S-transformations (1.45)

Z(⌧ ) = Z( 1/⌧ ) = e2⇡i⌧124c Z( 1/⌧ ),˜ (1.51)

where ˜Z( 1/⌧ ) = Tr e 2⇡iL0/⌧_.In the limit ⌧ ! 0, the factor ˜_{Z( 1/⌧ )} is dominated by

the contribution from the lowest eigenvalue of L0, which we take to be N = 0. Since the

ground state of the CFT is unique, the factor ˜Z approaches the value 1 as ⌧ ! 0. Hence, we see that in the high-temperature regime the leading behaviour of Z(⌧) is determined by the term e2⇡i_⌧1₂₄c _{, i.e. by the central charge c.}

Thus, approximating Z(⌧) by its leading behaviour leads to ⇢(N )⇡ Z C d⌧ exp  2⇡i ✓ ⌧⇣N c 24 ⌘ ₁ ⌧ c 24 ◆ . (1.52)

We can now evaluate this expression by making use of a saddle point approximation. The idea behind the saddle point approximation is that the dominant contribution to the integral in the limit ⌧ ! 0 will come from the stationary point – called “saddle point” generically – of the exponent. Hence, we will approximate the integral by ⇢(N) ⇠ e 2⇡if (⌧0), where ⌧

0 is the value for ⌧ at the stationary point of f(⌧). In this case the

stationary point is a minimum of f(⌧), and the value of ⌧ at this minimum is ⌧0= i

r _c

24(N c/24). (1.53)

Note that the high-temperature limit ⌧ ! 0 corresponds to N c/12. Hence the saddle point approximation is only valid in the regime where the energy level N is much larger than the central charge. By substituting (1.53) back into the integral, we obtain

⇢(N )⇡ const. ⇥ exp  2⇡ r c 6 ⇣ N c 24 ⌘ . (1.54)

The constant in front of the exponent is very small in the regime N c/12, and hence does not contribute to the entropy at leading order in N.7 _{Thus, if we restore the}

anti-holomorphic part, the entropy becomes SCardy= 2⇡ r cL 6 ⇣ NL cL 24 ⌘ + 2⇡ r cR 6 ⇣ NR cR 24 ⌘ . (1.55)

This is the famous Cardy formula. Interestingly, the entropy is only a function of the central charges and the energy eigenvalues. It is important to emphasise again that the Cardy formula only holds in the regime NL cL/12, NR cR/12.

7_{This constant does provide higher-order corrections to the Cardy formula. These high-order}

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1.5. THERMAL ENTROPY: A DERIVATION OF THE CARDY FORMULA 26 Lastly, we can also write the Cardy formula in the canonical ensemble by using the definition of the left and right temperatures

Lcir 2⇡ @SCardy @NL NR = 1 TL , Lcir 2⇡ @SCardy @NR NL = 1 TR , (1.56)

where Lcir is the circumference of the cylinder on which the CFT lives. The factor

Lcir/2⇡ ensures that the temperatures have the correct dimensionality. Hence the

left-and right-temperatures are TL= 1 Lcir s NL cL/24 cL/24 , TR= 1 Lcir s NR cR/24 cR/24 (1.57)

In terms of these temperatures the Cardy formula becomes SCardy=

⇡Lcir

6 (cLTL+ cRTR) , (1.58) which is now valid for TL 1, TR 1.

Match between Cardy formula and Bekenstein-Hawking entropy. Strominger [63] has famously matched the horizon entropy of a large BTZ black hole with the ther-modynamic entropy in the CFT through the Cardy formula. In order to demonstrate this match, all we need to do is insert the Brown-Henneaux formula for the central charge, c = 3L/(2GN), and the eigenvalues of L0 and ¯L0 for the BTZ geometry in the

Cardy formula. The energy eigenvalues can be obtained from equations (1.39) and (1.40): NL c/24 = 1₂(M L + J), and NR c/24 = 1₂(M L J). We assume here that the

left-and right-moving central charges are equal, i.e. cL= cR= c. The entropy from Cardy’s

formula (1.55) then becomes SBH = 2⇡ r L 8G(M L + J) + 2⇡ r L 8G(M L J) = 2⇡r+ 4GN . (1.59)

This is precisely the Bekenstein-Hawking entropy (1.36) of a BTZ black hole. Thus, we find exact agreement between the thermodynamic entropy calculated via the Cardy formula and the black hole entropy calculated via the Bekenstein-Hawking formula. Since the Cardy formula is a measure for the microscopic degeneracy associated to a thermal state, we conclude that the black hole entropy should also be interpreted as a measure of the number of microstates in the CFT. Recently, however, it has been proposed that black hole entropy should be interpreted as an entanglement entropy [18, 19, 32, 61]. In section 4.4 we will further discuss this proposal. The upshot of this discussion is that the Bekenstein-Hawking entropy always measures the microscopic degeneracy of the black hole system. Only for a very special case, the so-called “thermofield double system”, is the entropy of a black hole precisely equal to an entanglement entropy.

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Chapter 2

Holographic Entanglement Entropy

In this chapter we will review the proposal by Ryu and Takayanagi that entanglement entropy in the CFT is proportional a minimal surface area in AdS. First, in section 2.1 we will define entanglement entropy in quantum mechanics. Then, we will point out the diﬀerences between entanglement entropy and thermodynamic entropy in section 2.2. In section 2.3 we move to quantum field theory, and we will discuss the famous area law for entanglement entropy. Furthermore, we also explain how entanglement entropy can be calculated in quantum field theory by using the replica trick. In section 2.4 we will finally give the Ryu-Takayanagi formula, and in the last section 2.5 we interpret its implications for the emergence of the connectedness of space.

2.1 Definition and properties of entanglement entropy

Density operators. In quantum mechanics any state can be described by a density matrix ⇢. A density matrix is a self-adjoint and positive operator with unit trace, i.e. Tr ⇢ = 1. A pure state | i in a Hilbert space H of dimension N can be described by the following density operator

⇢pure=| ih |. (2.1)

The density operator of a mixed state, however, is by definition not given by a one-dimensional operator. In terms of its eigenstates {| ii} the density matrix takes a

diag-onal form ⇢mixed = N X i=1 i| iih i|, (2.2)

where i’s are the eigenvalues of ⇢. How should one interpret these quantum mechanical

mixed states? It is important to distinguish between two kinds of mixed states: proper and improper mixtures. When a system is really in some pure state, but we don’t know which one, then it is said to be properly mixed. The eigenvalue i represents in this case

the probability p(i) of the system being in the state | ii. Since the real state of the

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2.1. DEFINITION AND PROPERTIES OF ENTANGLEMENT ENTROPY 28 system is unknown to the observer, the density operator is said to have an ignorance interpretation. As we will see below, thermal density matrices are important examples of proper mixtures.

There are also quantum systems which do not have an underlying pure state at all, but can only be described by a mixed density operator. These systems are improper mixtures. For these mixtures an ignorance interpretation is impossible. A typical example of an improper mixture is the reduced state of an entangled system. Consider an entanglement between two disjoint subsystems A and B. These subsystems are disjoint whenever the total Hilbert space can be written as a tensor product H = HA⌦ HB. An entangled

state | iAB 2 H can then be written by the Schmidt decomposition as

iAB =

X

i

p

i| iiA⌦ | iiB, (2.3)

where {| iiA} is an orthonormal basis of HA and {| iiB} is an orthonormal basis of

HB. Since the entangled system is – by definition – not a product state, there will be

no pure states for the subsystems. However, we can assign a reduced density operator to subsystem A by tracing over the Hilbert space HB:

⇢A= TrB⇢AB =

X

i

i| iiA Ah i|, (2.4)

and analogously we can define a reduced state for subsystem B. It follows from expres-sion (2.4) that ⇢Ais a mixed state, and this is the only possible description of subsystem

A in standard quantum mechanics. Thus reduced states of entangled systems have no ignorance interpretation.

Definition of Von Neumann entropy. Suppose we want to perform a quantum measurement of observable O on a physical state ⇢ of the form (2.2). We assume that the measurement is maximal, meaning that it has N diﬀerent possible outcomes (where N =dim H). And we assume for simplicity that ⇢ and the observable O have a common set of eigenstates {| ii}. Prior to the measurement we only know the probabilities

p(i) = i for the possible outcomes. After the measurement one of these outcomes

materializes, and our uncertainty disappears. The Von Neumann entropy – which is the quantum analog of the Shannon entropy – provides a measure of our ignorance about the outcome, prior to the measurement. From another point of view, the entropy gives a quantitative measure for the average amount of information that we expect to gain in this measurement. It is defined by

S(⇢) = Tr ⇢ log ⇢, (2.5)

where log ⇢ is a natural logarithm (information theorists use base 2 instead of e for the logarithm). We have set the Boltzmann constant equal to one, kB = 1, but in

thermodynamics an extra factor of kB is included in the definition of the entropy.

If we choose the orthonormal basis {| ii} that diagonalises ⇢, then the entropy