Berry picking in the holographic forest

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MSc Physics

Track: Theoretical Physics

M

ASTER

T

HESIS

Berry picking in the holographic forest

by

Dimitrios Patramanis

11641371

October 2019

60 ECTS Main Supervisor: Dr. Ben W. Freivogel Daily Supervisor: Dr. Claire Zukowski Second reviewer: Prof. dr. Jan de Boer

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Abstract

The goal of this thesis is to make progress towards elucidating the nature of the interplay between geometry and quantum mechanics in the context of holography. In particular, we investigate the recently proposed notion of the modular Berry phase [1,2] using a number of different approaches and techniques. Initially, we review the notions of the Berry phase and kinematic space, as an example of a non-holonomic process in non-relativistic quantum mechanics and the auxiliary space of AdS3 geodesics respectively. Subsequently, we proceed to study them from the point of view

of coadjoint orbits, which are mathematical objects arising in the study of Lie groups and their representations. This process results in solidifying the validity of the proposed interpretation of the kinematic space of AdS3 as a coadjoint orbit [3]. More specifically, we explicitly compute the

associated modular Berry phase, using exclusively information arising from the orbit geometry, reproducing the same results that had been previously established by different means. In the final part of this work, we take a step towards the generalization of these results for cases other than pure AdS, by considering the Ba˜nados geometries, since they constitute the most general solutions of asymptotically AdS3 spacetimes. To this end, we perform explicit calculations, in order to obtain

the related Berry phase and support the claim that it is associated with a coadjoint orbit of the Virasoro group.

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Introduction

1.1 Introduction to holography

This is by no means meant to be a complete review on the subject of holography. However, it is essential that a few of the elementary notions are laid out early on, for the sake of completeness. We will be mostly interested in the conjectured AdS/CFT duality, but we will take a historical tour on the topic, so as to gain a well-rounded perspective.

The idea of holography, as a tool to describe the universe, was born in an attempt to secure unitarity as a fundamental law of nature. More specifically, in 1975 Hawking published one of his most seminal papers, titled “Particle creation by black holes” [4], where, by using the tools of quantum field theory in curved spacetime, he showed that black holes evaporate emitting thermal radiation (later named Hawking radiation). Therefore, slowly but surely, a black hole will radiate all its mass away and vanish, thereby irreversibly destroying the information that had been stored in it during its lifetime. At least, that is the view that Hawking held, which he expressed in his follow-up paper “Breakdown of predictability in gravitational collapse” [5]. This was (and would still be) a very radical idea, given that it implies a violation of one of the most fundamental laws of nature as we currently understand it. Namely, it implies the loss of information, or, in other words, the breakdown of unitarity. Before we present the arguments of the opposing side, it is instructive to understand better why the black hole evaporation seemingly leads to information loss.

We begin our discussion with a brief review of some aspects of black hole mechanics. In particular, we will interested in black hole thermodynamics, which is a term stemming from the resemblance of the relations between the black hole mass M , its horizon’s area A and the surface gravity κ, and the thermodynamic quantities: energy E, entropy S and temperature T . More specifically, by performing the identifications1:

E ↔ M, S ↔ A 4, T ↔

κ

2π (1.1)

one can write down the first law of black hole thermodynamics as [6]: δM = 1

8πκδA + ΩδJ (1.2)

where ω is the angular velocity of the horizon, defined by the existence of the killing vector ξα = tα_{+ Ωφ}α _{and J is the angular momentum of the black hole. One can compare it with the first law}

1

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of thermodynamics dE = T dS − pdV to verify their similarity given the above identifications [7]. Accordingly, one can think about the zeroth law of thermodynamics which states that for a system in thermal equilibrium the temperature is constant everywhere. The analogue for black holes implies that the surface gravity is constant on the entire horizon, which can be shown to be indeed true, under some reasonable assumptions. Finally, and more importantly, the second law implies that the black hole horizon area cannot decrease, much like the entropy of a closed system. This statement was used by Bekenstein who proposed the generalized second law, which interprets the horizon area as an actual entropy and states that the combined entropy of matter and black holes never decreases.

δ S + A 4 ≥ 0 (1.3)

In the context of classical GR, it would seem that this is just taking a simple resemblance between two different sets of equations too literally. After, all the no-hair theorem states that black holes are described by just three quantities; the mass, the charge, and the angular momentum. However, the area of a black hole is quite the big number, so if we were to interpret it as an entropy, that would imply that there is an equally big number of associated microstates, seemingly contradicting the no-hair theorem. By showing that black holes evaporate, Hawking, proved that they do indeed have a temperature that matches the aforementioned assumptions, thereby providing a more concrete basis for the interpretation of black hole mechanics as thermodynamics.

Returning to the topic of information loss, one can consider the possibility of taking two different quantum states A and B and collapsing them into two black holes that are macroscopically iden-tical. This means that they are characterized by the same mass, charge and angular momentum (obviously these quantities have to be the same for the initial A and B). The two black holes are indistinguishable as far as their macroscopic degrees of freedom are concerned, but if they lived eternally one could argue that the information relating to the differences of A and B would be safely stored somewhere in the interior of the black holes, which cannot be probed. It is at this point that the evaporation of black holes introduces the issue of information loss. Now, instead of the information being stored permanently somewhere inside the black hole, one would have to come up with a mechanism that allows it to escape through the Hawking radiation. However, the fact that the latter is thermal and uncorrelated [5], means that the radiation escaping black hole A and black hole B also has to be indistinguishable. Hawking naturally concludes that the extra information differentiating between the initial states A and B is truly lost and that unitarity breaks down. Of course, this claim was confronted with great resistance, which as we shall see eventually led to Hawking admitting defeat. This was due to the developments in the field strongly suggesting that the theory describing this process has to be unitary after all. Despite the fierce debate that ensued, the issue has not been definitively resolved to this day, not for lack of proposed solutions, and has been coined the term: “The black hole information paradox”.

The pioneers among the advocates of unitarity, who came up with holography as a convincing alternative to Hawking’s arguments, are Susskind and ’t Hooft who firmly believed that the loss of information could not be the answer to Hawking’s riddle. Their first line of defense was black hole complementarity [8,9], which will not be discussed in detail here, even though the core of the idea is presented. The main claim of the proposal has to do with the experience of an observer falling into a black hole as opposed to that of a static observer living far away from the horizon. Due to the equivalence principle (and if we assume, without loss of generality, that the black hole is big enough), the infalling observer should not notice anything peculiar when they cross the horizon, whereas what the static observer would perceive is drastically different. More specifically, what they would see, is the infalling observer moving increasingly closer to the horizon yet never reaching it (due to the gravitational time dilation), while also becoming more and more flat. Naturally, these

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two perspectives seem completely incompatible, and yet, black hole complementarity suggests that they are both true, similarly to the way that the electron can be both a wave and a particle, even though these two qualities are inherently thought of as incompatible.

Soon after, in their respective most seminal papers on the topic [10,11] Susskind and ’t Hooft formulate the holographic principle, which states that all the degrees of freedom in a volume of spacetime can be encoded on its boundary. Therefore, much like a hologram, all the information for the bulk of the spacetime is encoded in a surface of 1 less dimension. This profound statement, even though it challenges the most fundamental notions of our intuition about spacetime, has probably been the most influential idea in theoretical physics since its conception. Once again, a detailed description of the internal parts of the original construction of Susskind and ’t Hooft is beyond the scope of this work, but we shall attempt to illustrate how one may, heuristically, arrive at their conclusion by means of a simple thought experiment.

Let us think, for simplicity, about a spherical region in space containing matter with total entropy S0. Obviously, the total mass of the system must be less than that of a black hole that would fill

the same region. Let us now imagine that a spherical shell of matter collapses around the system with just enough mass to turn it into a black hole of radius R. Then, it is well known that the entropy of the resulting black hole is given by:

Sf =

A 4 =

4πR2

4 (1.4)

The value for the final entropy has to be at least equal or larger than the initial S0, otherwise,

the second law of thermodynamics would be violated. Therefore, we conclude that the maximum entropy that a region of space can contain is given by the so-called holographic bound [12]:

Smax ≤

A

4 (1.5)

Throwing more information in the black hole will increase its horizon’s area, thus extending it beyond the original region under consideration. As a result, strangely enough, one finds out that the maximum number of degrees of freedom that a region of space can contain is not proportional to its volume, but rather to its area.

However, the above construction lacks in mathematical rigor and is dependent on certain assump-tions that need not be true. This was mended in 1998 when Maldacena established a scenario in which the holographic principle manifests, based on the rich mathematical framework of string the-ory. This is of course the widely known AdS/CFT correspondence [13], which Maldacena showed that is true for the specific case of AdS5× S5 and N = 4 SYM (for a reader who’s unfamiliar with

either sides of the duality, some pieces of literature that might be of interest are [7,14–16]). It has been conjectured that the duality is true for other dimensions as well, and a lot of work has been done, especially in the case of AdS3/CFT2 and more recently AdS2/CFT1.

We shall now attempt to capture the basic features of the correspondence, without delving too deep into the intricate mathematical structures or the formal arguments of string theory. The main result of AdS/CFT is that the physics in an anti-de Sitter spacetime of d + 1 dimensions is dual to that of a conformal field theory in d dimensions. In other words, in this context, a quantum field theory admits a gravitational description and vice versa, a spacetime with gravity admits a field theory description. This is, of course, far from a trivial statement since it implies a deep connection between gravitation and quantum mechanics, which are generally thought of as incompatible. So what is special about the two sides of the duality that make the correspondence possible? A good place to start is the realization that the isometry group of AdSd+1 is SO(d, 2), which is the same

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as the conformal group in d dimensions. Furthermore, the boundary of the Euclidean AdSd+1

(Lobachevsky) space in Poincare coordinates is Rd _{and in global coordinates R}

t× Sd−1, which

is conformal, and as such equivalent, to Rd [14]. Therefore, it might be considered natural that gravitational theories in AdS are holographic and equivalent to the field theories defined on its boundary. A more deep argument that hints towards the existence of the correspondence is the fact that an AdS spacetime is much like a quantum mechanical box [14,17]. This means that light signals can reach the boundary in a finite time, thus causally connecting the bulk and boundary degrees of freedom. This serves as a strong indication that the theory might be holographic. To clarify, since the boundary of AdS lies at infinity with respect to the radial coordinate, this implies that light can travel an infinite distance in a finite time. This might be a little unsettling, even though we know that for light, our common intuition about how spacetime distances work, breaks down. However, one can work out that this is true by showing that the Klein-Gordon equation (for massless fields) can be rewritten as a 1-dimensional Schrodinger equation for a particle in a box [14].

In more concrete terms, the essence of AdS/CFT is captured by the following relation:

ZCF T = ZAdS (1.6)

Meaning that the partition functions of the two sides of the correspondence are the same. This implies that observables, which are given by correlation functions, are also the same, as one obtains them from functional derivatives of the partition function. This was first worked out in detail in [18,19]. Let us elaborate on the content of the above equation. The partition function for the CFT has the form:

Z[φ∆i] ≡ he

R dd_xφ

∆i(x)O∆i(x)i_{CF T} (1.7)

where φ∆ are sources and O are primary operators. As expected the above expression is invariant

under conformal rescalings, meaning that: Z ddxφ∆(x)O∆(x) = Z dd(λx)φ∆(λx)O∆(λx) = λd−∆ Z ddxφ∆(x)O∆(x) (1.8)

On the AdS side, any generally covariant function of fields φ(z, x) (using Poincar´e coordinates) will be conformally invariant. However these fields live in one more dimension, so we want to restrict to fields on a d-dimensional subspace of AdSd+1. The boundary ∂AdS ' {z = 0, ~x} is special

because the group of conformal isometries acts on it the same way as the conformal group acts in d-dimensional spacetime, as we mentioned previously. Therefore, boundary values of functions on AdS transform as representations of the conformal group on spacetime and as such, the partition function can be any generally covariant function of the boundary values. More concretely, the partition function can be written as a path integral over boundary field configurations as:

Z[ ¯φ(~x)] = Z

φ|∂= ¯φ

Dφ(z, ~x)e−S[φ(z,,~x)] (1.9)

Comparing this expression to the CFT partition function one can infer that ¯φ ≡ ¯φ∆ is the source

for a primary operator O∆. Here we have used scalar fields, but the same results can be shown to

be true for other types of fields as well.

Before we move on, a couple of remarks are in order. First, one could be skeptical as to why holography is the right tool to approach the problem of quantum gravity. Obviously, research is not a one-way street and as such, there is no single “correct” way to approach a problem, but holography, as a proposal, exhibits many promising features that make it a prime candidate. One can argue that the most prominent of these features is the mathematical consistency that follows from the

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well-studied string theory framework on which it relies. It is evident that this is a key property that has to be satisfied by any theory of quantum gravity and it is thus pivotal that the AdS/CFT duality satisfies this criterion. Next, it is very important that AdS/CFT and holography in general, do not inherently rely on techniques of perturbative nature to produce results. It would seem fair to say that our current understanding of physics through the standard model, is almost exclusively reliant upon such perturbative results stemming from quantum field theory. This does not, in any way, lower the value of said results, given that they have proved to be extremely useful tools that have contributed both in advancing our fundamental understanding of the universe and in developing applications that improve (or may do so in the future) the average person’s life. However, one type of problem that seems out of reach for these techniques is that of strongly correlated systems, which have become increasingly more important and equally popular among the scientific community in the last few years. As a result, in this instance, it seems imperative to develop new methodologies that are well suited to tackle problems in a non-perturbative manner and holography has a structure that allows endeavors such as this. Finally, one last quality that renders holography a topic that is worth delving in, is its wide range of applicability. Even though it was inspired and developed for a particular purpose, which was described earlier, things have changed drastically during the last two decades. The concepts and way of thinking that holography first introduced have now been adopted in the study of other fields, such as condensed matter and statistical physics.

The second remark has to do with the relevance of holography to this thesis. In the following chapters, it will become clear that this work draws from several different areas of physics and mathematics which do not necessarily fall under the umbrella of holography as it has been described so far. So why was this topic chosen as a place to start, rather than one of the alternatives that appear later on? The answer is that, apart from the overarching theme, the ultimate goal of this thesis is also related to holography. Namely, our purpose will be to study some of the aspects of the AdS/CFT correspondence, mainly for AdS3/CFT2. More specifically, we shall attempt to establish

a relation between objects pertaining to the two sides of the correspondence, through the novel use of mathematical constructions. The most prominent example of this is the relation between the length of curves in AdS3 and the entanglement entropy of intervals in the CFT, which we will

explore in later chapters, through the mathematics of both integral geometry and coadjoint orbits. In conclusion, as this work progresses we hope to shed some light on the interplay between geom-etry and quantum mechanics, be it in the form of non-holonomic processes in standard quantum mechanics, the interrelation of boundary and bulk quantities, or the study of relevant holographic quantities through the geometry of an auxiliary space.

1.2 Motivation

Having introduced some of the elementary notions of holography, we can now elaborate on the recent advances in the field leading up to this work. As we mentioned above, the central tenet of holography is the ability to describe the bulk of spacetime and its dynamics using a quantum field theory of 1 less dimension (which one can identify with the boundary of the spacetime at least in the context of AdS/CFT) and vice versa. The typical notion of a hologram, from which this type of dualities inherits its name, involves a 2-dimensional surface on which we shine light to produce a 3-dimensional figure. That is precisely the relationship between the two sides of a holographic theory and thus, one should be able to construct the bulk of spacetime and its dynamics using the data from the boundary theory. In the context of AdS/CFT, this process is simply referred to as bulk reconstruction and it has been a topic that has attracted significant attention throughout the years.

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Even though up to this day, this has not been accomplished for the AdS/CFT correspondence, various constructions have been devised to relate objects of the bulk with boundary quantities, such as the HKLL construction [20], which provides the instructions to construct free local bulk fields from operators in the boundary theory.

One recent development on the topic of bulk reconstruction was given in [21], where the authors propose a mathematical framework that allows the description of geometric concepts in the bulk of AdS, using the entanglement entropy of boundary intervals. These concepts were later used to develop tools for constructing local bulk operators [22]. However, it was not long before another class of interesting phenomena arose through the study of the above. Namely, it was argued in [1] that there exists a quantity reminiscent of a Berry phase, called the modular Berry phase, that appears to be connected to the aforementioned framework. A reader who is unfamiliar with the term “Berry phase”, should not be alarmed, as we will discuss in detail its origins, derivation, and generalization to holography in the following chapters. For the moment one can simply think of it as the quantum analog of the angle difference between a vector and itself after it has been parallel transported along a loop on a curved space. This example should be familiar from introductory general relativity, as it is most commonly used to illustrate that the familiar notion of parallel transport in Euclidean space does not survive the generalization to spaces with curvature. This constituted a new entry in the AdS/CFT dictionary, which is an accomplishment in itself, but it also opened the path to the exploration of holography from novel mathematical perspectives. In conclusion, all the above fall under the umbrella of the more general entanglement=geometry proposal. Namely, one of the most prominent views on holography in recent years is that the entanglement structure of the boundary CFT contains the instructions for the geometric features of the bulk. However, the big question that remains, for the time, unanswered is how can this be performed in practice. So, even though this work is multifaceted, our primary objective will be to explore aspects of this question.

1.3 Summary of results

Having explained the motivation for this work, it is deemed useful to also give an overview of the main results that we have obtained. As we mentioned in the previous section the goal of this thesis is to study a holographic generalization of the Berry phase. One of our main results is the calculation of the latter in the context of AdS3/CFT2 (specifically for an AdS3 timeslice),

utilizing the mathematics of coadjoint orbits. More specifically, we show that the modular Berry phase is given as the flux of the Kirillov-Kostant symplectic form through a closed surface on a coadjoint orbit of SL(2,R). Furthermore, we show that our result matches the literature and thus, it constitutes an alternative interpretation of the same phenomenon. As this is just a summary, we will not provide additional technical details; instead, we shall illustrate some geometric aspects of the above statement.

As we show in later chapters, the relevant calculation is related to an integral over a surface area on a dS2 spacetime. We view the latter as a coadjoint orbit of SL(2,R) which, consequently, is

automatically endowed with a symplectic structure. This is depicted schematically in figure 1.1. Thus, the final expression for the modular Berry phase that we obtain is:

B = −a λ Z Σ 1 sin2(t)dtdθ (1.10)

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appro-Figure 1.1: The modular Berry phase associated to an AdS3 timeslice is computed by the surface

area enclosed by the red loop, weighted by a constant.

priate volume form.

One more important result is given in the final chapter, where we consider a generalization of the modular Berry phase for asymptotically AdS3 geometries. The most general metric for these

spacetimes is given by: ds2 = ` 2 r2dr 2₋ rdx+−` 2 r L−dx − rdx−−` 2 r L+dx + (1.11) where L+, L− are periodic functions of the coordinates x+, x− respectively. The geometries

gen-erated by the above metric are asymptotically AdS3 spacetimes that respect the Brown-Henneaux

boundary conditions and are usually referred to as the Ba˜nados geometries. These include, apart from pure AdS3, BTZ black holes and conical defects. In order to study the modular Berry phase in

this case, the idea is the following: We start from pure AdS3, where we have a good understanding

of the modular Berry phase and subsequently, we perform an infinitesimal conformal transformation of the form:

h(x±) = x±+ ζ(x±) (1.12)

where ζ(x±) is a periodic function of x±. This moves us away from pure AdS3 and somewhere in the

range of the Ba˜nados geometries. Then, we go back to the definition of the modular Berry phase, which we discuss in detail in a later section, and we investigate how this conformal transformation affects our results. What we find, is that if we simply shift the geometry and then perform the process that leads to the Berry phase, the answer that we get is the same as that for pure AdS3.

However, if we let the conformal transformation itself vary cyclically (remember that the Berry phase arises as the quantum analogue of a parallel transport process around a loop) then we obtain a non-trivial result. As an example, we consider the simple case where the conformal transformation has the form:

h(x±) = x±+ (λ) sin (x±) (1.13) where λ parametrizes the aforementioned cyclic variation. This leads to the calculation of the parallel transport operator, which is the operator generating the parallel transport parametrized by λ. The result that we obtain is the following:

V(1)= ˙π 3 L3 (e 2πimλ_L _L m− e−2πi mλ L L_−m) (1.14)

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The important point on the above equation is that it depends on Lm, L−m, which are the well

known Virasoro algebra generators satisfying:

[Ln, Lm] = (n − m)Ln+m+

c 12(n

3_{− n)δ}

n+m,0 (1.15)

As we will argue in more detail in the appropriate section, this illustrates that the associated Berry phase is related to the larger Virasoro group, rather than the SL(2, R) group that is relevant in the case of pure AdS.

1.4 Outline

We conclude this chapter with the layout for the rest of this thesis. Our starting point is the articles by Czech, et al. [1,21] and Oblak [23]. In the former, the holographic notion of kinematic space is first introduced and studied in detail. We will repeat part of this analysis in the next chapter, with the goal of defining and computing the modular Berry connection. The term stems from the usual Berry phase that one encounters in quantum mechanics, but as we shall see the analogy, even though it is accurate, has its limitations. In chapter 2, one can find a review of the Berry phase in quantum mechanics, along with a few different mathematical frameworks which can be utilized to compute it. This provides the necessary background before the holographic version is introduced in chapter 3.

Once we have arrived at some of the key expressions for the modular Berry connection, we will aim to provide a link between it and the mathematics of coadjoint orbits, based on the work of Penna and Zukowski [3], which will be briefly discussed in chapter 3. This work establishes a novel description of kinematic space in terms of the coadjoint orbits of the isometry group of AdS. In turn, this provides a natural stage to apply the technology developed in [23] for computing Berry phases, using the pullback of the Maurer-Cartan form of an orbit. This procedure is initially sketched in the following chapter, while the full calculation of the Berry connection that we are interested in, takes place in chapter 4. We shall see that the result one obtains with this method is the same as in [1], thus providing an alternative description and an independent verification of the same phenomenon. Finally, in chapter 5 we discuss the prospect of generalization of these results beyond pure AdS, expanding on the results of [2]. More specifically, we will be concerned with Ba˜nados geometries, which capture a wide range of spacetimes that are asymptotically AdS3. After introducing some

key notions relating to these geometries and their respective boundary duals, we will show how one can obtain a more general version of the modular Berry connection, which is associated with the coadjoint orbits of the Virasoro group, rather than those of its SL(2, R) subgroup.

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Chapter 2

The toolkit

2.1 Review of fibre bundles

In this chapter we introduce the necessary tools that we will be using for the rest of this thesis. Namely, we will review some aspects of the Berry phase, we will discuss the notion of kinematic space and define the coadjoint orbits as mathematical objects, while also looking at some simple examples for all of the above.

One of our main tools in this endeavor is going to be the mathematics of fibre bundles and since a lot of what follows relies on it, it is deemed useful to begin this chapter by reviewing the fundamentals of the topic. For the definition of fibre bundle, following [24], we require a base space M and a total space E, along with a bundle projection which is a surjective map π : E → M . We also need a Lie group G, called the structure group acting on F by the left action and a set of open covering {Ui} of M with a diffeomorphism φi : Ui× F → π−1(Ui) such that π ◦ φi(p, f ) = p. This map is

called a local trivialization and it captures the essential feature of fibre bundles, which is that they are product spaces of the form M × F locally, but usually not globally. The standard example to discern between the two cases is that of a space that is locally R × S1. If the same holds globally, then this space has the topology of a cylinder; this type of fibre bundle is referred to as trivial. However, one can also consider the case of the Mobius strip which is locally R × S1_{, but is not}

an orientable surface like the cylinder and thus this property is not retained as a global feature. Finally, If we write φi(p, f = φi,p(f )) the map φi,p : F → Fp is a diffeomorphism. On Ui∩ Uj 6= ∅,

we require that tij(p) ≡ φ−1i,p ◦ φj,p: F → F be an element of G. Then φi, φj are related by a smooth

map tij : Ui∩ Uj → G as:

φj(p, f ) = φi(p, tij(p)f ) (2.1)

where the maps tij are called transition functions. This last requirement essentially ensures that

two local trivializations of nearby, overlapping neighborhoods Ui, Uj are compatible with each other;

this is made certain by requiring the existence of a smooth map tij that takes us from one to the

other. In order to gain some geometric intuition, we include a pictorial representation of a fibre bundle with a two-dimensional base space and a one-dimensional fibre in figure 2.1.

Even though we included all the above formal requirements for the sake of completeness of the definition, we will be primarily interested in the simple case where the fibre is the same as the structure group. In this case one obtains a special kind of fibre bundle, called a principal bundle. A notion that is going to be of particular interest later on is that of a section, which is the map

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Figure 2.1: A pictorial representation of a 3-dimensional fibre bundle. The 2d surface represents the base space and the lines, the fibres at different points. The planes intersecting the fibre represent the horizontal subspaces (defined in the text) at different points.

defined by:

s : M → E, π(s(p)) = p (2.2)

This can be seen schematically in figure 2.2. Such maps always exist locally, being defined only in

Figure 2.2: A pictorial representation of the maps defined on a fibre bundle. The map π takes us from a point on the total space to a point on the base manifold, whereas σ does the inverse process. the neighborhood of p, but global sections might also exist. A good indicator for whether this is the case, is whether the space under consideration can be covered with only one coordinate chart or not. If the latter is true, then it is evident that global sections cannot exist since the coordinate chart labeling the points of the space is not well defined everywhere.

The final object we will be interested in is the connection defined on a (principal) fibre bundle. The bundle connection is a choice we make as to how the individual points between neighboring fibers are connected. In more abstract language the connection can be defined as follows: Let P (M, G) be a principal bundle. A connection on P is a unique separation of the tangent space TuP , where

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• T_uP = HuP ⊕ VuP

• A smooth vector field X on P is separated into smooth vector fields XH _{∈ H}

uP and X ∈ HuV

as X = XH + XV

• H_ug = Rg∗HuP for arbitrary u ∈ P and g ∈ G

The first two conditions are self-explanatory and the third implies that the horizontal subspaces HuP and HugP on the same fibre are related by a linear map Rg∗ induced by the right action

of the Lie group. In essence, this choice holds the instructions for parallel transporting objects between different points of the fibre bundle. Once we have made that choice we can define a Lie algebra valued one-form, commonly referred to as the connection one form, which encodes our choice by specifying a way to decompose vectors in a horizontal and a vertical subspace1. First, let us provide a more rigorous construction of the vertical and horizontal subspaces; given an element u of a principal bundle P (M, G) and Gp the fibre at p = π(u), we define the vertical subspace VuP to

be a subspace of TuP that is tangent to Gp at u. This means that the vertical direction coincides

with the fiber direction, but the horizontal is up to us to choose (and that is precisely the choice we are making); this is visualized in figure 2.1. We construct VuP , by taking an element A ∈ g and

by the right action:

R_exp(tA)u = u exp(tA) (2.3)

we specify a curve through u. However, acting by a group element only pushes u along the fiber and therefore, π(u) = π(u exp(tA)) = p. This implies that this curve lies entirely within the fibre Gp and thus we can define a vector A#f (u) ∈ TuP , where f (u) is an arbitrary function defined in

P , such that:

A#f (u) = d

dtf (u exp(tA))|t=0 (2.4)

A#is tangent to Gp at u and hence A#∈ VuP . Working in a similar manner one can define A#at

every point, thus constructing the fundamental vector field generated by A. This provides a vector space isomorphism # : g → VuP given by A → A#. One can now proceed in defining the connection

one-form as follows: A connection one-form ω ∈ g ⊗ T∗P is a projection of TuP onto the vertical

component VuP u g. This is summarized in the requirements:

• ω(A#_{) = A,} _{A ∈ g}

• R∗_gω = Ad_g−1ω

It follows that the horizontal subspace is simply defined as the kernel of ω:

HuP ≡ {X ∈ TuP |ω(X) = 0} (2.5)

For physical applications, however, we are usually interested in the base manifold; as an example of this one might consider general relativity, where spacetime is the base manifold and the total space is given by the bundle of morphisms that one can perform on it. As such the connection that is really of use to us is defined on the base manifold, using a local chart, rather than abstractly on the total space. The way to obtain it, is by pulling back the bundle connection by a local section, as seen in figure 2.2. Here we emphasize the local nature of the section, not only for the reasons that were discussed previously, but also from a physical point of view, we require that the language

1_{Even though the connection and the connection one-form are defined separately, they are very closely related and}

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we use to formulate laws is a local one. Once more, let us formulate the above in a more formal manner. Let {Ui} be an open covering of M and let σi be a local section defined on each Ui. The

local Lie algebra valued one-form is then given by:

Ai ≡ σi∗ω (2.6)

where ∗ denotes the pullback. In a physical context, this is usually referred to as a gauge potential, or in the case that the group is non-abelian (i.e. most groups considered beyond U (1)) as a Yang-Mills potential [25].

We conclude this section with a brief mention to the notion of holonomy, which will play a central part throughout the thesis. Before we can discuss the notion of holonomy itself, we have to define horizontal lifts and discuss their relevance in parallel transporting an element of a fibre bundle. The horizontal lift is defined as follows [24]: Let P (M, G) be a principal G-bundle and let γ : [0, 1] → M be a curve in the base space M . A curve ˜γ : [0, 1] → P is said to be a horizontal lift of γ if π ◦ ˜γ = γ and the tangent vector to ˜γ(t) (where t is a choice for the parametrization of the curve) always belongs to H˜γ(t)P . If we denote by ˜X the tangent vector to ˜γ, then this by the definition of the

connection one-form ω, the above implies that ω( ˜X) = 0.

Parallel transport of an element of a principal bundle along a curve in M is provided by the horizontal lift of the curve. A theorem that we will invoke without proof (for the proof refer to [24]), is that for any point belonging to the curve, the horizontal lift to any other point in P is unique and as such parallel transport is defined in a unilateral manner. Let us attach a more rigorous mathematical notation to the above; let γ : [0, 1] → M be the curve of interest and take a point u0 ∈ π−1(γ(0))]. There is a unique horizontal lift ˜γ(t) of γ(t) through u0, and hence a unique

point u1 = ˜γ(1) ∈ π−1(γ(1)). The point u1 is called the parallel transport of u0 along the curve

γ. This defines a map Γ(˜γ) : π−1(γ(0)) → π−1(γ(1)) such that u0 → u1. In terms of the local

connection one-form this map becomes: u1= σi(1)Pexp − Z 1 0 Aiµ dxµ(γ(t)) dt dt (2.7) where P denotes path ordering, which is essential for the exponential to be well defined, but in practice we will not have to deal with it explicitly.

Given the above, it follows that for two different curves α, β : [0, 1] → M , with α(0) = β(0) = p0

and α(1) = β(1) = p1, if we choose their horizontal lifts ˜α, ˜β such that ˜α(0) = β(0) = u˜ 0, then

˜

α(1) is not the same as ˜β(1). This serves to show that if we consider a loop γ : [0, 1] → M at p = γ(0) = γ(1), in general it is the case that ˜γ(0) 6= ˜γ(1). Thus a loops defines a transformation tγ : π−1(p) → π−1(p) on the fibre, which is compatible with the right action of the group:

τγ(ug) = τγ(u)g u ∈ P, g ∈ G (2.8)

Then, in order to define the notion of holonomy, we simply need to consider a point u ∈ P with π(u) = p along with the set of loops Cp(M ) ≡ {γ : [0, 1] → M |γ(0) = γ(1) = p)}. The set of

elements:

Φu≡ {g ∈ G|τγ(u) = ug, γ ∈ Cp(M )} (2.9)

is a subgroup of the structure group G, which is called the Holonomy group at u. The concept of holonomy is represented pictorially in figure 2.3.

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Figure 2.3: Pictorial representation of holonomy. The plane represents the base space, the blue curve is a loop, the red dashed line is its horizontal lift and the straight black line corresponds to the fiber direction. The holonomy element associated to the loop is the one inducing a right action on the fibre from point A to point B.

2.2 Generalities on the Berry phase

In this section we provide the basics regarding the Berry phase and associated concepts. Ultimately, the aim is to establish a connection between the elementary case where Berry phases arise in quan-tum mechanics and more sophisticated versions of the same phenomena encountered in holography. To this end, after a brief mention to the adiabatic theorem, we consider the appearance of nonholo-nomic processes in quantum mechanics, i.e. processes during which a system does not return to its original state after being transported around a closed loop.

Before we are able to confront the actual problem of nonholonomic processes, it is useful to shortly mention the adiabatic theorem along with its implications in standard quantum mechanics. For a given system with a time dependent Hamiltonian, giving rise to the Schrodinger equation:

dΨt

dt = −iHtΨt (2.10)

there is, generally, no stationary solution. However, if we consider the limit where Ht changes

infinitely slow, then we can make the assertion that at any given moment the system will be in an eigenstate En(t) of the Hamiltonian at that time. This statement is known as the adiabatic theorem,

which even though, to a certain degree, is intuitively obvious, proving it in a rigorous manner is in any case non-trivial. Therefore, we are going to omit the proof and instead, only discuss the relevant details with regards to nonholonomic processes.

The reason why the adiabatic theorem is important, is because it allows for the decoupling of the fast and slow degrees of freedom of a system thus making it possible to study them separately. This holds either in the case of intrinsic degrees of freedom, as in the vibration of a diatomic molecule where we can treat the motion of the nuclei as slow compared to that of the electrons (Born-Oppenheimer approximation), but it is also true for extrinsic degrees of freedom, such as a temporally varying potential that we impose on a system and whose time dependence can be treated as slow compared to the characteristic timescale of the system. These concepts will become more clear later on, when we consider some simple examples.

This is also a convenient point to introduce the notion of a nonholonomic process in a slightly more concrete manner, by considering a simple example coming from classical physics [26]. Let us imagine

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a frictionless pendulum located at the north pole of the earth and oscillating in a plane of our choice (in reality we do not even need the pendulum, just a vector pointing along a certain direction would do just as fine). We would like to start moving this pendulum along the surface of the earth keeping the plane in which the pendulum oscillates fixed. One can easily convince themselves that if the pendulum is transported to the equator followed by a subsequent movement for a certain distance along the equator and then returned to the north pole, the plane of oscillation that we were keeping fixed throughout this process will, generally, not coincide with the one we started with. This might seem strange at first, as though we somehow failed in our task of keeping the plane of oscillation fixed at all times while we were carrying the pendulum around, but in reality this is just a consequence of parallel transport on manifolds with non-flat geometry. So, how does the adiabatic theorem enter this example? In order to be able to keep the plane of oscillation fixed, or at least parallel to the initial, it is essential that we move the pendulum adiabatically (as compared to the frequency with which it is oscillating), otherwise a sudden movement is liable to also radically change the plane of oscillation. Moreover, by thinking in terms of the adiabatic theorem we are able to recognize the slow degrees of freedom as the movement of the pendulum on the surface of the earth and the fast degrees of freedom as its oscillation. We therefore see that the decoupling of these degrees of freedom can lead to a nonholonomic process when a loop in parameter space is traversed; this is an important realization which will be generalized in the following paragraphs.

It is well known that a quantum mechanical system with a time dependent Hamiltonian is described by the Schrodinger equation (2.10). A natural guess for the form of the wave function is:

Ψ(x, t) = ψn(x, t)e− i ~ Rt 0En(t 0_)dt0 (2.11) Where −1 ~ Rt 0En(t

0_)dt0 _{is the familiar dynamic phase obtained by time evolving the state and which}

henceforth will be labelled as θn for convenience. However, one can easily check that this is in fact

not correct, since it does not satisfy (2.10). This is due to the fact that the time derivative will not only act on the exponential producing the desired eigenvalue, but also on the state itself producing an extra term. To mend this situation we can add an extra phase term, which is referred to as the geometric phase, to obtain:

Ψ(x, t) = ψn(x, t)e−iθneiγn(t) (2.12)

Now by taking the time derivative we get: i~[∂Ψ ∂t − i ~Enψne −iθn_eiγn_{+ i}dγn dt ψne −iθn_eiγn_{] = E} nψne−iθneiγn (2.13)

And therefore we can now demand that: ∂Ψ

∂t + i dγn

dt ψne

−iθn_eiγn _{= 0} _(2.14)

In order for the Schrodinger equation to be satisfied. Taking the inner product of this expression with the state ψn yields:

dγn

dt = ihψn| ∂ψn

∂t i (2.15)

Which in turn implies:

γn(t) = i

Z Rf

Ri

hψ_n|∂ψn

∂RidR (2.16)

Where R(t) is some parameter that is varying with time. It is evident that for the case of a single parameter R, if the Hamiltonian traverses a closed loop in R space (Ri = Rf), then the geometric

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phase is simply zero. However, if one considers a parameter space with more dimensions, then the above expression can be rewritten as follows:

γn(t) = i

I Rf

Ri

hψ_n|∇_RψnidR (2.17)

This is the definition of the Berry phase, which as can be seen is given by a line integral in parameter space that is generally non-zero. An important point that needs to be stressed is the fact that our interpretation of γnas a phase implies that it is real. Differently it would just lead to an exponential

dependence and thus the change in the normalization of the wavefunction, (a unitarity violation) which would definitely be a bad sign. It is comforting that a rather simple calculation found in [26] shows that Berry phase is indeed real.

At this point it is reasonable to ask at which point does the adiabatic approximation enter in the above. In reality, it is not essential that the Hamiltonian is varied adiabatically; however, assuming that this is the case ensures that a cyclic variation of the parameters leads to a cyclic evolution . It is important to mention that the procedure followed above to derive the Berry phase is not the most rigorous and although the final result is correct there are certain subtleties which have not been dealt with. More information on the matter can be found in [26].

Let us put what we have described so far in use, to study a simple example which is known in the literature as the Aharonov-Bohm effect. We are going to consider a charged particle moving on a ring at the center of which there’s a solenoid, as depicted in figure 2.4 [26]. We assume that the solenoid is long enough such that the magnetic field is uniform inside the solenoid and zero outside. The vector field defined by B = ∇ × A outside the solenoid is given by2:

A = Φ

2πrφˆ (2.18)

where Φ is the magnetic flux through the solenoid, r is the distance and ˆφ is the unit vector in

Figure 2.4: Setting of the Aharonov-Bohm effect. A charged particle is moving on a ring of radius b through which a long solenoid passes.

the direction of azimuthal angle (it is easier to think about this problem in cylindrical coordinates). The scalar potential is zero, since the solenoid is uncharged. Given the above, the Hamiltonian of the problem becomes:

H = 1 2m[−~

2_∇2_{+ q}2_A2

+ 2i~qA · ∇] (2.19)

2_{We are not going to be proving these claims concerning electrodynamics. Pedagogical reviews on the topic can}

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and because the vector potential’s only component is in the φ direction, Schrodinger’s equation is: − 1 2m[ ~2 b2 d2 dφ2 + ( qΦ 2πb) 2₊i~qΦ πb2 d dφ]ψ(φ) = Eψ(φ) (2.20)

The solutions of this differential equation are of the form ψ = Aeiλφ, where λ = _2π~qΦ ± b

~

√

2mE. As usual, the boundary conditions ψ(0) = ψ(2π), imply that λ is an integer which we will denote n and as such the energy is given by:

En= ~ 2 2mb2(n − qΦ 2π~) 2_{, n ∈ Z} _(2.21)

As one might recall, the energy of a free particle on a ring is twofold degenerate, which is manifestly not true in the case that a vector potential is introduced. Also, it is important to note, that even though the magnetic field, which we regard as the physically relevant quantity, at the location of the particle is zero, the latter’s energy is still affected.

Having set the stage, we can now consider the case where a particle is moving through a region where the magnetic field is zero, but the vector potential is not. The time dependent Schrodinger equation for such a scenario is:

[ 1 2m( ~ i∇ − qA) 2 + V ]Ψ = i~∂Ψ ∂t (2.22)

which we can simplify by seeking solutions of the form Ψ = eigΦ, where Φ is the solution to the Schrodinger equation without the electromagnetic potential. It can be explicitly shown that [26,29]:

g(r) = q ~

Z r

O

A(r0)dr0 (2.23)

where O is some arbitrarily chosen reference point. It is now evident that by moving a charged particle in a circle around the solenoid, the phase that it picks up is:

g = q ~ Z 2πb 0 A · dr = qΦ 2π~ Z 2π 0 (1 r ˆ φ) · (r ˆφdφ) = ±qΦ ~ (2.24)

where the + sign applies to particles moving in the direction of A and the - sign to particles moving in the opposite direction. This is the Berry phase that arises in this problem, but the Aharonov-Bohm effect refers to a slightly different setting. Namely, instead of moving particles in a loop around the solenoid, one uses a beam of particles that is split in two, and the resulting beams are driven through opposite sides of the solenoid as shown in figure 2.5 [26]. The two beams are then recombined and interference patterns are observed, due to the difference in phase between the particles of each beam. This difference in phase is, as expected, equal to the Berry phase and is the result of the particles of one beam going in the direction of the vector potential, and the particles of the other going in the opposite. Even though the Aharonov-Bohm effect was theorized and observed well before the conception of the Berry phase, here we see how the two are related as the outcome of a non-holonomic process.

Having introduced some of the key concepts, we can now consider a more mathematically rigorous approach to the Berry phase, which will also enable us to generalize these concepts outside the framework of quantum mechanics. This approach consists of a change of formalism relying on the translation of the above in topological terms, which will ultimately allow us to express the previous results in a more general manner. More specifically, we can consider a Hamiltonian H(R) depending on the set of parameters R and subsequently define a manifold M , whose geometry describes the

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Figure 2.5: The experimental setting with which the Aharonov-Bohm effect is observed. The figure depicts the top view of the solenoid and the vector potential. The dashed lines represent the two particle beams mentioned in the main text.

space of said parameters. In other words, the parameters R form a coordinate system on the manifold M . More concretely, one can think of such a parameter as, for example, a magnetic field that varies in time, thus changing the Hamiltonian and its corresponding eigenstates. This enables us to think of the physical states as points on this manifold and moreover, since we know that these states are determined up to a phase, we can claim that there exists a U (1) degree of freedom on each point of M . This in turn implies the existence of a U (1) bundle over M .

Going back to the topic of the Berry phase, the bundle structure defined on the parameter space is accompanied by the existence of a connection 1-form which will be denoted as A = AµdRµ and

whose field strength Fµν = ∂µAν − ∂νAµ is the curvature. It is important to mention that the

connection A is local and therefore it comes with the choice of a gauge for different points on the manifold. In the specific context that is examined here these quantities are referred to as the Berry connection and Berry curvature respectively. The Berry phase is then simply the line integral of the connection around a closed loop and is thus given by:

γ = i I R A = i Z S F (2.25)

where the two integrals above are related by Stokes theorem. It is important to note that this approach allows us to think of the Berry connection, in geometric terms, as the natural way of comparing the phase of two states at nearby points on M [30].

In an attempt to provide a more intuitive understanding of the Berry connection, it is instructive to compare it with the, hopefully, more familiar notion of the spacetime connection one encounters in general relativity. The latter relates the Lorentz frames of nearby tangent spaces of the space-time manifold and endows it with curvature. In other words, the connection is what contains the information that allows us to discover the global features of a spacetime manifold, by instructing us how to glue together small patches of flat (Minkowski) space [2]. This means that parallel trans-porting an observer’s local frame of reference around a closed loop is the same as taking a Lorentz transformation [31]. In a similar fashion, the Berry connection instructs us how to relate the phases of states, by gluing nearby tangent spaces of the parameter space (manifold).

It is a well known fact that any smooth manifold such as the ones we considered above are related to Lie groups and their corresponding algebra. Naturally, one might ask whether it is possible to

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consider the Berry phase in the language of group representation theory. The answer is of course affirmative and we explore the details below.

The necessary ingredients for this endeavor are a connected Lie group G and its associated Lie algebra g. We want to interpret G as a group manifold which describes the space of parameters of the Hamiltonian of some physical system. Therefore, in this context we will assume that G contains a one-parameter subgroup generated by an element of the Lie algebra X0 ∈ g such that all group

elements etXo _{can be viewed as “time translations” [}₂₃_{]. Where as per usual we have used the}

exponential map to relate the elements of the group and the corresponding algebra.

We are going to consider a unitary representation U of G, which associates with each f ∈ G a unitary operator U [f ]. In light of this, the time translations discussed previously will give rise to a an evolution operator U [etX0_{] = e}tu[X]_{, where u is the differential of U at the identity (or in other}

words the Lie algebra representation associated with U ). Thus we are led to the conclusion that the choice of Hamiltonian is not unique, since it relies on the choice of a reference frame. More specifically, for each element h ∈ G we have the unitary transformation: U [h]HU [h]−1, which can be interpreted as a change of reference frame.

With the above in mind, we only need to consider the case of a path h : [0, T ] → G : t → h(t) such that h(0) = h(T ) in order to investigate the existence of a Berry phase. Furthermore we impose that the process of traversing this path is adiabatic to ensure that the assumptions mentioned previously are satisfied. Therefore, some state U [h(0)] |φi can only differ by a phase which implies that U [h(0)] |φi = eiθ(T )U [h(T )] |φi. Where the phase θ is given by [23]:

θ(T ) = −ET ~ + i Z T 0 dt hφ| U [h(t)]†∂ ∂tU [h(t)] |φi (2.26) The first term is the familiar dynamical phase and the second term is the geometric phase, which in this context coincides with the Berry phase. Thus, using the fact that U is a unitary representation we can rewrite the Berry phase as follows:

Bφ[h] =

I

h

i hφ| U [h(t)]−1dU [h(t)] |φi (2.27) Furthermore, we can write:

U [h(t)−1]∂ ∂tU [h(t)] = u[ ∂ ∂t|t=τ(h(t) −1 h(τ ))] (2.28)

One can recognize the argument of u as the Maurer-Cartan form: Θ_h(t)( ˙h(t)) ≡ d

dτ|t=τ[h(t)

−1_{· h(τ )]} _(2.29)

which is an isomorphism between the tangent spaces ThG and TeG [32] where we know the latter

to be the Lie algebra of G. In other words the Maurer-Cartan form associates an element of the tangent space at some point h on G with an element of the tangent space at the point identified with the identity. Notice the similarity between the Maurer-Cartan form and a fibre bundle connection; indeed the Maurer-Cartan form satisfies the properties of a connection one-form, but furthermore it is an inherent property of the group and thus provides a natural group theoretical way to think about the bundle connection. Therefore, we can write the Berry phase in terms of the pullback of the Maurer-Cartan form by a section as follows:

Bφ[h] = I h i hφ| u[Θ] |φi ≡ I h Aφ (2.30)

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where Aφ is the Berry connection which in this case is given by:

hφ| iu[Θ] |φi = A_φ (2.31)

Where Θ now denotes the pullback in an abuse of notation. Before we move on, there are two important remarks.

Remark 1: Given a group G describing the space of parameters of the Hamiltonian of a physical system and its corresponding Maurer-Cartan form, one is able to evaluate the Berry phases of the group’s unitary representations.

Remark 2: It is important to realize that the group representation approach to obtaining the Berry phases is in essence no different from using arguments concerning the geometry of the manifold we are interested in. Hence, the only difference is in some sense linguistic and as such one can use one approach or the other depending on the specifics of each case. However, having knowledge of both is paramount in understanding the Physics in a more rigorous and holistic manner and enables us to circumvent the usual problems that arise during a calculation by utilizing the appropriate tools. We will delay an explicit demonstration of the practical use of the above formalism, until chapter 4, where this framework is used to perform a novel calculation. However, one can find a simple example that makes use of the above in [23].

2.3 Kinematic space

The term kinematic space in the context of holography, and specifically AdS3/CFT2, was first

introduced in [21], where it is defined as an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual information and which organizes the entanglement pattern of a CFT state. This definition might not give a clear and intuitive understanding of kinematic space as a concept, but luckily due to the theory being holographic, there is a dual version of the above, which stems from the bulk point of view. In this picture, the kinematic space is simply the space of bulk (AdS3) geodesics and the main tool that was initially used to study it, is a corner of math called

integral geometry. In the following paragraphs, we will introduce some key concepts relating to the above, but before we do so, it is important that the framework in which everything takes place is clear. To this end, let us imagine this space of AdS3 geodesics; it is going to be some space, in

the broad sense of the term (for now let us forget the claim that it essentially is some Lorentzian spacetime), where every point corresponds to a specific geodesic. This in itself might appear to be a rather mundane construction and it would indeed be so if it were not for the Ryu-Takayanagi proposal [33], according to which, there is a deep connection between minimal surfaces in AdS and entanglement entropy. Specifically for the case of AdS3/CFT2 it implies that the length of a bulk

geodesic is equal to the entanglement entropy of the subregion in the CFT, defined by its endpoints. This is illustrated in 2.6, where, for simplicity, we consider a geodesic lying on a timeslice of AdS3.

The latter is represented by a circle, given that there exists a conformal mapping between the complex plane, where it is natural to define a 2d CFT, and the cylinder.

It is therefore evident that in this holographic setting, there is a much richer structure than what one would naively expect, since the Ryu-Takayanagi proposal allows us to relate geometric notions of kinematic space to information theoretical quantities, such as the entanglement entropy. In fact, we shall see later on that the natural volume form defined for the kinematic space associated with

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Figure 2.6: A geodesic on a timeslice of AdS3 with endpoints u, v. The CFT subregion is defined

by the arc uv and the bulk dual is the area enclosed by the geodesic and the boundary of the circle.

a timeslice is given by:

ω = ∂

2_S ent

∂u∂v (2.32)

Let us now follow the process described in [21] in order to establish the basic features of kinematic space. We will first try to understand a simpler version of this problem, by considering the Euclidean plane, instead of AdS. Of course, we know that there is not a holographic prescription in the case of the Euclidean plane and as such it is not of much use for our purposes, but it is a good starting point to gain some intuition.

Let us start with some convex closed curve in Euclidean space; we remind that a set of points K on the Euclidean plane is called convex if for each pair of points A ∈ K,B ∈ K, it is true that AB ∈ K, where AB is the line segment joining A and B [34]. In geometric terms this means that if a curve is convex there is no line segment with endpoints in the interior of the curve that intersects the boundary; this is illustrated pictorially in figure 2.7. Suppose we want to measure the circumference of such a convex curve; there are many different ways of doing so, but let us try to think of the simplest way possible. Since, this is the familiar to everyone Euclidean plane we can draw this curve on a piece of paper and ask a 6-year old child to measure the circumference of the shape for us. Chances are that they are going to grab a ruler, stick it to the side of the curve and slide it across its boundary until they get a number. Silly as that may seem, if we take the main idea and perform a few substantial upgrades to our ruler, this can turn into a very useful mathematical tool. Namely, we can substitute the ruler with the tangent line at some point of the curve, which we will move around such that it remains tangent to the curve until we get back to the starting point. In other words, we will consider the support function of the curve we are interested in. Any straight line on the plane can be written as:

x cos φ + y sin φ − p = 0 (2.33)

where φ is the angle of the normal with the x-axis, p is the distance from the origin and x,y are the coordinates of the contact point with the curve. One should note, that the support function is a well defined object only for convex curves and for that reason it is imperative to work with such objects for the moment. Thus, we can obtain the equations for the coordinates of the points lying

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Figure 2.7: a) The cardioid is an example of a non-convex closed curve since there is a line segment AB that does not satisfy the condition specified in the main text. b) The circle is an example of a convex closed curve since all line segments AB belong in the interior of the curve.

on the curve in a parametric form by considering a family of lines: x = p(φ) cos φ −dp(φ)

dφ sin φ (2.34)

y = p(φ) sin φ − dp(φ)

dφ cos φ (2.35)

where now we consider the distance from the origin p as a function of the angle φ; this is the support function of our curve. One can easily show that for the simple example of a circle of radius r, we just need to consider a support function p(φ) = r. One can then show that the circumference of the curve is given by the integral of the support function with respect to the angle (we will not prove this claim here, but one can find more information in [34]):

L = Z 2π

0

p(φ)dφ (2.36)

which for the case of a circle becomes simply: L =

Z 2π

0

rdφ = 2πr (2.37)

An alternative way to think about this method is to imagine that the curve we are interested in is rolling without slipping on a line (this is particularly intuitive in the case of the circle). Therefore, a measurement of the circumference can be performed by simply allowing the curve to perform one full rotation and measuring the distance it travelled. However, this language is not well suited for generalizations and thus, it is not of particular use for our purposes.

Therefore, inevitably, we have to switch to the more abstract language of Lebesgue measures; instead of lengths or areas we consider the notion of a measure of a set (for a pedagogical review on the

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topic we refer the reader to the lecture notes [35], which also contain some enlightening points on integration generally). In Euclidean space, which we have been considering so far, the intuitive way to think about measures of sets are simply as their volumes. For example in R the set [0, 1] has simply a measure of 1, which coincides with its length. Thus, we can define the measure of a set of lines X as:

m(X) = Z

X

f (p, φ) dp ∧ dφ (2.38)

where p and φ were defined in (2.33). We can immediately recognize the wedge product dp ∧ dφ as a differential form, which should be familiar from the definition of integrals in differential geometry. Therefore, the density of sets of lines on the Euclidean plane is given by:

dG = dp ∧ dφ (2.39)

it is thus intuitively clear why one would integrate over it to obtain the measure of a set of lines. However, the most important ingredient in (2.38) is the function f (p, φ) that holds the instructions for any particular set we are interested in, which is related, in this case, to some curve of interest on the plane. It should now be evident that one can manipulate this function to study cases that go beyond convex curves on the Euclidean plane; we will not elabarote further on this topic, since it goes beyond the scope of this thesis; however, one can find a detailed review in [34]. Furthermore, this function is subject to a condition of immediate interest to our purpose. Namely, f (p, φ) should be chosen such that the measure is invariant under the group of motions on the plane. This justifies the definition of the space generated by φ ∈ [0, 2π] and p ∈ [−∞, ∞] as the kinematic space [21]. Essentially, this provides yet one more alternative description of the same problem. Namely, Crofton’s formula translates the length of a curve into a volume in the space of straight lines.

Given what we have discussed so far, there should exist generalizations that allow us to consider a number of different cases where the curves of interest are not closed or convex (in this case the support function produces lines that intersect the boundary of the curve; it is easy to convince one’s self that the number of intersection points has to be even as is already hinted in figure 2.7), or the lines are simply intersecting the curve and are not necessarily solutions of the support function. This enhancement is encoded in Crofton’s formula (it should be evident that this formula is produced by considering the aforementioned measures), which we quote without proof [21]:

L = 1 4 Z 2π 0 dφ Z ∞ −∞ dp n(φ, p) (2.40)

where n, is the number of intersection points between the line and the curve. In a similar manner, one can extract results about the surface area of curves and generalized volumes.

To review, we have contemplated the problem of computing the lengths of curves on the Euclidean plane and we have concluded that one way to do so is through the notion of kinematic space. Namely, we consider the space of straight lines, whose volume form is the appropriate differential form to integrate over, in order to compute the quantities we are interested in. The fact that the volume form (otherwise referred to as Crofton’s form) is invariant under the group of motions on the plane is crucial, as it holds the key to generalizing this construction to cases other than the Euclidean plane. More specifically, we will be interested in the hyperbolic plane, where the natural objects to consider are the corresponding geodesics rather than straight lines. We can thus define kinematic space, in a similar manner, as the space of geodesics; yet, it is not trivial to find the corresponding Crofton form. As we have already mentioned, the latter has to be invariant under the group of motions of the hyperbolic plane; we now make this more precise be stating that the group of motions coincides with the isometry group of the hyperbolic plane, which is the well studied

(28)

SO(1, 2). The non-triviality of this endeavor stems from the fact that there is no standard procedure for determining the Crofton form; rather, one can make an educated guess and then check whether it satisfies the aforementioned condition. One choice that conforms to this, is the following:

ω = 1

2 sin2(v−u₂ )du ∧ dv (2.41)

where u, v label the endpoints of the geodesic (these two parameters uniquely define the geodesic), as shown in figure 2.6. In [21] it is shown that ω is indeed invariant under the isometries of the hyperbolic plane. For future reference it is useful to define the opening and midpoint angles, in terms of the endpoints of the geodesic as:

α = v − u

2 (2.42)

θ0 =

v + u

2 (2.43)

In terms of those, the Crofton form changes into: ω(θ, α) = 1

2 sin2αdα ∧ dθ (2.44)

Given the above, we can now return to the starting point of this section, concerning the kinematic space as a tool to explore the AdS/CFT correspondence. We are going to start with an AdS3

spacetime with metric:

ds2= L2(− cosh2ρdt2+ dρ2+ sinh2ρdφ2) (2.45) It is easy to check that by choosing a timeslice t = constant we are left with the metric for the hyperbolic plane ds2 = dρ2+ sinh2ρdφ2. Therefore, we can apply what we have formulated so far to study the physics of an AdS3 timeslice (it is not in any way mandatory to constrain ourselves

to a timeslice, but we do so merely for the sake of simplicity). Going back to the Ryu-Takayanagi proposal, we know that the entanglement entropy is given by [21,33]:

Sent(u, v) =

c 3log

sin (v−u₂ )

µ (2.46)

for a subregion in the CFT defined by the endpoints of a geodesic as shown in figure 2.6. One can readily check that indeed

ω = ∂

2_S ent

∂u∂v (2.47)

as foreshadowed at the start of the section. Furthermore, one can also argue that there is an associated causal structure which can be encoded in a metric of the form:

ds2 = ∂

2_S ent

∂u∂v dudv =

1

2 sin2(v−u₂ )dudv (2.48) Which is the metric describing the geometry of a dS2 spacetime, which relates to the claim the

the space of geodesics is a Lorentzian spacetime. We shall postpone a more rigorous proof of this claim until later chapters, but it is useful to keep this information in mind, as it is going to be a recurring topic throughout the rest of this thesis. Moreover, it is useful for visualization purposes as demonstrated in figure 2.8.

With this, one can go a step further and compile a more detailed dictionary between the quantities of kinematic space and those of information theory. We are not going to repeat this analysis here, but

Berry picking in the holographic forest

MSc Physics

M

T

Berry picking in the holographic forest

Dimitrios Patramanis

October 2019

Contents

Chapter 1

Introduction

1.1

Introduction to holography

1.2

Motivation

1.3

Summary of results

1.4

Outline

Chapter 2

The toolkit

2.1

Review of fibre bundles

2.2

Generalities on the Berry phase

2.3

Kinematic space