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Natalie Gibson
Supervisor: Niko Jokela Examiners: Mark Hindmarsh and Niko Jokela Master’s thesis UNIVERSITY OF HELSINKI
Department of Physics Helsinki, May 2023 HELSINGIN YLIOPISTO — HELSINGFORS UNIVERSITET — UNIVERSITY OF HELSINKI
Tiedekunta — Fakultet — Faculty Laitos — Institution — Department Faculty of Science Department of Physics
Tekijä — Författare — Author Natalie Gibson Työn nimi — Arbetets titel — Title Exploration of the de Sitter/Conformal Field Theory Correspondence
Oppiaine — Läroämne — Subject Master’s Programme in Particle Physics and Astrophysical Sciences
Työn laji — Arbetets art — Level Aika — Datum — Month and year Sivumäärä — Sidoantal — Number of pages Master’s thesis May 2023 61
Tiivistelmä — Referat — Abstract The search for a profound connection between gravity and quantum mechanics has been a longstanding goal in theoretical physics. One such connection is known as the holographic principle, which suggests that the dynamics within a given region of spacetime can be fully described on its boundary surface. This concept led to the realization that string theory provides a lower-dimensional description that encapsulates essential aspects of spacetime. While the “AdS/CFT correspondence” exemplifies the success of this holographic theory, it was discovered soon after that the Universe has a positive cosmological constant, Λ. This immediately sparked interest in a potential correspondence centered around de Sitter (dS) space, which is also characterized by a positive cosmological constant.
This thesis comprehensively explores the de Sitter/Conformal Field Theory (dS/CFT) correspondence from various perspectives, along with the unique challenges posed by the distinct nature of dS space. The original dS/CFT duality proposes that a two-dimensional Conformal Field Theory resides on the boundary of three-dimensional asymptotic dS space. However, the definition and interpretation of physical observables within the dS/CFT framework remain open questions. Therefore, the discussions in this thesis not only cover the original dS/CFT conjecture, but also extend into more recent advancements in the field. These advancements include a higher-spin dS/CFT duality, the relationship between string theory and dS space, and the intriguing proposal of an “elliptical” dS space.
While the dS/CFT correspondence is still far from being well-defined, there have been extensive efforts devoted to shedding light on its intricate framework and exploring its potential applications. As the Universe may be evolving towards an approximately de Sitter phase, understanding the dS/CFT correspondence offers a unique opportunity for gaining fresh insights into the link between gravity and quantum field theory.
Avainsanat — Nyckelord — Keywords de Sitter space, holography, dS/CFT correspondence, higher-spin dS/CFT
Säilytyspaikka — Förvaringsställe — Where deposited Muita tietoja — Övriga uppgifter — Additional information
1 Introduction 4
2 de Sitter and Anti-de Sitter spacetimes 6
2.1 General relativity, briefly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 de Sitter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 de Sitter Penrose diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Anti-de Sitter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Constructing a holographic theory 19
3.1 Holographic principles of dS space . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Quantum field theory
4.1 Free fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Conformal field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Conformal fields in d ≥ 3 dimensions . . . . . . . . . . . . . . . . . . . . 4.2.2 Conformal fields in d = 2 dimensions . . . . . . . . . . . . . . . . . . . . 4.2.3 Operators and the correlation function . . . . . . . . . . . . . . . . . . .
24 25 28 29 30 32
5 The dS/CFT correspondence
5.1 Gravity in dS space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Understanding the Hilbert space . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Boundary conditions for dS space . . . . . . . . . . . . . . . . . . . . . . 5.2 Green’s functions in dS space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Bridging the gap between dS and CFT . . . . . . . . . . . . . . . . . . . . . . .
34 34 34 36 38 39
6 Further advancements in dS/CFT
6.1 Higher-spin gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Bulk correlators: from EAdS to dS . . . . . . . . . . . . . . . . . . . . . 6.1.2 Boundary correlators: CFT 3 . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Is there any room for string theory? . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 An unconventional approach to string theory in dS space . . . . . . . . . 6.3 Advancements towards a working model for dS 4 /CFT 3 . . . . . . . . . . . . . .
42 43 44 45 46 47 49 50
7 Conclusion 52
A Conformal algebra 55
Acknowledgements 55 3
The theory of general relativity, published by Einstein in 1915, became an important pillar in the foundation of fundamental physical theories. Its description of how spacetime behaves upon interacting with matter has stood the tests of many scientific experiments that took place thereafter. However, in the case of black holes, the curvature of spacetime is so extreme that it diverges and forms a singularity. Thus, the theory of gravity falls apart. Another pillar of fundamental physics is quantum mechanics, which was fully developed in the 1920’s. The idea here was to describe the physical properties of sub-atomic particles. In the 1940’s, quantum field theory combined special relativity and quantum mechanics to explore matter and its nongravitational interactions. Here, particles are roughly described as excited states of a quantum field, and are even more fundamental than particles themselves. In an attempt to explore the intersection between gravity and quantum physics, and understand the physics happening at Plank-scale spacetimes, a theory of quantum gravity was suggested. The idea of this theory is to quantize the gravitational field itself. As of today, there is still no firmly established theory on how to accomplish this, but there have emerged a few solid interpretations that are widely acknowledged. One approach came about in the 1970’s and is known interchangeably as string, superstring, or M-theory, where point-particles in quantum field theory are replaced by one-dimensional “strings” [1].
In 1972, Jacob Bekenstein used the 2nd Law of Thermodynamics to postulate that black holes should have entropy [2]. Steven Hawking developed this idea further by proposing a relationship between temperature, energy, and entropy and by showing that black holes emit what is known as “thermal Hawking radiation” [3]. The formula for the entropy of a black hole is called the Bekenstein-Hawking formula, which brought about a fascinating discovery. Bekenstein realized that the entropy of the black hole must not be smaller than the original entropy, otherwise the 2 nd Law of Thermodynamics would be broken [4]. Therefore, he concluded that the maximal entropy of a black hole is proportional to its area rather than its volume. Gerard ’t Hooft extended this discovery to spacetimes, suggesting that all phenomena within a given spatial region can be described by the set degrees of freedom which reside on the bounding surface of that region. In other words, if the boundary of a surface is taken to be a flat plane, then information in a three-dimensional space can be projected onto a two-dimensional space without losing any of the information [5]. ’t Hooft pointed out that this description was analogous to that of a hologram [6].
In 1994, Leonard Susskind expanded on this idea, arguing that the correct interpretation of a quantum theory of gravity is a holographic one, and that a non-perturbative string theory in the light front gauge 1 is an exact realization of this same holographic interpretation. Therefore, what is now known as the holographic principle, first suggested by ’t Hooft, can also be interpreted through string theory [4]. Before the holographic principle was even proposed, Charles Thorn noticed that string theory realizes a lower-dimensional description in which the gravitational dynamics of a full spacetime emerge holographically [7, 8]. Following these findings, in 1998, Juan Maldacena put forward the idea that a string theory on the product of a d + 1-dimensional Anti-de Sitter space with a compact manifold, which is a universe with a negative cosmological constant, can describe a non-gravitating conformal field theory (CFT) in d dimensions [9, 10]. The AdS/CFT duality has become a landmark case of the holographic principle, highlighting the desire that many have to uncover a theory of quantum gravity.
1 Perpendicular rays that hit the screen during one moment in time form a three-dimensional “light front”.
4 Very soon after the publishing of [9], Maldacena and Andrew Strominger published a paper on the statistical entropy of de Sitter space, a spacetime where the cosmological constant, Λ, is positive [11]. Although this paper did not tackle the question of whether a CFT in d dimensions could also be described by a string theory in d + 1-dimensional de Sitter space, it came at an equally pivotal time. This is because, only a few months later, astronomical observations revealed that the Universe was accelerating rather than decelerating, which meant that the sign of Λ was positive [12]. While the idea of the “Anti-de Sitter/Conformal Field Theory” (henceforth shortened to AdS/CFT) continued to garner interest, some began to shift their focus towards a hypothetical “de Sitter/Conformal Field Theory” (dS/CFT). After all, if the real Universe does not have that much in common with AdS space, then it would be beneficial to have a holographic theory that can describe it as accurately as possible.
This thesis aims to explore the developments made towards a viable dS/CFT correspondence, while also examining the challenges encountered by numerous authors in this pursuit. By the end, it will become clearer where the dS/CFT correspondence currently stands within the scientific community, and some ideas for future progress will also be investigated. Chapter 2 will discuss de Sitter and Anti-de Sitter spacetimes, including their respective metrics and other pertinent concepts in general relativity. The goal will then be to analyze the causal structure of these spacetime solutions through their corresponding Penrose diagrams. This will offer valuable insights into the impact of the sign of Λ and distinguish the dS/CFT correspondence from the AdS/CFT correspondence. Building on this foundation, Chapter 3 serves as an introduction to holography and the necessary conditions that must be met in order to propose a dual holographic theory. Key properties of de Sitter space will be looked into, including its entropy, temperature, and central charge. The dimensionality of the Hilbert space is an essential concept that is briefly discussed here, but it receives further comprehensive examination in Chapter 6. It is important to cover these topics chronologically, as they were the first steps taken in the pursuit of an operational dS/CFT correspondence.
Chapter 4 presents the intriguing topic of Quantum Field Theory and its relevance to the dS/CFT correspondence. The discussion begins by focusing on the significance of free field theories in QFT, before exploring their connection to the Feynman propagator and the subsequent derivation of the n-point correlation function. Field theories that involve interactions are excluded from the conversation due to the symmetries of dS space, which will be understood later. This discussion on QFTs can be found in Section 4.1. Then, Section 4.2 ventures into the realm of Conformal Field Theories–what they are, their symmetries, and the discernible differences between a CFT in d = 2 versus one in d ≥ 3 dimensions. Special attention is paid to the conformal algebra of a CFT, because it offers a way to find observables of a physical theory. Furthermore, studying the correlation functions of operators in the CFT gives valuable insight that sets the stage for a deeper understanding of the relationship between QFT and the dS/CFT correspondence, which unfolds in subsequent chapters.
Chapter 5 undergoes a comprehensive exploration of the intricate framework of the dS/CFT correspondence. Section 5.1 starts by examining the discourse surrounding the dimensionality of Hilbert space, a subject which requires significant attention. Then, the focus shifts to the examination of boundary conditions in three-dimensional dS space, which will play an essential role in understanding Strominger’s groundbreaking dS/CFT conjecture. Furthermore, Section 5.2 illuminates the pivotal role played by Green’s functions in comprehending the behavior 5
of correlation functions of quantum fields in dS space and its dual CFT. In short, they will serve as a bridge connecting the two realms of the dS/CFT correspondence. Finally, Section 5.3 largely follows Strominger’s proposal, highlighting the correspondence between spherical correlators in the CFT and gravity correlators in the bulk. Collectively, Chapter 5 will shed light on the important aspects of the dS/CFT correspondence and the intricacies that accompany it.
Chapter 6 opens with a discussion of Vasiliev’s higher-spin gravity theory. Specifically, Section 6.1 explores the correlation between higher-spin gravity accompanied by Neumann (Dirichlet) boundary conditions in Euclidean Anti-de Sitter space and a free (critical) interacting Sp(N ) CFT on the boundary—a relationship that extends through an analytical continuation into dS space. Section 6.2 then moves into the intriguing connection between string theory and dS space, where recent work has suggested the possibility of circumventing “no-go” theorems through “metastring” theory. Lastly, Section 6.3 concludes the chapter by discussing the proposal of an “elliptical” dS space. This offers a unique lens through which one might explore the physics within the causal patch of dS space. As a whole, Chapter 6 will capture the more modern approaches to the dS/CFT correspondence.
A seemingly basic question one might ask about the dS/CFT correspondence is, “If the difference between de Sitter space and Anti-de Sitter space is simply a change in sign of Λ, then why is it so much less understood than the AdS/CFT correspondence?” To fully appreciate the complexity of the situation, one should first understand the differences between asymptotically dS and AdS spacetimes. These differences play a big role in the resulting holographic theories, which will be made more clear in the following chapters. The main task for this chapter will be to explore Carter-Penrose diagrams (sometimes just referred to as “Penrose diagrams”), which were originally proposed by Roger Penrose in 1962 and employed by Brandon Carter in 1965. These diagrams provide a clever way to illustrate the conformal structure of a spacetime by making coordinate transformations to the metric [13, 14]. Mapping a metric with conformal coordinates means that the formally infinite regions of space are now expressed as finite regions. This works because light propagates along a metric as ds 2 = 0, so the conformal transformations can bring infinite regions to be expressed as finite regions, and any conformal factors are just dropped. Penrose diagrams provide a visual interpretation of this conformal compactification, and are discussed in Section 2.3.
Before diving in, it is essential to understand a few fundamental ideas of spacetime metrics in terms of general relativity. Describing gravity on a universal scale requires the use of a mathematical structure known as a “differentiable manifold”, where locally the spacetime will look flat, but globally there is some curvature due to gravity. In other words, a d-dimensional space will have some complicated topology on a global scale, but locally the space always looks like R d . Specifically, every point on a Lorentzian manifold, M , has a metric, g µν , that will smoothly connect each of the local regions together. An example of a manifold that will prove relevant to the conversation is the d-sphere, S d . It is defined as the locus of the set of points in R d+1 at some distance from the origin. As the name suggests, the dimension of the d6 sphere manifold is d, but it is describing the surface or boundary of a d+1-dimensional ball [15].
A spacetime metric is a symmetric, non-degenerate tensor of rank 2 (N ×N matrix) that has some important qualities worth noting. 2 Firstly, the metric gives some meaning to the terms “past” and “future.” It also determines the motion of particles, or “observers.” The metric will also be able to determine the causality of a region [15]. This is quite important, as no information can travel faster than the speed of light! The most generalized form of a metric is written as
where µ, ν = 0, 1, . . . , d. The last important bit of information to acknowledge is the notion of a “spherically symmetric” spacetime. This means the isometry group that defines it is isomorphic to the “special orthogonal” group SO(3) [14], which describes the rotational symmetries of an object. In other words, SO(3) leaves the three-dimensional length dx 2 -invariant. Therefore, a spherically symmetric metric is invariant around such rotations.
In general relativity, the curvature of the manifold can be fully described by the Riemann µ . This is just one of two ways to write the Riemann tensor. The definition curvature tensor, R
νλσ of the “first kind” has all 4 indices lowered and is defined in terms of the “second kind”,
(2.2)
It is found by taking the inner product of the metric with the second kind of Riemann tensor [16]. By assigning a tensor to each point on a Riemannian manifold, the full region will form a tensor field. Spacetime is considered to be flat if and only if the value of the Riemann tensor is equal to zero everywhere on the manifold [15].
The Ricci tensor, R µν , is the trace of the Riemann tensor, found by contracting the first and ρ third indices of R
µρν . Physically, it describes the change in the volume element of the spacetime due to gravitational curvature. Taking the trace of the Ricci tensor (contracting indices) gives the Ricci scalar, R = g µν R µν = R µ µ , which measures the curvature of a Riemannian manifold in the form of a real number. Using these equations, the Einstein tensor
(2.3)
can be found [14]. This can essentially be thought of as the trace-reversed Ricci tensor. Einstein’s significant realization in regards to the curvature of spacetime was that it must be related to the matter content on it. This led to the proposal of the energy-momentum (or stress-energy) tensor, T µν . Famously, Einstein’s field equation relates the energy-momentum tensor to a spacetime metric, given as
(2.4)
The importance of including this equation in this thesis is that it has solutions for black
2 Non-degenerate just means that the determinant of the metric, g = |g
µν | ̸ = 0, and it is sometimes referred to as a pseudo-Riemannian manifold.
7 holes, as well as static and maximally symmetric metrics, which will give information about the sign of Λ and the scale factor. In a spacetime described as a vacuum where the right-hand-side of the equation is zero, T µν vanishes, leaving
(2.5)
Then, the spacetimes are solutions to this form of the Einstein equation, G µν = −Λg µν [17]. It is now time to see how the solutions for when Λ > 0 (de Sitter) and Λ < 0 (Anti-de Sitter) behave! Relating curvature terms G µν , R µν , R, and Λ via Einstein’s field equation allows one to determine the curvature of the spacetime, which will be calculated for de Sitter spacetime below.
In 1917, William de Sitter found the solution to Einstein’s equation for a vacuum with a positive cosmological constant, Λ [14]. Named after him, the de Sitter metric (dS) is maximally symmetric with constant positive curvature and topology R × S d−1 . This means that it has coordinates that cover the entire manifold. The term “maximally symmetric” involves concepts of homogeneity and isotropy, which can be thought of as invariant under translations and rotations, respectively [15]. Similarly to a “spherically symmetric” metric, the symmetry group
space can be thought about in terms of the quotient between Lie groups [18]:
(2.6)
Choosing coordinates in dS d such that g µν is dimensionless, the Riemann tensor of the first kind can be calculated from the metric by using the equation [18]:
(2.7)
A popular choice for defining the Ricci tensor R µν and Ricci scalar R is to include a normalization factor that will hold for every coordinate system. Following the work of [18], R µν = d−1 g l 2 µν
Plugging in these new definitions for the Einstein tensor in Eq. (2.3), the result is
(2.8)
Therefore, in terms of the scale length factor l, the cosmological constant is
(2.9)
2 which
qis Λ = 3/l when considering d = 4 dimensions, for example. Then, l is related to Λ by l = 3/Λ, which provides a good description of the positive curvature in this space.
3 Here, “pure” is referring to the time at which the scale factor will grow exponentially, and neither matter nor radiation will be contributing factors.
8 Considering spatial and timelike coordinates given as X 0 , X 1 , X 2 , . . . , X d , these form a d+1- dimensional hyperboloid, defined by
In the context of dS spacetime, the spatial and timelike coordinates of dS d can be embedded onto a d + 1-dimensional Minkowski (flat) space, where the term on the right-hand-side of the equation is the length scale given by l. For dS space in d dimensions, its spacetime solution is a subset of Minkowski spacetime,
where the dS radius is given by l. A huge connection to make here is that this is equivalent to the description for a hyperboloid in d + 1 dimensions! The first convenient parameterization to highlight is one that transforms the metric into global coordinates: 4
X 1 = l cosh ( tl ) cos θ 1
where the coordinate t is defined from −∞ < t < ∞, the spatial coordinates θ 1 , θ 2 , and θ d−2 range from [0, π], and θ d−1 ranges from [0, 2π]. The resulting global dS d metric is [15, 16]
with topology dS d ≈ R×S d−1 . The term dΩ 2 d−1 contains the foliation of dS space with the d−1- sphere. The spatial surfaces have positive curvature and are maximally symmetric. Figure 1 illustrates dS d embedded in the d + 1-dimensional hyperboloid.
Upon observation, one can see that dS d is a d − 1-sphere which contracts to a finite radius, l cosh ( tl ), at t = 0 and then expands again as t → ∞. The coordinates cover the entire sphere [19,20]. While this is a useful way to visualize de Sitter spacetime in global coordinates, this is not the desired coordinate transformation that one should use to draw Penrose diagrams. One reason for this is that null lines (or lightlike lines) do not travel at ±45-degree angles, which is an important property of these diagrams. Additionally, the coordinates must be sure to cover the entire spacetime. So, the next place to look at is the flat slicing of de Sitter spacetime, also known as planar (inflationary) coordinates. This particular coordinate transformation offers a lot of information about dS space in relation to cosmology.
The Friedmann-Robertson-Walker (FRW) universe is a spatially homogeneous model that evolves with time and has hypersurfaces at constant cosmic time. In this universe, matter is
4 Global coordinates refer to coordinates that cover the entire hyperboloid, up to the poles.
9
assumed to have a preferred rest frame at fixed comoving coordinates but is growing apart at a proper distance. The growth (scale) factor is exponential, a(t) = e (t/l) , which will become relevant shortly. So, the FRW property of spatial homogeneity means that there is a preferred slicing of the spacetime, which is the same idea behind planar coordinate transformations. The future inflationary coordinates form the dS d planar metric. They are given as
where the parameter r = (X i /l), and i = 1, 2, . . . , d − 1 correspond to the flat line element.
In this case, the global coordinates have been mapped to the future inflationary coordinates by (t, θ) ↔ ( t̃, r), where t̃ is a different time component than global time, t. It is important to understand that, since −X 0 + X d = −le (− t̃/l) ≤ 0, the inflationary coordinates cover only the upper-left half of dS d , as seen in Figure 2. Additionally, r 2 = X i 2 , where dX i 2 is a flat metric on R d−1 [17]. Then, the dS spacetime with flat slicing is expressed as
This metric expresses the lower, flat half of dS d space. This discovery reveals that timelike worldlines will diverge from some point in the infinite past! To quickly summarize some of the points made here, the spacetime in this metric is described as flat with an exponential growth factor of e (2 t̃/l) . This means that half of dS space looks like a FRW universe eternal to the past 10
q and future, with a constant expansion rate given by the Hubble parameter, H = Λ/3, where H ≡ 1/l. In other words, the growth factor can also be expressed as e (2H t̃) . Once the Λ term dominates the Universe, the spacetime will become dS.
In both the global dS space and flat slicing dS space, one may recognize that the metrics still retain a time-dependent coordinate. Furthermore, dS space has observer and event horizons, both of which are quite strange to think about conceptually. By switching to a set of coordinates which describe the static metric, the time coordinate can be ignored, and horizons can be studied in more depth. This dS metric describes a static (another word for “causal”) patch shown by Figure 3 [19], where it is possible for an observer to both send and receive signals. The X 0 and X 1 components are inspired by Rindler coordinates, which aim to boost coordinates embedded in the d + 1-dimensional hyperbola [21]. The components of this static metric are as follows:
where i = 2, . . . , d, and z̄ parameterizes S d−2 . The radial distance r ranges from [0, l]. Here, r = 0 is the South Pole, and the observer’s horizon is defined by the singularity, r = l [17]. 5 The static dS metric is found to be
This is at the equator of the t = 0 slice seen in global coordinates.
11 (2.17)
It is important to note that this metric does not cover the entire region of dS space, but only a patch. The Penrose diagram of Figure 3 is a square, where the rightmost triangle represents the static region, and the other three triangular regions are covered by other (static and independent) coordinate systems. More will be said about this Penrose diagram shortly, in Section 2.3. First, there should be one more coordinate transformation mentioned, but not without a good cause. Such a coordinate system is known as conformal dS space. This is an incredibly important metric representation to look at, because it covers the entire dS d space, is essentially flat (aside from one conformal factor), and allows one to study dS infinities.
The conformal embedding of dS space in a static universe found by using the global metric from Eq. (2.13) and defining a new time coordinate, T :
The conformal metric in dS d is:
where −π/2 < T < π/2 and 0 < χ < π. So dS space is conformal to the Einstein static universe, which appears in the metric as (−dT 2 + dΩ 2 d−1 ) [22]. For visual aid, the conformal embedding of a d-dimensional dS space onto static Einstein space is shown in Figure 4 and Figure 5, where χ is the conformal boundary, and I ± show the boundaries at past and future null infinities. The surface in dS d again has the topology S d−1 .
12
The coordinate transformations emphasized in the last section offer valuable insights now that they have been familiarized. This section gives what is promised, showing two Penrose diagrams for de Sitter space that will really pin down what it would look like for a traveling observer to be stuck in such a universe. These diagrams will both acknowledge past and future null 13 infinites, which were introduced earlier as I ± [14, 20]. Continuing from the planar dS d metric in Eq. (2.15), and referring back to Figure 2, the Penrose diagram of an observer on the South Pole is given by Figure 6. One can see that a timelike observer is trapped on a flat region of space in their past observer horizon, because future inflationary coordinates do not cover the entire dS region. So in this Penrose diagram, it is not possible to approach the boundary at I + . Then, Figure 7 offers a different interpretation of the conformal dS Penrose diagram by including the static patches given by the FRW and static metrics.
There are several noteworthy conclusions that can be made from looking at Figures 6 and 7. First, the observer horizon is defined as the boundary of a region containing particles that have already been seen (or are yet to be seen), as one approaches past (or future) infinity. Additionally, the event horizon, which describes the boundary on which the observer can no longer send or receive signals, is set by future and past null cones, and can be further split into either future or past observer horizons. An observer who dares to cross their future event horizon will be redshifted 6 infinitely. This is interpreted as approaching I + , seen in Figure 7 [15]. Furthermore, a second observer would be able to watch the first observer be infinitely redshifted in their horizon, too! In other words, in de Sitter spacetime, spatial infinity is irrelevant to any observer, as they are all trapped inside their own horizons of finite area, which will increase in size as time increases. Causal signals will take an infinite amount of time to travel a finite distance, which leaves most of the dS boundary inaccessible [24].
6 This means that the wavelength of light increases and its energy decreases as it moves away from an observer.
14
After a long and successful journey towards understanding de Sitter space, one can try to repeat these steps in Anti-de Sitter space in an attempt to compare the spacetimes to one another. After all, the AdS/CFT duality has much more success in the scientific community, but the reason for this has not been fully recognized yet. The next section will briefly highlight specific coordinate transformations of the standard AdS metric, revealing how the resulting Penrose diagram is different from those in this section.
The AdS metric was coined by William de Sitter in 1917 as a solution to Einstein’s equation of a vacuum, and used to be called “de Sitter spacetime of the second kind”. Like dS spacetime, this spacetime is maximally symmetric (the curvature is constant everywhere), has no matter in it, and has topology R × S d−1 . However, the value of the cosmological constant Λ is negative, which means that the curvature, is negative. Thus, the name “Anti-de Sitter” is used today to describe its “opposite” geometry to dS space. Before trying to embed a d-dimensional AdS spacetime, AdS d , onto a d + 1-dimensional flat space, the implications of negative curvature should first be addressed. The geometry of AdS space is hyperbolic, resembling what is often
curvature of AdS is equivalent to the radius of a circle for each corresponding section [18, 25].
15
Using the same coordinate notation as before, then AdS d embedded in R 2,d−1 has a metric written like:
AdS space is a d-dimensional subset in this space (a hypersurface), resembling the hyperboloid [15, 18]:
Note that there is a significant difference seen in both the metric and hyperboloid, which is the extra minus sign in front of the X 1 coordinate. AdS space is maximally symmetric and has exactly one timelike direction, but the embedding spacetime must have two timelike directions. This means the flat metric is not Minkowski space, and the hyperboloid can be interpreted as
that the topology is R × S d−1 , but more importantly, it shows that this metric does not require the use of R d+1 to explore coordinate transformations on the metric.
For the time being, it will be easier to switch to AdS 4 space, and the results will be more concrete. Following the notation of [18] (see more in [15, 16, 22]), choosing a time coordinate and three spatial angular coordinates (t ′ , ρ, θ, ϕ) to be
X 3 = l sinh ρ sin θ cos ϕ X 4 = l sinh ρ sin θ sin ϕ (2.21)
gives the static AdS metric:
2 ds 2 = − cosh 2 (ρ)dt ′ + dρ 2 + sinh 2 (ρ)(dθ 2 + sin 2 θdϕ 2 ). (2.22)
Here, dΩ 22 = dθ 2 + sin 2 θdϕ 2 gives a two-sphere at every point, ρ is confined to the range 0 ≤ ρ < ∞, and t ′ has no singularities for any value, i.e. −∞ < t ′ < ∞. 7 This metric covers the entire AdS spacetime, where t ′ = const on the surfaces. This transformation of the AdS metric will be the building blocks with which the Penrose diagram is later constructed, but it is not yet in an acceptable compactified form. Before that, however, it is worthwhile to quickly cover the hyperbolic foliation of the AdS metric.
While it is nice to have a static (causal) metric that covers the entire AdS space, the coordinate time, t ′ , runs at a different rate at different points on the hyperboloid, defined by the part of the metric in Eq. (2.22): dρ 2 + sinh 2 (ρ)(dθ 2 + sin 2 θdϕ 2 ) ≡ H 3 . By making a small change in coordinates, the AdS metric can be expressed in terms of a time-dependent 7 The coordinate t ′ corresponds to physical points, as there are no closed timelike curves in this AdS space.
16 hyperbolic foliation,
(2.23)
where the metric looks very similar to Eq. (2.13), the dS metric in global coordinates. 8 More specifically, this metric is a FRW metric with positive curvature. However, remember that the negative spacetime curvature in AdS d means that space is decelerating in AdS d−1 rather than accelerating, like in dS d−1 . The discovery of coordinate singularities in this metric implies that there is a Big Bang discovered at t = − 2l π and a Big Crunch at t = 2l π [22].
Now, turning the attention back to the metric which covers the full AdS space in Eq. (2.22), one last coordinate transformation must be made to construct the Penrose diagram. The problem lies with t ′ and ρ, whose values range from −∞ < t ′ < ∞ and 0 ≤ ρ < ∞. A conformal Penrose diagram must express finite regions only, so the new coordinate transformation will need to compactify all of the coordinates. Making a transformation to the radial coordinate, ρ ′ , then
so that now the conformal coordinate ρ ′ ranges as 0 < ρ ′ < π 2 , then the new conformal AdS metric can be written as [15, 18]:
where the part of the metric in brackets is the Einstein static universe [15, 18]. Although the radial coordinate has now been compactified, it is actually impossible to also compactify t ′ in a way that keeps light traveling at 45-degree angles. The fact that t ′ still ranges from −∞ < t ′ < ∞ will be reflected in the Penrose diagram. Additionally, one can see that the universal AdS space is now conformal to the region 0 < ρ ′ < π 2 on the Einstein static cylinder (half of the Einstein static space), so this form of the metric no longer covers the whole space [22]. Figure 8 illustrates the AdS Penrose diagram, where future and past infinities are timelike, represented as disjoint points, and there are singularities at ρ ′ = 0. The topology of this diagram is R × S 2 , where each point represents a two-sphere except for those at the spatial origin (left side of the diagram). One difference from the conformal dS Penrose diagram is that no Cauchy hypersurface exists in conformal AdS space, meaning that it is not globally hyperbolic [15]. Furthermore, its causal structure is such that observers are always allowed to both send and receive signals to one another, as well as send signals to and from spatial infinity [16].
After finally comparing dS and AdS Penrose diagrams and their conformal structures, one might start to address the question asked at the beginning of this chapter. Essentially it was, “What is the big deal with changing the sign of Λ?” 8 This hyperbolic space where t = const. is also known as Lobachevsky space, H 3 [26].
17
Well, in asymptotically dS space, 9 each observer O traveling on a worldline resides in their horizon only, which can be defined by some finite area. This makes the concept of spatial infinity meaningless when talking about observable physics, because any signal will take an infinite amount of time to travel a finite distance. The boundaries (asymptotes) of dS space are at past and future infinity, which is essentially a pair of pointlike times. All of these features arise due to the fact that Λ is positive, so the dS universe is accelerating. When crossing over from dS to AdS space, the space and time roles are essentially inverted! This is because Λ is now negative, meaning that the AdS universe is contracting. Then in asymptotically AdS space, signals take a finite amount of time to travel infinite distances, with the conformal boundary at spatial infinity. Thus, it is a physically meaningful place. In fact, observers are not trapped within their own horizons, but can get arbitrarily close to the boundary. So, the sign of Λ has major implications on the global structure and topology of a universe.
This leads to the second part of the question asked earlier,
“Why is the AdS/CFT correspondence so desirable?”
It turns out that the timelike nature of infinity in Figure 8 means that spatial infinity for
9 Asymptotically dS space is important to think about rather than just “pure” dS space, because it considers a cosmological model that is approximately dS on large scales, but may look different on small scales. This can be thought of as adding some boundary conditions, or matter and radiation, to the spacetime metric.
18 the AdS universe (Λ < 0) is well-defined, so descriptions of quantum-gravitational observables calculated by the AdS/CFT are well-understood and make sense physically. The AdS/CFT is a feature of string theory that tries to equate quantum gravity on an AdS background to a non-gravitational CFT defined on the boundary [15]. Although the AdS universe does not match the real Universe, it provides an excellent example of how one can fix the classical geometry at infinity to precisely measure quantum fluctuations of the bulk spacetime [24]. Unfortunately, trying to repeat this setup in the dS universe only brings about more questions than answers. If there are two boundaries at future and past infinities, then what does that even mean physically? How can someone find a well-defined observable if the spacelike boundaries are outside of every observer’s horizon? How does the CFT fit into this mess? Alas, Chapter 3 will start to look at the decades-long journey of many researchers as they try to untangle the knot of questions the dS/CFT correspondence brings. Quite a few of these attempts have been used as steppingstones for others thereafter, so there is still much to explore before one decides to give up on the matter.
Until now, only a vague connection between the holographic principle and string theory has been suggested. Bekenstein deduced that the maximal entropy of a black hole is proportional to its area rather than its volume. ’t Hooft expanded on this notion by saying that all phenomena inside a given spatial region can be described by the set degrees of freedom residing on the bounding surface of that region. In this context, the degrees of freedom represent the number of ways in which information can be stored. The name “holographic principle” comes from the idea that information from a three-dimensional space can be projected onto a two-dimensional space without losing any information, much like that of a hologram. Although the holographic principle is necessary for a desirable holographic theory, it is not a sufficient theory on its own [27]. Similarly, while string theory naturally unifies gravity and matter fields, it cannot fully explain the behavior of cosmological spacetimes and singularities without incorporating the holographic principle. A holographic theory must meet some conditions to satisfy the holographic principle, mainly that the laws of thermodynamics are not broken, and the principles of general relativity and quantum mechanics are followed.
There are a couple of popular ways to go about forming a holographic theory that connects the vast number of degrees of freedom of a local physical theory to the holographic principle. The first method is to find a gauge invariance which relates the local and holographic theories, and decides the number of degrees of freedom that are relevant to the holographic theory. The Bousso bound places a constraint on this number, because it sets a limit on the number of degrees of freedom that can be stored on a given surface. 10 Therefore, preserving locality is the main priority with this method. With that being said, it is difficult but necessary to figure out a way to eliminate most of the remaining degrees of freedom. The second method is to consider holographic data as the most important factor, and say that locality is simply emergent. Then, the goal is to understand how the holographic data can be translated into a classical spacetime
10 The entropy bound, first suggested by Bekenstein, sometimes fails in cosmological scenarios, such as the entropy of a star that is falling into a black hole. Bousso suggested a new bound [28, 29], also known as the “covariant entropy bound”, by introducing a light sheet. A light sheet is a region that is traced out by non-expanding light rays orthogonal to a surface, which means there is no particular direction that the light rays diverge (or converge). A light sheet accounts for all of the degrees of freedom on a surface, and forms a holographic screen upon being inverted.
19 with local quantum fields. The greatest difficulty with this approach is grasping how locality can become emergent. An explicit example of the second method of the holographic theory is the AdS/CFT correspondence, because it contains a hypersurface where holographic data is stored and can be evolved forward by a conformally-invariant field theory [27].
The unexpected success of the Bousso bound lends to the idea that the holographic principle helps to push dualities between seemingly different physical theories forward, although it has an important condition in that supersymmetry must always hold. This discounts many spacetimes, but there are still some that can both be described by string theory and have geometry that is approximately classical. 11 Then, the question is if the number of degrees of freedom from the string theory description are set by the surface areas [27]. Susskind [4] revealed that the entropy bound of one bit per Planck area is violated with perturbative string theory, but luckily, there are some types of spacetimes that have been shown to work with non-perturbative string theories. So, authors searching for a working (A)dS/CFT correspondence hope to find a situation where data on the boundary of the spacetime describes all physics of its interior. This will be achieved by a dual CFT that “lives” on the holographic boundary (or screen) of the spacetime and holds the number of degrees of freedom that would be contained in a boundary area.
The holographic screen in any universe is found by slicing the spacetime into light cones, where the total entropy of each light cone is then stored on the surface which embeds it [27]. These surfaces together form a hypersurface on the boundary. As long as the number of degrees of freedom does not exceed that of the boundary area in Planck units, then it can be said that the holographic bound manifests itself in the dual CFT. In a general sense, quantum field theories can be derived by taking different limits of different string theories. It is also important to point out that the proper area on the conformal boundary of some spacetimes, along with the number of degrees of freedom, is divergent. Taking the limit here will give the holographic boundary for the entire light cone. It was then proposed by [31] that the bulk spacetime should be truncated, which is referred to as the “infrared (IR) cutoff”. This means that the spacetime only needs to be considered up to a certain distance scale, after which the description breaks down. Fascinatingly, it was also realized that this IR cutoff of the bulk spacetime influences the CFT. Specifically with the AdS/CFT correspondence, the physical IR energy effects on the bulk translates to ultraviolet (UV) energy effects on the corresponding boundary! This stems from the fact that the IR scale in the bulk is inversely proportional to the UV scale on the boundary. Therefore, the IR cutoff in the bulk is equivalent to a UV cutoff on the boundary, which has important consequences for the dual CFT [31]. This concept is brought up again in Section 4.1, so it is worth introducing now.
Having introduced the main concepts that make up any dual holographic field theory, it is time to think about how this applies to the dS/CFT correspondence specifically. So, what topics does one need to know? Covering the holographic properties of dS spacetime, such as its entropy, temperature, and other thermodynamic properties, will undoubtedly be an important first step. Next, while Quantum Field Theory (QFT) is a broad subject, there are certainly aspects of it that will be necessary to talk about. Furthermore, topics known as “supersymmetry” and “supergravity” will arise when discussing string theory and Conformal Field Theory, because string theory itself is a supersymmetric theory and its low energy limit is supergravity. The 11 Null hypersurfaces and global boundaries may have special roles where classical geometry emerges, which is being explored in twistor quantum gravity [24, 30].
20
next section will cover ideas from the earliest papers regarding the dS/CFT, before it was even clear how to actually make the connection to a dual CFT. The topics presented are focused mainly on the relation between entropy, temperature, and the number of degrees of freedom for dS spacetime.
Andrew Strominger’s 2001 paper [23] is widely-regarded as the first attempt to establish a dS/CFT correspondence. Coming shortly after Maldacena’s famous paper on the AdS/CFT correspondence, Strominger’s proposal caught a lot of attention, too. Moreover, Maldacena and Strominger collaborated on some research in the hopes of promoting this emerging concept. At this time, there was no known way of connecting any string theories with de Sitter space, so Strominger’s paper focused on defining its entropy and thermodynamic properties. However, he explicitly commented on the fact that his research was incomplete, and ideally there should be a way to embed dS space as a solution of some string theory that can then be used to find a microscopic description [23].
Strominger’s attempt at uncovering a dS/CFT was to work parallel to earlier research done by Brown and Henneaux [32], who found an AdS/CFT duality by studying the asymmetries of AdS space and calculating the resulting central charge of the CFT. An area-entropy law was later derived from their work, and has since been studied quite heavily due to its universality. This semi-classical law, also known as the Bekenstein-Hawking formula, is often written as
(3.1)
where A is the area and G N is Newton’s constant. However, it is important to note that in the context of natural units, the Boltzmann constant, k B , the speed of light, c, and the reduced Planck constant, ℏ, have been set to unity. Thus, the complete expression of the formula is [2]
Written in this form, one can see that ideas from statistical mechanics, quantum mechanics, and gravity are combined to provide a remarkable understanding of the geometry of spacetime.
Bekenstein and Hawking’s formula was first used to relate the entropy of a black hole to the area of its event horizon, but it turns out that it also works for Rindler and dS spacetimes. In dS space, A is related to the horizon entropy, which is based on the experience of an observer who will never be able to see the entire space. Therefore, there is no clear physical interpretation of dS entropy. However, the entropy is related to the central charge in the dS/CFT correspondence by the Cardy formula. The central charge is an important coefficient in understanding the holographic duality between dS space and a CFT. In the case of dS 3 /CFT 2 , it is given by c = 2G 3l
N[23, 33]. So, it is clear that learning more about the thermodynamic properties of dS space is worthwhile.
How can one go about this? The answer is to look more closely at the Bekenstein-Hawking 21 entropy for the event horizon in dS space [27],
(3.2)
or one quarter of the horizon area, A = 4πl 2 . While the physical interpretation of this result may not be clear at first, examining the equation [20]
(3.3)
can provide some insight. Using the First Law of Thermodynamics, the change in entropy for Schwarzschild-de Sitter (SdS) space is related to the change in energy for dS space. Now, SdS space is actually a solution to Einstein’s field equations, and is used to study black holes in dS space. The geometry of the event horizon in SdS space is equal to the event horizon in dS space, so making this relation means that the temperature of the SdS black hole event horizon can be determined. This temperature is also the temperature of the event horizon in dS space, and is given by T = 12 πl [34].
Additionally, one can try to understand the entropy of everything beyond the event horizon in dS space. While in theory this entropy may be unobservable, General Relativity says that the energy on a surface is equal to the surface integral on the boundary of said surface. This means that the energy vanishes on the closed surface of dS space. So, the resulting entropy in dS space can be determined solely by considering the negative energy at the singularities at the North and South poles. According to [20], this implies that there is no positive conserved energy in all of dS space. Applying Eq. (3.3) with respect to the negative energy yields the resulting entropy of Eq. (3.2). Thus, the First Law of Thermodynamics and the SdS solution can be used to understand the entropy and temperature of dS space and its singularities.
Next, although it has already been discussed that the number of degrees of freedom in a local physical theory relate to the holographic principle, it is perhaps time to provide a more rigorous definition for degrees of freedom. In a quantum mechanical system, the number of degrees of freedom, N , is given by the dimension of the corresponding Hilbert space, H, which is used to describe possible states and evolutions of the system. 12 Specifically,
where N is the total number of states in the system. In this system, N is also equal to the number of bits of information necessary for a state to be fully specified [27,35]. A local quantum field theory on a background spacetime will have harmonic oscillators everywhere in that space, and each harmonic oscillator has N = ∞ degrees of freedom. Naively, one might think that the total number of degrees of freedom for the local field theory is also infinite. However, by including gravity it is possible for that number to be finite. If each point in space is instead defined by the length of one Planck volume, then the total number of harmonic oscillators can be described by V , and the total number of degrees of freedom in a local field theory is given by:
In his 2002 paper, Bousso remarked, 12 This is the notation used by Raphael Bousso in his paper covering the holographic principle [27].
22 “Complexity grows with volume.” The number of states, referred to as macrostates in thermodynamics, 13 can be realized at a microscopic level. In other words, if S is the entropy of a thermodynamic system, then N = e S describes the independent number of quantum states congruent to the corresponding macrostates. This means that entropy in a statistical sense is a measurement of how little is known about a system’s microstate.
Furthermore, while Eq. (3.5) implies a finite number of degrees of freedom for a local quantum field theory in a spacetime, the maximum number of states in asymptotic dS space is finite, bounded by N = e S dS = e (3π/Λ) . This is known as the spherical entropy bound. This suggests that, if Hilbert space in dS space is truly finite, then Einstein’s theory (see more in Chapter 5.1) must be quantized for G N and Λ. 14 The Bousso, or covariant entropy bound, starts with a two-dimensional surface to find a one-dimensional null hypersurface. This onedimensional surface is a light sheet, mentioned at the beginning of this chapter. A light sheet is described by the light rays that come from the two-dimensional surface, and its entropy is again bounded by one quarter of the surface area. So, the total number of degrees of freedom on the light sheet,
is bounded by the spatial surface area with a light cone consisting of two separate light sheets. This bound on the number of degrees of freedom is generally only specific to some local surface. However, in dS space, N has an absolute entropy bound, where the global light sheet (the future event horizon) cannot exceed this value. If this were to happen, it would physically mean that the matter density in the space was outweighing that of Λ, which would make the entire spacetime collapse into the “Big Crunch” [15].
For a timelike geodesic observer on the holographic boundary of dS space, the number of degrees of freedom of the CFT living on the boundary is bounded by this absolute entropy bound. The topology of the surface is analogous to the BTZ black hole case, 15 which allows one to choose specific gauge fields and determine boundary conditions for the CFT. So, by understanding the properties of the boundary, one can “locate” and interpret the CFT which lives on the boundary of dS space. Although many of the authors studying the dS/CFT correspondence have paved their own paths, so to speak, understanding how the holographic principle applies to dS spacetime has undoubtedly played a big role in their decisions. Those who want to learn more about the thermodynamic properties of dS space can take a look at any of the following papers: [20, 33, 39–41].
The next chapter will introduce relevant concepts of Quantum Field Theory, starting with free QFTs, before moving on to Conformal Field Theory. As mentioned earlier, perturbation theory suggests the existence of a fundamental theory to describe certain properties of dS space.
13 In thermodynamics, examples of macrostates are temperature, volume, energy distribution, etc.
2 Expanding in powers of G
N Λ
(d−2)/d means that N ∼ e 1/G N . However, in perturbation theory, the Hilbert (d−2)/d space is not finite, as N diverges as G
N Λ approaches zero. This could imply that perturbation theory breaks down in the weak coupling limit. Then, there must be a fundamental theory that can find values for
15 The “BTZ black hole” case is a 2 + 1-dimensional black hole solution that comes from a 2 + 1-dimensional spacetime with a negative cosmological constant, Λ. The name comes from the initials of the authors who made the discovery [37, 38].
14 23 However, as one will discover in Section 5.1, approaching dS space classically leads to a Hilbert space that is finite-dimensional, which poses a significant challenge. This is because authors Juan Maldacena and Carlos Nuñez introduced a “no-go” theorem 16 which states that string theory in d = 10 or d = 11 dimensions cannot be compactified to form a classical dS space of any dimension N [42]. Then, the questions asked by Strominger, Witten, Susskind, and others, were “What values can N have? What are the ways that one can define quantum gravity in dS space?” It had already been established by the 1990’s that AdS space has quantum gravity with a holographically dual CFT living on the boundary. Given the proposal that dS space has a finitedimensional Hilbert space, one can then search for some principle or theorem that generates quantum gravity for when the cosmological constant is positive.
Just as string theory seeks to unify general relativity and quantum mechanics, Quantum Field Theory aims to connect special relativity and quantum mechanics. Special relativity is used to study the motion of an object, while quantum mechanics can use tools like the Schrödinger equation and the uncertainty principle to study certain aspects of a particle. However, quantum mechanics alone cannot account for particle creation and annihilation, even though it is a naturally-occurring phenomenon. By incorporating the principles of special relativity, one can describe the conversion of energy into matter, and vice versa. In nature, there are many “classical-like” fields, such as the electromagnetic field, that can be quantized into an infinite number of harmonic oscillators. This approach leads to the identification of creation and annihilation operators for the quantum excitations of the field (photons) [43]. Quantum Field Theory can generalize fields such as this one to the quantum realm, which sounds like a promising idea. Although it will be discussed later, it is still worth mentioning here that gravity is an example of a “QFT failure”, as it has evaded all attempts of quantization.
There is another great reason to discuss QFT. If a particle created in some initial state, |I⟩, propagates in its environment for some time, the final state of the particle, ⟨F |, can be described by the superposition of all possible outcomes. The probability amplitude of each outcome is determined by the path integral, which sums over every possible path it could have taken. Feynman observed that there are infinitely many choices [44]. The states are often carefully chosen such that one can find a quantity of interest while being low-cost in computational effort. Minimally excited, or “vacuum” states, 17 are easiest to work with and can be used to find the ground state energy of a system. In this previous case, creating a particle in some initial state and destroying it later is a way of exciting the vacuum with a “source” and a “sink” [43]. In QFT, the path integral representation can be used to describe the behavior of interacting particles. The classical action for the field, S, can be found by summing over every probability amplitude over the paths between two fixed points. The excitations in the field correspond to the particles, and the action is derived from the time evolution of the Lagrangian.
17 A vacuum state is defined as a state that has been annihilated by annihilation operators, or a |0⟩ = 0.
24 It is important to note that the path integral is just a tool for calculating probabilities of particle interactions, but it does not provide a complete picture of the quantum behavior of particles.
Before anyone had come up with a way to make string theory useful for de Sitter space, the main ideas from authors like Witten, Maldacena, and Strominger were to start by studying free scalar fields from QFT, then find a description where scalar correlators have a holographic dual with correlators on the S d−1 sphere [23, 39, 45]. 18 Thus, a good direction to take for this chapter is to start by introducing free (massive and massless) scalar field theories, which will later become quite relevant in discussions about the n-point correlation function as it pertains to QFT, a recognized dual of dS. The second part of the chapter will focus on Conformal Field Theory, an incredibly important branch of QFT. This discussion will give one the tools necessary to tackle the dS/CFT correspondence head-on.
A scalar field assigns a scalar quantity to every point in a space, and it is invariant under Lorentz transformations. This is perhaps the most simple type of free field theory. The path integral for a real relativistic scalar field theory in d + 1-dimensional spacetime looks like:
where J(x) is the source function, included to account for the disturbance of a system “turned on” at time t = 0. The action S in Eq. (4.1) is a functional integral,
which means that it is a function of every possible arrangement of the scalar field, ϕ. It can also be rewritten in terms of the Lagrange density,
This functional integral is usually very hard, or even impossible, to do, other than when [43,46]
(4.4)
The equation of motion can be found from this Lagrangian the same way as in classical mechanics:
The result is known as the Klein-Gordon (KG) equation, which is a second-order partial differential equation given by (∂ 2 − m 2 )ϕ = 0. The KG equation is linear, meaning that if ϕ 1 and ϕ 2 are solutions, then any linear combination of them are also solutions. Here, the source term J(x) = 0, and m is the mass of a relativistic particle. This is a typical quality of a free field theory with a mass term, such as the Klein-Gordon theory [46]. The general solution of the KG equation is a linear superposition of simple harmonic oscillators which all vibrate at
18 Later, Strominger [23] suggests that the dual CFT lives on one Euclidean sphere, even though global dS
dspace implies that there are two spheres.
25
different frequencies and amplitudes.
Now, consider a situation where there are two time-ordered scalar field operators, ϕ(x 1 ) and ϕ(x 2 ), and the path integral is described as:
The functional integral is evaluated over some boundary conditions at different times, ϕ(−T, x) and ϕ(T, x). Taking the limit as T → ∞ will allow this to be expressed as a time-ordered vacuum correlation function:
with ϕ̂ H being the Heisenberg field operator [46, 47]. The general form,
is known as a time-ordered Green’s function. Green’s functions are, by definition, equivalent to the n-point correlation function:
This last form comes in quite handy because the scalar fields, now denoted by ϕ(x i ), can now be expressed as a discrete Fourier series in k-space:
where V is the volume of the system. The term V 1 is a normalization factor, ensuring that the Fourier transform of the field is well-behaved in the limit where the volume of the system becomes infinitely large. Taking this limit,
After skipping some steps for the sake of being concise, the result is exactly the Feynman propagator [43, 47]:
There are other forms of the n-point function in addition to the Feynman propagator that are worth acknowledging: retarded, Hadamard, advanced, and more. These can be obtained by taking the real or imaginary part from Eq. (4.9) and considering a step function in time.
One might be thinking,
“That’s a nice result, but what does this really mean?” Well, D(x − y) and D(y − x) are the quantum transition amplitudes for some particle to 26
travel from x → y and y → x, respectively. In other words, the goal is to find the transition amplitudes by evolving an initial state, |0⟩, up to some later time, y. This can be done by mixing the vacuum with the particles and then evolving using the time-ordered evolution operator to see what happens [48]. Furthermore, the term iϵ is one that is added in by hand, such that m 2 → m 2 − iϵ. This is to avoid the integral diverging for ϕ and hitting a pole. The goal is to instead nudge the pole slightly so that it is further into the complex plane, or position the contour further into the complex plane. D(x) can then be evaluated by using contour integration.
qThe real and complex solutions to the KG equation are the two poles of the integrand, ± ⃗k 2 + m 2 − iϵ, where ±iϵ correspond to the upper and lower halves of the plane, respectively [43, 46]. By somewhat generalizing Eq. (4.9) into the form
for better understanding, then Figure 9 serves as a visual interpretation of what the contour and poles would look like.
In the context of the dS/CFT, taking the large N limit 19 of free scalar fields leads to the emergence of classical physics in bulk dS space (while remaining weakly coupled) by more easily summing over an infinite number of Feynman diagrams. Free fields are known to exactly 19 N can be thought of as the number of fields used, the number of degrees of freedom in the field theory, or the rank of the gauge symmetry.
27 preserve the symmetries of de Sitter space, while interacting fields introduce additional complexities that make it difficult to fully understand the behavior of fields in the bulk. This is still an open problem in understanding the dS/CFT, because the quantization of gravity in dS space would need to be well-defined before being able to understand the behavior of interacting fields on the bulk. The good news is, working with free fields allows one to use perturbation theory, which is a powerful tool for studying correlation functions.
Furthermore, the large N limit of the dS/CFT corresponds to IR divergence, 20 which expresses the classical limit of dS space. This divergence stems from the contribution of modes with low energies or long wavelengths. In dS space, the divergence is mild because the curvature of dS space is smaller than the Planck scale. This leads to the decoupling of higher-spin fields from low-energy physics. On the other hand, the AdS/CFT uses strongly coupled, interacting theories, such as N = 4 super-Yang-Mills theory and Type IIB string theory. In this case, the large N limit is a UV limit, where the contribution of modes with very high energies or short wavelengths dominates. This leads to a strong UV divergence, which is often dealt with using renormalization [9, 40, 42].
Conformal Field Theory (CFT) is a very broad topic that not only plays a significant role in describing almost all natural phenomena, but is also a crucial aspect of this thesis. Before the dual correspondence of gravity on d + 1-dimensional dS space with a d-dimensional CFT can be truly appreciated, it is important to clearly present the definition of a CFT. The next section will aim to capture the most relevant parts, including the differences between a CFT in d = 2 dimensions versus one in d ≥ 3 dimensions.
In QFT, the idea of scale invariance is an important one. A CFT is a QFT that is both scaleinvariant and angle-preserving under transformations. 21 This means that, while the distances of a spacetime can be altered at will, its local physics must always look unchanged. When considering the effects of scale and conformal invariance in a local field theory, it is important to think about the number of dimensions in which the theory is formulated. In particular, in two dimensions, scale invariance becomes conformal invariance. In four dimensions, the Yang-Mills theory is an example of an interacting theory that is conformally-invariant due to its supersymmetry, while the Dirac free field is a non-interacting theory that is also conformally-invariant. These examples of how scale invariance and conformal invariance are relevant to certain field theories in specific dimensions can help to provide a deeper understanding the correspondence between d + 1-dimensional dS space and a d-dimensional CFT.
Most of this section will focus CFTs in d ≥ 3 dimensions, with d = 2 being a special case.
20 To make it clear what this means, IR divergence is the divergence of an integral due to contributions that become significant at very low energy or long length scales. On the other hand, UV divergence is the divergence of an integral due to infinities appearing in the perturbation calculations, or by considering physical phenomena at extremely small length scales. Divergences like these have been discussed at length in the study of inflationary cosmology. While a workaround for UV divergence is through renormalization, a workaround for IR divergence is by imposing an “IR cutoff”, which is to ignore contributions below a certain energy value, or larger than a certain length scale [49].
21 The Poincaré group describes a set of transformations that leave a flat spacetime metric scale-invariant. These are a combination of Lorentz transformations and translations. The conformal group are a set of transformations which include Poincaré group, but with more symmetries.
28 This is because the concepts from the d ≥ 3 case are more general, so it is a natural starting point. Generally speaking, conformal transformation can be accomplished by transforming a metric such that
term is the scale factor, and µ, ν are the spacetime coordinates [50, 51]. Ω(x) can vary, which leads to different conformal transformations.
4.2.1 Conformal fields in d ≥ 3 dimensions
Since the conformal group in d ≥ 3 dimensions is a Lie group, 22 then the transformations can be linearized and represented by generators of infinitesimal translations [50,52]. By considering a flat metric, g µν , in d dimensions given by
then an infinitesimal coordinate transformation looks like:
where ϵ ≪ 1. This makes it so that
Now, for the transformation to be conformal to the first order, the term in parentheses must be (∂ µ ϵ ν + ∂ ν ϵ µ ) = f (x)g µν , where the right-hand-side is some arbitrary function. 23 Following the notation of [51], the transformation can be written as:
(4.13)
This expression can eventually be worked into the form
where □ ≡ ∂ µ ∂ µ . The significance of this result is that, for d ≥ 3 dimensions,
and for d = 2 dimensions, An ansatz can now be made regarding Eq. (4.15), which will be used to make three classifications of conformal infinitesimal transformations. The fact that (∂ µ ∂ ν )(∂ µ ϵ µ ) = 0 implies (∂ µ ϵ µ ) = a µ + b µ x µ , which is linear (at most) in x µ . Then, ϵ µ can be quadratic (at most) in x µ , which gives ϵ µ in the form:
The terms a µ , b µν , c µνρ are constants ≪ 1. Now, below are the three classifications for infinitesimal 22
23 A Lie group is a smooth (differentiable) manifold whose symmetries are continuous. Taking the trace with respect to g µν gives f (x) = d 2 ∂ µ ϵ µ .
29 transformations: [50, 51] (i) Translations are given by ϵ µ (x) = a µ , where ϵ µ (x) is a scalar quantity. The generator
that corresponds to this is the momentum operator, P µ = −i∂ µ .
(ii) Scale transformations and rotations correspond to the symmetric and antisymmetric parts
of b µν x ν . This time, ϵ µ (x) is linear, and b µν + b νµ = ( d 2 )b ρρ g µν . The scale transformation can be written as x µ → λx µ with the generator D = −ix µ ∂ µ , and the rotation transformation
the angular momentum operator.
(iii) Special conformal transformations come from the term ϵ µ (x) = c µνρ x ν x ρ , where ϵ µ (x) is
quadratic. It takes a little bit of work, but eventually one can find that the transformation takes the form
There is always a point at x ′µ = b 1 2 b µ that will lie at infinity, which is why infinities must be included on a compactified space (such as the Penrose diagrams from Chapter 2). The
Now, the generators, P µ , D, L µν , and K µ have all been explicitly written in terms of the conformal group. The point in doing so is to obtain the conformal algebra, which is expressed by taking the commutation relations of the generators with one another. To avoid discussing this topic in excessive length, the conformal algebra have been written down in Appendix A. One can additionally count number of generators that are found within this algebra by following the formula [51]:
It turns out that this result is precisely equal to the number of generators found in a SO(d + 2) symmetry group! Alternatively, on a d-dimensional Euclidean manifold, the conformal group
4.2.2 Conformal fields in d = 2 dimensions
In contrast to higher-dimensional CFTs, two-dimensional CFTs are described by a symmetry group with Virasoro generators, 24 which is an infinite-dimensional, complex Lie algebra [53]. In other words, the conformal algebra contains an infinite number of generators, rather than the number given by Eq. (4.19). An important property of CFTs with Virasoro algebra is that they have a central charge, c. The central charge is a number that measures the extent of conformal symmetry in a CFT, where the larger c is, the more symmetry the CFT possesses [51].
The first step in analyzing this conformal field is to apply a conformal transformation to the metric in Euclidean space, which is done in a similar fashion as before. Given the metric
24 (4.20)
Virasoro algebra is an extension of Witt algebra, a complex Lie algebra covering vectors that are holomorphic everywhere on a Riemann sphere except for the two poles.
30
then making a coordinate transformation z µ → w µ (z) will cause the metric, g µν , to change by
(4.21)
To satisfy the condition that this transformation is conformal, it must be that ∂ z,0 w 1 = ±∂ z,1 w 0 and ∂ z,0 w 0 = ∓∂ z,1 w 1 . The form of these conditions are known as holomorphic and antiholomorphic Cauchy-Riemann equations [51]. With this new insight, one can define a set of complex variables that will satisfy this condition:
resulting in the metric ds 2 = dzdz̄. The holomorphic functions z and z̄ are also said to be “left-moving” and “right-moving” functions [50]. Corresponding derivatives are defined as
(4.23)
The Cauchy-Riemann equations, ∂ z,0 w 1 = ±∂ z,1 w 0 and ∂ z,0 w 0 = ∓∂ z,1 w 1 , can now be expressed as
The solution is simply z → w(z), which corresponds to any/all holomorphic maps. So, if the solution for a d = 2 conformal group is the set of all holomorphic maps, then one can see why there are infinitely-many generators in this case [51].
The second step is to consider what infinitesimal transformations will look like. The functions f (z) = z + iϵ(z) and f ¯ (z̄) = z̄ + iϵ̄(z̄) can be used to describe general infinitesimal transformations that will transform the Euclidean metric from Eq. (4.20) by
It follows from Eq. (4.11) that the scale factor can be defined as Ω 2 (z) = |∂ z f | 2 . By Laurent expanding ϵ(z) around z = 0 to avoid singularities, then the general holomorphic map can be written as:
where ϵ n are constant infinitesimal transformation parameters, and the same goes for f ¯ (z̄) [50, 52]. By inverting this map using contour integration so that the transformation can be written as a power series in z, then performing some algebraic manipulations, the generator is eventually found to be
where n ∈ Z. The generator corresponding with f ¯ (z̄) is the same, and is denoted by ¯ l n . The commutation relation of these generators 25 make up the Witt algebra.
The last thing that was discussed for the d ≥ 3 conformal field model was the symmetry The commutation relations are given by the following: [l m , l n ] = (m − n)l m+n , [ l ¯ m , l ¯ n ] = (m − n) l ¯ m+n , and
25 31 group that described the number of infinitesimal generators found by the conformal algebra. This time, there are an infinite number of infinitesimal generators and the Witt algebra is infinite-dimensional, so the conformal group in d = 2 can be described only by global conformal transformations [52]. This symmetry for the global conformal group is approximately SO(3, 1) ≈ SL(2, C), 26 where SL(2, C) is defined by Möbius transformations that map each point of a flat plane to a sphere, and vice versa [54].
4.2.3 Operators and the correlation function
Now that the basic structure of conformal fields has been discussed, how can they be “actually applied” to physics to obtain meaningful results, such as for the AdS/CFT or dS/CFT correspondence? Well, to find a description for desired physical observables, one will need to work with operators and their algebra, which are differentiable polynomials that describe local fields. 27 Using the conformal algebra, one should have the tools needed to derive the properties of a CFT. Thus, the conformal generators from earlier, P µ , D, L µν , and K µ will come in quite handy. A desirable feature of operators is that they can be described in terms of their scaling properties. More specifically, they are eigenfunctions of the scaling operator, D, whose eigenvalue is −i∆. Here, ∆ is known as the scaling dimension for a field [40]. It can also be defined by ∆ = h + + h − , where h + , h − are conformal weights. Both of these terms describe real numbers, and all unitary CFTs have operators with weights greater than zero. In Minkowski space, the scaling transformation is of the form
so the scalar operator, O(x), will undergo a transformation that looks like
By observing the conformal algebra in Appendix A, one can arrive at the conclusion that the momentum operator, P µ , will raise the dimension of O(x), and the special conformal generator, K µ , will lower it. It is important to note that there is a minimum dimension that unitary fields can have. The operator which resides at this lowest dimension is known as a primary operator, and it is annihilated by K µ at O(x = 0) [40, 55, 56]. From here, it follows that the action on O(x) is given by the commutation relations [56]:
(4.27)
The term Σ µν represents matrices that come from a finite-dimensional Lorentz group.
Next, how does one find the n-point correlation function for conformal observables, O(x), and what does this mean in CFT? To start, consider the functional path integral given by Eq. (4.6) from the free scalar field. The correlation function for two time-ordered scalar fields was 26 C is the complex plane.
Note here that “field” has a slightly broader meaning in CFT than in QFT. Not only does a field include the objects in the action, such as ϕ, but it can also include its derivative, ∂ n ϕ [50].
27 32 related to the path integral by:
Similarly for conformal operators, the time-ordered correlation function can be written as:
For the two-point correlation function, operators O 1 (x 1 ) and O 2 (x 2 ) also have some scaling dimensions to consider, ∆ 1 and ∆ 2 . The properties of rotation and translation invariance (x µ → x µ + a µ ) fix the form of the two-point and three-point correlation functions such that
which means that the correlation function depends on the separation distance between two points, x 1 and x 2 . 28 Furthermore, including a scaling invariance (x µ → λx µ ) fixes f (x) so that
The two-point function will scale as well:
Finally, one can write f (x) as:
(4.33)
where C 12 is a coefficient from the Operator Product Expansion. To obtain the final form of the correlation function, there is still one more transformation to consider. From the last section, Eq. (4.18) gives the form of the special conformal transformation. Applying this here, the result is
Some simplifications can still be made, because the correlation function will equal zero unless the scaling dimensions (or weights) are equal to each other. So, from ∆ 1 = ∆ 2 one can obtain the desired form of the two-point correlation function [50]:
Sometimes, the coefficient C 12 can be normalized to 1, but that choice is left up to the reader.
Now that the most relevant introductory concepts of Quantum Field Theory and Conformal Field Theory have been addressed, it is time to go into more detail. While a free scalar field 28 This also makes sense, because the Operator Product Expansion (OPE) says that two local operators that are close together will create a small disturbance that can be described by an expansion of other local operators at that point. The OPE will appear inside of the time-ordered correlation function [40].
33 with the implicit assumption of a flat Minkowski background is the simplest example of a QFT that one can look at, the focus of this paper is on de Sitter space. Therefore, it will be more insightful to learn about the applications of QFT and CFT for a curved, dS background. The next chapter will start by examining gravity in dS space, and then move on to uncovering the asymptotic behavior at future and past null infinities that result from dS metrics in various coordinate systems. The dS/CFT will hopefully become quite apparent by then.
Due to the current open-endedness of the formulation of a dS/CFT correspondence, there have been unique approaches proposed by numerous authors throughout the years. However, since his work is still inspiring much of the progress made today, this chapter will pay special attention to Strominger’s initial dS/CFT proposal [23], which was to analyze the asymptotic symmetries of dS space and properties of the corresponding CFT. The goal of subsequent work has been to take a different route in order to describe the dS/CFT in more detail. A select few of the most promising ideas will be highlighted in Chapter 6.
The goal in this section will first be to explain the dimensionality of Hilbert space, and what effect this has on the dS/CFT correspondence. Then it will be shown that a two-dimensional Euclidean CFT, introduced in Section 4.2, can be used to describe quantum gravity on dS 3 .
5.1.1 Understanding the Hilbert space
Most of the challenges that will be presented in this chapter stem from the rationalization that General Relativity cannot be quantized, at least not macroscopically [36]. This conundrum is due to the uncertainty of whether Hilbert space is finite or infinite-dimensional. Furthermore, the isometry group, SO(d, 1), of dS space in d dimensions places doubt on the existence of a finite-dimensional unitary representation. Thus, a finite Hilbert space would present difficulties for the group SO(d, 1) acting on it. So, while an infinite-dimensional Hilbert space would be good news for dS d space, it has been shown that the phase space of General Relativity with asymptotic dS boundaries at future and past infinities is compact [19]. By definition, a compact space has a finite Hilbert space upon quantization. Additionally, there are several other arguments in favor of a finite-dimensional dS Hilbert space [57–60]. With all of this considered, how can one find a way around this problem to support the dS/CFT correspondence?
One strategy by Witten [36] is to insert operators on future or past conformal infinities and examine the resulting “meta-observables”. 29 From Chapter 3, the dS entropy for N -dimensional Hilbert space is given by S = ln N , which suggests it is finite. However, considering perturbative dS space, one would see that it is infinite-dimensional. There is still no dilemma here, because N diverges as G N Λ (d−2)/d → 0 in the weak coupling limit [27,36]. This is yet another argument in favor of quantum dS Hilbert space being finite-dimensional. It does, however, mean that the 29 “Meta-observables” is a term coined by Witten to describe observables that work mathematically, but can never actually be measured by an observer in dS space.
34 Einstein-Hilbert action,
(5.1)
cannot be quantized. Instead, there must be a more fundamental theory that can derive the action and find values for G N Λ (d−2)/d . Here, the Lagrangian, L m , is the matter term and the quantity representing the dS metric, g, is defined in the Euclidean signature.
Perhaps there is a non-perturbative theory one can find that will give some meaningful workaround to all of this. But first, it is important to discuss some observations regarding perturbation theory in dS space. Considering a matter field theory in its free form provides a valuable starting point because it produces a corresponding Hilbert space when quantized. Following the notation of Witten [36], a Gaussian wavefunction with scalar fields can be written in coordinate space as:
(5.2)
where T describes spatial dS space, and likewise the “equator” on the d-sphere. The product T ×T represents the collection of all possible pairs of points on this spatial section. Additionally, the wavefunction is dS-invariant for a unique D(x − y), and there is an invariant Gaussian state which can be described by the two-point function from Eq. (4.7). It is found by a path integral that covers the entire d-sphere, where the path on one hemisphere finds the state ⟨Ψ| and the path on the other hemisphere finds |Ψ⟩. The other states that are not the vacuum can be expressed as
where f (x (1) . . . x (n) ) is a function that depends the points in T and is used to construct quantum states in perturbation theory, and the points x (1) . . . x (n) are on the d-sphere. Most of these states will end up being discarded in the context of quantum gravity, due to the condition of dS-invariance. The expectation is that the quantum states constructed using the function f will have an infinite-dimensional structure, but authors Banks and Fischer note that a full theory should combine matter and gravity, which gives a finite-dimensional result [59].
Now, defining a non-perturbative dS space should be thought of in a different manner than its perturbative counterpart. By looking at the how the field behaves near past and future infinities of dS space, then the lack of microscopic knowledge (the type of fields, local topology, degrees of freedom, etc.) will not be a hindrance. This makes it possible for one to derive important insights and understand the implications of the non-perturbative theory. One of these implications includes meta-observables, which define the Hilbert space and play a role in the interpretation on the sign of Λ [36].
Next, consider the planar dS spacetime metric similar to the one from Eq. (2.15):
where Ω 2 d−1 is the d − 1-sphere. There will be a conformal metric that applies to each boundary 35
at t → ±∞. They can be written as:
(5.4)
The superscripts i and f correspond to the initial or final state, denoted by |i⟩ and ⟨f |. Then, ⟨f |i⟩ is a meta-observable defined by the path integral. More specifically, ⟨f |i⟩ is the probability amplitude for the transition between two states. The significance of this result is that ⟨f |i⟩ is mathematically represented as an ∞ × ∞ matrix, which refers to the number of possible states in the transitions. However, this infinite-dimensional matrix can have finite rank, which represents the number of relevant, linearly-independent states [19, 36]. It is believed that the rank is the dimension, N , of the Hilbert space!
5.1.2 Boundary conditions for dS space
So far, examining the properties of dS space in d dimensions has been the main focus of this thesis. However, the end goal of this chapter is to explore Strominger’s 2001 dS 3 /CFT 2 conjecture, so it will be best to look at boundary conditions for dS space in three dimensions, specifically. This will allow the interplay between dS 3 space and the corresponding CFT 2 to be more insightful and concrete. Now, the goal in understanding gravity in dS is to define the boundary conditions which will lead to a symmetry group corresponding to the two-dimensional Euclidean CFT. While the gravity theory of dS space is strongly coupled, making it harder to probe, the non-gravitational CFT is weakly coupled, making the task much easier. This is precisely the reason why the dS/CFT is such a nice tool to have.
However, there are still a couple of things to think about first. A two-dimensional CFT will have correlators on I ± which correspond with the correlators of three-dimensional dS gravity, so the theory is diffeomorphism-invariant [20]. 30 This means that diffeomorphisms in asymptotic dS 3 should be chosen such that the boundary conditions on the metric are preserved at I ± without being too restrictive. Otherwise, the theory will become trivial. Therefore, the best choice one can make is to make the spatial fall off of the metric as weak as possible while still preventing symmetry generators from diverging. This concept will be explained in what follows.
Authors Brown and York [61, 62] proposed that there is a stress-energy tensor that lives on the timelike boundary of AdS spacetime and captures diffeomorphisms of the bulk. For dS space in three dimensions, the Brown-York energy-stress tensor lives on the spacelike boundary, and can be written as
where γ µν represents the metric at the I − boundary and K µν is the extrinsic curvature. If the planar metric of dS 3 is considered, this time including first-order perturbations, then it can be written as:
It can further be expressed by a lapse and shift function to obtain the form:
30 (5.7)
Diffeomorphism invariance is another way to say that the underlying physics of a theory will be the same regardless of the coordinates used.
36 where the N terms and metric γ ij can be broken up into [20]:
By expressing the extrinsic curvature, K µν , in terms of K zz , K zz̄ , and its trace K, then one obtains the components of the Brown-York stress tensor [23]:
(5.8)
The boundary conditions can then be found by requiring this result to be finite. For asymptoticallypast dS 3 space, t → −∞, they are given by: 31
(5.9)
Here, O − is the region of the causal past for a timelike observer [23]. There are now diffeomorphisms, ζ, which satisfy the symmetries of the boundary conditions. They are also known as “asymptotic symmetries”. By considering the most general type of diffeomorphism,
1 ζ = U (z)∂ z + U ′ (z)∂ t + O(e 2t ) + complex conjugate
2 (5.10)
then taking the Lie derivative, L, with respect to the vector field, ζ, allows the components of the metric to undergo a coordinate transformation:
Applying this to Eq. (5.8) results in
(5.12)
The boundaries are preserved, so this diffeomorphism does indeed generate the asymptotic symmetry group of dS 3 . Now the question is, “What is the symmetry group?” To uncover the answer, first recall the definition of a conformal transformation from Section 4.2 (basically a diffeomorphism + a scaling transformation). The first term of Eq. (5.10) will produce a holomorphic diffeomorphism on the plane, while the second term is the shift of t in the metric. 32 Therefore, the diffeomorphism ζ is the sum of a tangential term acting as a coordinate transformation and a normal term acting as a Weyl transformation, both on the complex plane. This is the equivalent to the description for a d = 2 CFT.
It has now been argued that the finite-dimensional asymptotic symmetry group of dS 3 can 31
32 These are the non-zero terms. It will look like a scaling transformation from the z-plane point of view [23].
37 be described as a two-dimensional CFT on the boundary. Furthermore, the isometry group of dS 3 is a subgroup of the asymptotic symmetry group, which is SL(2, C) [20, 23]. The next section will present a general overview of Green’s functions, as they can be used to analyze the behavior of field theories and their interactions. Then in Section 5.3, these concepts are applied specifically to the dS 3 /CFT 2 conjecture.
Green’s functions play a crucial role in understanding the correlation properties of quantum fields in both dS space and the dual CFT, and therefore serve as a bridge between the two sides of the dS/CFT correspondence. First, one can consider the action of a free (massive) scalar field in d dimensions, specifically in the context of de Sitter space. This will provide important background information necessary in understanding the Hadamard function and Green’s functions. The action is:
(5.13)
where g ≡ det(g αβ ), and the partial derivative from Eq. (4.2) is now a covariant derivative upon the introduction of gravity. This captures the dynamics of the scalar field, ϕ, in a curved spacetime background. Then, evaluating the path integral will lead to the two-point correlation function, also known as a Wightman function [63]:
This can also be written as the Hadamard two-point function [20, 64, 65]:
in which |0⟩ is dS-invariant under the full disconnected group O(1, d) 33 and the two-point function is symmetric. To interpret this further, refer back to Eq. (2.10) from Chapter 2, where
The equation looks like:
By defining the geodesic distance between two points as D(X, X ′ ) = l cos −1 P , then the Minkowski space can be rewritten as:
The invariance of the vacuum state, |0⟩, implies that the Hadamard two-point function depends solely on the geodesic distance between two points, X and X ′ , in dS space. This distance, denoted as D(X, X ′ ), is the only factor that affects the two-point function. For a massive scalar field, the Hadamard two-point function obeys the Klein-Gordon equation [20]:
where ∇ 2 X is the Laplacian in dS d . Staying away from singularities, one can write G(X, X ′ ) as 33 The Orthogonal group, O(n), preserves distance under transformations in Euclidean space.
38 G(P (X, X ′ )) to obtain a differential equation in terms of P :
There is a one-parameter family of dS-invariant Green’s functions, 34 G α , which provide two linearly independent solutions due to the fact that Eq. (5.18) has symmetry under P → −P . In other words, G α (X, X ′ ) = ⟨α| ϕ(X)ϕ(X ′ ) |α⟩. Out of the two solutions, one has a singularity at P = 1 that occurs when points X and X ′ are separated by a null geodesic, or light cone, , one gets the hypergeometric equation: D = cos −1 P . By making a change of variables, z = 1+P 2
with a solution of the form [65]:
(5.20)
Here, F is the hypergeometric function, and the conformal weights, 35 ∆ + and ∆ − , are defined by [23, 64]:
There is an antipodal solution to Eq. (5.20) which exists at P = −1 [20]:
. This singularity in the Green’s function of dS space comes from the where this time z = 1−P 2 presence of the observer horizon. Because the mathematical nature of the singularity is not physically accessible, it can never be measured or experimented on by an observer. However, this result is not one that should immediately be disregarded, because the unique characteristics of dS space allows “unphysical” results to still have significance. So, although the singularity and dS-invariant vacua here are not physical, they remain of conceptual importance for understanding the strange nature of dS space. Now it is finally time to discuss in detail the connection between gravity in dS 3 and CFT 2 .
To summarize the end result of Section 5.1.2, the boundary at I − on dS 3 is acted on by the Euclidean CFT 2 . Strominger argued that the same logic could then be applied to gravity correlators on I − . Still working with the planar dS metric from Eq. (5.6), consider the wave equation given by a massive scalar field:
34 (5.22)
A dS-invariant vacuum state is obtained from dividing the modes in such a way that the invariance is preserved under SO(d, 1).
35 The conformal weights ∆
± are eigenvalues of the Virasoro generators, L n .
39 Near the boundary, as t → −∞, √ the solutions behave like ϕ ≈ e ∆ ± t , where the conformal weights are equal to ∆ ± = 1 ± 1 − m 2 l 2 [23]. The following boundary conditions can be imposed, where ∆ ± are chosen to be real:
(5.23)
Now, the proposition for the dS 3 /CFT 2 correspondence is that the source term on the boundary, ϕ − , is dual to the ∆ + -dimensional operator, O ϕ , of the Euclidean CFT. Just like in Section 4.2, all of the observables of the CFT can be found by the correlation function: 36
(5.24)
On the right-hand-side of this equation, the derivative of the action on the bulk gravity is taken two times. The action can be written as:
(5.25)
where only the quadratic terms of the Lagrangian have been considered. The expression
is a two-point correlator of O ϕ near the boundary, I − , and is also analogous to the expression used in the AdS/CFT. G(t, z; t ′ , v) is the dS-invariant Green function, whose behavior near the boundary of I − is of interest. The discussion in Section 5.2 provides relevant information on this concept. Taking the limit of Green’s function as t → −∞, one can observe its behavior to
where C ∆ ± are fixed constants.
By plugging Eq. (5.23) and Eq. (5.27) back into Eq. (5.26), the result is the action
The derivative of this action with respect to the source, as seen in Eq. (5.24), gives the two-point correlation function describing the dual operator, O ϕ , with conformal dimension, ∆ + :
(5.29)
If one were to look at lower-order behaviors, there is a solution 37 for which the behavior of ϕ ≈ e ∆ + t ϕ + (z). In that case, the correlation function turns out to be
40 (5.30)
where the conformal dimension is now lower, ∆ − .
So far, only the boundary of I − has been considered, since planar coordinates do not include the region at I + . However, both boundaries I ± can be considered by switching to global spherical coordinates. With this coordinate transformation, Green’s functions of dS space are represented by a complex metric on an expanding and contracting two-sphere, S 2 . The metric is [23]:
where w is a complex coordinate living on the sphere, defined as w = tan 2 θ e iϕ . Similarly to Figure 1, global spherical coordinates describe spacelike slices with I − and I + boundaries. A visual representation of this is given by Figure 10. An interesting question to consider now is, “What will happen by taking the correlation function when considering both correlators to be somewhere on I − ?”
Using the hypergeometric function in global spherical coordinates to obtain ∆(w) = 2(1 + ww̄) −2 , the asymptotic behavior at I − for the spherical correlators is of the form [23, 64]:
A helpful notation is to let “in” and “out” represent this behavior on I − and I + , respectively. That way, the behavior near the boundary I − can be expressed as:
Given the two-point correlation function, denoted by f , for a ∆ ± -dimensional conformal field 41 on a two-sphere,
(5.34)
the behavior of Eq. (5.32) is found to be proportional to
This is indeed quite a bit of heavy lifting, but a huge connection can now be made. It looks like the spherical correlators from Eq. (5.32) are related to the conformal correlators of Eq. (5.35)!
The planar dS metric from Eq. (5.6) allows the transformation formula, G(z, v), to be evaluated as:
Using the notation of Strominger [20, 23] for the divergence of t, t ′ → −∞, this simplifies to:
where the relation between spherical and planar coordinates are defined as:
Although tedious, by finding the behavior of P (τ, w, w̄; τ ′ , v, v̄) at I − , one can then use the fact that P → −P to immediately understand the behavior at I + . Therefore, with just one more important equation,
it can be reasoned that P → ∞ for one point each on both I − and I + , with a singularity occurring at P = 1. Now, drumroll please: CFT correlators on the sphere correspond to bulk gravity correlators for every point in I − !
As advertised at the beginning of Chapter 5, this chapter will discuss the advancements and proposals made after 2001, many of which aim to answer some of the unanswered questions first left by Strominger. For instance, string theory techniques have been used to shed light on the formulation of the dS/CFT correspondence in higher dimensions. Some recent and notable papers on this topic are [66–70]. Additionally, there have been proposals that the dS/CFT correspondence should include additional fields and symmetries [71]. Other authors, 42 such as [24], have tried to obtain a more precise link between bulk dS space and a CFT by relating the geometry of causal patches to the entanglement structure of the CFT. While research on the dS/CFT has gone through periods of more and less community interest, there has been a noticeable increase in activity in recent years. With that being said, there are many fresh ideas on the subject for those who are interested. This discussion will only be able to cover the areas which have currently been studied the most, starting with higher-spin gravity theory.
Without going into too much detail about why string theory is incompatible with dS spacetime, it turns out that such incompatibilities played a role in the development of a higher-spin dS/CFT correspondence. While the structure of dS space shares a maximal supersymmetry algebra with Minkowski space, the presence of unbroken supersymmetry in Minkowski space and the broken supersymmetry in dS space lead to different implications for string theory. More specifically, string theory heavily relies on unbroken supersymmetry, which poses a challenge for incorporating it into dS space [24]. 38 Additionally, the timelike infinities of dS space imply that complex conformal weights are associated with the boundary of dS space, consistent with previous chapters in this thesis. As a result, discussions from authors [73–75] highlighted the skepticism towards reconciling the dS/CFT duality with string theory. Therefore, one idea is to develop a theory that circumvents the requirements of supersymmetry and massive fields altogether [24].
A decade after Strominger’s initial paper, authors Anninos, Hartman, and Strominger [71] proposed a model of dS/CFT where Vasiliev’s higher-spin gravity in four-dimensional dS space corresponds to a three-dimensional Sp(N )-invariant CFT on the boundary at I + . The idea of higher-spin gravity is that an infinite tower of massless particles with increasing spins (greater than 2) will interact with one another in a way that corresponds to an infinite-dimensional gauge symmetry, or “higher-spin symmetry” [24, 76]. These interactions all act at the same time, which suggests that they are nonlocal at the cosmological scale. Furthermore, an unbroken global higher-spin symmetry is extended in the bosonic direction and can only be realized in the boundary of a free CFT. Although this theory should be thought of as a “toy model”, it is still a great way for one to observe the properties of higher-spin correlators in dS space and their relationship to the corresponding CFTs, all the while avoiding string theory.
Generally speaking, there is another property of higher-spin gravity that makes it appealing to study. While string theory faces challenges in dS space, it is well-established and successful in AdS space. On the other hand, higher-spin gravity works equally well in both AdS and dS spacetimes, and there are no complex conformal weights, ∆ ± , to deal with. So, by starting in Vasiliev AdS 4 gravity, one can analytically continue to Vasiliev dS 4 gravity by simply changing the sign of Λ from Λ → −Λ while fixing Newton’s constant, G N . The conjecture that a bulk theory with higher-spin fields in AdS 4 space is dual to an O(N )-invariant CFT model in d = 3 dimensions is known as the Giombi-Klebanov-Polyakov-Yin (GKPY) duality. This duality involves a coefficient N ∼ ΛG 1
N[71]. Upon the change of the sign in front of Λ, the result is that N → −N , which means O(N ) → O(−N ) ≡ Sp(N ). The GKPY duality comes in two realizations, which leads to two dS/CFT correspondences. This means that there are two
38 In the context of string theory, supersymmetry refers to the idea that particles, specifically bosons and fermions, have “superpartners” whose spins differ by a value of 12 [72].
43 bulk realizations of the boundary CFT 3 , free O(N ) and critical O(N ). These are dual to bulk theories in dS 4 which have their own distinct boundary conditions at I + , known as Neumann scalars and Dirichlet scalars, respectively. The goal of this section is to show that this holds.
There are two “types” of Vasiliev’s gravity theory, known as Type A and Type B [77]. Type A theory is the primary consideration for the authors in [71] because it has parity-invariant, minimal bosonic gauge fields of all even spins, s = 0, 2, 4, . . .. Furthermore, this Type A theory is related to the free O(N ) realization in the dS/CFT correspondence. In contrast, Type B theory deals with odd spins and is associated with a different set of boundary conditions that give rise to a bulk theory with critical O(N ) symmetry. Type A theory can be applied to dS 4 space as well as Euclidean AdS 4 space, both of which possess O(4, 1) symmetry.
The first important thing to uncover is that the Vasiliev dS 4 boundary correlators can be found by the free action [78, 79]:
(6.1)
Here, Φ = φ, h µν , w µνρσ , . . . are the even, higher-spin fields, and G N is Newton’s constant. With this result, one can find the corresponding linear equation of motion:
The choice of a plus sign in front of the ΛΦ term is associated with the real form for the higher spin algebra, as used in [71]. The action must also include the interacting parts, which come from an expansion in ∇ 2 /Λ. With this in mind, one can write the action [71, 78] for spins
(6.3)
where h refers to the metric fluctuation for spin s = 2. The cubic interactions are given by
From this, one can see that the action has an overall factor of ΛG 1
N, which describes quantum fluctuations [81]. This result is given by the coefficient, N , mentioned earlier.
6.1.1 Bulk correlators: from EAdS to dS
A thoughtful next step would be to try to show that the analytic continuation of AdS 4 onto dS 4 is plausible. The hyperbolic Euclidean AdS 4 metric on the Poincaré patch can be written as:
where dt E is the Euclidean time component. This metric can be continued into dS 4 space by making a double Wick rotation, l EAdS → −il dS and z → −iT [82]. The resulting dS metric is 39 The notation (2, s, s) describes linear interactions involving a spin-2 field and quadratic interactions involving two spin-s fields [80].
44 in conformal planar form:
Now, the correlation functions in EAdS 4 can be compared to those in dS 4 .
(6.5)
The two-point correlation function of the bulk scalar, ϕ, shows the behavior of EAdS 4 at short distances, 40 and can be written as:
(6.6)
where C 2 is a dimensionless coefficient proportional to N 1 ∼ −ΛG N . C 2 is also denoted as the overall “strength factor” of the quantum corrections [71]. Analytically continuing into dS 4 by applying one Wick rotation from earlier, z → −iT , there is an undesirable overall minus sign that appears. To fix this, the coefficient C 2 can be given a minus sign, so that the Lorentzian signature will have the right sign in the end. Therefore, C 2 → −C 2 , which is essentially the same as making the transformation Λ → −Λ and N → −N . From this result, one can show a generalized transformation rule for n-point correlation functions from EAdS 4 → dS 4 with the following:
The bulk correlators derived here can now be used in comparison with the non-unitary Sp(N ) CFT 3 correlators on the boundary.
6.1.2 Boundary correlators: CFT 3
It will now be shown that the free and critical (interacting) Sp(N ) vector models have n-point correlation functions that are equivalent to free and critical O(N ) vector models, simply by making the transformation N → −N .
To begin, one can consider an N -component anti-commuting scalar field, χ a , whose action is given by the free Sp(N ) vector model:
Ω ab = 0 −1 N × N
2 2 1 N × N
2 2 0 !
(6.8) N ×N
where Ω ab is the anti-symmetric symplectic form [83]. 41 Since the spins in this case are even, currents J (s) 42 are conserved, and there is a scalar operator of ∆ = 1 dimensions whose current is found by taking the inner product, J (0) = χ · χ ≡ Ω ab χ a χ b [71]. This implies the presence of
These currents will now come in handy because the correlation functions in the free theory can be represented as one-loop diagrams. 43 In these diagrams, the current is inserted n times, which applies to both the Sp(N ) and O(N ) models. By inserting real anti-commuting scalars, χ(φ 1 , φ 2 ), and performing the transformation N → −N , then the free Sp(N ) correlators can
The symplectic form, Ω ab , is a mathematical object that encrypts the geometrical structure of a symplectic manifold.
42 In free theory, these currents have dimension ∆ = s + 1. 43 One-loop diagrams are a useful took for calculating the scattering amplitudes of higher-spin particles.
45
be related to the O(N ) correlators by:
Now including interactions, the action for the Sp(N ) model can be written as:
(6.10)
where the coefficient, λ, refers to a quadratic coupling constant [78]. To demonstrate the relation between the interacting Sp(N ) theory and O(N ) correlators by the transformation N → −N , one needs to introduce a commuting auxiliary field, α(x), into the Lagrangian. Then, the action is of the form:
(6.11)
The Feynman diagrams representing the interacting theory now indicate higher-order contributions, which are depicted as closed loops of fields χ connected by the auxiliary field, α. The anti-commuting property of χ means that each loop contributes a minus sign. Thus, it can be said that the correlators of Sp(N ) correspond, up to the order 1/N , to the correlators of O(N ) through the transformation N → −N [71].
While the anti-commuting scalar fields of Sp(N ) are non-unitary [83] due to the unbounded energy from fields in the bulk theory, the symplectic model Sp(N ) is considered to be pseudounitary [84–86], a weaker form of unitarity. This means that the Hamiltonian is pseudohermitian, 44 which allows for non-unitary states but still requires the preservation of certain properties of the theory, such as the conservation of probabilities [84].
Everything that has been discussed so far in this chapter can now be put together to show that Vasiliev gravity in EAdS 4 with Neumann (Dirichlet) boundary conditions is correlated to a free (critical) interacting Sp(N ) CFT 3 on the boundary. This correspondence can be extended analytically into dS space.
This correspondence can be established by first taking the bulk EAdS 4 correlators to the boundary CFT 3 and rescaling them. Consider a bulk scalar, ϕ(z, x), near the boundary, z = z c → 0. z c denotes the “cutoff scale”, or the distance to the boundary at which the scalar field is evaluated, and x is a quantity which scales linearly as x n are rescaled in the boundary CFT [71]. Then, by performing dimensional analysis and forcing O(4, 1) symmetry invariance, the full n-point correlation function expressed in Eq. (6.6) can be generalized to:
(n)
(6.12)
Here, the constant, A C 2 , in the boundary correlators represents the normalization factor for the n-point correlation functions in the boundary CFT. The subscript, C 2 , means that A (n) depends on C 2 , a parameter that is involved in the rescaling of the bulk field. By introducing
44 Pseudo-hermiticity gives the relation H † = ηHη −1 , where the unitary operator η satisfies η 2 = 1.
46 a rescaled operator O on the boundary given by:
it follows that the boundary correlators are
In the classical limit when C 2 is small, this equation scales as C 2 −1 , which signifies the emergence of GKPY duality.
Now it is time to analytically continue into dS 4 space, which is done by referring back to the EAdS 4 → dS 4 transformation from Eq. (6.7). The correlation function for bulk dS 4 space can be written as:
(6.14)
where C 2 → −C 2 . Ensuring that the boundary operator O has been appropriately rescaled in the context of dS 4 , it is now defined such that
The boundary correlation functions look like:
Therefore, the correlators in bulk Vasiliev dS 4 gravity are mapped one-to-one with the correlation functions of CFT 3 [71]! In other words,
This is exactly the result one would hope to find.
As discussed in the previous section, a higher-spin dS/CFT duality suggests that the infinite tower of higher-spin fields from Vasiliev’s gravity theory in dS space is dual to a CFT with an infinite number of higher-spin symmetries. However, this proposal was also a big motivating factor in the search for a holographic dual description involving string theory. String theory offers a useful framework, as it accommodates the cosmological constant Λ, which accounts for the vacuum energy of the Universe. This is an important condition when searching for a complete theory of quantum gravity, especially if the Universe is approximately de Sitter.
It turns out that higher-spin gravity is compatible with string theory in several ways, with symmetry being a key aspect. In string theory, the symmetries of a string’s world-sheet 45 play a big role in determining the string’s various vibrational modes, while the symmetries of the un-
45 A world-sheet refers to a two-dimensional surface that a string traces out while moving through a spacetime.
47 derlying spacetime determine the properties acquired by the resulting particles. Furthermore, these symmetries are in close relation to gauge fields that emerge in the low-energy limit of a given spacetime. In regards to higher-spin gravity theory, particles with higher-spin states can be described in terms of higher-spin fields, which naturally arise as excitations of the string within string theory.
Not only are the symmetries of the string’s world-sheet compatible with the symmetries of higher-spin particles, but the connection between Vasiliev’s gravity theory and string theory also leads to some intriguing insights. For instance, higher-spin states possess the property of non-locality, which offers a unique perspective on their physics and how they interact with gravity. Also, since the number of higher-spin states is infinite, it is theoretically expected that the bulk string theory in dS space is dual to an infinite set of higher-spin operators living in the boundary CFT. This abundance of observables holds the potential to establish a more welldefined dS/CFT correspondence. While it is important to mention that finding a string theory solution in dS space remains an ongoing research endeavor, there has already been significant progress made in this area. What follows in this section will be a discussion that highlights the key ideas stemming from this research.
It has been mentioned that several “no-go” theorems hinder the existence of a stable and viable string theory solution in dS space. One such no-go theorem, introduced by Gibbons [87], proposes that a theory of quantum gravity in dS space must have a negative sign for Λ to avoid breaking the supersymmetric ground state. However, having a negative cosmological constant corresponds to AdS space rather than dS space. Thus, the plausibility of dS space within string theory comes into question. In later work, Maldacena and Nuñez [42] presented their own no-go theorem, which examines string theory compactifications in d dimensions through the study of supersymmetry transformations of scalar fields. Several of the claims made regarding this theorem include the absence of higher curvature corrections in the gravitational action, a negative scalar potential, and the lack of conserved charges. Collectively, these results argue against the consistent compactification of string theory in dS space [19].
However, these no-go theorems are by no means the final say on the matter. Since the original dS/CFT proposal, there have been several suggested ways to circumvent these limitations. One notable counter-argument was put forward by Hull [88], who introduced the concept of a a Type IIB* string theory. In traditional Type II string theories, which are typically formulated in ten-dimensional flat spacetime, there are no open strings whose endpoints are restricted to hypersurfaces, called D-branes. Instead, Type IIB strings are closed, meaning that they are able to move freely throughout a given spacetime without boundary conditions. Type IIB* string theory extends this idea by introducing a twisted torus proportional to 1/Λ, resulting in a compactification to nine dimensions rather than ten. In Type IIB* theory, there are E-branes analogous to D-branes, except they experience timelike T -duality and impose Dirichlet boundary conditions in both spacelike and timelike directions [88]. The solution for such a theory involves the product of a five-dimensional hyperboloid with dS 5 . Intriguingly, there is a duality between this solution and the Euclidean N = 4 super-Yang-Mills theory. Unfortunately, Hull’s work contains kinetic terms with the wrong sign, which raises concerns about the presence of ghost fields. This is a generally undesirable characteristic.
48 6.2.1 An unconventional approach to string theory in dS space Another recent discussion by [66] aims to evade the no-go theorem of Maldacena and Nuñez [42] by considering a dS space modeled with quintessence. “Quintessence” is a parameterized model that explains the expansion of the Universe and is allowed to evolve with time to account for various cosmological observations [66]. This choice does, however, require a considerable amount of fine-tuning to ensure consistency with known cosmological parameters of the Universe. In their theory, these authors propose the existence of a “metastring”, a T -duality symmetric, doubled, non-commutative formulation of string theory that is connected to a positive cosmological constant in dS space. The zero modes of metastring theory are “metaparticles”, which possess an additional local symmetry and can be used to explore underlying geometric properties of the modular spacetime that emerges within the theory [89]. The goal of metastring theory is to generalize the principles of string theory so that it can be a tool for describing gravity [90].
Furthermore, it is argued that a theory involving higher-spin gravity and metastring theory in dS space requires the consideration of two entangled CFTs living on the future and past null boundaries, I ± , instead of just one. This way, the finite entropy of dS space can be interpreted as the entanglement entropy of these non-canonical 46 CFTs [66]. The idea is that the entangled dS “wavefunction”, which describes the entanglement of the CFTs, bears resemblance to the BCS wavefunction for superconductivity [91], where certain materials have been observed to conduct electricity with zero resistance at low temperatures. By exploring this analogy further, one may find a connection between metastrings and dS holography.
This connection might still seem somewhat ambiguous, so the BCS/dS wavefunction analogy will be examined in some more detail. When working in a large dS 3 space dually described by two-dimensional CFTs, the wavefunction can be expressed as follows:
In the context of BCS
qsuperconductivity, the Hamiltonian can be minimized to obtain the gap equation, E k = e 2 k + ∆ 2 k . Drawing an analogy to the cosmological constant, given that the acceleration in dS space is a dS = understood as [66]:
q a 2 + a 20 , the non-perturbative gap equation can be
46 (6.18)
“Non-canonical” here refers to the specific algebraic structure of the CFTs, which are different from what has been discussed in this thesis.
49 where V represents a positive interaction, 47 D AdS represents the density of states in a CFT in AdS space, and M cAdS is a “critical” energy scale associated with the ground state of AdS space. In reference to BCS superconductivity, the “critical” energy scale represents the energy scale associated with the ground state of the superconductor, which is vital in understanding the system’s behavior and important properties.
Now, how can this result be interpreted? Well, the vanishing resistance in the final form of the BCS gap equation corresponds directly to the weakening of the Newton constant (gravity) in dS space. Furthermore, the critical energy scale is associated with the AdS vacuum energy, and can be on the order of 10 11 GeV. This critical energy scale naturally has Planckian units, which aligns with the notion that the phase-space spacetime features of metastring theory have holographic characteristics [66]. This relationship resembles the seesaw formula, where large scales are suppressed by a significantly smaller scale.
In conclusion, while exploring dS space with quintessence may not be the most promising approach, the investigation of dS holography and its connection to metastring theory holds potential. The dS holography analogy offers a quantum gravitational and non-perturbative perspective, suggesting that supersymmetry breaks on cosmological scales, and that dS spacetimes should have a small value for the cosmological constant, Λ [66]. Pursuing this line of research not only promises a more comprehensive understanding of the dS/CFT correspondence, but also presents an opportunity to address the cosmological constant problem, a significant theoretical challenge in the real Universe. Such progress could provide evidence that the Universe is heading towards an approximately dS space. As of now, this area of study remains open and eagerly awaits new ideas.
The formulation of string theory in the context of the dS/CFT correspondence is just one avenue of research that has been explored in recent years. In this final part of the discussion, the focus will be on the work presented by [24]. They postulate that there is a specific geometry of dS space that offers a potential solution for developing a working dS/CFT model within the observer’s horizon. This pursuit is particularly important because, as mentioned earlier, quantum observables on the dS boundary are often considered as meta-observables, which lack a physical interpretation. However, if observables are identified within the observer’s horizon, then they could be measurable, or have physical significance. This approach combines elements of the dS/CFT correspondence, Vasiliev’s higher-spin gravity, and the utilization of global coordinates to work in “elliptical” dS space.
To learn more about this topic, it is essential to understand the geometry of dS space that is being analyzed. Elliptical dS space can be thought of as a global dS space that has been folded in half, resulting in the emergence of antipodal points. 48 Specifically, the four-dimensional global dS metric from Eq. (2.13) undergoes transformations under the antipodal map, t → −t and θ → π − θ, and the two-sphere part of the metric undergoes the transformation Ω → Ω A . Here, Ω A refers to the angular components. This folding process results in a dS spacetime with a topology defined as dS 4 /Z 2 [24]. In elliptical dS 4 space, the symmetry group O(1, 4) is
47 This is also the case in BCS superconductivity, where V is positive for the electron-phonon interaction.
From Chapter 2, dS space can be conceptualized as being embedded within flat Minkowski space. Folding global dS space in half corresponds to passing through a point X i and its antipodal point −X i in the space [92].
48 50
preserved, but there exists only one boundary, since the future and past conformal boundaries of global dS space coincide at antipodal points.
It is also important to recognize that the observer’s horizon covers the entire space, but not the entire spacetime. While an observer within their causal patch does not have a sense of spatial direction, the timelike orientation is preserved. This has significant implications for the behavior of quantum mechanics, such as the sign conventions associated with the operator algebra. However, according to the rule of complementarity, while each observer’s patch may be described differently, all measurements will yield equally valid results. Therefore, it is possible to obtain a theory that is well-defined and complete [92].
To be able to define the physics inside the causal patch from the Euclidean CFT living on the boundary, the first step is to obtain the operator algebra in dS 4 /Z 2 , where the symmetry of the spacetime, O(1, 4), is broken aside from the time translations and rotations, R × O(3) [24]. In the context of the Lorentzian signature and the consideration of higher-spin gravity, the bulk fields on the boundary are massless. This implies that the modes of the field, represented as a lm (ω) and a ∗ lm (ω), form a basis of harmonic oscillators everywhere throughout the spacetime. Likewise, there are modes, b lm (ω) and b ∗ lm (ω), which lie on the antipodal conformal boundary of dS space. The operator algebra is then found by a one-to-one mapping of the modes:
where a lm is the same as b lm aside from N l (ω), which are some coefficients for b lm [93]. The 1 goal is then to find the coefficients that make up the normalization factor,
|N (ω)| 2 . l Referring back to the discussion of higher-spin gravity, bulk fields are massless and free, and have two types of boundary conditions, Dirichlet and Neumann scalars. This allows the operator algebra to be found by looking only at the CFT. The normalization coefficients turn out to be of the form
where C l (ω) are coefficients that stem from the Gaussian part of the CFT partition function [24]. This part of the partition function can be seen as:
(6.21)
Although C l (ω) are not explicitly stated by [24, 93], these coefficients give rise to Eq. (6.20), which then leads to the establishment of a dictionary of operator algebra from the Euclidean CFT residing on the causal boundary.
Next, the authors of [24] acknowledge that there should be a way to work with higher-spin gravity in a more generalized way, without relying on R × O(3) symmetry. It turns out that higher-spin gravity can be reformulated in such a way that it appears “non-gravitational” on a fixed, pure dS 4 or AdS 4 geometry. Doing so does not negatively affect higher-spin gauge symmetry. The significance of this is that there is now a fixed geometry containing fields of all spins, and the spacetime metric no longer has to be a spin-2 field. However, this setup puts constraints on the background, making it one that is unrealistic. Nevertheless, it is argued that 51 the fascinating properties of higher-spin gravity make it possible to work with an interacting theory on a fixed background geometry. In this context, the pure dS 4 /Z 2 geometry, with its causal structure and symmetries, is particularly tractable and suitable for further investigation.
The main approach of [24] was to connect bulk interactions in the higher-spin theory with correlators on the boundary of a free CFT. This was accomplished by introducing Penrose’s twistor theory as a theoretical framework for connecting bulk interactions with correlators on the boundary [94]. 49 This theory provides a Penrose transform which shares properties of higher-spin gravity on a fixed dS 4 spacetime, and is a holographic dual to a twistor form of the operators of the CFT. This takes the form:
where F (Y ) represents the twistor function, P (x; Y ) is a generating function, δ x (Y ) denotes derivatives of the fields at point x in the bulk, and ⋆ denotes the star product of higher-spin gravity [95]. This expression arises from the Penrose transform, and right-hand-side of the equation corresponds to a linear solution in the bulk [24].
Although the result of Eq. (6.22) may be one that is specific to this discussion and thus unrealistic in its current form, it still offers insights into the physics within the causal patch of dS space. It may also point towards a finite-dimensional Hilbert space. As of now, this avenue of investigation remains open to interpretation.
In this thesis, the main concepts, properties, and ongoing investigations surrounding the dS/CFT correspondence have been explored. This field of research is one that was built upon decades of fascinating discoveries, all driven by the quest to understand quantum gravity in a spacetime which closely resembles the universe. While many questions remain unanswered, there are plenty of active efforts towards the development and exploration of the dS/CFT in a unique and inspiring way.
In Chapter 2, the discussion focused on the fundamental concepts of the dS/CFT correspondence, beginning with an exploration of the geometric properties of dS space. By subjecting the dS spacetime metric to various coordinate transformations, including global, planar, and conformal, intriguing insights into its geometry were revealed. Notably, the utilization of Penrose diagrams was invaluable for understanding the conformal structure of dS spacetime at its infinities. Considering the success of the AdS/CFT correspondence, the geometry of AdS space was also introduced to serve as a comparative framework. This chapter established a solid foundation for comprehending the challenges and opportunities inherent to the dS/CFT correspondence, which set the stage for further investigation.
Chapter 3 summarized the ways in which one can formulate holographic theories, which establish a connection between local physical theories and the holographic principle. Then, the fundamental concepts of the holographic principle were laid out, shedding light on their specific relevance to dS space. Some significant aspects of the discussion included the Bekenstein-
49 Twistors are known as the spinors of the O(1, 4) symmetry group and play a significant role in higher-spin gravity.
52 Hawking entropy-area law, which unveils the deep connection between the entropy and central charge of dS space, among other relevant thermodynamic properties. This highlighted the intriguing parallels between holography and the rich thermodynamic nature of dS space. The next part of the discussion was directed towards the Hilbert space and its associated states. The relationship between the number of states within the Hilbert space and the degrees of freedom of a local physical theory was carefully examined. Overall, the goal of Chapter 3 was to establish a solid comprehension of the holographic nature of the dS/CFT correspondence.
Chapter 4 explored quantum field theory and its significance in understanding the n-point correlation functions central to the dS/CFT conjecture. The discussion began by analyzing free scalar fields, which provided the foundations for grasping the Feynman propagator and how it transforms into the n-point correlation function. It was essential to focus exclusively on free fields rather than interacting fields, as they exactly preserve the symmetries within dS space, making it easier to comprehend field behavior in the bulk. On the other hand, interacting fields introduce complexities that hinder a complete understanding of bulk field behavior. The subsequent section introduced conformal field theories in two and higher dimensions, emphasizing the role of conformal symmetries and operators within the framework of the dS/CFT. By understanding the correlation function of these operators within the CFT, a powerful tool emerges for comprehending the relationship between the boundary and bulk descriptions of dS space. This chapter served to deepen one’s understanding of the dS/CFT correspondence by examining free fields in QFT and the role of conformal symmetries in CFT.
Chapter 5 went into the intricacies of the dS/CFT correspondence, shedding light on several of its fundamental aspects. First, since the dimensionality of Hilbert space is a very important topic, there was further discussion which revolved around whether it is finite or infinitedimensional. The prevailing conclusion was that dS space likely exhibits a finite-dimensional Hilbert space, which is a remarkable characteristic of this spacetime. In the following section, boundary conditions in three-dimensional dS space were examined, as proposed by Strominger in his dS 3 /CFT 2 conjecture. The role of Green’s functions was also explored, illustrating their significance in comprehending the correlation properties of quantum fields in dS space and its dual CFT. Specifically, these functions act as a bridge that connects the two sides of the dS/CFT correspondence. In the final section, Strominger’s conjecture came to the forefront of the conversation. The striking result was the correspondence between spherical correlators in the CFT and bulk gravity correlators on every point of the past conformal boundary. Overall, Chapter 5 aimed to illuminate critical aspects of the dS/CFT correspondence.
Chapter 6 embarked on an explorative journey of two fascinating topics within the dS/CFT correspondence framework. The first section focused on Vasiliev’s higher-spin gravity theory, exploring the idea that there is a correlation between higher-spin gravity with Neumann (Dirichlet) boundary conditions in EAdS 4 and a free (critical) interacting Sp(N ) CFT 3 on the boundary. Then, EAdS 4 can be analytically continued into dS 4 space. The next section shifted gears to investigating the relationship between string theory and dS space. There have recently been more ideas exploring the intriguing possibility of circumventing no-go theorems through metastring theory, so this was discussed in some detail. The intriguing analogy between the dS holography and the BCS wavefunction of superconductivity was also highlighted, shedding some light on the quantum-gravitational and non-perturbative nature of dS holography. In the final section, the proposal for an “elliptical” dS space offered a unique perspective on exploring physics within the causal patch of dS space. Although acknowledging its current impracticality, 53 the potential of this theory to unveil new physics within dS space while hinting at the existence of a finite-dimensional Hilbert space was thoughtfully discussed. Collectively, Chapter 6 detailed some of the interconnections between higher-spin gravity, string theory, and modern approaches towards the dS/CFT correspondence.
The hope of this thesis is to successfully capture the intricate interplay between dS spacetime, holography, and other fundamental theories, shedding light on their complex relationships. In addition, recent work has highlighted the new innovative approaches aimed at understanding the challenges inherent to the dS/CFT correspondence, so there will undoubtedly be continued progress made in this dynamic field.
54
This appendix contains the full conformal algebra for the d ≥ 3-dimensional CFT discussed in Section 4.2.1. As a reminder, the infinitesimal generator given by translations is P µ = −i∂ µ , and
arise from scale transformations and rotations, respectively, and L µν is known as the angular momentum operator. Finally, performing a special conformal transformation results in the
Together, these conformal generators make up a conformal algebra, which can be fully described by the following commutation relations:
(A.1)
Any other commutation relations have vanished. Using these properties of the conformal algebra, one is able to calculate the correlators and correlation functions of the CFT.
I want to start by thanking my supervisor, Niko Jokela, for introducing me to this topic and giving me valuable insights every step of the way. I am also incredibly grateful for my grandparents, Ray and Nancy, my dad, Greg, my husband, Christian, my mom, Susan, and my good friend, Zhang Na (张娜). Their love and support has gotten me through this amazing chapter of my life. Finally, I must offer an additional thanks to my dad for introducing me to Star Trek at a young age and indulging me in endless conversations on space. Without him, I would not be where I am today.
55
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