Gravity from Entanglement in Holographic Theories with Defects

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MSc Physics & Astronomy

Theoretical Physics

Master Thesis

Gravity from Entanglement in

Holographic Theories with Defects

by

Huy Hoang Vu

10336818

August 2020

60 EC

Supervisor:

Dr. Jay Armas

Examiners:

Dr. Jay Armas

Dr. Alejandra Castro

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Abstract

In [1] conformal field theories with a boundary were considered which are holographic duals to AdS spacetimes that are terminated by a brane. Allowing the brane to be dynamical, it was shown that the dynamics of the brane are constrained by the quantum entanglement in the field theory. Building upon this idea, we considered conformal field theories with a co-dimension one defect. Given that the defect conformal field theories are dual to AdS spacetimes that are divided into two regions by a brane that acts as an interface, we obtained constraints on the dynamics of these branes.

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Appendix

60

A The embedding formalism 61

A.1 Correlation functions in embedding spacetime. . . 62

Example 1:h𝜙₁(𝑥₁)𝜙₂(𝑥₂)𝜙₃(𝑥₃)i. . . 63

Example 2:h𝑇𝐴𝐵(𝜉1)𝑇𝐶𝐷(𝜉2)i . . . 63

A.2 Modular flow on𝐷[𝐵] . . . 64

B Covariant phase space formalism 67 The generator of infinitesimal diffeomorphisms . . . 69

The(𝑑 −1)-form𝝌 . . . 70

C Junction & Neumann conditions 72 C.1 Israel junction conditions . . . 72

C.2 Linearized Neumann condition . . . 73

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Introduction

1

Many breakthroughs in physics found their origin in the understanding of the underlying principles of nature. Two important principles that come to mind are the principle of relativity and the (gauge) symmetry principle. The former principle resulted in the formulation of special relativity and a better understanding of electrodynamics, whereas understanding of the latter principle lead to the formulation of the Standard Model.

The existence of another principle was hinted by the following two discoveries. First, the entropy of a black hole is proportional to the area of the black hole rather than the volume. And second, given a finite region of space with a finite amount of energy, there is an upper bound on the entropy of that region. Remarkably, the upper bound is saturated by black holes. These discoveries led to the idea that the description of a spacetime region is encoded by the area rather than the volume. This is known as the holographic principle.

Unlike the two principles mentioned above, the holographic principle has not been resolved into a well-established theory (yet). Therefore, it is natural to question: What kind of theory requires the holographic princi-ple? This question can be answered by another question. Considering the thermodynamic properties of a black hole, it can be thought of as a statistical ensemble of micro-states. Is it then possible to compute the entropy by counting the micro-states that will agree with the gravitational computation that gave the area law of the black hole entropy? If the holographic principle would ever be incorporated into a theory it should be a theory that unites a gravitational theory with quantum mechanics. That is, the framework should be a quantum theory of gravity.

One framework that tries to encapsulate the holographic principle into a theory of quantum gravity and which plays a central role in this work is the AdS/CFT correspondence. This is a conjecture which states that Anti de Sitter spacetimes are dual to a conformal field theory in one dimension lower. Although not proven yet, the evidence has mounted in favor of the duality in the past two decades and has convinced many that the correspondence holds. Instead of working out the validity of the correspondence, the mindset gradually shifted towards thinking about what can be learned from the duality. As per definition of a duality, it allows the study of quantum systems by studying the dual gravitational theory and vice versa. This work will be primarily engaged in understanding the dynamics in the gravitational theory using the knowledge from the quantum field theory.

The main narrative of the thesis concerns understanding how quantum entanglement can lead to the emergence of gravitational dynamics.

Chapter 2 contains a short overview of conformal field theories. To make the discussion even more interesting, extended objects will be introduced in the conformal field theory inChapter 3, these extended objects are also known as defects. InChapter 4the AdS spacetimes will be

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1 Introduction 2

considered followed byChapter 5in which a more detailed description of the correspondence will be given. The important tools that will be necessary for the analysis are discussed inChapter 6andChapter 7. Then, the gravitational dynamics will be considered for field theories without defects inChapter 9, and with a co-dimension one defect inChapter 10.

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1: Evidently, the different renormalisation

schemes should all give the same physics

in the end.

Conformal field theories

2

What are conformal field theo-ries? . . . . 3 More symmetry: From QFT to CFT. . . . 4 The algebra of the𝐶𝑜𝑛 𝑓 . . . 5 The operator product expansion7 Why study conformal field theo-ries? . . . . 8

In this chapter we review some essential features of a conformal field theory, most of what is presented here can be found in [2–4].

What are conformal field theories?

We begin with a generic quantum field theory and focus on what is known as the partition functional

𝑍[𝐽] = ∫ D[Φ] exp 𝑖 ∫ 𝑑𝑑 𝑥 L[Φ, 𝜆] , (2.1)

whereLis the Lagrangian (density), {Φ} is the set of fields that are present in the theory and{𝜆} denotes the set of sources, such as the masses and coupling constants. The partition function contains all the information of the physical system and once the values of the sources are defined we obtain a specific theory. The path integral is taken over all possible field configurations and, in general, this will lead to complications involving divergences. One possible occurrence of divergence is due to the high energy modes of the field configurations, known as a UV divergence. For instance, the computation of Feynman diagrams may involve loop diagrams that are associated with integrals over all possible momenta 𝑘. If the integrals do not yield a finite result due to the high momenta, we have to regulate them by introducing a cut-off |𝑘| < Λ. However, the introduction of a cut-off is a mathematical convenience and physical observables should not depend on this quantity. There are various ways1 to deal with this seemingly arbitrariness and the process of eliminating the cut-off is called renormalization, a few of these methods are discussed in [5].

A profound consequence of renormalization is the running of the coupling constants. This means that coupling constants are in fact functions of the energy scale. To elaborate on this insight, we consider the Wilsonian formulation. In this picture we renormalize the theory by integrating out small momentum shells and perform a re-scaling of the spacetime 𝑥 → 𝑥/𝑠such that we can compare the old action𝑆and new action𝑆0. During this process we have effectively changed the relevant energy scale 𝜇 of the theory. The coupling constants might change after integrating out the momentum shells and as a result we obtain a different theory. Another possibility is that the coupling constants do not change at different energy scales. How the coupling constants change as functions of the energy scale is given by the𝛽-function

𝑑𝜆 𝑑log𝜇

=𝛽(𝜆).

In the Wilsonian renormalization approach it is apparent that different theories on different scales are connected, this is the principle of the

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2 Conformal field theories 4

2: Although proven only in a few

dimen-sions,we assumethatscaleinvariance

always implies conformal invariance.

3: This is achieved by introducing a

rele-vant operator in the theory.

Figure 2.1: An illustration of the RG flow in the space of theories. The purple and red points indicate a CFT in the UV and IR, respectively. The dotted lines connect-ing the two fixed points denote various pathways of the RG flow.

4: To preserve the non-degeneracy of the

metric we requireΩ(𝑥)> 0. renormalization group flow. We can imagine the sources𝜆 to span an

infinite dimensional parameter space where each point defines a theory, seeFigure 2.1. The points in the space of theories where the𝛽-function vanishes are called fixed points. It is exactly at the fixed points where the conformal field theories live. To be more precise, the fixed points in the space of theories give us a quantum field theory with scale invariance, upon promoting scale invariance to conformal invariance2we obtain a conformal field theory (CFT).

Now we know what a CFT is, the next step would be to consider the symmetries of the theory, but before we move on it is worth commenting on the fixed points in the space of theories. The fixed points can be classified as IR or UV fixed points and accordingly we have IR and UV CFT’s. The IR CFT’s have more high energy modes integrated out compared to UV CFT’s, implying a more detailed short-distance description in the latter. Then, the renormalization group flow states that breaking the conformal invariance3of the UV CFT will trigger a flow from the UV to the IR fixed point, seeFigure 2.1. Intuitively, we may expect that the number of physical degrees of freedom decreases under an RG flow. A function that measures the degrees of freedom is broadly termed a𝑐-function and in𝑑 = 2, 4 it has been shown that such a function indeed is a monotonically decreasing function under the RG flow [6,7].

More symmetry: From QFT to CFT

We know that quantum field theories are invariant under diffeomor-phisms𝑥 → 𝑥0that preserve the form of the metric𝑔0

𝜇𝜈(𝑥0) = 𝑔𝜇𝜈(𝑥). The

global symmetries of a quantum field theory are given by the Poincaré groupP, which consists out of spacetime translations and Lorentz trans-formations. Each symmetry under the action ofP yields a conserved current, which is encoded by the stress-energy tensor𝑇_𝜇𝜈.

CFT’s are invariant under a larger symmetry group consisting out of diff_{oWeyl transformations, naturally we have P ⊂ diffoWeyl. The} diffeomorphisms that are allowed preserve the form of the metric up to a conformal factor4 𝑔_𝜇𝜈0 (𝑥0) = Ω(𝑥)𝑔_𝜇𝜈(𝑥). On the other hand, Weyl transformations only affect the metric 𝑔_𝜇𝜈0 (𝑥0) = Ω(𝑥)𝑔_𝜇𝜈(𝑥), such that the line element transforms as(𝑑𝑠0)2= Ω(𝑥)𝑑𝑠2. The action of an element of diff_{oWeyl can be summarised as follows}

𝜙(𝑥) → 𝜙0_(𝑥0 ) = Ω−Δ𝜙(𝑥)𝜙(𝑥) 𝜕𝜇𝜙(𝑥) → 𝜕0𝜇𝜙0(𝑥0) = 𝜕𝑥 𝜈 𝜕𝑥0_𝜇𝜕𝜈 Ω −Δ_𝜙₍_{𝑥)𝜙(𝑥)} 𝑔_𝜇𝜈(𝑥) → 𝑔0 𝜇𝜈(𝑥0) = 𝑔𝜇𝜈(𝑥),

where the conformal factorΩ−Δ𝜙(𝑥)is due to the Weyl transformation andΔ_𝜙is the scaling dimension of the field𝜙(𝑥) (see next section). The global symmetries of a CFT are the global symmetries ofP enhanced by dilatations and special conformal transformations. These symmetry transformations form the conformal group𝐶𝑜𝑛 𝑓. The larger symmetry

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5: Denoting the spacetime as the manifold 𝑀, we have𝑀 ∪ {∞}.

group of a CFT allows us to determine the 2- and 3- points functions (up to a constant) h𝜙1(𝑥1)𝜙2(𝑥2)i =      𝐶12 𝑥2Δ1 12 if Δ₁= Δ₂ 0 if Δ₁≠ Δ₂ h𝜙₁(𝑥₁)𝜙₂(𝑥₂)𝜙₃(𝑥₃)i = 𝐶123 𝑥Δ1+Δ2−Δ3 12 𝑥Δ2+Δ3−Δ1 23 𝑥Δ1+Δ3−Δ2 13

where 𝑥𝑖𝑗 = |𝑥𝑖−𝑥𝑗|. These expressions can be obtained by solving

the differential equations that arise by applying the symmetry trans-formations. However, there is a more economical approach to obtain the correlation functions by realising that the conformal group in 𝑑 dimensions𝐶𝑜𝑛 𝑓_𝑑is locally isomorphic to𝑆𝑂(𝑑,2), the Lorentz group in𝑑 +2 dimensions. The idea is to embed the𝑑dimensional theory into a𝑑 +2 dimensional spacetime such that the nonlinear transformations of𝐶𝑜𝑛 𝑓𝑑become linear transformations of𝑆𝑂(𝑑,2). This procedure is known as the embedding formalism, the reader is referred to appendix

Afor a review of this formalism.

The algebra of the

𝐶𝑜𝑛 𝑓

In this section we will briefly discuss the algebra associated to𝐶𝑜𝑛 𝑓𝑑. We consider infinitesimal transformations of the form𝑥𝜇→𝑥𝜇+𝜖𝜇. The form of the vector𝜖𝜇is constrained by the conformal Killing equation

(L_𝜖𝑔)_𝜇𝜈=𝜎𝑔_𝜇𝜈,

whereL_𝜖denotes the Lie derivative with respective to the vector field𝜖. After some manipulations we obtain the following equations

𝜕𝜇𝜖𝜈+𝜕𝜈𝜖𝜇= 2

𝑑(𝜕 · 𝜖)𝑔𝜇𝜈 (2.2)

_𝑔

𝜇𝜈 + (𝑑 − 2)𝜕𝜇𝜕𝜈𝜕 · 𝜖 = 0. (2.3) The second equation simplifies when𝑑 = 2, we will consider this special case subsequently. For𝑑 > 2 we observe that 𝜖𝜇can be at most quadratic in the coordinates. The allowed transformations are

𝜖𝜇₌              𝑎𝜇 Spacetime translations 𝜔𝜇𝜈𝑥𝜈 Lorentz boosts 𝜆𝑥𝜇 Dilations

2(𝑥 · 𝑏)𝑥𝜇−𝑥2𝑏𝜇 Special conformal transformations (SCT).

We apply the exponential mapping of the infinitesimal transformations to obtain the global transformation. For example, the finite SCT becomes

𝑥0𝜇

= 𝑥

𝜇+𝑏𝜇𝑥2

1+ 2𝑏 · 𝑥 + 𝑏2𝑥2

.

Notice, that this transformation is ill-defined when the denominator is zero. We may demand that the denominator is nonzero, however this will restrict the symmetry group. A second option is to add the point infinity5to the topology of the spacetime such that the symmetry group

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6: That is, an element𝑔 ∈diff_oWeyl. is extended to a larger set.

Let us now consider the case𝑑 = 2 for which (2.3) becomes the Cauchy-Riemann equations

𝜕1𝜖1=𝜕2𝜖2 and 𝜕2𝜖1= −𝜕1𝜖2.

This highly suggest that we work with complex coordinates:

𝑧 = 𝑥 + 𝑖𝑦 and 𝑧 = 𝑥 − 𝑖𝑦¯ 𝜖(𝑧) = 𝜖1+𝑖𝜖2 and 𝜖(¯𝑧) = 𝜖¯ 1−𝑖𝜖2.

The Cauchy-Riemann equations tell us that𝜖𝜇can be any infinitesimal transformation as long as it is (anti-)holomorphic. Thus, for𝑑 = 2 the global conformal transformations can be identified with the analytic coordinate transformation𝑧 → 𝑓 (𝑧), with holomorphic inverse 𝑓−1(𝑧). However, just as for𝑑 > 2, a similar issue arises when we do not add the point infinity: The exponential mapping to global transformations must correspond to entire functions [8], then the only allowed transformations are of the form

𝑓 (𝑧) = 𝛼𝑧 + 𝛽, 𝛼, 𝛽 ∈ℂ.

We observe a discrepancy between the local and global transformations in𝑑 = 2, the latter transformations being less constraining. Again, we can extend the global group by adding{∞}, i.e. by compactifying the complex plane. Then the allowed global transformations are

𝐶𝑜𝑛 𝑓∞ 2 = ( 𝑓 (𝑧) 𝑓 (𝑧) = 𝛼𝑧 + 𝛽 𝛾𝑧 + 𝛿, 𝛼, 𝛽, 𝛾, 𝛿 ∈ℂ, 𝛼𝛿 − 𝛾𝛽 ≠ 0 ) .

We will now consider the action of the symmetry group on the fields for 𝑑 > 2, to obtain the elements of the algebra

𝛿𝜔𝑎𝜙(𝑥) = −𝑖𝜔𝑎𝐺𝑎𝜙(𝑥), (2.4)

where{𝜔_𝑎} is the set of infinitesimal parameters and𝐺_𝑎are the generators of the𝐶𝑜𝑛 𝑓𝑑. As an example, we consider a dilatation transformation 𝑥0𝜇

= 𝜆(𝑥)𝑥𝜇. This conformal transformation6 induces the following action on a scalar field

𝜙0₍

𝑥) = 𝜆−Δ_𝜙_𝜙(𝜆−1_𝑥).

We assume𝜆 = 1 + 𝜔_𝑎, such that we can expand the field 𝜙0_{(𝑥) ≈ (1} − Δ_𝜙𝜔𝑎)𝜙([1 − 𝜔𝑎]𝑥) ≈ (1 − Δ_𝜙𝜔𝑎) 𝜙(0) + 𝑥𝜇_𝜕 𝜇𝜙(0) + 1 2 𝑥2𝜕2𝜙(0) + . . . −𝜔𝑎𝑥𝜇𝜕_𝜇𝜙(0) − 𝜔𝑎𝑥2𝜕2𝜙(0) + . . . ≈𝜙(𝑥) − Δ_𝜙𝜔𝑎𝜙(𝑥) − 𝜔𝑎𝑥𝜇𝜕_𝜇𝜙(𝑥) + . . .

From (2.4), we obtain the generator of dilatations −𝑖𝜔𝑎𝐺𝑎𝜙(𝑥) = −𝜔(Δ_𝜙+𝑥𝜇𝜕_𝜇)𝜙(𝑥)

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7: Also known as the Cartan subalgebra.

8: There are exceptions, see [10] and refer-ences therein.

9: In analogy to group theory, we have a

subset of elements{O_𝛼} that generate all fields in the theory{Φ, O_𝛼}.

Following similar steps we can obtain the complete set of generators of 𝐶𝑜𝑛 𝑓_𝑑, which is given by 𝑃_𝜇= −𝑖𝜕_𝜇 (Spacetime translations) 𝐾_𝜇_{= −𝑖(2𝑥}_𝜇𝑥𝜈_𝜕_𝜈−𝑥2 𝜕𝜇) (SCT) 𝐿_𝜇𝜈 _{= −(𝑥}_𝜇_𝜕_𝜈−𝑥_𝜈𝜕_𝜇) (Lorentz boosts) 𝐷 = −𝑖𝑥𝜇_𝜕 𝜇+ ˜Δ (Dilations).

Having obtained the generators of𝐶𝑜𝑛 𝑓_𝑑it remains to obtain the max-imal set of commuting generators7.We refrain from presenting all the commutators here, instead we mention that the maximal set is{𝐷, 𝐿_𝜇𝜈}. It is preferred to work with the irreducible representations of the Lorentz group and in these representationΔ will simply be a number, the scaling˜ dimension of the field𝜙 [3]. Thus, the fields are labelled by the scale dimensionΔ and spin 𝑠 which are the eigenvalues of 𝐷 and 𝐿_𝜇𝜈, respec-tively. In this representation, the generators𝑃_𝜇and𝐾_𝜇act as raising and lowering operators of the scaling dimension, respectively. The fields that are annihilated by𝐾_𝜇are known as primaries and the raising operator 𝑃_𝜇can act upon the primaries, these fields are known as descendants. To conclude this section, we mention that the Casimir invariant of the Poincaré group,𝑃2, does not commute with the generator𝐷[9]

𝑒𝑖𝛼𝐷𝑃2𝑒−𝑖𝛼𝐷

= 𝑒2𝛼𝑃2

𝛼 1.

Since 𝑃2 is associated with the mass (squared), the non-commutivity implies that conformal field theories either contain particles with a continuous mass spectrum or contain massless particles only. In the realms of physical theories, the first option makes less sense and therefore it is almost always8assumed that conformal fields theories only contain massless particles.

The operator product expansion

We have mentioned that a generic quantum field theory contains a set of fundamental fields{Φ}, these fields are still present in a conformal field theory. However, we have to take into account an additional set of fields {O_𝛼} that appear in the computation of the correlation functions, where the index𝛼 = {Δ, 𝑠} denotes the set of quantum labels that characterise the fields. The fields{Φ, O_𝛼} are related by means of the operator product expansion (OPE) which we will denote symbolically as [11]

lim 𝑥 1→𝑥2 Φ(𝑥1)Φ(𝑥2) = X 𝑛 𝑎𝑛[O_𝛼]𝑛.

The OPE reveals that we do not have to compute the correlation functions of all the fields in the theory; Computing the correlation functions of the conformal fields{O_𝛼} is sufficient to also determine the correlation functions of the fundamental fields9.The strength of a CFT resides within this detail, since the symmetries of a CFT are strong enough to constrain the two- and three-point functions. In other words, a CFT is completely defined by the OPE coefficients𝑎_𝑛and the values of the quantum labels of the operators in the CFT, this information is known as the CFT data.

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10: For the present discussion.

Why study conformal field theories?

The first motivation is given in the context of the discussion above, where we have discussed the space of theories in which the fixed points are the conformal field theories. We introduced the space of theories as a result of renormalization, which in turn was introduced to remove the UV divergences in the quantum field theory. Any quantum field theory that does not possesses these UV divergences, the so-called UV complete theories, can be viewed as effective theories at a point along an RG flow. The reason is because a UV complete theory must be a conformal field theory, since at high energies we can neglect any mass scale in the theory. From this point of view, it is very natural to study conformal field theories to obtain any well-defined quantum field theory that has the hope of being UV complete.

The scale invariant property of a conformal field theory makes these theories also applicable in systems with characteristic length scales that are either zero or infinite. Therefore, conformal field theories are suitable to describe the critical behaviour of condensed matter systems at second order phase transitions.

Another reason to study conformal field theories is within the context of string theory. The fundamental objects in string theory are one-dimensional extended objects, i.e. strings. The time evolution of these one-dimensional objects sweeps out a two-dimensional manifold, known as a sheet, within the spacetime. It turns out that the world-sheets are invariant under diffeomorphisms and Weyl transformations. The combination of these transformations exactly form a conformal transformation, i.e. the world-sheets display conformal invariance and it is therefore natural to study these world-sheets with conformal field theories in two dimensions.

The most important reason10to study conformal field theories is within the realisation of the holographic principle in the form of the AdS/CFT correspondence, where conformal field theories are dual to a gravitational theory in one dimension higher. We will discuss this correspondence in more detail inChapter 5.

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1: For example, kinematics of rotating

ob-jects are constraint by the conservation of

angular momentum.

Conformal field theories with

defects

3

Less symmetry: From CFT to DCFT . . . . 9 A DCFT example . . . . 10

InChapter 2we have seen that conformal field theories possess enough symmetries to constrain the two- and three-point functions. The con-formal symmetries become even greater in two dimensions, where the conformal algebra is infinite-dimensional. These examples illustrate that having a greater number of symmetries in the theory can aid in solving said theory. From a more physical perspective, symmetries dictate what is physically allowed and what is forbidden1.However, more symmetries also means more constraints to the physical phenomena that we would like to observe; These constraints might not reflect the reality in which we live. To still enjoy the mathematical niceties of having a great number of symmetries a good approach would be to start with an idealized theory, study the physics under these idealized circumstances and then relax the number of symmetries to construct a more realistic theory. We can apply this strategy to conformal field theories by introducing extended objects into the theory. The application of conformal field theories to real world experiments would necessarily require the understanding of the behaviour of these theories in the presence of extended objects such as defects, interfaces and boundaries. Boundary conformal field theories (BCFT’s) and defect conformal field theories (DCFT’s) have found many application within different areas of physics, a summary of applications is given in [12]. Here, we are primarily concerned with conformal field theories with defects and in the next section we discuss the symmetries of these theories.

Less symmetry: From CFT to DCFT

In the broadest sense, a defect can be viewed as a non-local operator, i.e. an extended operator. An example in a quantum field theory would be the Wilson loop: An operator introduced in gauge theories to study the confinement mechanism of quarks [13]. The Wilson loop is defined as the integral over a closed path and therefore we may classify this operator as a defect. There are other ways to introduce defects into the theory and, as a general guideline, we can distinguish three types of defects [14]:

1. Defects constructed by localising ambient fields at the location of the defect. The Wilson loop is an example of this type of defect. 2. Defects formed by imposing boundary conditions on the ambient

fields around the defect.

3. Defects viewed as a localised system with additional degrees of freedom on the defect.

The presence of a defect will break some of the symmetries in a quantum field theory and the surviving symmetries depend on the shape of the defect. Since we are interested in conformal field theories, we shall only consider defects that preserve a subgroup of the conformal symmetries. With the exception of the example given in the next section, we will consider the defects as a localised system. That is, suppose that we have

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3 Conformal field theories with defects 10

2: There are a few subtleties when𝑛 = 𝑑 −1, these subtleties will not be relevant for the present discussion, see for example

[15].

3: This is an important point: To define

a DCFT werequire a higher dimensional CFT.

4: We assume that we are in the weak

coupling regime, such that𝜆 1. a𝑑dimensional conformal field theory defined on a manifold𝑀. Then,

a defect is a localised system defined on a submanifold of𝑀. We will refer to the𝑑dimensional conformal field theory as the bulk CFT. We shall be primarily concerned with planar defects of spatial co-dimension2𝑛 < 𝑑 − 1. The surviving symmetries of a conformal field theory in the presence of a planar defect can be understood intuitively as follows. If we want to preserve as many conformal symmetries as possible then the theory living on the defect must inherit the conformal symmetries of the bulk CFT3, albeit in a lower dimension. Furthermore, the system should also be invariant under the rotations in the directions transverse to the defect. Therefore, for a co-dimension𝑛 < 𝑑 − 1 planar defect the symmetry group becomes𝑆𝑂(𝑑 − 𝑛,2) ×𝑆𝑂(𝑛). The genera-tors of this symmetry group leave the position of the defect unaltered. The symmetry group of a DCFT is a subgroup of the conformal group, nonetheless it is large enough to constrain the correlation functions in a conformal field theory with planar defects.

A DCFT example

In this section we will consider a simple set-up of a conformal field theory with a defect, namely that of a free scalar field minimally coupled to a𝑝-dimensional planar defect. This example serves to highlight some features that are inherent to DCFT’s.

We consider a single scalar field𝜙 defined on a 𝑑-dimensional spacetime 𝑀, such that the action of the free theory is given by [16]

𝑆₀₌ ∫ 𝑀 𝑑𝑑_{𝑥 L} 0= 1 (𝑑 −2)Vol(_𝕊𝑑−1) ∫ 𝑀 𝑑𝑑_𝑥1 2𝜕𝜇𝜙𝜕 𝜇_𝜙, where Vol(_𝕊𝑑−1) = 𝜋𝑑−21/Γ 𝑑+1 2

is the volume of a (d-1)-sphere. The normalisation factor is chosen such that the two-point function has the following form

h𝜙(𝑥)𝜙(𝑦)i0 =

1

|𝑥 − 𝑦|2Δ, (3.1)

here the subscript 0 indicates that these functions are weighted with respect to the free scalar action𝑆₀.

We now introduce4a planar defect by localising the field𝜙 on a subman-ifold𝑁 ⊂ 𝑀: D𝑝 = exp       𝜆 ∫ 𝑁 𝑑𝑝_𝑥 k𝜙(𝑥k)       , where𝑥_k= {𝑥𝑎

k|𝑎 = 0, . . . , 𝑝} denotes the coordinates on the defect and

we place the origin𝑥 = 0 within the defect. For conformal field theories we requireD_𝑝to be conformal, which is the case when𝑝 = Δ = (𝑑 − 2)/2. The action of the free theory including the defect can be written as 𝑆 = 𝑆0+𝑆D, where𝑆D= − log D𝑝. Applying the variational principle

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3 Conformal field theories with defects 11 5: Defined as 𝑇𝜇𝜈= 𝜕L0 𝜕𝜕𝜇𝜙𝜕 𝜈_{𝜙 − 𝛿}𝜇𝜈_L 0.

on𝑆, we obtain the equation of motion

𝜙 = −𝜆(𝑑 − 2)Vol(𝕊𝑑−1)𝛿(𝑛)_𝑁 , (3.2) where𝛿

(𝑛)

𝑁 is a delta function with support on the defect and𝑛 = 𝑑 − 𝑝 is the co-dimension of the defect. We can observe that the defect behaves as a (non-local) source term for the field𝜙.

Since D_𝑝 is an operator, we are now required to include this operator in the correlation functions. As a consequence, one-point functions are no longer forced to vanish by symmetry and the coefficients of these correlation functions become part of the CFT data. For instance, the one-point function of the scalar field𝜙 is defined as

h𝜙(𝑥)i =

h𝜙(𝑥)D𝑝i0

h D𝑝i0

.

We can observe that this one-point function is non-zero as follows

h𝜙(𝑥)D𝑝i0 h D𝑝i0 = h𝜙(𝑥)i₀+𝜆∫ 𝑑𝑝𝑥_kh𝜙(𝑥)𝜙(𝑥_k)i₀_{+ O(}𝜆2₎ 1+𝜆 ∫ 𝑑𝑝_𝑥 kh𝜙(𝑥k)i0+ O(𝜆 2₎ =𝜆 ∫ 𝑑𝑝_𝑥 kh𝜙(𝑥)𝜙(𝑥k)i0+ O(𝜆 2₎ = 𝑎_𝜙 |2𝑥_⊥|Δ,

where𝑥⊥= {𝑥𝑖|𝑖 = 𝑝 + 1, . . . , 𝑑} denotes the coordinates orthogonal to

the defect. Here, we usedh𝜙i₀= 0 in going from the first to the second line and (3.1) to evaluate the integral. In a similar manner, we can show that the two-point function becomes

h𝜙(𝑥)𝜙(𝑦)i = h𝜙(𝑥)𝜙(𝑦)D𝑝i h D𝑝i = h𝜙(𝑥)𝜙(𝑦)i0 + h𝜙(𝑥)ih𝜙(𝑦)i (3.3) = 1 |𝑥 − 𝑦|2Δ+ 𝑎2 𝜙 |𝑥⊥|Δ|𝑦⊥|Δ . (3.4)

Comparing this to (3.1), we observe that the defect two-point function obtains an additional contribution from the one-point function. As we move far away from the defect we expect a negligible effect of the defect and, indeed, in the limit𝑥⊥ → ∞ the one-point function vanishes and

we retrieve the two-point function (3.1). In other words, far away from the defect we have a homogeneous conformal field theory.

We now turn our attention to the stress tensor of the free theory, however instead of working with the canonical stress tensor5we work with the modified stress tensor

Θ_𝜇𝜈= 𝑇_𝜇𝜈− 1

4(𝑑 −1)Vol(_𝕊𝑑−1)

(𝜕_𝜇𝜕_𝜈−𝛿_𝜇𝜈_)𝜙2, 𝜕

𝜇Θ𝜇𝜈 on-shell= 0. (3.5)

The modified stress tensor has the nice property of being traceless and it allows us to write the Noether current associated to (infinitesimal)

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3 Conformal field theories with defects 12

dilatation transformations as

𝑗𝜇

𝐷 = Θ𝜇𝜈𝑥𝜈. (3.6)

The tracelessness ofΘ𝜇𝜈ensures that the conformal field theory is scale invariant. In fact, under certain conditions, we can reverse the argument to show that scale invariance implies conformal invariance; If we are able to construct a modified stress tensor such that the dilatation current has the form of (3.6), then conformal invariance follows from scale invariance, for a detailed discussion see [3].

In the presence of the defect, the modified stress tensor (3.5) is not conserved nor traceless:

𝜕𝜇Θ𝜇𝜈= 𝜕 𝜈_𝜙𝜙 (𝑑 −2)Vol(_𝕊𝑑−1) = −𝜆𝛿 (𝑛) 𝑁 𝜕𝜈𝜙 Θ𝜇_𝜇= 𝜙𝜙 2Vol(_𝕊𝑑−1) = − (𝑑 −2) 2 𝜆𝛿 (𝑛) 𝑁 𝜙,

where we used the equation of motion (3.2). Recalling that we have to view these equations as operator equations, we may define the operator 𝐷_𝜇 ₌ _𝜆𝜕_𝜇_{𝜙. The orthogonal part of this operator is known as the} displacement operator

𝐷𝑖 =𝜆𝜕𝑖𝜙.

This operator appears as a delta function contribution to the full stress tensor of the conformal field theory with defect. To be more precise, the displacement operator encodes the breaking of translational invariance in the directions orthogonal of the (planar) defect. The displacement operator is a primary and, as such, the coefficient of the two-point function becomes part of the CFT data

h𝐷𝑖(𝑥)𝐷𝑗(𝑦)i = 𝜆2𝜕𝑖𝜕𝑗h𝜙(𝑥)𝜙(𝑦)i _𝑥 ⊥=𝑦⊥=0 =𝛿𝑖𝑗 𝑎𝐷 |𝑥_k−𝑦_k|2Δ+2,

where we used (3.4). The displacement operator highlights a prominent feature of a conformal field theory with a defect: The set of field operators consists out of bulk fields and fields localised on the defect. Thus, in addition to bulk correlation functions, we now also have bulk-defect and defect correlation functions. An example of a bulk-defect correlation function is the two-point function of the the displacement operator𝐷𝑖

with the field𝜙

h𝐷𝑖(𝑥)𝜙(𝑦)i = 𝜆𝜕𝑖h𝜙(𝑥)𝜙(𝑦)i _𝑥 ⊥=0 = 2Δ𝜆 𝑦_𝑖 |(𝑥 − 𝑦)_k+𝑦⊥|2Δ+2 . (3.7)

For more computationally demanding correlation functions we can, again, resort to the embedding formalism, which is now constrained by 𝑆𝑂(𝑑 − 𝑛,2) ×𝑆𝑂(𝑛)symmetry. The reader is referred to [16,17] for a more detailed discussion on the displacement operator, the embedding formalism for defects and other examples of conformal field theories with defects.

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1:Here we assume that the variations of

the fields vanish at the boundary.

Anti de Sitter spacetime

4

Maximally symmetric space-times . . . . 13

Poincaré patch of AdS_𝑑+1 . . 14 Global cover of AdS𝑑+1 . . . 14

Throughout this work we will be spending a considerably amount of time in Anti de Sitter spacetimes. It plays an important role in the gauge/gravity duality, which will be discussed in the next chapter. For this reason, we will consider some aspects of this spacetime in this chapter. Most of the material in this chapter can be found in [18].

Maximally symmetric spacetimes

The Lagrangian description of general relativity is given by the Einstein – Hilbert action plus a term for matterL𝑀

𝑆 = ∫ 𝑑𝑑+1𝑥 𝑅 16𝜋𝐺 (𝑑+1) 𝑁 + L𝑀 ! ,

where𝑅is the Ricci scalar and𝐺

(𝑑+1)

𝑁 is the𝑑+1 dimensional gravitational constant. We assume that the matter term only contains a cosmological constantΛ, such that the action becomes

𝑆 = 1 16𝜋𝐺 (𝑑+1) 𝑁 ∫ 𝑑𝑑+1𝑥 (𝑅 −2Λ) .

By the variational principle1we obtain the Einstein field equations 𝐺_𝜇𝜈 = 𝑅_𝜇𝜈−1

2

𝑅 𝑔_𝜇𝜈+ Λ𝑔_𝜇𝜈 = 0 (4.1)

where𝑅_𝜇𝜈is the Ricci curvature tensor and𝐺_𝜇𝜈 is the Einstein tensor (plusΛ𝑔_𝜇𝜈). We are interested in spacetime solutions to (4.1) that admit the maximal number of Killing vectors:(𝑑 +1)(𝑑 +2)/2. A maximally symmetric spacetime necessarily has constant curvature and the Riemann curvature tensor can be written in terms of the metric and Ricci scalar

𝑅_{𝜇𝜈𝜌𝜎}₌ 𝑅

𝑑(𝑑 +1)(𝑔𝜇𝜌𝑔𝜈𝜎−𝑔𝜇𝜎𝑔𝜈𝜌). (4.2) From (4.1) we have

𝑅 = 2Λ(𝑑 + 1)_{𝑑 −} 1

and depending on the sign of Λ we can distinguish three maximal symmetric spacetimes. The most familiar solution is the flat Minkowski spacetime whereΛ = 0 = 𝑅. The positive curvature 𝑅 > 0 solutions are called de Sitter (dS) spacetimes and the solutions with negative curvature𝑅 < 0 are called — as an evil counterpart — anti de Sitter (AdS) spacetimes. Although it is believed that we live in a Universe with a positive cosmological constant, i.e. a dS spacetime, in this work we are exclusively interested in AdS spacetimes.

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4 Anti de Sitter spacetime 14

Figure 4.1: The Penrose diagram of the Poincaré patch of AdS𝑑+1. The spacetime

is conformally equivalent to Minkowksi spacetime. Each point in this diagram

con-tains𝕊𝑑−2. We can realise a𝑑 +1 dimensional AdS𝑑+1spacetime by an embedding

as a hypersurface in a𝑑 +2 dimensional Minkowksi spacetime,ℝ𝑑,2, with metric 𝑑𝑠2 = −(𝑑𝑋−1)2− (𝑑𝑋0₎2₊ 𝑑 X 𝑖=1 (𝑑𝑋𝑖)2

and the embedding is given by

−(𝑋−1 )2− (𝑋0₎2₊ 𝑑 X 𝑖=1 (𝑋_𝑖)2 = −𝐿2, (4.3)

where𝐿2is the radius of curvature. Notice that (4.3) reveals that AdS𝑑+1

spacetimes are invariant under the group𝑆𝑂(𝑑,2).

There are various ways to parameterise (4.3) and, by means of the pullback, each parameterisation will yield a different metric on AdS𝑑+1. We will consider two different parameterisations to illustrate this feature.

Poincaré patch of AdS

_𝑑+₁

One particular useful parameterisation — although not an obvious one — is given by 𝑋−1 = 𝐿𝑡 𝑧 (4.4a) 𝑋0 = 𝑧 2 " 1+ 1 𝑧2 𝐿2₊ 𝑑−1 X 𝑖=1 (𝑥𝑖₎2₋𝑡2 ! # (4.4b) 𝑋𝑖 = 𝐿𝑥 𝑖 𝑧 (4.4c) 𝑋𝑑 = 𝑧 2 " 1− 1 𝑧2 𝐿2₋ 𝑑−1 X 𝑖=1 (𝑥𝑖₎2₊𝑡2 ! # , (4.4d)

where−∞< 𝑡, 𝑥𝑖 < ∞ and 0 < 𝑧 < ∞. The metric of the Poincaré patch of AdS𝑑+1is 𝑑𝑠2 = 𝐿 2 𝑧2 𝑑𝑧2₋𝑑𝑡2₊ 𝑑−1 X 𝑖=1 (𝑑𝑥𝑖)2 ! . (4.5)

This parameterisation reveals that the metric is conformally flat with a conformal factor 𝐿

2

𝑧2. The Poincaré patch of AdS𝑑+1admits a conformal

boundary and the causal structure is equivalent to the Minkowski spacetime, see Figure 4.1. As we move to the boundary 𝑧 → 0 the conformal factor diverges, whereas as𝑧 → ∞the metric has a coordinate singularity, the Poincaré horizon. This horizon is merely an artifact of the parameterisation, since — as the name already suggested — this coordinate system only covers a patch of the AdS_𝑑+1spacetime.

Global cover of AdS

_𝑑+₁

We now consider a coordinate system that covers the whole AdS𝑑+1

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2: Since 0 ≤ 𝜃 < 𝜋

2

, we only cover the

northern half of𝕊𝑑. is given by

𝑋−1

= 𝐿 cosh𝜌 cos 𝜏 (4.6a)

𝑋0

= 𝐿 cosh𝜌 sin 𝜏 (4.6b)

𝑋𝑖

= 𝐿 sinh𝜌 ˆ𝑥𝑖 (4.6c)

where 𝑥ˆ𝑖 denotes a parameterisation of a unit-radius(𝑑 −1)-sphere,

𝕊𝑑−1, and where the range of the coordinates are 0 ≤ 𝜌 < ∞ and 0≤𝜏 < 2𝜋. Notice that we have a periodic time-like coordinate 𝜏, which is not preferable in view of closed time-like curves. We can avoid having periodic time by considering the universal covering of AdS𝑑+1instead. In the universal covering we unwrap the𝜏-direction and we extend the range to−∞< 𝜏 < ∞. The global coordinates give the metric

𝑑𝑠2 = 𝐿2_{(− cosh}2𝜌 𝑑𝜏2₊𝑑𝜌2_{+ sinh}2𝜌 𝑔 𝕊𝑑−1), (4.7) where 𝑔 𝕊𝑑−1 is the metric of𝕊 𝑑−1

. The global covering of AdS_𝑑+1also admits a conformal boundary which can be seen by using

tan𝜃 = sinh 𝜌, (4.8)

where 0≤𝜃 < 𝜋

2

. The metric takes the form

𝑑𝑠2 = 𝐿 2 cos2𝜃 (−𝑑𝜏2₊ 𝑑𝜃2_{+ sin}2 𝜃𝑑𝕊2 𝑑−1) (4.9)

and we identify the conformal factor 𝐿

2

cos2𝜃. This form of the metric reveals

that the global covering of AdS_𝑑+1is conformal to2ℝ_𝜏×𝕊𝑑. Furthermore, the AdS_𝑑+1boundary at𝜃 = 𝜋

2

is conformal toℝ_𝜏×𝕊𝑑−1, we can view this as a flat spacetime where the spatial dimensions are compactified.

Figure 4.2depicts the Penrose diagram of the universal covering of global AdS𝑑+1spacetime.

Figure 4.2: The Penrose diagram of the global cover of AdS_𝑑+

1. The interior of

the spacetime is represented by𝜃 = 0, whereas the (timelike) boundary is located at𝜃 = 𝜋/2. The dashed lines indicate trajectories, amongst which a light-like geodesic is shown to be reflected at the

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In view of the next chapter it is insightful to highlight several charac-teristics of AdS_𝑑+1. The above two parameterisations reveal that AdS_𝑑+1 spacetimes admit a conformal structure on the boundary, i.e. on the boundary we have an equivalence class of conformally related metrics. The boundary is time-like and we have seen that the spacetime on the boundary is conformally flat, albeit compactified in the global coordi-nates. Moreover, fromFigure 4.2we can observe that light rays are able to reach the boundary in a finite coordinate time. This signifies that the future evolution of the fields in spacetime cannot be determined by their initial data only. As a consequence we cannot take Cauchy slices of the AdS𝑑+1spacetime and have a well-defined initial value problem; The existence of the boundary requires that we impose boundary conditions on the fields.

Since light rays can move to and return from the boundary in a finite time this reveals something more profoundly about AdS_𝑑+1spacetimes: Physics in the bulk can be holographically projected onto the boundary. And indeed it has been conjectured that AdS𝑑+1spacetimes aredual to 𝑑dimensional conformal field theories, this is known as the AdS/CFT correspondence. In the next chapter we will consider this correspondence in more detail.

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1:A comprehensive review of the

AdS/CFT correspondence can be found

in [19].

2: Although known for the given claim,

generalisations to other dimensions were

already explored in the original paper.

3: In our notation the dimension of the

AdS spacetime is𝑑 +1.

4: The lack of experimental evidence and

predictive power make for a great debate

whether superstring theory describes our

Universe or not.

The AdS/CFT correspondence

5

AdS/DCFT correspondence? 18 The holographic dictionary 18 States ⇔ Geometries. . . . . 18

The AdS/CFT correspondence has a central role in this work and here we will consider this duality in more detail. However, rather than unpacking the mathematical content that lead to the conjecture of the correspon-dence, we take the AdS/CFT correspondence as given1and instead we give a short descriptive summary.

We begin by stating the initial claim of the AdS/CFT correspondence that gives the equivalence between the following two theories2

Strongly-coupled 4-dimensional gauge theory

dual

=

Gravitation theory in 5-dimensional AdS spacetime.

The equivalence of such two theories might seem impossible at first sight, since we may expect that a higher dimensional theory contains more degrees of freedom. This apparent discrepancy in number of degrees of freedom can be resolved by realising that the physics of the gravity theory is completely encoded on the boundary of its spacetime. Therefore, the AdS/CFT correspondence is a realisation of the holographic principle.

In the present discussion, we will not restrict ourselves to the case3 of𝑑 = 4 and we assume that the correspondence holds for arbitrary 𝑑. The AdS/CFT correspondence in arbitrary dimensions are commonly referred to as gauge/gravity dualities. In addition to being holographic in nature, the gauge/gravity dualities all share the following characteristics [20]:

I There is a mapping between a quantum gravity theory and a non-gravitational quantum field theory.

I The correspondence is a strong/weak coupling duality. A strongly-coupled gauge theory is equivalent to a weakly-strongly-coupled quantum gravity theory.

The first characteristic might come as a surprise, since we have not made any mention of quantum gravity theories. However, the AdS/CFT corre-spondence has its origins in superstring theory, whichis4a candidate for being a quantum gravity theory. The assumption is that the duality holds for spacetime geometries that become AdS spacetimes in the asymptotic limit of moving towards the boundary of this spacetime. The second characteristic has a practical implication — besides of being conceptually interesting — since the computations of a weakly coupled theory are, in general, less difficult compared to the equivalent computation in the strongly-coupled dual theory.

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5 The AdS/CFT correspondence 18

AdS/DCFT correspondence?

Soon after the inception of the AdS/CFT correspondence questions arose whether the duality could be extended to conformal field theories with defects. One of the first realisations of a possible AdS/DCFT correspon-dence was given in [21] where an AdS₅spacetime was constructed that included an AdS₄brane. It was conjectured that this spacetime construc-tion would be dual to a conformal field theory with a co-dimension one defect. This conjecture was put on a stronger footing in [22] where the Lagrangian for the dual field theory was constructed and the degrees of freedom on the defect where identified with the degrees of freedom on the AdS₄ brane. These constructions where then generalised to de-fects with arbitrary co-dimension in [23]. The important feature that where shared among these constructions is that the𝑆𝑂(𝑑 − 𝑛,2) ×𝑆𝑂(𝑛) global symmetry group of the DCFT is encoded in the dual spacetime geometry. Since a necessary condition for the DCFT is the embedding of the theory in a bulk CFT, we assume that the remaining characteristics of the AdS/CFT correspondence naturally carry over to a AdS/DCFT correspondence.

The holographic dictionary

Now that we have a conceptual idea of the AdS/CFT correspondence we will consider the set of maps that establishes this duality. The maps between the quantities that characterises both sides of the duality are the entries of the so-called holographic dictionary.

From the previous chapters we can immediately identify two entries of the holographic dictionary. First, the Hilbert space of physical states in the bulk is identical to the Hilbert space of the field theory. This follows from the property of AdS spacetimes to have a time-like boundary, such that additional boundary conditions have to be imposed on the fields. Second, the𝑆𝑂(𝑑,2) global symmetry group of the field theory in𝑑dimensions is to be identified with the𝑆𝑂(𝑑,2) isometry group of AdS𝑑+1. This also implies that the Hamiltonian is the same on both sides of the duality. The next entry in the holographic dictionary is an illuminating example that captures the essence of the AdS/CFT correspondence, namely as a duality between geometry and quantum mechanics.

States ⇔ Geometries

The AdS/CFT correspondence can be applied to solutions to the Einstein equations that admit a conformal structure on the boundary. The general solution that has this property can be written in the following form [24]

𝑑𝑠2 = 𝐺_𝜇𝜈𝑑𝑥𝜇𝑑𝑥𝜈 = 𝐿 2 𝑧2 " 𝑑𝑧2₊𝑔 𝑖𝑗(𝑥, 𝑟)𝑑𝑥𝑖𝑑𝑥𝑗 # (5.1) where 𝑔(𝑥, 𝑧) = 𝑔(0)(𝑥) + 𝑧 2𝑔 (2)(𝑥) + . . . + 𝑧 𝑑_𝑔 (𝑑)(𝑥) + . . . . (5.2)

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5 The AdS/CFT correspondence 19

5: We consider the perturbations as being

parameterised by𝜆, such that |Ψ(𝜆)i = |Ωi +_{𝜆|𝜙i + . . .}

6: That is,𝐻_𝜇𝜈→ 0 as𝜆 → 0.

7: For an overview of other real-world

applications of the correspondence, see

[20]. These solutions are known as asymptotic AdS𝑑+1 (AAdS) spacetimes,

since the curvature tensor𝑅_{𝜇𝜈𝜌𝜎}approaches the form (4.2) as𝑧 →0. By substituting this general solution into the Einstein equation we can solve for the coefficients𝑔(𝑖)and it has been shown that all coefficients𝑔(𝑖)can be written as functions of 𝑔₍₀₎ and𝑔_(𝑑)[24]. Therefore, specifying the system amounts to specifying 𝑔₍₀₎and 𝑔_(𝑑). More specifically,𝑔₍₀₎can be identified as the boundary metric. Indeed, assuming 𝑔₍₀₎(𝑥) = 𝜂_𝜇𝜈all other coefficients vanish and we obtain the metric of pure AdS_𝑑+1in the Poincaré patch (4.5).

Knowing that pure AdS_𝑑+1spacetimes are maximally symmetric space-times it is reasonable to assume that other AAdS𝑑+1spacetimes exhibit less symmetries. Moving to the field theory side, we know that sym-metries do not preserve all states. The only state that is invariant under all symmetries is the vacuum state|Ωi. This highly suggests that the vacuum state of a𝑑dimensional conformal field theory is dual to the Poincaré patch of pure AdS_𝑑+1.

Let us now consider an infinitesimal perturbation to the vacuum state5 |Ψ(𝜆)i. By rewriting (5.1) as perturbations away from pure AdS_𝑑+1as

𝑑𝑠2 = 𝐿 2 𝑧2 " 𝑑𝑧2₊ 𝜂𝜇𝜈𝑑𝑥𝜇𝑑𝑥𝜈+𝑧𝑑𝐻𝜇𝜈(𝑥, 𝑧)𝑑𝑥𝜇𝑑𝑥𝜈 # , (5.3)

we can observe that|Ψ(𝜆)i will be dual to geometries that are perturba-tions6to pure AdS_𝑑+1. Generalising this observation, we obtain the first entry of the dictionary:

Quantum states

dual

= Spacetime geometries.

We can also consider thermal states, which are characterised by their thermodynamic properties as temperature, energy and entropy, which all depend on the Hamiltonian of the system. Knowing that the Hamil-tonians are the same on both sides of the duality, this implies that the thermal states are dual to geometries that also display these thermody-namic properties. The geometries that have a notion of thermodythermody-namic properties are the black hole spacetime geometries and we conclude

Thermal states

dual

= Black hole spacetime geometries.

It is interesting to mention that this allows us to use black hole physics to study a field theory at finite temperature. For example7, the processes of thermalisation and dissipation can be translated into the creation of a black hole and falling into a black hole, respectively.

In the next chapter, we will consider the dual of the entanglement entropy of a quantum system. This entry of the holographic dictionary will proof to be extremely useful for the analysis in the subsequent chapters.

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Entanglement entropy

&

Ryu-Takayanagi formula

6

Density matrix & entanglement entropy . . . . 20 The Ryu-Takayanagi formula 22

Quantum entanglement can be viewed as the key feature that separates quantum theories from classical theories. Because the notion of inde-pendence is affected by the presence of quantum entanglement, we are required to change the way we consider physics in a profound way. This can be already seen in the simple example of two qubits. The total Hilbert space is the tensor product of two single qubit Hilbert spaces H = {|0i ⊗ |0i, |1i ⊗ |0i, |0i ⊗ |1i, |1i ⊗ |1i}. We consider the following state

1 √ 2

(|0i ⊗ |0i + |1i ⊗ |1i).

This state does not admit a factorisation which would have allowed us to identify single qubit systems. We cannot make a meaningful separation of the two qubits and, as a consequence, knowing the state of the first qubit determines the state of the second qubit. Therefore, the qubits are said to be entangled.

This example illustrates how we cannot study the system considering only the building blocks of the quantum theory. Since the building blocks are entangled we have to consider the whole system at once, i.e. there is no notion of independence. Quantum entanglement has been a great source of theoretical interest and it will play a major role in this discussion. Therefore, we will consider some aspects of quantum entanglement that will be useful for the subsequent chapters.

Density matrix & entanglement entropy

Consider a quantum field theory defined on a𝑑-dimensional spacetime manifold 𝑀 = ℝ1,𝑑−1 which contains a set of fields{Φ}. In order to define a state we have to pick a Cauchy slice C, here we will consider C at𝑡 = 0. A priori we do not know the nature of the state, it can be a pure state or a mixed state. A pure state is an element of the Hilbert spaceH, whereas a mixed state is a statistical ensemble of pure states. There is an efficient way to account for both kinds of states in a single operator: the density matrix𝜌. This operator has a number of properties [25]:

𝜌†

=𝜌 (6.1a)

𝜌 is positive semi-definite (6.1b)

tr 𝜌= 1 (6.1c)

tr 𝜌2 ≤ 1. (6.1d)

The equality in (6.1d) is saturated only for pure states.

The density matrix can be formulated as a path integral and in this formulation it is natural to work in Euclidean time𝜏𝐸 = −𝑖𝑡. First we

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6 Entanglement entropy & Ryu-Takayanagi formula 21

Figure 6.1: An (imaginary) bi-partition of a Cauchy sliceC by an entangling surface Σ, indicated by the yellow line.

1:One difference between (classical)

en-tropy and entanglement enen-tropy would be

that the latter is not an extensive quantity.

2: From (6.1c) and (6.1d), for pure states we have𝜌2_𝐴=𝜌𝐴⇒𝑆𝐴= 0.

consider a transition amplitude between two states|𝜙₁i, |𝜙₂i ∈ H given by h𝜙₂|𝑒(𝜏2−𝜏1)𝐻|𝜙 1i = 𝜙(𝜏𝐸=_𝜏2)=𝜙2 ∫ 𝜙(𝜏𝐸=𝜏1)=𝜙1 D[Φ] 𝑒−𝑆𝐸[Φ]_,

where the limits of integration are the boundary conditions imposed on the fields at a fixed time. We recognise the operator as the thermal density matrix𝜌_𝛽= 𝑒−𝛽𝐻associated to the inverse temperature𝛽 = 𝜏₂−𝜏₁. Then, the density matrix𝜌 of the vacuum state is obtained by taking the limit 𝛽 → ∞

h𝜙2|𝜌|𝜙1i = lim

𝛽→∞h𝜙2|𝑒𝛽𝐻|𝜙1i.

Suppose that we divide C into a region 𝐴 and its complement𝐴, such¯ that𝐴 ∪ ¯𝐴 = C. The boundary that separates these two regions will be called the entangling surfaceΣ, seeFigure 6.1. The requirement of the quantum theory to be local, factorises the Hilbert space,H = H𝐴⊗ H_𝐴¯, such that it gives a restriction on the domain of support of the fields Φ(𝑥) = {Φ𝐴, Φ𝐴¯}. Suppose now that we are confined to region𝐴and only have access toH𝐴. We can construct a reduced density matrix𝜌𝐴

that only acts onH𝐴by tracing out the degrees of freedom of region𝐴¯ 𝜌𝐴= tr𝐴¯ 𝜌= ∫ D[Φ𝐴] ∮ D[Φ𝐴¯]𝑒−𝑆𝐸[Φ]∝ ∫ D[Φ𝐴]𝑒−𝑆𝐸[Φ𝐴].

To define the matrix elements of𝜌_𝐴we have to introduce a branch cut on region𝐴, such that the transition amplitude between two states is well-defined on C. The boundary conditions become the limits of the fields as we approach the branch cut from either side and we have

hΦ0_𝐴|𝜌𝐴|Φ𝐴i = 1 𝑍 Φ(𝜏𝐸=0+ )=Φ0𝐴 ∫ Φ(𝜏𝐸=0−_)=Φ 𝐴 D[Φ𝐴]𝑒−𝑆𝐸[Φ𝐴],

where𝑍is a normalisation factor.

It seems that by constructing𝜌𝐴we are all settled and we may completely dismiss𝐴. However, from the discussion at the beginning of this chapter,¯ we know that we cannot simply dismiss𝐴. The process of tracing out¯ degrees of freedom does not eliminate the entanglement between the two regions𝐴and𝐴¯and the inaccessibility of region𝐴¯leaves us with an amount of ignorance in the system. This ignorance can be quantified by the von Neumann or entanglement entropy

𝑆𝐴= −tr𝐴 𝜌𝐴log𝜌𝐴. (6.2) Similar to the entropy in classical thermodynamics1, the entanglement entropy is non-negative𝑆_𝐴≥ 0 and is zero only for pure states2. Computing the entanglement entropy of a generic quantum field theory proofs to be difficult. Moreover, entanglement entropy suffers from UV divergences due to the short-distance behaviour around the entangling surfaceΣ and it requires a short-distance cut-off𝜖 to regulate the diver-gent behaviour. However, computations become more manageable for conformal field theories that admit a holographic dual. The AdS/CFT

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Figure 6.2: The entanglement entropy of an interval of 2𝑅in the two dimensional CFT is given by a minimal surface area (dashes line) which is anchored on the

en-tangling surface (yellow dots) and extends into the AdS3spacetime.

3: The central charge of a CFT2 counts

the degrees of freedom in the theory; The

so-called c-function mentioned inChapter 2.

correspondence has an entry in the dictionary stating that the entan-glement entropy of the field theory is dual to extremal surface areas of co-dimension two in the bulk. We will discuss this entry in the next section.

The Ryu-Takayanagi formula

We consider𝑑 dimensional conformal field theories that are dual to AdS_𝑑+1spacetimes. The field theory has the spacetime set-up as discussed in the previous section and here we emphasise the role of the entangling surfaceΣ. The AdS/CFT dictionary tells us that the entanglement entropy associated toΣ has a geometrical description in the bulk, namely it is dual to a co-dimension two extremal surface (divided by 4𝐺(𝑑+1)_𝑁 ) which is anchored on the boundary and extends into the bulk [26]:

𝑆𝐸𝐸 dual = Aext 4𝐺(𝑑+1) 𝑁 . (6.3)

For completeness, the area functional is given by

A =

∫

𝑑𝑑−1_𝜎√_𝛾, (6.4)

where𝛾 is the induced metric on the surface. If (6.4) admits more than one extremal surface solution, then the entanglement entropy is given by the surface of least areaA_min. The extremal surfaceA_minis homologous to the entangling surface,𝜕A_min= Σ. The homology constraint implies that there exist a region in the bulkΞ that is bounded by A_minand the region𝐴on the boundary, such that𝜕Ξ = 𝐴 ∪ A_min. The minimal surface area acts like a holographic screen that separatesΞ from the remaining region of the bulk spacetime. This concurs with the intuitive notion of how the entanglement entropy encodes the amount of ignorance in the system.

We end this chapter with an application of the Ryu-Takayanagi pre-scription to compute the entanglement entropy of a known (and solved) problem. Consider a two dimensional conformal field theory onℝ1,1. We choose a Cauchy slice at𝑡 = 0 and the region 𝐴 is given by an interval of size 2𝑅centered around the origin, seeFigure 6.2. It is known that the entanglement entropy associated to this interval is given by [27]

𝑆𝐸𝐸= 𝑐 3

log 2𝑅

𝑎 , (6.5)

where𝑐is the central charge of the conformal algebra3and𝑎is the UV cut-off.

To compute the entanglement entropy holographically, we consider the dual spacetime in the Poincaré patch of AdS₃

𝑑𝑠2 = 𝐿 2 𝑧2 " 𝑑𝑧2₋𝑑𝑡2₊𝑑𝑥2 # .

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The RT prescription states that we have to extremise the area functional

A = ∫ 𝑑𝑠 = ∫ _𝐿 𝑧 √ 𝑑𝑧2₊𝑑𝑥2₌ ∫ 𝑑𝜆𝐿_𝑧 s _𝑑𝑥 𝑑𝜆 2 + _𝑑𝑥 𝑑𝜆 2 ,

where we parameterise the surface by𝜆. Varying this functional yields the equation for a hemi-sphere

𝑧2₊𝑥2

= 𝑅2 _⇒ 𝑧 = 𝑅 sin 𝜆 𝑥 = 𝑅 cos 𝜆.

Inserting these equations into the area functional we obtain

A𝑚𝑖𝑛= 𝐿 𝜋 ∫ 0 𝑑𝜆 1 sin𝜆 .

Since the integral is divergent, we cannot compute this integral as presented. This was anticipated because the entanglement entropy is a divergent quantity due to the short-distance behaviour. In this case, the divergent origin of the minimal surface area is due to the infinite proper distance from any point in the bulk spacetime to its boundary and is therefore due to the long-distance behaviour of the bulk spacetime. This is an example of an UV/IR divergence relation that is inherent to the AdS/CFT correspondence.

To regulate the integral, we have to restrict the domain of the AdS₃ spacetime to𝑧 > 𝜖 = _𝑅𝑎 [26], this will change the domain of integration to [𝜖, 𝜋 − 𝜖]. We can now evaluate the integral and obtain the entanglement entropy by (6.3) 𝑆_𝐸𝐸dual = 𝐿 2𝐺(3) 𝑁 log 2𝑅 𝑎 .

We can observe that this expression is equivalent to (6.5) by using the Brown-Henneaux relation between the curvature of the AdS₃spacetime and the central charge [28]

𝑐dual

= = 3𝐿 2𝐺(3)

𝑁

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1: The notation here differs fromChapter 3, where the defect was indicated by the dimension𝑝.

Figure 7.1: A Cauchy slice at𝑡 = 0 bi-partitioned into region𝐵and𝐵¯by a spher-ical entangling surface (yellow line). The causal development of region𝐵is indi-cated by the dotted lines.

CHM formalism

7

A map of mappings . . . . . 24 Spurionic invariance . . . . . 26 The CFT story . . . . 27 The AdS story . . . . 30 Combining the two stories . 32 The DCFT story . . . . 33

The CHM formalism [29] provides us with the necessary tools and insights to explore the gravitational dynamics in the bulk. The goal of the formalism is to obtain the entanglement entropy of a(𝑑 −2) dimensional spherical entangling surface in Minkowski spacetime. Via a conformal mapping from Minkowksi to a hyperbolic spacetime it was shown that the entanglement entropy is equivalent to the thermal entropy of the latter spacetime. According to the AdS/CFT dictionary this thermal entropy can be computed as the horizon entropy of the (topological) black hole in the bulk. The horizon entropy was found to agree with the result obtained by using the RT formula providing a non-trivial confirmation on the validity of the formula. In this chapter, we will review the formalism as well as extending the analysis to include defects [15].

A map of mappings

In [29] a series of diffeomorphisms were applied such that different spacetime geometries could be related. In this section we will consider the various diffeomorphisms and we will keep track of the defect along the way.

We start with a conformal field theory in Minkowski spacetime

𝜂𝜇𝜈𝑑𝑥𝜇𝑑𝑥𝜈 = −𝑑𝑡2₊𝑑𝑟2₊𝑟2𝑔

𝕊𝑑−2,

where𝑥0= 𝑡, 𝑟2= 𝑥𝑖𝑥𝑖(𝑖 ≠ 0) and 𝑔

𝕊𝑑−2 is the metric of a(𝑑 −2)-sphere

𝕊𝑑−2of unit radius. A planar defect1 D_𝑛of co-dimension𝑛is positioned at

D𝑛 = {𝑥𝑑−𝑛 = 𝑥𝑑−𝑛+1 = . . . = 𝑥𝑑−1= 0}

and we will place the origin𝑟 = 0 within D𝑛. We pick a Cauchy slice C at𝑡 = 0 and take as our entangling surface Σ a (𝑑 − 2)-sphere of radius 𝑅 centered on the defect

Σ = {𝑡 = 0, 𝑟 = 𝑅}.

The entangling surface divides the space into two regions: a(𝑑 −1)-ball denoted by 𝐵 and its complement 𝐵, such that¯ 𝐵 ∪ ¯𝐵 = C. The set of points that are in causal contact with points in 𝐵is known as the causal development 𝐷[𝐵]of region𝐵, which is the spacetime region {𝑥+ _≤_{𝑅} ∩ {𝑥}− _≤_{𝑅}, where we introduced null coordinates}

𝑥±

= 𝑟 ± 𝑡 , seeFigure 7.1.

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7 CHM formalism 25

Figure 7.2: The figure depicts the Rindler spacetime for a CFT with a planar defect of

co-dimension one, located at𝑥𝑑−1= 0. The entangling surface (yellow line) is mapped to the plane𝑋+ = 𝑋− = 0. The Rindler wedge on the co-dimension one defect is given by the shaded area.

2: In the Rindler coordinates there is an

ambiguity in the position ofΣℍin the𝜏 direction. Following [14] we also choose the reference point𝜏 = 0.

Consider the following diffeomorphism [14] (a special conformal trans-formation followed by a translation)

𝑋𝜇_{= 4} 𝑥𝜇− (𝑥 · 𝑥)𝐶𝜇 1− 2(𝑥 · 𝐶) + (𝑥 · 𝑥)(𝐶 · 𝐶) + 𝑅 2 2 𝐶𝜇 _{𝐶 = −}1 𝑅 𝜕1, (7.1)

which maps𝐷[𝐵]to the right Rindler wedgeR R = {𝑋±≡𝑋1_±𝑋0 ≥ 0}.

More specifically, the boundaries of 𝐷[𝐵]are mapped to the Rindler horizons,𝑋±= 0. The defect is mapped to

D𝑛R= {𝑋𝑑−𝑛 = 𝑋𝑑−𝑛+1= . . . = 𝑋𝑑−1= 0}.

Notice that a co-dimension one defect itself is also a Rindler wedge, see

Figure 7.2. We should expect that the entangling surface is mapped to the plane𝑋+= 𝑋−= 0 and indeed

ΣR= ( 𝑋0 = 0, 𝑋1 = 4 " 𝑥1₊𝑅 1+ 2𝑥 2 𝑅 + 1 − 𝑅 2 # = 0 ) ,

since(𝑥 · 𝑥)|_Σ = 𝑅2. The metric inR is conformally flat 𝜂𝜇𝜈𝑑𝑥𝜇𝑑𝑥𝜈= Ω(𝑋)2

𝜂𝜇𝜈𝑑𝑋𝜇𝑑𝑋𝜈,

where the conformal factorΩ(𝑋) can be eliminated by a Weyl transfor-mation such that the transfortransfor-mations together form an element of diff_o Weyl.

It is natural to considerR via a family of uniformly accelerating observers by introducing Rindler coordinates

𝑋0

= 𝑧 sinh_𝑅𝜏 and 𝑋1 = 𝑧 cosh 𝜏

𝑅. (7.2)

In the new coordinates the metric becomes

− 𝑧 2 𝑅2𝑑𝜏 2₊𝑑𝑧2₊ 𝑑−1 X 𝑖=2 (𝑑𝑋𝑖₎2 = 𝑧 2 𝑅2 " −𝑑𝜏2₊𝑅 2 𝑧2 𝑑𝑧2₊ 𝑑−1 X 𝑖=2 (𝑑𝑋𝑖₎2 ! # .

This metric is conformally equivalent toℝ×ℍ𝑑−1, whereℍ𝑑−1is a (d-1)-hyperboloid of unit radius. The resulting hyperbolic spacetime, after eliminating the conformal factor 𝑧

2

𝑅2 by another Weyl transformation, will

be denoted byℍ. The defect is extended along an equatorial hyperboloid inside ofℍ𝑑−1

Dℍ𝑛 = {𝑋𝑑−𝑛 = 𝑋𝑑−𝑛+1= . . . = 𝑋𝑑−1= 0},

whereas the entangling surface is mapped to2 Σℍ= {𝑧 = 0,𝜏 = 0}.

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7 CHM formalism 26

We have considered a set of diff_{o Weyl transformations that relate three} geometries𝐷[𝐵], R andℍ, seeFigure 7.3.

Figure 7.3: From left to right: the causal de-velopment𝐷[𝐵], the right Rindler wedge R and the hyperbolic spacetimeℍ. The entangling surface is highlighted in yel-low. The modular flow in each geometry is depicted by the blue lines.

The final diffeomorphism we consider directly maps𝐷[𝐵]toℍ 𝑟 = 𝑅sinh𝑢 cosh𝑢 +cosh(𝜏/𝑅) (7.3a) 𝑡 = 𝑅sinh(𝜏/𝑅) cosh𝑢 +cosh(𝜏/𝑅) , (7.3b)

such that the metric becomes

−𝑑𝑡2₊𝑑𝑟2₊𝑟2𝑔 𝕊𝑑−2 = Ω(𝑢,𝜏)2₋𝑑𝜏2₊𝑅2 𝑑𝑢2_{+ sinh}2𝑢 𝑔 𝕊𝑑−2 = Ω(𝑢,𝜏)2₋ 𝑑𝜏2₊𝑅2𝑔 ℍ𝑑−1 _, where−∞ < 𝜏 < ∞, 0 < 𝑢 < ∞ and 𝑔 ℍ𝑑−1 is the metric ofℍ 𝑑−1 . After eliminating the conformal factorΩ(𝑢,𝜏) by a Weyl transformation, we again recognise the resulting geometry asℝ×ℍ𝑑−1. It is noteworthy to mention thatℍdoes not contain any information on the complement region 𝐵. This can be easily seen, because¯ Σℍ is located at 𝑧 = 0 (or 𝑢 = ∞), which is the location of the boundary of the hyperbolic geometry. Once we consider the bulk we will have a better understanding of the fate of this region𝐵.¯

The reason for this rigmarole of transformations will become apparent when we consider the reduced density matrix in the different geometries. Before we continue, it is important that we justify that the analysis in [29] — which was done in the absence of defects — is still valid when defects

are present.

Spurionic invariance

Imagine for the moment that we consider a quantum field theory with a Lagrangian description as given by (2.1). This theory may allow for symmetries which will leave the generating functional invariant up to anomalies. The symmetry transformations may change the sources, but still leave the generating functional invariant. These specific symmetry transformations are called spurionic transformations. Another subset of symmetry transformations, known as global transformations, leave the generating functional invariant and, in addition, the sources. We may take the following analogy in calculus to obtain a better intuition of these

Gravity from Entanglement in Holographic Theories with Defects

MSc Physics & Astronomy

Theoretical Physics

Master Thesis

Gravity from Entanglement in

Holographic Theories with Defects

by

Huy Hoang Vu

10336818

August 2020

60 EC

Supervisor:

Dr. Jay Armas

Examiners:

Dr. Jay Armas

Dr. Alejandra Castro

Abstract

Contents

Appendix

60

Introduction

1

Conformal field theories

2

What are conformal field theories?

More symmetry: From QFT to CFT

The algebra of the

𝐶𝑜𝑛 𝑓

The operator product expansion

Why study conformal field theories?

Conformal field theories with

defects

3

Less symmetry: From CFT to DCFT

A DCFT example

Anti de Sitter spacetime

4

Maximally symmetric spacetimes

Poincaré patch of AdS

Global cover of AdS

The AdS/CFT correspondence

5

AdS/DCFT correspondence?

The holographic dictionary

States ⇔ Geometries

Entanglement entropy

&

Ryu-Takayanagi formula

6

Density matrix & entanglement entropy

The Ryu-Takayanagi formula

CHM formalism

7

A map of mappings

Spurionic invariance