ds/cft duality in cosmology
TRANSCRIPT
dS/CFT Dualityin
Cosmology
Ramtin Amintaheri
Supervisor: Dr Toby Wiseman
Submitted in partial fulfillment of the requirements for the degree of Master of Science in Quantum Fieldand Fundamental Forces of Imperial College London
September 2015
Contents
1 Introduction 11.1 motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Conformal Field Theory 42.1 Conformal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Conformal Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Special Conformal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.3 Conformal Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.4 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Conformal Fields in Two Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Complexification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 The Witt Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.3 Global Conformal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Conformal Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5.1 Two-Point Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Anti de Sitter Space 113.1 Constant Curvature Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Einstein Static Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Geometry of Anti de Sitter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3.1 Global Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.2 Poincare Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4 Matter Field in AdS Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 AdS-CFT Duality 174.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 The Field - Operator Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3.1 Bulk-to-Bulk Green Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3.2 Bulk-to-Boundary Green Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4.1 Two- Point Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.5 Mass-Dimension Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.6 Radial Direction-Energy Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.7 Holographic Renormalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.7.1 Domain Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.8 Example of the Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.8.1 Supergravity and P-Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.8.2 String Theory and D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
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4.8.3 Maldacena’s Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.8.4 Limits of Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 De Sitter Space 295.1 Geometry of de Sitter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2.1 Global Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2.2 Planar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2.3 Hyperbolic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.4 Matter Field in de Sitter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6 dS-CFT Duality 346.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.1.1 Asymptotic Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2 The Field - Operator Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.3 Mass-Dimension Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.4 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.5 Green Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.6 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.7 Example of dS-CFT Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.7.1 Higher-Spin Gravity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.7.2 EAdS - dS Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.7.3 GKPY Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.7.4 O(N) - Sp(N) Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.7.5 dS4 / Sp(N) CFT3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7 Cosmology 457.1 Inflationary Epoch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.1.1 Metric Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.1.2 gauge transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.1.3 Statistics of Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.1.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.1.5 Cosmological Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.2 RG-Flow Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.3 Holography for Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.3.1 Domain-Wall / Cosmology Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.3.2 Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.3.3 CFT Two-Point Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.3.4 Weak versus Strong Gravitational Coupling . . . . . . . . . . . . . . . . . . . . . . . . 53
7.4 Holographic Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.4.1 Prototype CFT Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.4.2 Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.4.3 Cosmological Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8 Conclusion 598.1 Futer Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.2 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
A Anti de Sitter Equation of Motion a
B Anti-de Sitter Bulk-to-Boundary Propagator c
C Anti de Sitter Two-Point Function e
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D De Sitter Green Function gD.1 Asymptotic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g
E De Sitter Two-Point Function iE.1 Asymptotic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . j
F Holography for Cosmology k
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Abstract
In this dissertation, gauge/gravity correspondence in a de Sitter space, and their application in the fieldof cosmology is studied. Basic material necessary to understand the topic including conformal field theory,and geometrical properties of AdS and dS spaces are reviewed. The well-constructed counterpart theory ofAdS/CFT duality, and the renowned Maldacena’s conjecture are also explained. dS/CFT correspondencehas some special features, time is holographically emergent, and non-unitary conformal field lives in theasymptotic future of the space time. In dS/CFT theory universe is an RG flow from the far future UVpoint to the far past IR fixed point. We show that it is possible to formulate a holographic framework forthe inflationary universe, despite the fact that the direct map between cosmology and dual gauge field is notunderstood. A holographic model is proposed that is able to provide predictions that are compatible withrecent observations. It is also argued that predictions of this model in the perturbative regime can describestrongly-coupled quantum gravity state of the early universe, and offer some solutions for the initial conditionproblem.
Chapter 1
Introduction
Gauge/gravity duality plays an important role in our understanding of modern physics as it connects twosupposedly different types of physical theories. On one side is the gauge theory which deals with quantumfields, and on the other side is gravitational theories like general relativity. The duality states that anytheory of gravity in a given space-time can be completely described by a conformal field theory living onthe boundaries of the space-time. [28] Since field theory is in one dimension lower, it therefore functions asa hologram encoding the information about higher-dimensional gravity formalism. [40] Any quantity in onetheory has a corresponding counterpart in the other, as if there is a dictionary translating one formalism tothe other. [39]
There are two known classes of dualities: the first one is the well-understood AdS/CFT correspondencewhich deals with spaces of negative curvature, and the second is the newer version dS/CFT which is topicof this report. Differernt aspects of AdS/CFT connection have been studied for a long time, and it is alsofound to have realisations in different field of physics in particular quantum chronodynamics and condensedmatter. [54, 53] Nevertheless anti-de Sitter does not support expansion of space-time while all observationalevidence indicates that expanding universe has a positive cosmological constant. In addition at early times,during inflationary epoc universe was a Sitter space, and in far future it will turn into a de Sitter again.[17, 18] Therefore there is a need for formulation of the dS/CFT correspondence to apply gauge/gravityduality to our universe.
Notion of principle of holography stemmed from G. t’Hooft, and later L. Suuskind works on the blackhole thermodynamics in 1993. [28, 29] Later in 1995 Polchinski showed that p-branes are the same objectsas the D-branes [27]: field theory describing dynamical endpoints of open strings on the brane correspondsto the supergavity solutions in the space-time. Researches conducted following Polchinski’s work finally ledinto discovery of the correspondence between gauge theories and gravity.
In 1997 Maldacena published a paper on his conjecture about AdS/CFT correspondence that is regardedas the birth of gauge/gravity duality. [31] He could use type IIB string theory on a AdS5 × S5 backgroundto describe the dual N = 4 super Yang Mills theory living on a four-dimensional D3 brane. His work excitedgreat interest among string theorists, and later was completed by S. Gubser, I. Klebanov, Polyakov, andWitten [32, 33].Soon after a complete gauge/gravity dictionary was developed.
However all of the mentioned realisations was centred on space-times of negative cosmological constant,and left this curious question unanswered that how we can apply the holography to the real universe withpositive cosmological constant. Despite considerable research there was no success in solving string theory inde Sitter space as a conformal field on the boundary. [56] It was not until Strominger became interested in theearly works of Brown and Henneaux. They used asymptotic symmetries and central charges to demonstraterelation between AdS3 and CFT2 without any reference to string theory [45]. In 2001 Strominger finallypublished his paper and dS/CFT duality was brought to the physics’ world attention. Along the lines ofAdS/CFT correspondence, he could show that quantum gravity on dS3 is related to a Euclidean conformalfield of one lower dimension.
In the same year he wrote another paper about inflation and dS/CFT correspondence showing thatthe main inspirations to find de Sitter analog of the duality stem from cosmology [75].His work was mademore precise by other scientists in particular K. Skenderis who could extend the holographic renormalisation
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formalism to de Sitter space. He even could manage to offer a phenomenological prototype that providespredictions consistent with latest observational data. [76, 77]
However in absence of string theory there was no example of the correspondence at the fundamental level,and this problem somehow stymied the progress in the subject. Finally recently in 2011 Strominger couldexplain that candidate theory in de Sitter space is higher spin gravity introduced in early 90’s by Vasiliev[62,63]. By an analytical continuation of Giombi-Klebanov-Polyakov-Yin (GKPY) duality [69, 66, 71, 72] hecould establish that higher spin gravity in dS4 is dual to a fermionic Sp(N)-invariant CFT3 on the Euclideanboundary space. [61]
1.1 motivation
From simple Newtonian models of mutual attractive forces through to Einstein’s geometrodynamical treat-ment and more modern notions of quantum gravity and string theory, gravitational theories are becomingmore and more sophisticated. In general current gravity theories are complicated and not fully understoodin all details, many gravitational systems therefore can not be studied analytically. In addition numericalapproximations can not cover the whole physics, and this imposes a limitation on the scope of our knowledge.Gauge gravity duality opened totally new avenues in fundamental physics as quantum fields has long beenstudied, and well understood. It provided new tools to solve problems in different theoretical areas of gravitythrough the corresponding field theories. [19]
Apart from this, AdS/CFT duality has significant physical interpretations: One of the fundamental forcesof the nature that is gravity is a phenomenon emerging from the underlying quantum fields. Furthermoreone of our microscopic dimensions is not fundamental and is just emergant. In other words the world can bethought of as three-dimensional information. But dS/CFT correspondence even brought in more unprece-dented results, ’time’ is emerging from more fundamental temporally-frozen quantum theories. Universe is afluid flowing from the far future infinity back to the far past in the space of renormalisation group theories.[29, 76]
Inflation theory is undoubtedly one of the cornerstones of the contemporary cosmology. It has providedthe best solutions to the classical cosmological problems about horizon, flatness and monopole issues in the oldbig-bang theory, and made correct predictions about formation of the large structures. Current observationsconfirm that universe is spatially flat, homogeneous, and isometric with small nearly Gaussian perturbationswhich has an almost scale invariant spectrum. Nevertheless it turns out that inflation is not a completetheory yet as it suffers from some problems with cosmological constant, super-planck scales, and particularlyinitial singularity. This theory does not remove the original singularity and can not provide any value forinitial conditions of the inflationary epoc. [80, 81]
Gauge-gravity correspondence, on the other hand, is a strong-weak duality which means when one theoryis strongly interacting the other one is weakly coupled. Accordingly the unknown area of strongly coupledquantum gravity corresponds to a weakly interacting quantum field theory which is theoretically very wellunderstood. So holographic model can proceed beyond the standard inflation formalism, and potentiallycan provide unrivalled predictions about initial values , and hopefully it will shed light on mysterious pre-inflationary era of the universe. [39, 20]
Furthermore it is desirable that a theory of inflation does not suffer from ultra-violet divergences. Asa result considerable amount of work in the current research is focused on expressing the inflation in thelanguage of string theory. The holographic model explained here features a three-dimensional dual quantumfield and is super-renormalisable and hence UV complete. [76, 77]
1.2 Outline
Gauge/gravity correspondence draws upon various areas of theoretical physics. It is assumed that readerhas an introductory knowledge of quantum field theory, general relativity, string theory, super-symmetry,black holes, and cosmology. Nevertheless, a brief review of conformal field theories, spaces of constantcurvature particularly de Sitter, and Anti-de Sitter, and inflationary cosmology will be provided in thefollowing chapters.
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We start with conformal fields which are a generalisation of quantum fields with an additional scalesymmetry. It is aimed to provide a proper description of geometrical and physical properties of CFT bystudying the structure of conformal groups, algebra, and associated symmetries. At the end, effect of con-formal symmetry on the general form of the correlation functions as the main physical observables will beinvestigated.
In the third chapter we start studying study space forms in details because holographic framework appliesto the manifolds that asymptotically have a constant curvature. Geometrical properties of anti-de Sitterwhich is a space with negative constant curvature are reviewed. Most commonly used coordinate systemsare introduced, and it is shown how they can describe topology and different symmetries of AdS. Generalsolutions to the equation of motion that are derived in the last section will be referred to frequently in thispaper.
Having reviewed conformal properties of the quantum fields on one hand, and geometrical and physicalaspects of anti-de Sitter, on the other hand,in chapter 4, we proceed to discuss about AdS/CFT correspon-dence as the first and best understood realisation of gauge/gravity duality. Correspondence is motivated byidentification of isometry group of AdS with conformal group of the dual field. The one-to-one map betweenoperators of QFT, and the fields in the bulk is explained. Subsequently green functions of the theory arederived which enable us to find an explicit expression for the correlation functions. In the last section, bymaking a comparison between predictions of the Type IIB supergravity and N = 4, SYM superstring theoryon parallel D3 branes, the renowned Maldacena’s conjecture is explained in details.
In chapter 5 we start studying spaces of positive curvature by introducing geometrical properties of deSitter manifold. Common coordinate systems are introduced, and their applications are explained. Specialfeatures of de Sitter space including existence of a horizon, problem of observables, and different asymptoticboundaries are explained. In the last section dynamics of scalar matter field are studied, and equations ofmotion in this space are solved.
Up to this point in the report, all the discussions about the gauge/gravity duality centred around thespaces of negative curvature which based on observational data do not apply to the real universe. In chapter6, therefore, we start studying the dS/CFT correspondence, and explore the similarities and differencescompared to the more familiar AdS/CFT counterpart. At first it is explained how asymptotic isometrygroup of a de Sitter space is identified with conformal symmetry group on the boundary. This acts as themain motivation to define the correspondence between conformal theory and de Sitter fields. Next differententities on both sides of the correspondence are compared and a duality map is constructed. In particularit is stressed that in dS/CFT time is holographically emergent, and thus notion of unitary time evolution isno longer valid in this theory. In the last section we investigate the correspondence at a more fundamentallevel by introducing higher spin formalism of gravity and defining the dual anticommuting Sp(N)-invariantconformal field theory.
In the last chapter we begin exploring the dS/CFT in the context of cosmology as the main topic ofthis report and the most important application of the duality. In the first section an introductory review ofperturbation theory during inflation is provided. Then the duality between time evolution of the universe,and renormalisation group flow is explained. In the first step towards formulation of a holographic frameworkfor our universe, the correspondence between domain-wall spaces and cosmology is defined. Subsequentlyusing the well-established AdS/CFT formalism observables are mapped to the equivalent quantum fieldcorrelators of the domain-wall, and finally by analytical continuation we find an expression for the QFT dualthe original cosmology. We finish this report by a detailed study of a proposed holographic prototype, anddemonstrate that how predictions of this quantum model complies with latest observational data.
Before proceeding to detailed scientific discussions it is worth clarifying some points about the notation.Summation convention is used in this article, that is repeated indices are summed over. Also note
that repeated covariant (contra-variant) indices are also summed, and it should be interpreted as they arecontracted with a Kronecker delta function, i.e. xµxµ ≡ δµνxµxν , and xµxµ ≡ δµνxµxν .
Space-time coordinates have Greek indices while for spatial dimension Latin indices are used.We adopt the notation that coordinates on the boundary manifold xµ are written in bold compared to
the bulk space-time coordinates xµ.
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Chapter 2
Conformal Field Theory
Conformal symmetry is a generalisation of Poincare invarience. In a simple language, when working withconformally invariant fields we are interested in theories that are the same for observables at different scales.Some field theories like four dimensional Yang-Mills theory are conformally invariant, in general all fieldsliving at fixed points of renormalisation group flow are conformal. Most quantum fields have a UV unstablefixed point, flowing towards another trivial IR fixed point in the space of renormalisation groups. In thispaper we are interested in conformal fields because they correspond to the gravitational theories in spacesthat are asymptotically (anti) de Sitter.
In the first part of this chapter conformal transformation in an arbitrary dimension is defined, then themathematical prescription of conformal groups, generators of infinitesimal transformation, conformal algebra,and finally associated symmetries is provided in the following sections. Subsequently we specialise to thecase of infinite dimensional field theory in a two-dimension space. Then the main properties of a conformalfield are reviewed, and finally correlation functions under conformal invariance restriction are computed.
2.1 Conformal Transformation
Conformal transformation is a map between two manifolds such that: [11]
g′µν(x′) = Ω2(x)gµν(x) (2.1)
Where gµν and g′µν are original and transformed metrics. Scale factor Ω is a continuous, non-vanishing,finite, real function. In this paper always two manifolds are the same which implies g′ = g. Note in case of aflat metric, given a unity scale factor Ω(x) = 1, conformal transformation turns into Poincar transformationwhich is essentially Lorentz transformation and translation.
Geometrically a conformal transformation can be defined as a map which locally preserves the anglebetween any two curves on the manifold (figure 2.1). [10]
Conformal transformation 2.1 consists of a Weyl rescaling gµν(x)→ Ω2(x)gµν(x), followed by a coordinatetransformation (diffeomorphism) x→ x′ so that Ω2(x)gµν(x)→ g′µν(x′). It can be shown as: [10]
g′αβ(x′)∂x′
α
∂xµ∂x′
β
∂xν= Ω2(x)gµν(x) (2.2)
A significant feature of conformal maps is that they preserve the casual structure of space-time. For anarbitrary tangent vector uµ, it is clear that:
gµνuµuν → g′µνu
µuν = gµνΩ2(x)uµuν (2.3)
So if a curve is originally time-like/null/space-like with respect to g, it remains the same with respect tog′; hence invariance of casual structure under conformal mapping. Also null geodesics in the original metricare transformed into null geodesics. However it is not necessarily true of time-like/ space-like geodesics, e.g.geodesics of a massive particle dose not correspond to its geodesic after conformal transformation. [3, 11]
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Figure 2.1: Conformal transformation in 2-dimensional Euclidean surface. (reproduced from [10])
2.2 Conformal Group
Conformal group is the group of all invertible and finite conformal transformations, i.e. transformationswhich preserve conformal structure of the manifold. Global conformal group like any other connected Liegroup can be built upon generators of infinitesimal transformation. In the following sub-section relevantinfinitesimal transformations will be computed.
2.2.1 Generators
Conformal fields considered in this dissertation live on a a flat pseudo-Riemannian manifold with metricηµν = diag(−1, ...,+1, ...). Given a general infinitesimal coordinate transformation to the first order:
x′µ = xµ + εµ(x) (2.4)
It turns out that εµ can be at the most quadratic in xµ for a conformal scaling of the form 2.1: [10]
εµ(x) = aµ + bµνxν + cµνλx
νxλ (2.5)
Where aµ, bµν , and cµνλ are arbitrary coefficients and cµνλ is symmetric in the last two indices. There arethree possibilities available. If εµ(x) is a scalar then εµ = aµ, and we have infinitesimal translation:
xµ → xµ + aµ (2.6)
with momentumPµ = −i∂µ (2.7)
as the corresponding generator. [5]If εµ(x) is linear, then bµν can be divided into a symmetric part ηµν , and an anti-symmetric part wµν such
that bµν = αηµν + ωµν . Then symmetric part gives infinitesimal scaling x′µ = (1 + α)xµ with generator [5]
D = −ixµ∂µ (2.8)
Finite scaling can be written as:xµ → λxµ (2.9)
Where λ = 1 + α+O(α2)The anti-symmetric part describes infinitesimal rotation x′µ = (δµν + ωµν )xν which has angular momentum
[5]Lµν = i (xµ∂ν − xν∂ν) (2.10)
as the generator. The finite rotation is usually noted as:
xµ →Mµν x
ν (2.11)
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Figure 2.2: Special Conformal transformation in 2-dimensional Euclidean surface
and rotation matrix is expanded like Mµν = δµν + ω(x)µν +O(ω2)
Finally with εµ(x) being quadratic in xµ, one can reach to the restriction [10] cµνλ = ηµνcλ+ηµλcν−ηνλcµwhere cµ = 1
dcνµν . This will give rise to infinitesimal form of so-called special conformal transformation
x′µ = xµ + 2(ηνλx
νcλ)xµ + cµx2. The associated generator is: [10]
Kµ = −i(2xµx
ν∂ν − x2∂µ)
(2.12)
This type of transformation will be studied in more details below.
2.2.2 Special Conformal Transformation
Special conformal transformation (SCT) is a type of spherical wave transformations (transformations whichleave a spherical wave invariant), and as mentioned above, one of the generators of conformal symmetry.SCT’s could be obtained by a composition of space-time translation, and inversion. One can easily checkthat an inversion xµ → xµ/x2, followed by a translation→ x′µ = xµ/x2+cµ, and another inversion→ x′µ/x′2
give rise to the following expression for finite SCT (figure 2.2) : [10]
xµ → xµ + x2cµ
1 + 2xµcνηµν + c2x2(2.13)
Where expansion of denominator for small values of cµ results in infinitesimal version of transformationdiscussed earlier. Note that for an arbitrary vector cµ, there is always a point xµ on the manifold suchthat the denominator vanishes. It means that xµ is conformally mapped to infinity. In order to avoid thisproblem, it is required to define conformal fields on a compactified space which includes points at infinity.This will become more clear when CFT’s on compactification of space-time boundaries are discussed in thefollowing chapters. [10, 37]
2.2.3 Conformal Algebra
To summarise the results so far, we introduced the finite conformal transformations and the related generatorsas follows:
1. Pµ generator of transaltion xµ → xµ + aµ
2. Mµν generator of the Lorentz transformation xµ → Λµνxµ
3. D generator of scaling xµ → λxµ
4. Kµ generator of special conformal transformation xµ → (xµ + x2cµ)/(1 + 2xµcνηµν + c2x2)
6
Generators of infinitesimal conformal transformation constitute the conformal algebra which admits thefollowing commutation relations: [10]
[Mµν , Pρ] = −i(ηµρPν − ηνρPµ), [Mµν ,Kρ] = −i(ηµρKν − ηνρKµ),
[Mµν ,Mρσ] = −i(ηµρMνσ − ηµσMνρ − ηνρMµσ + ηνσMµρ), [Mµν , D] = 0, (2.14)
[D,Pµ] = −iPµ, [D,Kµ] = −iKµ, [Pµ,Kµ] = 2iMµν − 2iηµνD
and all other commutations vanishes.
2.2.4 Symmetries
In order to find the dimension of the algebra, one needs to count the number of generators. Total number ofgenerators for Lµν , Pµ, Kµ, and D are:
1
2d(d− 1) + d+ d+ 1 =
1
2(d+ 2)(d+ 1)
which equals the dimension of SO(p+ 1, q + 1) symmetry group in a Mp,q manifold where p+ q = d,.
The conformal group on a Mp,q manifold is SO(p+ 1, q + 1).
If inversion xµ → xµ/x2 is included, then conformal group will be O(p+1, q+1) with connected subgroupSO(p+ 1, q + 1). Special cases of interest are d-dimensional Laplacian space Md−1,1 where conformal groupis SO(d, 2), as well as Euclidean d-dimensional manifold with a SO(d+ 1, 1) conformal group.
Familiar generators of special orthogonal group can be defined via: [10]
Jµν = Mµν , Jµd =1
2(Kµ − Pµ), Jµ(d+1) =
1
2(Kµ + Pµ), Jd(d+1) = D (2.15)
with µ, ν = 0, . . . d− 1. In matrix form it is clear that Jmn is antisymmetric with a Lorentz subgroup:
Jmn =
Mµν12 (Kµ − Pµ) 1
2 (Kµ + Pµ)− 1
2 (Kµ − Pµ) 0 D− 1
2 (Kµ + Pµ) D 0
(d+2)×(d+2)
The usual commutation relation is defined as: [11]
[Jµν , Jρσ] = i (ηµσJνρ + ηνρJµσ − ηµρJνσ − ηνσJµρ) (2.16)
2.3 Conformal Fields in Two Dimension
In this section we specialise to the case of conformal symmetry in two dimensions. It is of special importancebecause the algebra is infinite-dimensional in this case. Consequently two-dimensional conformal fields havean infinite number of conserved quantities, which means symmetry considerations alone suffice to make themcompletely solvable. [10]
2.3.1 Complexification
Introduce complex variables:z = x+ iy, z = x− iy (2.17)
which implies:
∂z =1
2(∂x − ∂y) , ∂z =
1
2(∂x + ∂y) (2.18)
7
Also metric will be:ds2 = dx2 + dy2 = dzdz (2.19)
In two dimensions, the restriction that ε(x) be at the most quadratic in x no longer applies. In fact itturns out that ε must be either an arbitrary function of z or of z. So the complexified version of infinitesimaltransformation 2.4 can be shown by a holomorphic function f(z) = z + ε(z) or f(z) = z + ε(z). That isinfinitesimal two-dimensional transformation is an analytic coordinate transformation: [10]
z → f(z), z → f(z) (2.20)
Which implies metric will transform like:
ds2 → ∂f
∂z
∂f
∂zdzdz (2.21)
So scale factor in two dimensions is Ω = |∂f∂z |2.
Minkowski Manifold
The complexification above was done on an Euclidean manifold, however one can generalise it to a Minkowskispace-time by defining the new coordinates u and v:
u = −t+ v, v = t+ v (2.22)
which changes the metric to:ds2 = −dt2 + dx2 = dudv (2.23)
The infinitesimal transformation is therefore given by:
u→ f(u), v → g(v) (2.24)
Euclidean and Minkowski versions are related to each other by a Wick rotation. So without loss ofgenerality we continue the discussion on a Euclidean manifold, and treat z and z as two independent variables.
2.3.2 The Witt Algebra
In order to obtain infinitesimal transformation 2.4 in two dimensions, one needs to expand ε around zero inLaurent series:
z′ = z + ε = z +∑n
an(−zn+1
),
z′ = z + ε = z +∑n
an(−zn+1
)The corresponding generators of infinitesimal transformation is thus given by: [10]
ln = −zn+1∂z, ln = −zn+1∂z (2.25)
where ln is the generator of transformation with parameter ε = an(−zn+1
). As usual to define the
algebra, it is required to work out the commutators of generators: [10]
[lm, ln] = (m− n)lm+n,[lm, ln
]= (m− n)lm+n, (2.26)[lm, ln
]= 0
This algebra is known as Witt algebra. Since n ∈ Z the algebra is infinite dimensional which is a uniquefeature of conformal group in two dimensions. 1
1In fact each commutation relation is an independent copy of Witt algebra because these two copies commute. This alsojustifies our assumption about considering z and z as two independent variables.
8
2.3.3 Global Conformal Transformation
The Witt algebra is generated by infinitesimal conformal transformation, in this section we proceed to considerthe global transformations in two dimensions. Note that generators of Witt algebra are singular at certainpoints:
1. Consider 2.25 the generator ln = −zn+1∂z is non-singular at z = 0 only if n ≥ −1.
2. In order to study the behaviour near z → ∞, we perform change of variable w = 1/z, and investigatew → 0. It can be seen that ln is non-singular at w = 0 only if n ≤ 1.
It can be concluded that the only globally defined generators of conformal transformation are l−1, l0 and l1.Therefore conformal group in two-dimensions is locally described by Witt algebra, and has infinite dimensions,in contrast, globally it is generated by the mentioned three generators and thus has six dimensions.
2.4 Conformal Fields
All relativistic quantum fields in the Standard Model of particle physics have Pioncare group ISO(3, 1)symmetry. Conformal Field is a relativistic quantum field (in d-dimension) which is invariant under conformalgroup SO(d, 2) not only Pioncare group ISO(d− 1, 1).
Physicists are interested in representations with operators that have well-defined scaling properties whichmeans operators that are eigenfunction of scaling operator D (with eigenvalue −i∆). ∆ is refereed to asscaling dimension or conformal dimension. Under the scaling xµ → λxµ the operator is charged like:
O(x)→ λ∆O(λx) (2.27)
Back to commutation relations 2.2.3 it can be noticed that action of momentum Pµ increases the scalingdimension while Kµ decreases the dimension. It turns out that there is a minimum value of conformaldimension, and the corresponding operator which is annihilated upon action of Kµ at the origin x = 0 iscalled primary operator. 2 Commutation relations of the conformal operator with generators of conformalgroup are given by: [37]
[Pµ,O(x)] = i∂µO(x), (2.28)
[Mµν ,O(x)] = i(xµ∂ν − xν∂µ) +AµνO(x), (2.29)
[D,O(x)] = i(−∆ + xµ∂µ)O(x),
[Kµ,O(x)] = i(x2∂µ − 2xµxν∂ν + 2xµ∆)− 2xνAµνO(x) (2.30)
where Aµν is a matrix in the finite representation of the Lorentz group. It is worth mentioning thatbecause representations of conformal group are eigenfunctions of scaling operator, they are not necessarilyeigenfunctions of the Hamiltonian or mass operators. [10]
2.5 Correlation Functions
Correlation functions are the building blocks of the physical observables. High symmetries of conformal groupin comparison with Poincare , imposes strong restriction on the correlation functions. General form of themost useful correlators that are two-point functions will be reviewed in this section.
2.5.1 Two-Point Correlation Function
Two-point correlation function of operators O1 and O2 with dimensions ∆1 and ∆2 is denoted as g(x1, x2) =〈O1(x1)O2(x2)〉.
Invariance under translation xµ → xµ + aµ means that the two-point function should depend on thedifference between the two points: [10]
g(x1, x2) = g(x1 − x2) (2.31)
2 For a scalar field the minimum bound is ∆ ≥ (d− 2)/2 which describes the free scalar. [11]
9
Invariance under rescaling xµ → λxµ implies:
〈O1(x1)O2(x2)〉 → 〈λ∆1O1(x1)λ∆2O2(x2)〉λ∆1+∆2g (λ(x1 − x2)) ≡ g(x1 − x2)
This amount to the following form:
g(x1 − x2) =d12
(x1 − x2)∆1+∆2(2.32)
where d12 is a structure constant. Finally correlation function is invariant under special conformal trans-formation which is composed of translation and inversion. Since restriction under translation was alreadyconsidered, we concentrate on inversion symmetry:
〈O1(x1)O2(x2)〉 → 〈 1
x2∆11
1
x2∆22
O1(− 1
x1)O2(− 1
x2)〉
=1
x2∆11 x2∆2
2
d12(− 1x1
+ 1x2
)∆1+∆2≡ d12
(x1 − x2)∆1+∆2
Last equality holds if scaling wights of the two operators are the same ∆1 = ∆2 ≡ ∆. So ultimatelytwo-point correlator will be fixed by conformal symmetry as:
〈O(x1)O(x2)〉 =d12
(x1 − x2)2∆(2.33)
10
Chapter 3
Anti de Sitter Space
Anti-de Sitter is a manifold with constant negative curvature analogous to Euclidean hyperbolic space.Spaces of constant curvature play a significant role in gauge/duality correspondence as existence of such acorrespondence depends on cosmological constant. It turns out that boundary of compactified anti-de Sitteris the compactified Minkowski space where the dual field theory lives. It is therefore important to be familiarwith interior and asymptotic regions of this space, in order to have a better understanding of correspondencebetween gravity and conformal field theory.
This chapter starts with a review of spaces of constant curvature which includes anti-de Sitter and deSitter. Following this introduction geometrical properties of anti-de Sitter space are studied. Then we proceedto introduce the most common coordinate systems and show how they provide a better understanding ofsymmetries and topology. Final section focuses on the presence of matter field in this space-time, and generalsolutions are derived. We will refer to these results frequently throughput this paper.
3.1 Constant Curvature Spaces
The spaces of constant curvature or space forms are the simplest examples of Riemannian manifolds wherethe sectional curvature K has the same value at any point on the manifold. As a result Ricci Scalar R whichis defined as a multiple of the average of the sectional curvatures at a point is constant and will be given byR = n(n− 1)K.[1]
In general for manifolds of more than three dimensions d > 3 neither Ricci scalar R or Ricci tensor Rµν issufficient to describe the curvature and the full Riemann tensor Rµνλσ is needed. The remaining componentsof curvature tensor is represented by so-called Wyle Tensor Cµνλσ. [2]
However one can show that for a space form Riemann tensor can be locally defined by Ricci scalar alonethrough the condition:
Rµνλσ =1
12R (gµλgνσ − gµσgνλ) (3.1)
This means that Wyle tensor vanishes and Cµνλσ = Rµν − 14Rgµν = 0. A straight forward calculation
using contracted Bianchi identity verifies that Ricci scalar is constant throughout the manifold. [2, 3]In order to get a physical interpretation of space-time one needs to calculate the Einstein tensor Gµν :
Gµν = Rµν −1
2Rgµν = −1
4Rgµν (3.2)
So spaces of constant curvature can be considered as the vacuum solutions to the Einstein equation withcosmological constant Λ = 1
4R. [3]Space forms are important since they are locally maximally symmetric i.e.they have 1
2n(n + 1) numberof local isometries.Also any maximally symmetric space has a constant curvature. These manifolds havetranslation invariance and thus are homogeneous.[1]
11
3.1.1 Einstein Static Universe
In the following sections, in many occasions, manifolds of constant curvature will be mapped to the staticuniverse, so it is useful to have an overview of this cosmological space-time at this point.
Einstein static universe is a stationary cosmological model being temporally infinite but spatially finite. Itis completely homogeneous with spatial spherical symmetry and topology R×S3.[3]. The metric is describedby: [26]
ds2 = −dt2 + a
(dr2
√1− r2
+ r2dθ2 + r2sin2θdφ2
)(3.3)
Note a is a constant independent of time. By change of variables r′ = Arcsin r metric can be re-writtenas:
ds2 = −dt2 + dr′2 + sin2r′(dθ2 + sin2θdφ2
)(3.4)
So one can regard Einstein static universe as an embedded cylinder:
x2 + y2 + z2 + w2 = l2 (3.5)
in a five-dimensional Minkowski manifold ds2 = −dt2+dx2+dy2+dz2+dw2. Where l is radius of cylinder,and has length dimension. In this paper for better visualisation, we usually suppress two coordinates θ andφ, and represent the model as a cylinder in a three dimensional space.
Einstein universe is also a solution of Friedmann-Robertson-Walker (FRW) metric with a positive cur-vature k = 1 if cosmological constant is at a critical value Λc = 3/a2 + 8πGρm where ρm is the matterenergy density. In this case the attractive force of the matter density is balanced by the repulsive force ofthe cosmological constant. But it turns out to be an unstable solution thus not describing a physical state.
3.2 Geometry of Anti de Sitter Space
Given a (d+ 1)-dimensional manifold Mp,q with metric:
ds2 = −dT idT i + dXjdXj i+ j = 0, . . . d (3.6)
Anti de Sitter space AdSd can be realised as a hyperboloid induced in one less temporal dimensionthrough:
−T iT i +XjXj = l2 i+ j = 0, . . . d (3.7)
Parameter l referred to as anti de Sitter radius has dimensions of length being the only scale of thegeometry. In physics we are interested in the special case where q = 2 resulting into an embedded space-timeAdSp,1. Anti de Sitter metric is nondegenerate and has Lorentzian signature, it has topology S1 × Rd−1. [1]
O(d− 1, 2) is the isometry group of anti de Sitter in d dimensions, and has SO(d− 1, 2) as the connectedsubgroup.
In addition, anti de Sitter is the vacuum solution to the Einstein equation with negative cosmologicalconstant: [2]
Rµν −1
2Rgµν + Λgµν = 0 Λ < 0 (3.8)
Riemann tensor in AdSd manifold is given by: [3]
Rµνλσ = − 1
l2(gµλgνσ − gµσgνλ) (3.9)
Ricci tensor is proportional to the metric, so anti de Sitter space is a an Einstein Manifold. [3]
Rµν = −d− 1
l2gµν (3.10)
Ricci scalar is given by: [3]
R = −d(d− 1)
l2(3.11)
12
Figure 3.1: AdS3 space mapped into the 3-dimensional Einstein universe. Boundary θ = π/2 is an S1 circle.(reproduced from [37])
and finally cosmological constant will be: [3, 56]
Λ = − (d− 1)(d− 2)
2l2= −d− 2
2dR (3.12)
In a 4-dimensional space-time we have the relation R = 12/l2 = 4Λ which is the same result as in section3.1.
3.3 Coordinate Systems
Due to geometrical symmetries, anti de Sitter space admits a number of coordinate systems. In this sectionwe review global coordinates and Poincare patch as the most commonly used systems. These coordinatesprovide a better understanding about topology and symmetries of the AdSd, and we frequently refer to themin this paper.
3.3.1 Global Coordinates
Global coordinates (τ, ρ, ωi) covers the whole anti-de Sitter space, and obtained via parametrisation: X0 = l cosh ρ cos τ,X1 = l cosh ρ sin τ,Xi = l sinh ρ xi i = 2, ..., d
(3.13)
where xi is coordinate on a unit sphere Sd−2 such that xixi = 1. The metric takes the form: [3]
ds2 = l2(− cosh2 ρ dτ2 + dρ2 + sinh2 ρ dΩ2d−2 (3.14)
where ρ ≥ 0 and 0 ≤ τ < 2π. It can be seen that the metric has closed time-like curves in τ direction onS1, so we need to unwrap this dimension such that −∞ < τ <∞ and thus space-time is casual. This spaceis called the universal covering of AdS, and has topology R4. [3]
Isometry group of AdSd has the subgroup SO(d−1, 2) ⊃ SO(2)×SO(d−1) where SO(2) is time evolutionin τ direction, and SO(d− 1) corresponds to rotations in Sd−2. [37]
In order to investigate casual structure of AdS, one can demand tg θ = sinh ρ where 0 ≤ θ < π/2:
ds2 =l2
cos2 θ(−dτ2 + dθ2 + sin2 θdΩ2
(d−2))
13
Figure 3.2: Penrose diagram for AdS2 in Poincare patch. (reproduced from [37])
Now we can do a conformal compactification, and rescale the metric by cos2 θ/l2, then metric takes theform of Einstein static universe 3.4: [3]
ds2 = −dτ2 + dθ2 + sin2 θdΩ2(d−2) (3.15)
However in since 0 ≤ θ < π/2, anti-de Sitter space is mapped to half of Einstein universe.Note that constant τ hyper-surfaces has topology of a (d− 1)-hemisphere. Boundary of AdS corresponds
to θ = π/2 on this surface which is a Sd−2(figure 3.1). So boundary of con formally compactified AdSd i.e.R× Sd−2 is the conformally compactified Minkowski space Md−1,1.
3.3.2 Poincare Coordinates
Poincare coordinates (t, u, xi) cover half of the AdSd space, and are given by the parametrisation: [1] X0 = (1/2u)(1 + u2(l2 + xixi − t2)
),
X1 = (1/2u)(1− u2(l2 − xixi + t2)
),
Xi = luxi i = 2, ..., d(3.16)
where u > 0, then the metric takes the form:
ds2 = l2(du2
u2+ u2(−dt2 + dxidxi)
)i = 2, . . . d− 1 (3.17)
Penrose diagram for AdS2 is illustrated in figure 3.2.In this system isometry group of AdS has subgroups SO(d − 1, 2) ⊃ ISO(d − 2, 1) × SO(1, 1) where
ISO(d− 2, 1) is the Poincare transformation on (t, xi) dimensions, and SO(1, 1) is scaling in the conformalsymmetry group SO(d− 2, 1).
By a change of coordinate z = 1/u the Poincare metric is commonly written as: [1]
ds2 =l2
z2
(dz2 − dt2 + dxidxi
)i = 2, . . . d− 1 (3.18)
In this metric boundary is approached via z → 0, while AdS interior is the region z →∞.Poincare metric can also be brought into radial coordinates by a change of variable z = le−r/l:
ds2 = dr2 + e2r/lηµν(dxµdxν) µ, ν = 1, . . . d− 1 (3.19)
In these coordinates boundary corresponds to r →∞, and the deep interior is approached as r → −∞.
14
3.4 Matter Field in AdS Space
Having discussed geometrical properties of Anti de Sitter manifold so far, in this section we study presenceof matter field in this space. The simplest model will be a single scalar field on a fixed AdSd background.The action in Poincare coordinates reads:
S =1
2
∫ √−ggµν∂µφ∂νφ−m2φ2
dd−1xdz (3.20)
Varying the action with respect to the field gives the Klein-Gorden equation:
(−m2) φ = 0 (3.21)
with usual definition of Laplacian:
φ = gµνφ,µ;ν =1√g∂µ (√ggµν∂νφ) (3.22)
Due to translation invariance on the x coordinates, we try separation of variables:
φ(z,x) = f(z)Φ(x) (3.23)
Following the steps specified in details in appendix A this leads to two independent equations for f(z)and Φ(x): [38]
(∂2x − k2)Φ(x) = 0 (3.24a)[
−zd∂z(zd∂z) +m2l2 + k2z2]f(z) = 0 (3.24b)
where k is momentum, and ∂2x indicates Laplacian on Minkowski x-plane. Solution to the Laplace equation
3.24a on the flat space are known to be the plane waves. Consequently the complete solution will be givenby superposition of plane waves and f(z):
φ(z,x) =
∫fke
ikx dd−1x
(2π)d−1(3.25)
This indicates that the solution is actually the Fourier transform of f(z). By inverting the Fouriertransformation above, one realises that fk is in fact, the solution in momentum space. So the momentum-space version of equation of motion 3.24b can be written as:
z2f ′′k − (d− 2)zf ′k − (m2l2 + k2z2)fk = 0 (3.26)
By a change of coordinates this equation amount to the modified Bessel equation (see appendix A) withthe solution given by: [23]
fk(z) = ak(kz)(d−1)/2Kδ(kz) + bk(kz)(d−1)/2Iδ(kz) (3.27)
where δ =√
(d− 1)2/4 +m2l2, and Iδ and Kδ are modified Bessel functions of first and second typerespectively. Imposing regularity in the interior and considering asymptotic behavior of Bessel functionsA.6, when approaching the boundary z → 0 the solution in the momentum space behaves like: [39]
fk ≈ φ−(k)z∆− + φ+(k)z∆+ (3.28)
where ∆± are given by ∆± = (d− 1)/2± δ that is:
∆± =(d− 1)
2±√
(d− 1)2
4+m2l2 (3.29)
We come back to this relation later. Finally applying the inverse Fourier transformation, the solution tothe equation of motion for a scalar field in an anti-de Sitter background will be given by:
15
limz→0
φ(z,x) = z∆−φ−(x) + z∆+φ+(x) (3.30)
The above asymptotic behavior of the field bulk when approaching the boundary will play a significantrole in gauge/gravity duality discussions in the following chapters.
16
Chapter 4
AdS-CFT Duality
Although the main focus of this report is on de Sitter spaces, understanding AdS/CFT correspondence isin fact a prerequisite in studying the positive curvature counterpart i.e. dS/CFT duality. Formulation ofgauge/gravity duality in de Sitter space was developed along the lines of the anti-de Sitter counterpart, andin many cases well-established results of AdS/CFT relation were used as a conjecture in dS/CFT version ofthe duality. 1
In preceding chapters, on one hand a conformally invariant description of quantum fields, and on theother hand gravitational properties of Anti-de Sitter geometry were studied. We are now ready to explainthe duality between these two fundamentally different theories. What is meant by duality is actually a one-to-one correspondence between fields and operators on both sides, and consequently a map between correlationfunctions in both theories.
Some literature start with an explicit example, usually the AdS5 × S5/N = 4, SYM correspondence dueto historical reasons, and then proceed to discuss the generalised prescription. In this dissertation, however,a top-to-bottom formalism is selected: we first explain the abstract AdS/CFT duality conjectures, and thenwill exhibit some examples of the correspondence.
In first section, we make a comparison between AdS space-time isometries and symmetries of conformalQFT as the initial motivation behind the duality. In the following section a precise abstract formulation ofgauge theory / gravity identity is discussed. In particular the correspondence between quantum field operatorson the boundary and gravity fields in the bulk is defined. Computation of correlation functions as the mainoperators of the theory is carried out in the next section. Subsequently a comprehensive mapping betweenboth sides of the duality will be exhibited in the following section. Next we proceed to give examples ofAdS/CFT conjecture, and start with the well-known Maldacena or AdS5×S5/N = 4, SYM correspondence.It is discussed how Typ IIB supergravity theory on AdS5×S5 geometry is identical to N = 4 super Yang-Milltheory on parallel D3 branes. In addition three different types of correspondence including weak, semi-strong,and strong forms will be reviewed.
4.1 Symmetries
Description of anti de Sitter space in radial coordinates 3.19 provides a suitable way to understand symme-tries:
ds2 = dr2e2r/lηµν(dxµdxν)
The boundary of AdS corresponds to the limit r →∞ where the field theory lives. The metric is clearlysymmetric with respect to the transformations r → r + a, and ηµν → e−2a/lηµν . Near the boundary thesecond term dominates, so ds2 = ηµν(dxµdxν) describes the flat metric of the quantum fields. But the fieldis invariant under the scaling of the metric η which means the field theory is conformal.
It is a necessary condition for AdS/CFT correspondence that unbroken symmetries on both sides of theduality should be the same. It was discussed that isometry group of AdSd is SO(d− 1, 1) which is identified
1In the last two chapters we will see explicit examples indicating how AdS/CFT prescription facilitates development ofdS/CFT formalism.
17
with the conformal symmetry group on a (d− 1)-dimensional flat space. This is the first step in establishinga duality map between gravity in the bulk anti-de Sitter and conformal field theory on the boundary region.
4.2 The Field - Operator Correspondence
The duality between two theories, in some sense, means that there exists a one-to-one mapping betweentheir observables. Since conformal field theories do not have an S-matrix or asymptotic state, we thereforeconsider operators, and related correlation functions as observables [10]. So it is expected that AdS/CFTduality shows a connection between fields in anti-de Sitter space, and operators of the conformally invariantfield theory.
The first step is to define a boundary condition. Recalling the general form of solutions in 3.30, wedemand that limz→0 φ(z,x) = z∆−φ−(x). This admits that near the boundary the bulk field factorises intoa product of a z-dependent part and a local operator φ−(x) ≡ φ0(x). It is prescribed that boundary valueof the local bulk field φ0(x) functions as the external source J(x) for the operators of CFT living on theboundary of AdS space: [37, 39]
φ0(x) = J(x) (4.1)
where x is to be understood as the coordinates on the boundary x ∈ ∂AdS. 2 Under this condition, it isconjectured that partition function of the bulk classical gravity Zcl equals the generating functional of theconformal field theory Z[J ]: [37, 38]
〈ei∫J(x)O(x)dd−1x〉 =
∫e−Scl[φ]Dφ
∣∣∣∣φ0(x)=J(x)
(4.2)
Left hand side describes CFT generating functional which is defined in a similar fashion to QFT via
Z[J ] = 〈e∫J(x)O(x)dd−1x〉,3 and right hand side is the classical gravity partition function Zcl =
∫e−Scl[φ]Dφ.
It immediately follows that there is a duality between the correlation functions in the two theories: [41]
〈O(x1)...O(xn)〉CFT ↔ 〈φ(x0,x1)...φ(x0,xn)〉AdS (4.3)
This is the main idea behind the gauge/gravity duality, meaning that any bulk fields φ is in a one-to-onecorrespondence with a conformally invariant operator Oφ.
The n-point correlation function for CFT operators is therefore given by: [40]
〈O(x1)...O(xn)〉 =−iδ
δJ(x1)...−iδ
δJ(xn)Zcl∣∣∣∣J=0
(4.4)
In this equation Zcl is classical gravity partition function.Looking at the left hand side of equation 4.2, because quantum generating functional Z[J ] includes
bubbles, the resulting correlation function will be disconnected. To have connected correlators we need todefine connected generating functional E[J ] via Z[J ] = e−iE[J].
On the right hand side, we use saddle point approximation for classical gravity theory which meansin the limit where action Scl is large, the integral is dominated by the points which extremise the action(saddle points) δScl/δφ = 0. So it is required to solve the equations of motion, and find the on-shellsolutions. Consequently classical gravitational partition function to the leading order is approximated by:Zcl ≈ e−Scl
∣∣on−shell.
Under these assumptions, equation 4.2 is simplified to: [37, 41]
iE[J ] = Son−shell[φ]
∣∣∣∣φ0(x)=J(x)
(4.5)
2The motivation behind this will become clear when the AdS5 × S5/N = 4, SY M example is explained in the last section.In this case boundary value of dilaton in the bulk corresponds to the coupling constant of super-Yang-Mill theory.
3Similar to the case for QFT, taking J(x)O(x) as interaction Lagrangian, Z[J ] will be sum of one-point vertices withoutexternal legs. Note J(x) acts as position-dependent coupling. Z[J] includes sum of infinite number of separate pieces (bubbles)which is simply Z[0].
18
Which means generator of connected correlation functions in the conformal theory is the on-shell classicalgravity action. Now quantum connected correlation functions can be obtained by taking functional derivativeswith respect to the on-shell gravitational action: [41]
〈O(x1)...O(xn)〉connected =δ
δJ(x1)...
δ
δJ(xn)Son−shell
∣∣∣∣J=0
(4.6)
In fact, the same reasoning applies to other types of fields. It turns out that bulk spinor field ψa is relatedto fermionic operator Oa on the boundary, bulk gauge field Aµ is dual to the boundary conserved currentJi, and bulk metric gµν corresponds to the energy-momentum tensor Tij : [20, 21]
φ(x0,x)↔ 〈O(x)〉 =δSon−shellδφ0(x)
(4.7)
ψa(x0,x)↔ 〈Oa(x)〉 =δSon−shellδψa(0)(x)
(4.8)
Aµ(x0,x)↔ 〈Ji(x)〉 =δSon−shellδAi(0)(x)
(4.9)
gµν(x0,x)↔ 〈Tij(x)〉 = 2δSon−shellδgij(0)(x)
(4.10)
where ψa(0)(x), Ai(0)(x) and gij(0)(x) are boundary values of the local spinor, gauge field and the metric.4
4.3 Green Functions
In this section, we review derivation of green function as the response of the the bulk system in presence of apoint source with a specified boundary condition. The results will be used in the next section in formulationof the correlation functions.
4.3.1 Bulk-to-Bulk Green Function
The bulk-to-bulk propagator is defined via:( +m2
)G(x; y) = − i
√gδd+1(x− y) (4.11)
We adopt the notation that xi (in bold) indicates the transverse (d−1)-dimensional coordinates, and xµ isd-dimensional bulk coordinates (z,xi). Recall that Klein-Gorden equation 3.21 had two linear independentsolutions 3.30 which were multiples of the modified Bessel functions of first and second type:
fk(z) = (kz)(d−1)/2Kδ(kz), Iδ(kz) (4.12)
Using these modes one can try to find the green function and make an ansatz of the form: [50]
G(x, y) =
∫z(d−1)/2h(k, z′)θ(z − z′)Kδ(kz)Iδ(kz
′) + θ(z′ − z)Kδ(kz′)Iδ(kz)e−ik(x−y) dd−1k
(2π)d−1(4.13)
One can check that this ansatz satisfy the boundary conditions at z = 0, and z →∞, and it is continuedat z = z′. By matching this two region at z = z′ we can find: h(k, z′) = −z′(d−1)/2. The expression abovethen can be integrated which gives the hypergeometric function: [39]
4Any field f has an asymptotic Fefferman-Graham expansion of the form f(z,x) =z2m
[f(0)(x) + f(2)(x) + . . .+ z2n
(f(2n)(x) + ln z2h(2n)(x) + . . .
)]with different values of coefficients f(i) and h(j), and
integers m and n. What we mean by local values of the field on the boundary is actually the leading term f(0)(x) inFefferman-Graham expansion. [41]
19
G(x, y) =C∆
2∆− d
(ξ
2
)∆
F
(∆
2,
1
2(∆ + 1),∆− d
2+ 1, ξ2
)(4.14)
the normalisation constant C∆ will be fixed later in 4.22, and
ξ =2zz′
z2 + z′2 + (x− y)2(4.15)
At short distance x− y → 0 bulk-to-bulk propagator behaves like: [39]
G(x, y) ≈ C∆z1z2
(z1 − z2)2 + (x− y)2(4.16)
4.3.2 Bulk-to-Boundary Green Function
When the point source is located on the boundary, we need to define another solution of Laplace equation 4.11known as Bulk-to-Boundary Green Function K(x0,x;y). For convenience we work in Poincare coordinate3.18. On the boundary x0 ≡ z → 0, and since 1/
√g ∼ z the right hand side of the equation vanishes and the
definition of green function is simplified to: [43](−m2
)K(z,x;y) = 0 (4.17)
As usual the response solution is given by the convolution with the source: [6]
φ(z,x) =
∫K(z,x;y)J(y)dd−1y (4.18)
As discussed source is local part of the field on the boundary J(x) ≡ φ0(x).The methodology used is to solve Laplace equation 4.11, investigate if it is singular at some points as
singularity implies sources, and finally identify the kind of source. Pick the bulk point to be at infinityz → ∞, then the transverse −dt2 + dxidxi part of metric 3.18 vanishes because l2/z2 → 0, so the spacebecomes a point and Laplace equation simplifies to : [43][
−zd∂z(z−d+2∂z) +m2l2]K(z) = 0 (4.19)
Try the ansatz K(z) = C∆z∆ which amounts to:[
∆(d− 1−∆) +m2l2]z∆ = 0 (4.20)
The solution to the above equation clearly has two roots ∆± = (d− 1)/2±√
(d− 1)2/4 +m2l2 which isthe same relation as 3.29.
Due to asymptotic behavior near the boundary larger root ∆+ is chosen. The green function ought to beinvariant under conformal transformations that are isometries of AdS space. Doing an inversion z → z
z2+x2 ,followed by a translation x→ x− y, the boundary-bulk propagator becomes: [39]
K(z,x;y) = C∆+
(z
z2 + (x− y)2
)∆+
(4.21)
where normalisation constant is fixed as (refer to appendix B):
C∆+=
Γ(∆+)
π(d−1)/2 Γ(∆+ − d−12 )
(4.22)
20
Accordingly K vanishes on the boundary z → 0 except at one point, it diverges where (x − y) → 0on the boundary. This suggests that propagator K is singular at x = y, and as shown in appendix BKz−∆− behaves like a delta function as z approaches the boundary. So the boundary condition for theboundary-to-bulk propagator will be: [37]
limz→0
K(z,x;y) = z∆−δ(d−1)(x− y) (4.23)
So using equation 4.18, near the boundary the field behaves like:
limz→0
φ(z,x) = z∆−φ0(x) (4.24)
That implies on the boundary, transverse dimension part of the field z∆− factorises, leaving a local fieldoperator φ0(x). [57]
Since the field φ0(x) is dimensionless, the boundary field φ(z,x) acting as the source J(x) has massdimension ∆−. Recall external source term in the action
∫J(x)O(x)dd−1x, counting dimensions yields that
the field in the bulk AdS is dual to an operator of dimension ∆+ on the boundary CFT.
4.4 Correlation Functions
Since all the observable can be expressed in terms of correlation functions, our knowledge of n-point correlatorsallows us to work out all physical quantities describing the system. The correlation functions are alsoimportant because they provide a way of testing the theoretical model.
It was shown earlier that how correlators could be computed by taking derivatives of supergravity actionwith respect to the source field (equation 4.6). In this section we use green function results, and start withtwo-point correlation functions which are the simplest example.
4.4.1 Two- Point Functions
In case of two-point function, only quadratic terms in the supergravity action are relevant since correlator isgiven by: [43]
〈O(x1)O(x2)〉 =−iδ
δφ0(x1)
−iδδφ0(x2)
Ssugra∣∣∣∣φ0=0
(4.25)
where source operator J(x) is the local part of the bulk field at the boundary J(x) ≡ φ0(x). In additioninteraction term or any other higher terms can be ignored as source φ0 is put at zero at the end. So theaction considered here is that of a free scalar field in AdSd 3.20:
S =1
2
∫ √−ggµν∂µφ∂νφ−m2φ2
dd−1xdz
After integration by part the on-shell supergravity action becomes: (refer to appendix C)
S =1
2
∫∂µ(√−ggµνφ ∂νφ
)dd−1xdz − 1
2
∫ √−gφ(−m2)φ dd−1xdz (4.26)
It can be shown that (appendix C) on the boundary, on-shell action reduces to:
limz→0
S = −1
2
∫∂AdS
z2−dφ ∂zφ dd−1x (4.27)
By finding the asymptotic behavior of ∂zφ, one finally finds:
S =1
2
∆+
C∆+
∫φ0(x)φ0(y)
|x− y|2∆+dd−1xdd−1y (4.28)
Inserting into the equation defining two-point correlator 4.25, we find:
21
〈O(x1)O(x2)〉 = − ∆+
2C∆+
1
|x− y|2∆+(4.29)
Comparison of the above relation with correlation function of a conformally invariant operator 2.33 showsthat the dual operator has dimension ∆+. This confirms the results obtained for this model in previous section4.3.1
4.5 Mass-Dimension Duality
There is a relation between mass of the bulk scalar field in AdSD+1, and scaling dimension of the dualoperator as shown in 3.29.
∆± =D
2±√D2
4+m2l2
That is due to the fact that mass and scaling dimension are different representation of the second Casimiroperator for SO(D, 2) symmetry group. Since the roots of a stable solution need to be real, positivity of thesquare root imposes the Breitenlohner - Freedman bound : [48]
m2l2 > −D2
4(4.30)
This means that unlike the flat space, in anti-de Sitter space certain negative mass square is allowed. Forvalues of mass beyond this limit, scaling dimensions are complex, and modes become oscillatory resulting inan unstable system. [37]
There are different mass-dimension conventions for other types of the field . Note however that in asupersymmetric field theory like N = 4 SYM , it is enough to consider scalars. Since when the chiralprimary field is known other fields could be obtained by application of the supersymmetric generators. [16]The conventions in AdSD+1 can be summerised as: [52]
spinors: ∆ =1
2(D + 2m/l) , (4.31)
spin-3/2: ∆ =1
2(D + 2m/l) , (4.32)
vectors: ∆± =1
2
(D ±
√(D − 2)2 + 4m2l2
)(4.33)
4.6 Radial Direction-Energy Duality
So far it was discussed that xi coordinates in the bulk are mapped to the same coordinates xi on the boundaryof anti-de Sitter, but not clear how the radial coordinate r is encoded in the gauge theory formalism. Theanswer as usual stems from symmetries of the system. Consider AdSd space in Poincare coordinates 3.19:
ds2 = dr2 + e2r/lηµν(dxµdxν) µ, ν = 1, . . . d− 1
The bulk metric is invariant under the transformation
r → r + r′ xi → e−r′/lxi ≡ axi
Due to identification of xi coordinates it yields xi → axi on the boundary as well. In the momentumspace this amount to rescaling of the energy E → E/a. It can be therefore inferred that radial coordinate inthe bulk is dual to energy scale of CFT on the boundary: [13]
22
E ↔ r ∼ 1/z (4.34)
The boundary of anti-de Sitter r →∞ or z → 0 corresponds to UV limits of energy scales E →∞ on theboundary, whereas deep interior of the space-time r → −∞ or z →∞ is identified with IR limits E → −∞in the energy spectrum of the conformal filed.
4.7 Holographic Renormalisation
In general a quantum theory depends on a set of coupling constants as free parameters. Following a trans-formation in energy scales of the system if the theory could be defined by a finite number of parametersthen it will be renormalisable. Renormalisation group transformation consists of course graining and scalingand allows investigation of the system at different energy scales. Changes of the parameters with respectto scaling is described by β-function which induces renormalisation group flow in the space of all theories.Fixed point correspond to conformally invariant fields, and are of great importance as they determine stablemacroscopic state of the system. [51]
Consider the AdS space in Poincare coordinates 3.18:
ds2 =l2
z2
(dz2 + ηµνdx
µdxν)
ν, µ = 1, . . . d
This metric is of particular use when studying scaling behaviour of the system in the asymptotic region.When the field φ(z,x) approaches the boundary z → εz with ε < 1, the solution scales like φ0(x)→ ε∆−φ0(x).As already discussed value of the bulk field at a certain point corresponds to the coupling of the dual operator.So it can be seen that change of the coupling with respect to the scale is geometrically identified with changingthe distance between the bulk field and the boundary. As the field approaches the asymptotic regions of theAdS space, the coupling runs to the UV limits. While when the field moves towards the interior of the space,coupling flows into the IR limits.
The metric diverges as z → 0 which is the result of the fact that volume of the interior is infinite andsuffers from infra red divergences. As explained earlier small scales in the bulk corresponds to large scaleson the boundary, so this will lead to ultra violet divergence for the field theory. Due to existence of duality,it turns out that renormalisation of one theory amounts to renormalising the other. [49]
Steps taken to regularise the gravity is similar to quantum field, at first a cutoff scale is introduced,divergence terms are subtracted and then the limit is taken to infinity. Note that cutoff in the bulk has thegeometrical interpretation of truncating the AdS at r = rΛ which is related to a UV cutoff Λ on the boundaryQFT.
To clarify the above statements, consider again the example of a massive scalar field in the AdSD+1
background, it was shown that near the boundary generic solution behaves like: 3.30
limz→0
φ(z,x) = z(D−∆)φ−(x) + z∆φ+(x)
where D(∆−D) = −m2l2. Depending on the mass of the field there are three different regimes: [39]
1. If −D2/4 < m2l2 < 0 then the scaling dimension is in the range D/2 < ∆ < D. It means conformaldimension could be ∆ or D − ∆, and either way it is positive, so near the boundary as z → 0, bothmodes vanish, and classical action 3.20 is finite. In this case solutions are normal modes, and both ofthem are valid. Depending on the boundary condition, if one mode is the source the other will be VEVof the field in deep interior.
The coupling constant dies out near the UV limit, and gravity and field theory decouple at high energies.This describes deformation by a relevant operator so the solution dose not change the boundary metricdrastically, and the space will be asymptotically anti-de Sitter.
2. In case m2l2 = 0 then one of the dimensions vanishes. The solution will be constant, and correspondsto a marginal dual operator. The dual quantum field is conformally invariant living at a fixed point in
23
the RG flow. However quantum correction may affect the dimension, making the operator relevant orirrelevant.
3. Finally if m2l2 > 0 then ∆ > D. In this case one of the solutions φ− blows up when approachingthe boundary space. The classical action will be asymptotically infinite, corresponding to a non-normalisable mode. This mode will not be a valid solution to the equations of motion, but it acts asthe external source for the dual quantum operator. Scaling dimension ∆ −D will be negative, whichmeans magnitude of the coupling increases with scale, and the theory has an IR fixed point. The dualoperator in this case will cause irrelevant deformation and the asymptotic space-time will be totallydisfigured.
Changes of the coupling with respect to the scale is described by beta function meaning that variation ofthe solution determines the beta function β ∼ ∂φ0/∂ ln ε ∼ ∆−φ0 [42]
Also anomalous dimension is defined as the changes of wave-function renormalisation with respect to thescale γφ ∼ ∂ε∆/∂ ln ε ∼ ∆. [42]
and r = 1/z plays the role of the cutoff Λc.It can be seen that AdS/CFT duality provides a gravitational description for renormalisation group flow
in the conformal field theory: [41]
r ↔ Λc η ↔ µ (scale) β ↔ ∆−φ0 ∆↔ γφ (4.35)
Finding an exact expression seems non-trivial as regularisation and renormalisation schemes in the bulkare not known, and it is not likely that it follows one of the recognised schemes like dimensional regularisation.[41] One of the generally accepted schemes is Hamiltonian formalism which is the same as usual formulationwith the radial coordinate playing the role of time. It turns out that φ+ and φ− are canonically conjugatepairs, and Hamiltonian equations corresponds to renormalisation group equations. [37] We will use thisscheme in the final chapter of this paper.
4.7.1 Domain Wall
In this section we briefly review an example of RG flows typically dealt with in the AdS/CFT context.Consider a single bulk scalar field in a four-dimensional flat background minimally interacting with gravity.The action takes the form:
S =1
2κ2
∫ √−g−R− gµν∂µφ ∂νφ+ 2κ2V (φ)
d4x (4.36)
The arbitrary potential V (φ) has one or more critical points (figure 4.1). The general solution to theequation of motion is known as domain wall metric, and can be written as: [39]
ds2 = dr2 + e2A(r)ηµνdxµdxν φ = φ(r) (4.37)
The coordinate are separated into a three-dimensional xi coordinate with Poincare symmetry, and a radialr coordinate. Clearly AdS4 metric in Poincare coordinates fit into the domain wall by requiring A(r) = r/l.the boundary region corresponds to r → ∞ limit, and deep interior is approached as r → −∞. It turnsout that local maximum of the potential corresponds to the boundary of AdS, while local minimum is thedeep interior. So solutions to the domain wall space-time interpolate between these two critical points of thepotential.[39].
Substituting the metric above back to the action, one can reach to the following set of equations of motion:[37, 42]
φ′′ + 4A′φ′ = ∂V/∂φ
A′2
= 16 (φ′
2 − 2V )(4.38)
Finding the exact solution to the above second order non-linear set of differential equations is non-trivial,but still we can understand main physical properties by linearising about the critical points of the potential
24
Figure 4.1: A typical potential for a domain wall solution.
V . Setting φ(r,x) = Φ(r,x) + ϕ(r), and A′(r) = 1/l + a′(r), and working with the potential to quadraticorder, one gets a similar equation of motion to that of AdS4: [42]
ϕ′′ +4
lϕ′ − m2
l2ϕ = 0
with the solution:ϕ(r) = Be(∆−4)r/l + Ce−∆r/l (4.39)
with the same mass-dimension relation ∆(∆− 4) = −m2l2.Based on what discussed earlier at the local maximum, the bulk field approaches the boundary and
causes a relevant deformation of the dual QFT at an unstable UV fixed point. On the other hand at thelocal minimum, as the field entres the deep interior of the AdS, on quantum side RG flows towards a fixed IRpoint. So the RG flows from the boundary to the interior of the dual domain wall space, and since potentialat UV turns out to be higher than IR the flow is irreversible. [39]
In the last chapter of this paper, it will be demonstrated that how identifying QFT dual for cosmologycan be eased using the domain wall solutions. The application of holographic renormalisation group theoryand holographic Hamiltonian formalism will also be explained in more details.
4.8 Example of the Duality
In this section Maldacena’s conjecture which was the first realisation of AdS/CFT correspondence is re-viewed. As will be explained, the duality is established through a comparison between Type IIB supergravitypropagating in an AdS5 × S5 space-time, and N = 4 super-Yang Mill theory on parallel D3 branes. In thefirst section, we introduce P -branes and consider supergravity viewpoint, then in the next section, the samesituation is described using the gauge fields living on Dp branes. Subsequently the precise expression for theduality is defined, and finally limits of validity and different forms of the conjecture are discussed.
4.8.1 Supergravity and P-Branes
Before continuing with description of AdS5 × S5/N = 4, SYM duality, it is important to discuss about p-branes in more details. P-branes are classical solutions of 10-dimensional supergravity which have non-trivial
25
charge under a p+ 1 form. The simplest example is R-R charged spherically symmetric black holes in 9− pdimensions which are objects localised in p spatial dimensions. [15]
It turns out that there is an inequality between black-hole charge N and its mass M : [37]
M ≥ N
(2π)pgslp+1s
(4.40)
where gs and ls are string interaction and string length receptively. We are particularly interested inextremal p-branes whose mass is at ground state for a given R-R charge. The metric of an extremal black-hole is given by: [15]
ds2 = H− 1
2p (r)
(−dt2 + dxidxi
)+H
12p (r)
(dr2 + r2dΩ2
8−p)
i = 1, .., p (4.41)
where Hp(r) = 1 +(r7−p+ /r7−p
), and it is related to vacuum expectation value of dilaton φ via eφ =
gsHp(r)(3−p)/4. Moreover parameter r+ is related to the charge through r7−p
+ = dpgsNl7−ps where dp is a
constant.5 Note that event horizon is located at r = 0.There is a curvature singularity at r = 0 but investigation of supergravity metric above shows that 3-brane
is a special case as:
1. when p 6= 3 the horizon and curvature singularity coincide which is referred to as null singularity. Thesupergravity approximation is only valid at some distance from r = 0.
2. when p = 3 then the factor of r in the denominator of H(r) cancels out with r2 coefficient in the S5
metric. So r = 0 is regular and we can use supergravity to explore the solution near this point.
There is always a solution to supergravity as long as Hp(r) is a harmonic function in 9 − p transversedirections. So it is possible to generalise to the case where there are more than one p-brane which amountsto multi-centred solution: [15]
H(r) = 1 +
N∑i=1
r7−pp
|r− ri|7−p, r7−p
i+ = dpgsNil7−ps (4.42)
which indicates N parallel extremal p-branes located at the positions ri, each with R-R charge Ni suchthat
∑iNi = N .
As mentioned in p = 3 case it is possible to take the limit to r → 0 smoothly, and then the metric willbehave like:
ds2 =r2
r2+
(−dt2 + dxidxi
)+r2+
r2
(dr2 + r2dΩ2
5
)(4.43)
By a change of coordinates z = l2/r where l is AdS length defined through:
l ≡ r+ =(4πgsα
′2N) 1
4 (4.44)
The metric reads:
ds2 =l2
z2
(−dt2 + dxidxi + dz2
)+ l2dΩ2
5 (4.45)
One can recognise the first term as the anti de Sitter metric AdS5 in Poincar coordinates. It means thateffect of gravity changes the space-time into a product manifold AdS5 × S5 near the horizon. We will comeback to this point when explaining the correspondence later.
Throughout discussion provided here we used the classical version of string theory. This classical approx-imation is only valid where space time curvature, given by AdS length l in this case, is much bigger thanstring length ls that is ls l. Alternatively using 4.44 this condition can e rewritten as: gsN 1. [15]
5This constant is given by dp = 25−pπ12
(5−p)Γ(
7−p2
)[37]. Note for p = 3 in case of D3 branes: d3 = 4π.
26
4.8.2 String Theory and D-branes
In string theory a Dp-brane is a p + 1 dimensional hyper-surface in the space-time where open strings canend [13].By calculating R-R charges and open string tensions, and matching with supergravity solutions onecan conclude that D-branes and extremal p-branes are actually the same objects [27]. So we can look at theconfiguration described in the previous subsection from a deferent viewpoint: consider N D3 branes on topof each other in a 10 dimension space-time.
1. Excitations of empty space-time generate closed strings which give rise to gravity super-multiplet. Inthe low energy limit the theory is described by type IIB supergravity in the bulk. [14]
2. Excitations of D3-branes causes open strings ending on the branes. They support an N = 4 vectormultiplet on the world-volume of the brane, and the low energy limit will be N = 4 super-Yang-Millstheory. 6 [14]
3. Interaction between closed and open strings which produce Hawking radiation.7 [47]
So the action will be composed of:
S = Sbulk + Sbrane + Sint (4.46)
We are interested in the low energy behavior of the system i.e. scales lower than string length 1/ls.Alternatively one can keep the energy fixed and instead take ls → 0.
In the bulk supergravity theory Sbulk, consider infinitesimal perturbation of the metric around the flatMinkowski g = η + κh. The action reads: [13]
Sint =1
2κ2
∫ √−gRd4x =
∫(∂h)2 + κ(∂h)2h+ κ2(∂h)2h2 + . . . (4.47)
since α′ = l2s , in the ls → 0 limit, universal Regge slope vanishes and so does coupling κ ∼ gsα′2 → 0. Sogravity turns into a free theory at long distances. [14]
As to the last term Sint in 4.46, interaction Lagrangian is obtained through perturbation in powers ofα′:
Lint ≈ α′R+ (α′)2R2 + . . . (4.48)
So contribution from interaction Lagrangian coupling the bulk to the branes in the α′ → 0 limit vanishes.In conclusion, in the low energy limit there are two decoupled systems: gravity in the bulk, and N = 4 super-Yang Mills gauge theory on the world-volume of the branes. Presence of Dp-branes breaks the space-timesymmetry into stability subgroup on the branes and R-symmetry which describes rotations on the 5-sphereS5.
It is important to investigate limits of validity of perturbative string theory approach discussed here.When N D-branes connected by open strings concise each open string loop contributes a product of Chan-Paton factor N , and string coupling gs. In order for the purterbative expansion to be valid, one needs towork with small overall coupling constant which is the regime Ngs 1. [14]
4.8.3 Maldacena’s Duality
Reviewing the results from point of view of supergravity and from string theory perspective, in both casesthere are two decoupled systems in the low energy limit. In both cases one of the mentioned decoupledsystems is gravity in the bulk space-time. So we can ignore supergravity part in this description, and thenwe are left with the same physical situation described by two different theories:
1. N = 4 super-Yang-Mills theories on (3+1) dimensions with SU(N) gauge group, and
2. Type IIB supergravity on the AdS5 × S5.
6When all N branes are coincident, all states go massless, and gauge group is enhanced from U(1)N to SU(N). [13]7In string theory picture one can explain Hawking radiation as a quantum process. Two open strings on the branes interact
and couple like a closed string. The resultant closed string is not bound to the brain and can propagate in the empty space timeas Hawking radiation. [47]
27
It is therefore reasonable to identify these theories and state Maldacena’s Conjecture: [31]
Type IIB supergravity on AdS5 × S5 ↔ N = 4 super-Yang-Mills in (3 + 1) dimensions
Note that symmetries on both sides are identical. In the gravity view-point there is a SO(4, 2)× SO(5)symmetry being isometry groups of AdS5 space and S5 sphere. This is isomorphic to the superstring theorySU(2, 2)×SU(4)R symmetry corresponding to symmetry group of N = 4 super Yang-Mills and R-symmetryin transverse dimensions.
Gauge fields on the world-volume of branes are generated by open strings ending on them, and it turnsout that open string coupling is proportional to super-Yang Mills coupling such that: [13]
g2YM = 4πgs (4.49)
We will need the identity above as well as 4.44 to analyse the limits and different forms of AdS5/N =4, SYM correspondence.
4.8.4 Limits of Validity
As already explained earlier, the perturbative field description is valid as long as the coupling is small. Forsuper-Yang-Mills theory this means: [39]
g2YMN = 4πgsN =
(l
ls
)4
1 (4.50)
,and classical supergravity is applicable when AdS length is much larger than string length:(l
ls
)4
= 4πgsN = g2YMN 1 (4.51)
It can be seen that two theories are reliable in different regimes. This is the reason why AdS5/N = 4, SYMis taken as a strong-weak duality. This means when gauge fields are strongly coupled the gravity theory hasweak interactions and vice versa.
Depending on the range of validity, three forms of correspondence are proposed: [37]
1. Weak form: This version is valid when gsN is large which is the example described in this section. Inthis case supergravity approximation is enough to explain the duality.
2. Semi-strong form: This is the ’t Hooft limit where N is large but gs → 0 so that λ = gsN is fixed. Itmeans α′ correction are applicable as α′/l2 = 1/
√gsN , but gs corrections may not be.
3. Strong form: This is the ideal duality which is conjectured to be true for all values of α′ and N . Howeverit is difficult to prove as the full gravity theory is not well-understood.
As discussed above string theory serves as a tool to provide concrete evidence for the gauge-gravityduality. For example consider deformation of the gauge theory by an external source which changes thecoupling constant. This will in turn change the string coupling via 4.49. But string coupling is related toexpectation value of dilaton on the boundary through gs = e〈φ〉. So finally boundary condition for the dilatonwill change. So it is natural to consider boundary value of dilaton as the source for the conformal field theoryliving there.
Following the steps explained about abstract AdS/CFT correspondence in earlier parts of this chapter,one can define a map that connects fields on the supergravity theory to the gauge invariant operators ofSYM, and hence a correspondence between correlators on both sides of duality.
28
Chapter 5
De Sitter Space
De Sitter has some special features like existence of different horizons for observables, two asymptotic bound-aries connected by null geodesics or the fact that purely spatial planes can reach to far past null infinity. Inaddition to mathematical interests, studying geometrical properties of the de Sitter space is the first step inunderstanding of gauge/gravity duality in a space of positive curvature.
In the first section a review of de Sitter geometry is provided, and some geometrical properties arealso displayed for reader’s convenience and further reference. In the following section, different commoncoordinates for de Sitter space are derived, and their applications are explained. Penrose diagrams are usedto illustrate the casual structure of the space-time, and problem of de Sitter horizons for observers is discussed.Geodesics of a test particle are explained in the next part, and finally dynamics of the scalar matter field indS space is studied, and general solutions to the equation of motion is derived.
5.1 Geometry of de Sitter Space
Consider a Minkowski manifold of (d+ 1) dimensions Md,1 with metric:
ds2 = −(dX0)2 + dXidXi i = 1, ..., d (5.1)
The d-dimensional de Sitter Space dSd can be defined as an embedded hyperboloid of one sheet realisedvia: [1]
−(X0)2 +XiXi = l2 i = 1, ..., d (5.2)
Parameter l has dimensions of length, and known as de Sitter radius the unique scale of the manifold.1
De Sitter has topology R× Sd−1, it is non-degenerate, and has Lorentzian signature. [1]Lorentz group O(d, 1) is the isometry group of de Sitter space. Excluding reflection, the connected
isometry group will be proper Lorentz group SO(d, 1). [3]It turns out that de Sitter is a vacuum solution of Einstein equation with a positive curvature, and thus
positive cosmological constant: [3]
Rµν −1
2Rgµν + Λgµν = 0 Λ > 0 (5.3)
It describes a universe of uniform positive energy density ρ = T00 = Λ, and negative pressure P = Tii =−Λ for i = 1, ..., d− 1. In this manifold the Riemannian tensor is given by: [3, 56]
Rµνλσ =1
l2(gµλgνσ − gµσgνλ) (5.4)
Ricci tensor is proportional to the metric: [3]
Rµν =d− 1
l2gµν (5.5)
1Note replacing l2 with −l2, gives a hyperboloid of two-sheet where each of them is a d-dimensional hyperbolic manifold. Inthis case metric is Riemannian i.e. non-degenerate and positive-definite. This definition is not used in this report.
29
(a)
(b)
Figure 5.1: (a)Penrose diagram for de Sitter space. (b) De Sitter space mapped to the region −π2 < T < π2
of Einstein universe.
hence de Sitter space is a kind of Einstein Manifold. Ricci scalar is given by: [3]
R =d(d− 1)
l2(5.6)
and finally cosmological constant will be: [3, 56]
Λ =(d− 1)(d− 2)
2l2=d− 2
2dR (5.7)
Especially in a 4-dimensional space-time we recover R = 12/l2 = 4Λ which is the result obtained insection 3.1.
5.2 Coordinate Systems
De Sitter space admits a number of coordinate systems which are obtained through different foliation of themanifold. So spatial part of the metric could be closed, open, flat etc in different coordinates. We discuss themost common systems that are global, planar, and hyperbolic coordinates, and explain why they are morefrequently used. Some of these coordinates cover the whole manifold but some only partially cover it.
5.2.1 Global Coordinates
A convenient choice (t, θi) is obtained by closed spherical slicing using the usual parameterisation of ahyperboloid as follows:
X0 = l sinh(t/l),Xi = l cosh(t/l) xi i = 1, ..., d
(5.8)
where xi is the coordinates of a unit Sd−1 sphere such that xixi = 1, as discussed earlier in section ??.The induced metric will be:
ds2 = −dt2 + l2cosh2(t/l) dΩ2d−1 (5.9)
This metric covers the entire hyperboloid. the spatial section is compact, and describes a Sd−1 spherewith a time-dependant radius of l2 cosh2(t/l). The spherical metric can be writen as:
dΩ2d−1 = dθ2 + sin2θdΩd−2 (5.10)
with −π < θ < π. The point θ = π is usually referred to as north poles, in contrast to the south polecorresponding to θ = −π
30
Figure 5.2: Regions O− corresponding to the casual past of the observer on the south pole (right), and O+
the casual future of that observer.
In order to construct the Penrose diagram for de Sitter manifold we need to define conformal timecoordinate T via T = 2Arctg et/l − π
2 which means dT 2 = dt2/[l2cosh2(t/l)
]. This amounts to the following
metric in conformal coordinates:
ds2 =l2
cos2T
(−dT 2 + dΩ2
d−1
)(5.11)
where conformal time is within the range T ∈(−π2 ,
π2
). The Penrose diagram which contains all the
information about casual structure of de Sitter space is shown in figure 5.1a. In this figure, upper and lowerhorizontal edges are future and past null infinities I+, I−, and left and right vertical edges are North andSouth poles. Light rays are ±45 lines, time-like curves are more vertical, and space-like curves are morehorizontal. Each point on the diagram represents a Sd−2 sphere.
Identifying r′ = T , de Sitter space can be mapped to the region −π2 < t < π2 in Einstein static universe,
figure 5.1b. It can be seen that de Sitter space is a finite slab in Einstein universe.One of the features of the de Sitter manifold is that no observer have access to the whole space-time. It
is in contrast with say Minkowski manifold where eventually one can have the whole information of historyin his casual past. In figure 5.2 region O− illustrates the whole casual past for the observer sitting on thesouth pole, whereas region O+ shows the casual future of this observer. The intersection O− ∪O+ is refferedto as southern diamond, and it is the only region that south pole observer has full access to (figure 5.1a).The lower diamond is where he can never send any information to, and upper diamond is the region wherein future no messages reaches the observer from. The northern diamond is not accessible to him at all.
Also in de Sitter space, different observers have different horizons. The observer sitting on the south polewill never see anything beyond the diagonal line of O−; however, diagonal line of O+ is the horizon for theobserver on the north pole.
5.2.2 Planar Coordinates
Planar coordinates (t, xi) are obtained by flat slicing of the dSd with spatial Euclidean planes of infinitevolume. Parameterisation goes as follows: [3]
X0 = l sinh(tl) + (1/2l)xixie−t/l,X1 = l cosh(t/l)− (1/2l)xixie−t/l,Xi = xie−t/l i = 2, . . . , d
(5.12)
where xixi = 1, and xi gives the coordinates of a Sd−2 sphere. then the metric reads:
ds2 = −dt2 + e−2t/ldxidxi i = 1, . . . d− 1 (5.13)
where asymptotic past I− corresponds to t → −∞. This coordinate only covers half of the dS spacewhich is the casual past of the observable at the South pole O− (figure 5.3). One of the features of the deSitter space is that spatial planes can reach to the asymptotic past.
It should be noted that time t is not a Killing vector in this coordinates. Space-time isometries arerotations and translations in xi coordinates. [3]
31
Figure 5.3: Dashed lines represent are spaces of constant t in planar coordinates. (reproduced from [56])
Introducing conformal time τ = et/l, one reaches to the metric:
ds2 =1
τ2
(−l2dτ2 + dxidxi
)i = 1, . . . d− 1 (5.14)
where asymptotic past I− is approached when τ → 0.In order to use the above planar metrics for the North pole observable one needs to take t → −t, and
τ → −τ . [58]
5.2.3 Hyperbolic Coordinates
In Hyperbolic coordinates (τ, ξ, θi) de sitter space is foliated by open spaces of negative curvature that arehyperbolic planes. Open slicing is achieved using the following parameterisation: [55] X0 = l sinh(t/l) cosh ξ,
X1 = l cosh(t/l),Xi = l zi sinh(t/l) sinh ξ i = 2, . . . , d
(5.15)
where zizi = 1, and zi gives the coordinates of a Sd−2 sphere. The induced metric reads: [55]
ds2 = −dt2 + l2sinh2(t/l) dH2d−1 (5.16)
here dH2d−1 describes the metric for a hyperbolic space.
5.3 Geodesics
When working in de Sitter space, there is a measure of distance between two points P (x, y) that is moreconvenient than the geodesics distance D(x, y). It is defined via: [1]
l2P (x, y) = ηijxixj (5.17)
One can write an explicit expression for P in any coordinate introduced in this chapter. In planarcoordinates it reads:
P (t,x; t′,y) = cosh(t− t′)− 1
2e−(t+t′)|x− y|2 (5.18)
In the casual diamond this invariant measure is related to the geodesics distance by: [3]
P = cos(D/l) (5.19)
If P = 1 then geodesics distance vanishes D(x − y) = 0 so two points x and y either coincide or areseparated by a null geodesics.
If P = −1 then x = −y so antipodal point of x lies on the light cone of y.one can generalise this argument and summarise that for P > 1 geodesics separation is time-like, and for
P < 1 it is space-like. For P < −1 geodesic separation between x snd anti-podal point is timelike. [1]
32
5.4 Matter Field in de Sitter Space
Consider the simple case of a single scalar field φ with mass m on a fixed dSd background with flat slicing5.14. The action reads:
S =1
2
∫ √−ggµν∂µφ∂νφ−m2φ2
dd−1xdτ (5.20)
Klein-Gorden equations of motion(−m2
)φ = 0 is obtained through variation of action where denotes
Laplacian with usual definition in 3.22. We follow similar steps as in anti-de Sitter case to find the solutions.Separation of variables φ(τ,x) = ϕ(τ)Φ(x) amounts to two independent equations of motion in τ and xcoordinates: [55]
(∂2x + k2)Φ(x) = 0 (5.21a)[
−τd∂τ (τd∂τ ) +m2l2 + k2τ2]ϕ(τ) = 0 (5.21b)
with k2 being modulus square of momentum, and ∂2x indicating Laplacian on Euclidean manifold. Solution
to Euclidean Laplacian 5.21a are plane waves Φ(x) = eikx/(2π)d−1 , so the full solution to the equations ofmotion is obtained through superposition:
φ(τ,x) =
∫ϕke
ikx dd−1k
(2π)d−1(5.22)
This shows that ϕk is the solution in the momentum space. Writing the second equation of motion 5.21bin Fourier space ends up with the Bessel equation of the form: [55]
(kτ)2ψ′′k + (kτ)ψ′k −(
(d− 1)2
4−m2l2 + k2τ2
)ψk = 0
where ψk = τ (1−d)/2ϕk. The solution is a linear combination of first and second kind Bessel functions Jσ,and Yσ:
ψk(kτ) = akJσ(kτ) + bkYσ(kτ)
where in dS space σ =√
(d− 1)2/4−m2l2. Solution in terms of ϕk can be written as: [58]
ϕk(τ) = ak(kτ)(d−1)/2Jσ(kτ) + bk(kτ)(d−1)/2Yσ(kτ) (5.23)
When approaching the origin, Bessel functions have the following behaviour:
limτ→0
Jσ =1
Γ(σ + 1)
(τ2
)σlimτ→0
Yσ = −Γ(σ)
π
(2
τ
)σ(5.24)
So near the boundary I− when τ → 0−, general form of the solution in momentum space can be writtenas:
ϕk(τ) ≈ τ∆− φ−(k) + τ∆+ φ+(k)
where mass-dimension relation in dSd is found to be:
∆± =(d− 1)
2±√
(d− 1)2
4−m2l2 (5.25)
In the real space, solution to the equations of motion when approaching past null infinity I−, in theconformal planar coordinate, will be written as:
limτ→0−
φ(τ,x) = τ∆−φ−(x) + τ∆+φ+(x) (5.26)
33
Chapter 6
dS-CFT Duality
So far in this report, all the discussions about the gauge/gravity duality focused on the spaces of negativecurvature; however, based on observational evidence, we know that universe does not have a negative cosmo-logical constant. This is the main motivation to generalise the correspondence to a de Sitter space. A primarydifficulty with the dS/CFT duality is the absence of any example of a de Sitter solution in the context ofstring theory; in spite of this, as will be explained, it is still possible to conjecture the duality along the linesof well-constructed AdS/CFT correspondence. At the end we review a recent example proposing that theactual gravitational theory is the higher spin formalism corresponding to a Sp(N)-invariant field theory.
In the first section asymptotic symmetries of de Sitter space are compared to those of conformal groupas the initial motivation for the duality. After defining the green functions in the bulk, correlation functionswhich are the main observables of the theory are computed. An example of the duality is presented in thelast section, and correspondence between four-dimensional theory of higher spin gravity in de Sitter dS4, andconformal fields in three dimension CFT3 is explained.
6.1 Symmetries
It was discussed in section 2.2.4 that conformal group on a D-dimensional Euclidean space is Lorentzgroup SO(D, 1) which is identified with the isometry group of the de Sitter space. This is basically themain motivation to establish a one-to-one mapping between gravity in the bulk dS, and field theory on theEuclidean boundary. In a theory of quantum field, gauge symmetries are diffeomorphism invariant, so it issufficient to find diffeomorphisms in de Sitter which can preserve the boundary conditions.
In the next chapter we consider the dS3 space related to the two-dimensional CFT , and at the endgeneralise the results.
6.1.1 Asymptotic Symmetries
Recall metric for an dS3 5.13 in planar coordinates. We need to define boundary conditions that metricshould satisfy in order to be asymptotically de Sitter . The metric of an asymptotically past de Sitter spacewill behave for t→ −∞ as: [56] 1
gzz = 12e−2t +O(1)
gtt = −1 +O(e2t)gzz = O(1)gtz = O(e2t)
(6.1)
and all other metric components vanish. Asymptotic symmetries are diffeomorphisms that satisfy theabove boundary conditions. Changes in components of a metric δgµν under an active coordinate transforma-tion ζV is given by the Lie derivative with respect to the vector field LV : [1]
δζg = −LV g (6.2)
1This asymptotic behavior can also obtained from analytic continuation of dS3 boundary conditions [45]
34
where Lie derivative of a metric is given by [1]
LV g = LV gµνdxµ ⊗ dxν + gµν (LV dxµ)⊗ dxν + gµνdxµ ⊗ (LV dxν) (6.3)
So it is required to parametrise the diffeomorphism ζV in terms of the vector field. One can check thatthe most general diffeomorphisms will be: [56]
ζ = V ∂z +1
2V ′∂t +O(e2t) + cc (6.4)
where prime denotes differentiation. Using definition 6.2 and 6.3, it is inferred that under this coordinatetransformation the metric changes like: [56]
δζgzz = − l2
2 V′′′
δζgzz = δζgtz = δζgtt = 0(6.5)
It is not hard to see that these changes preserve the boundary conditions 6.1, and thus diffeomorphism6.4 generates the asymptotic symmetry of dS3. Now we need to identify this symmetry group generatedby ζ. In section 2.1 it was explained that a conformal transformation is composed of a Weyl scaling anda diffeomorphism. Consider 1
2 (V ′∂t + V ′∂t) term in 6.4, it creates a time translation which is equal toa Weyl scaling on the boundary complex plane. The other term i.e. V ∂z + V ∂z is simply a coordinatetransformation. This means the diffeomorphism ζ is a direct sum of a tangential term acting as a coordinatetransformation, and a normal term which functions as a Weyl scaling on the complex plane. Therefore inconclusion, asymptotic symmetry group of de Sitter space is the conformal group on the boundary. [58]
In the special case where:V = α+ βz + γz2 (6.6)
the V ′′′ term in 6.5 vanishes, so metric is invariant under the diffeomorphism generated by V that isLV g = 0. In this case Killing vector V generates the isometries of dS3. Note there are three complex freeparameters α, β and γ which give rise to SL(2,C) symmetry group. It is inferred that isometry group of dS3
is SL(2,C) which is a subgroup of asymptotic conformal group symmetry.One can generalise the above argument to a d-dimensional de Sitter dSd, and conclude that related
isometry is SO(d, 1).
6.2 The Field - Operator Correspondence
There is an extension of the already discussed AdS/CFT theory to dS space. The conjecture is that corre-lation functions of the fields in the de Sitter space are in a one-to-one correspondence with the corelators ofthe Euclidean conformal field on the boundary.
〈O(x1)...O(xn)〉 ↔ 〈φ(t,x1)...φ(t,xn)〉 (6.7)
This means bulk theory of gravity, and spatial conformal field theory are in fact two different representa-tions of the same fundamental theory. Moreover it is conjectured that de Sitter space fields are in one -to-onecorrespondence with single trace operators of the conformal theory.
When the field approaches the boundary of dS at either I+, or I−, it acts as the external source foroperators of the QFT. Partition function of the gravity theory is identified with generating functional of thequantum field. Similar to AdS case, correlation functions are obtained by taking functional derivatives of thebulk action with respect to the source.
6.3 Mass-Dimension Duality
In this part of the report we have a closer look at the mass-dimension duality relation 5.25 for a scalar fieldin dSD+1:
∆± =D
2±√D2
4−m2l2
35
1. In case 0 < m2l2 < (D/2)4, both conformal dimensions are real and positive 0 < ∆− < D/2 < ∆+ < d.In this case both solutions vanish when approaching the boundary, so both are normalisable modes.There is a freedom to choose dimension of the dual operator depending on the boundary conditions. Ifboundary condition imposes the weight ∆+ (∆−) for the operator then the source will be φ−(φ+) andvacuum expectation value will be given by φ+(φ+).
2. On the other hand if m2l2 > (D/2)2, then scaling dimensions are not real any more. Modes will beoscillatory which means there is no stable scalar field on the de Sitter background. Complex dimensionsindicate that dual field theory is not unitary. Non-unitarity dose not rule out the gauge-gravity dualityas the field theory lives on a Euclidean space. In other words there is no reason that CFT must beunitary. If it is, then according to AdS/CFT correspondence, the same field theory should be dual toan AdS gravity which is puzzling.
6.4 Time Evolution
Recall a d-dimensional de Sitter space dSd is defined as a purely spatial submanifold of a bulk space-time of(d, 1) dimensions. So in dS/CFT correspondence time itself is emerging from conformal field theory comparedto AdS/CFT duality where radial direction is emergent. So notion of time in de Sitter version of dualitychallenges our quantum intuitions of unitary time evolution.
Consider the dSd metric in planar coordinates 5.13:
ds2 = −dt2 + e2t/ldxidxi
The metric is invariant under following transformation:
t→ t+ τ xi → e−τ/lxi (6.8)
This transformation generates time evolution in first term, and scaling in the second term. This meansthat time evolution in the bulk theory corresponds to the scaling transformation on the boundary QFT.
Physics at infinity is intriguing since infinite time in the gravity theory means infinite divergence in thefield theory. So introducing a cut-off in the energy scale of the quantum field stops time from extendingto infinity which is the starting point for holographic renormalisation formalism which discussed in earlierchapters. From Wilsonian point of view this relation indicates that renormalisation group flow in CFT isrelated to time-dependence of supergravity theory. Therefore infinite future in the de Sitter corresponds toUV divergences of the dual CFT, while infinite past is dual to IR in the boundary field.
6.5 Green Function
Wightman green function as usual is defined via:(−m2
)G(x− y) = δ(d)(x− y) (6.9)
where is Laplacian in dSd. It is convenient to use translation symmetry of de Sitter to fix one point atx = x0. then we can solve the Laplace equation above away from this point, and at the end check that thesolution behaves like a delta function when these two points coincide. In this situation green function onlydepends on a single parameter that is dS invariant length P already defined in section 5.3 so G(P ) ≡ G(x−y).Laplace equation simplifies to: [58] (
−m2)G(P ) = 0 (6.10)
Using the definition of Laplacian 3.22, one can compute it in dSd space as: [1]
x =x2
l2[(1− x2)∂2
x − d x∂x]
(6.11)
where d is simply dimension. Then equation 6.10 simplifies to a one-variable ODE: [56]
(1− P 2)∂2PG− d P∂PG−m2l2G = 0 (6.12)
36
A change of variables z = 12 (P + 1) amounts to so-called hypergeometric equation: [23]
z(1− z)G′′ + d(1
2− z)G′ −m2l2G = 0 (6.13)
with hypergeometric function solution of the form: [56]
G(P ) = C∆F
(∆+,∆−,
d
2,P + 1
2
)(6.14)
where the constant C∆ = Γ(∆+)Γ(∆−)
(4π)d/2Γ(d/2)as shown in D.3, ∆± are given by:
∆± =1
2
[(d− 1)±
√(d− 1)2 − 4m2l2
](6.15)
and hypergeometric function is defined as: [25]
F (a, b, c, z) =
∞∑n=0
(a)n(b)n(c)nn!
zn (6.16)
here (x)n ≡ x(x + 1) . . . (x + n − 1) = Γ(x + n)/Γ(x). Now we need to examine the behavior of thepropagator G(x − y) when two points coincide. The hypergeometric function has a singularity at z = 1[23], or equivalently P = 1 which yields D(x, y) = 0. So singularity occurs when two points coincide or areseparated by a null geodesics. Near the singularity hypergeometric function scales like (D/l)2−d [23] whichafter action of Laplacian operator gives the right behavior of a delta function. Therefore the green functionG(x− y) satisfies all the requirements of Laplace equation 6.9.
Equation 6.12 is symmetric under the change P → −P , consequently G(−P ) is also another independentsolution:
G(P ) = C∆F
(∆+,∆−,
d
2,−P + 1
2
)(6.17)
Singularity of the above green function is at P = −1. This happens when first point x is located on thelight-cone of its antipode −x. 2 The green function for a de Sitter space therefore is a linear combination ofthe above mentioned solutions G(P ) and G(−P ).
6.6 Correlation Functions
As discussed earlier, in the de Sitter space dSd near the boundary as t → −∞, the solution behaives likeφ ≈ e∆±t, and there is freedom in choosing either of these two boundary conditions. From this point forward,we impose boundary condition:
limt→−∞
φ(t,x) = e∆−tφ−(x) (6.18)
and φ−(x) functions as the source on the boundary.All observables of the system can be constructed from a complete set of correlation functions. The simplest
case two-point correlation function is defined by taking the derivative of the bulk action two times:
〈O(x)O(y)〉 =δ
δφ−(x)
δ
δφ−(y)Sgravity
∣∣∣∣φ−=0
(6.19)
So only quadratic terms in the Lagrangian are required, and free theory formalism suffices for this purpose:
S =1
2
∫ √−ggµν∂µφ∂νφ−m2φ2
dt dd−1x (6.20)
2This singularity is still physical because antipodal points in de Sitter space is always hidden beyond the horizon (section5.2). So an observer cannot detect this singularity.
37
Following similar steps to AdS space explained in appendix E in details, the action in planar coordinatescan be written as:
limt,t′→−∞
S = −∫I−e(d−1)(t+t′)/lφ(t,x)
↔∂tG(t,x; t′,y)
↔∂t′φ(t′,y)dd−1ydd−1x (6.21)
Near the past null boundary I−, the action becomes (refer to appendix E):
S ≈∫φ−(x)φ−(y)
|x− y|2∆+dd−1xdd−1y (6.22)
Using the definition of two-point function in dSd it can be seen that tow-point function is the quadraticcoefficient in the action above: (appendix E)
〈O(x)O(y)〉 =∆+
2C∆+
1
|x− y|2∆+(6.23)
which confirms the assumption that dual operator has conformal dimension ∆+.As discussed earlier in de Sitter space-time the bulk field can also asymptotically behave as φ(t,x) ≈
e∆+tφ−(x). In this case correlation function will be:
〈O(x)O(y)〉 =∆−
2C∆−
1
|x− y|2∆−(6.24)
which corresponds to a dual operator with scaling dimension ∆−. In fact this boundary condition canonly be imposed on the future infinity I+ which is not within the casual past region O−. [56]
6.7 Example of dS-CFT Duality
Over a decade after introduction of dS/CFT duality by Strominger, there was no example of the correspon-dence at the fundamental level.
It was not also possible to analytically connect the results of AdS/CFT correspondence to dS space dueto encountering complexification issues about the rank of the gauge group. Recalling the Maldacena’s dualityin AdS4 where it was shown that N2 ∼ l2/GN with l2 ∼ 1/Λ; one can attempt to analytically continue to dS4
by reversing the sign of cosmological constant Λ→ −Λ while keeping the Newton’s constant unchanged. Butthis yields N2 → −N2 meaning that rank of the gauge group is imaginary N → iN . This raises issues aboutimaginary scaling weights, imaginary fluxes, ghost fields and negative energies that are not well-defined. [61]
In this section we explain the recent hypothesis that higher spin gravity in dS4 is actually dual to an anti-commuting, Sp(N)-invariant conformal field in three dimensions. This conjecture rests on well-establishedGiombi-Klebanov-Polyakov-Yin (GKPY) duality which relates HS gravity on AdS4 to O(N)-invariant CFT3.In GKPY duality conformal operators transform in the fundamental representation leading to the specialfeature N ∼ 1/(ΛGN ), so one can continue to the dS4 space simply by N → −N which yields appearance ofsymplectic group O(−N) = Sp(N). 3 [69]
The results of GKPY duality are used to derive HS dS4/Sp(N) CFT3 duality, and the methodology isillustrated in figure 6.1. At first step we take an analytic continuation to move from higher spin theory inAdS4 to EAdS4. Then results of GKPY duality is used to map the HS gravity observables to correlators ofthe dual O(N)-invariant conformal field. It will be discussed that by reversing the sign of the fundamentalindex N one can switch to the fermionic Sp(N)-invariant field theory. At the end we are able to establishthe connection between HS dS4 and Sp(N) CFT3 models.
38
Figure 6.1: Schematic diagram for methodology used to establish duality between HS dS4 and Sp(N) CFT3
models.
6.7.1 Higher-Spin Gravity Theory
Higher spin gravity theory is a generalisation of general relativity which contains an infinite number of bosonicfields with all non-negative integer spins. 4 HS theory comes in two types: Type A theory has a minimalbosonic spectrum of gauge fields for each even spin whilst type B theory contains odd spin fields. In thispaper, we are interested in type A higher spin gravity in a background with constant curvature. [62]
To construct the algebra one needs to define star product of polynomials f and g:
f(x, z) ∗ g(x, z) =
∫eu.vf(x+ u, z + u)g(x+ v, z − v)d2u d2v (6.25)
which has the properties: [63]
xα ∗ xβ = xαxβ + εαβ , xα ∗ xβ = xαxβ + εαβ (6.26)
where εαβ is totally antisymmetric tensor with ε12 = 1. Then the Lie algebra of d-dimensional higher spintheory hs(d) is defined with respect to the Lie bracket:
[f, g] = f ∗ g − g ∗ f (6.27)
with reality condition f† = −f , and projection condition Π f(x) ≡ f(ix) = −f . [63]It is common to pack all of the gauge fields Aµ into a master field defined as:
Aµ(x, y, y) = εα1α1µ (x) yα1
yα1+ ωα1α2
µ (x) yα1yα2
+ ωα1α2µ (x) yα1
yα2
+ . . .+
∞∑n,m=0
Aα1...αmα1...αnµ (x) yα(1
. . . yαm yα1. . . yαn)
(6.28)
Imposing the constraint that only even spins are independent which gives rise to a tower of real fields ofeven spins which describe all degrees of freedom for the theory. 5 Similarly one can package scalars φ, andWeyle tensors φα1...αn into a scalar master field:
Φ(x, y, y) = φ(x) + φαα(x) yα1 yα1 + . . .+
∞∑n,m=0
φα1...αmα1...αn(x) yα(1 . . . yαm yα1 . . . yαn) (6.29)
with constraint φαα = ∂ααφ, . . . , φα1...αmα1...αn = ∂α1...αrα1...αsφαr...αmαs...αn .
3The bar in analytic continuation O(−N) = Sp(N) means Young tableux are transposed. This changes the symmetricrelations of bosons to anti-symmetric fermionic relations.
4The spectrum obviously includes a scalar of spin zero.5In even spin reality condition reduces to the fact that component fields are real fields. [66]
39
In classical higher spin gravity the free action in d dimensions can be written as: [64] 6
Sfree ≈1
GN
∫(∂Φ)2 + ΛΦ2ddx (6.30)
where GN being the Newton’s constant, and Λ cosmological constant. The linearised equation of motionis found to be:
∂2Φ + ΛΦ = 0 (6.31)
It turns out that interaction couplings are inverse powers of cosmological constant, so interacting actionwill be an expansion in ∂2/Λ. For instance for cubic interactions between spins (2, s, s) the action reads: [64]
S ≈ Sfree +1
GN
∫∂2hΦ2 + ΛhΦ2 +
1
Λ∂2h(∂Φ)2 + h(∂Φ)2 + . . . (6.32)
where h is the spin 2 metric fluctuation, and cubic interactions can be written as:
∂2h ≈ ΛΦ2 + Φ∂2Φ + (∂Φ)2 +1
Λ[∂3Φ∂Φ + (∂2Φ)2] + . . . (6.33)
So quantum corrections take the form of an overall factor of 1/(GNΛ) for the action. [65]
6.7.2 EAdS - dS Correspondence
In this section we establish the relation between Euclidean anti de Sitter EAdSd and de Sitter dSd spaces ind-dimensions. Consider the Poincare patch coordinates in EAdS manifold:
ds2 =l2EAdSz2
(dz2 + dxidxi
)This can be transformed into dS metric in the conformal planar coordinates 5.14:
ds2 =l2dSτ2
(−dτ2 + dxidxi
)by a double wick rotation: z → −iτ , and lEAdS → −ildS . Note the last map is the familiar relation
between cosmological constants Λ→ −Λ.Having established the continuation between two types of manifold, we need to find the relation between
correlation functions. It was already discussed that in HS theory of gravity quantum corrections appear asan overall factor of 1/(GNΛ) for the action. It turns out that this factor is proportional to the dimensionlesscoefficient of two-point functions C2 ∼ −GNΛ. 7 Consider bulk-to-bulk propagator in anti de Sitter space4.14. At short distances free anti de Sitter correlator behaves like 4.16: [66]
〈φ(z1,x1)φ(z2,x2)〉 ≈ C2z1z2
(z1 − z2)2 + (x1 − x2)2(6.34)
If only continuation z → −iτ is applied, we end up with a wrong overall minus sign. So in order tocontinue EAdS to dS it is required to take C2 → −C2. It can also be though of as taking Λ → −Λ whilekeeping the Newton’s constant GN fixed.
This can be generalised to find the transformation rule for n-point functions of master fields betweenEADSd, and dSd: [61]
〈Φ(z1,x1) . . .Φ(zn,xn)〉EAdS = 〈Φ(−iτ1,x1) . . .Φ(−iτn,xn)〉dS∣∣∣∣C2→−C2
(6.35)
6In fact action formalism is not known for higher spin theories. In order to derive the schematic form of the action it isassumed in this paper, that equations of motion follow integrability conditions. [61]
7Coefficient of two-point function in this section is denoted as C2 ≡ C∆.
40
6.7.3 GKPY Duality
Consider the O(N) vector model consists of N -component conformal fields ϕa transforming in the fundamen-tal representation. The theory has O(N) symmetry with N being an even integer.
Free Theory
The free action in a 3-dimensional flat space can be written as:
S =1
2
∫ ∂iϕ
a∂iϕa +m2ϕaϕad3x (6.36)
The only class of singlet operators in this theory corresponds to the conserved currents of the form: [69]
J(s)i1...is
= ϕa∂(i1 . . . ∂is)ϕa + . . . (6.37)
So that there is a different conserved current for each spin. GKPY duality conjectures that there is a one-to-one mapping between these single trace operators, and the type A higher spin fields in AdS4 with Neumannboundary condition. Accordingly connected generating functional of the boundary CFT3 in fundamentalrepresentation of O(N) are given by on-shell action of the HS gravity in the bulk AdS4. There is therefore aone-to-one correspondence between correlation functions on both sides.
Quantum field prescribes that correlation functions in the free theory are one-loop diagrams with theoperators running around the loop. Explicit calculations show that there is an overall factor of N , forinstance two-point correlators for conserved currents are given by: [66]
〈J (s)(x1)J (s)(x2)〉 ∼ N
|x1 − x2|2(6.38)
or three-point functions are:
〈J (s)(x1)J (s)(x2)J (s)(x3)〉 ∼ N
x12x13x23(6.39)
where xij ≡ |xi − xj |2. One can compare the coefficient N in the above relations with that of a genericconformally invariant correlator in section 2.5 to yield C2 ∼ 1/2N . Comparison with the relation obtainedon the gravity side C2 ∼ GNΛ yields the following identity in GKPY duality:
N ∼ 1
ΛGN(6.40)
Interacting Theory
The action for an O(N) invariant interacting theory is given by:
S =
∫ 1
2∂iϕ
a∂iϕa +1
2m2ϕaϕa +
λ
2N(ϕaϕa)2
d3x (6.41)
where λ is quadric coupling. This theory is known to have a critical IR fixed point. [41] The usual practiceis to introduce an auxiliary field σ(x), then the action takes the form:
S =
∫ 1
2∂iϕ
a∂iϕa +1
2m2ϕaϕa + σϕaϕa − N
2λσ2
d3x (6.42)
The action is quadratic in ϕ, and it can be shown that the introduced auxiliary field removes all theone-particle reducible diagrams in terms of σ(x). [69] 8
According to GKPY duality prescription the critical interacting O(N) CFT in three dimensions corre-sponds to the HS gravity theory in AdS4 with Dirichlet boundary conditions.
8We will make use of this method in the next sections to find the analytic continuation between O(N) and Sp(N) models.
41
At the end it is essential to explain the difference between interacting fields in the fundamental and adjointrepresentations. In the adjoint theory single-trace operators have the general form of tr (ϕ∂i1 ϕ . . . ∂in ϕ) withϕ being an N ×N matrix charged under the adjoint. The number of these operators exponentially increaseswith dimension. Higher spin theories do not contain sufficient number of fields to correspond to the dualoperators. In contrast to this, in the fundamental representation single trace operators take the generic formof ϕa∂iϕ
a where ϕ is an N -component vector. The operators do not grow in number with respect to thedimension, so it is possible to map them to the fields in an HS gravity theory. [69]
6.7.4 O(N) - Sp(N) Correspondence
In order to find the gauge theory dual to higher spin gravity, in de Sitter space, we need to define a newquantum field. The conformal field theory of interest studied in this section is usually referred to as Sp(N)vector model with N being an even integer. It consists of anticommuting scalar field transforming underfundamental representation of Sp(N), and thus can be represented as N-component vectors χa. It can beshown that the only single particle state of the theory corresponds to single-trace operator Ωabχ
a∂iχb where
Ω is the antisymmetric symplectic form.
Free Sp(N) vector model
It is more convenient to discuss the free conformal field theory at first step and then generalise the resultsto the interacting case. The free Sp(N) vector model can be described by a non-Dirac, two-derivative formaction, and in the free theory it reads: [67]
S =1
8π
∫ Ωab∂iχ
a∂iχb +m2χ.χdDx, Ωab =
(0 1N
2 ×N2
−1N2 ×
N2
0
)N×N
(6.43)
and inner product is defined as χ.χ ≡ Ωabχaχb. As usual, one can work out the conserved current by
varying the Lagrangian:Ji1...is = Ωabχ
a∂(i1 . . . ∂is)χb + . . . (6.44)
where s denotes all even spins. Using equation of motion ∂i∂iχ = 0 it is straight forward to check that
∂i1J(i1 . . . ∂is) = 0. In the free theory current has conformal dimension ∆ = D − 2 + s. [66]Similar to O(N) vector model, correlation functions in the free theory are one loop diagrams with operator
circulating around the loop. However due to Fermi statistics there is an extra minus sign. Again analogousto O(N) model, there is an overall factor of N in the explicit expression of correlators (c.f. 6.38, and 6.39).So by taking N → −N one can relate Sp(N) invariant free correlators to O(N) free functions: [61]
〈J (s1) . . . J (sn)〉Sp(N) = −〈J (s1) . . . J (sn)〉O(N) (6.45)
Interacting Sp(N) vector model
Action for the Sp(N) invariant interactive model can be written as: [68]
S =1
8π
∫ Ωab∂iχ
a∂iχb +m2χ.χ+ λ(χ.χ)2dDx (6.46)
where λ is quadric coupling constant. The interacting theory has a critical IR fixed point. Similar toO(N) case one can introduce an auxiliary field σ(x) so that the action changes to:[68]
S =1
8π
∫ Ωab∂iχ
a∂iχb +m2χ.χ+ 2α√λ(χ.χ)− α2
dDx (6.47)
Consequently the Feynman diagrams reduces to closed loops of χ field linked by propagators of α. It canbe shown that analogous to O(N) case there is a symmetry factor of N for each loop. [61] Since χ fields areanti-commuting there is minus sign contribution form each loop. So the correlators in Sp(N) vector modelare related to those of SO(N) through the continuation N → −N . [67]
42
〈J (s1) . . . J (sn)〉Sp(N) = 〈J (s1) . . . J (sn)〉O(N)
∣∣N→−N (6.48)
It is worth mentioning that Sp(N) scalar field is not unitary. First of all this does not affect the dualityas CFT lives on a Euclidean space. In addition Sp(N) model turns out to be pseudo unitary. It means thatHamiltonian is pseudo hermitian such that H† = CHC where C is a unitary operator and C2 = 1. It can bedemonstrated that such a pseudo-hermitian Hamiltonian satisfy desirable properties of a unitary quantumtheory, especially that probabilities are conserved and eigenvalues are real. [70] 9
6.7.5 dS4 / Sp(N) CFT3 Duality
At this stage we have all the ingredients in place to express the first example of dS/CFT duality. It wasexplained that GKPY duality prescribes that higher spin gravity in AdS4 with Dirichlet (Neumann) boundaryconditions is dual to interacting (free) O(N) conformal field theory with identification N ∼ 1/ΛGN . EAdSspace can be analytically continued to dS via Λ→ −Λ, and O(N) vector model is related to Sp(N) throughN → −N . So it can be conjectured that higher spin gravity in AdS4 with Dirichlet boundary conditionscorresponds to critical interacting Sp(N) invariant conformal field in three dimensional Euclidean boundary,and HS theory with Neumann conditions is dual to the free conformally invariant Sp(N) model in threedimensions. [61]
In order to show the statement explicitly, consider the scalar field φ(z,x) in the EAdS4 space. Near theboundary zc → 0 where zc is cutoff scale. Using simple dimensional analysis and imposing O(4, 1) symmetryconstraints one can generalise the expression for the propagator 6.34 to n-point correlation functions as: [66]
〈φ(z1,x1) . . . φ(zn,xn)〉EAdS = A(n)C2
(C2)n(zcx
)n∆+
(6.49)
where A(n)C2
is a constant, and x is a function of distance between point which scales linearly. 10 One canalways define an operator such that:
O =
(1
C2z∆+c
)φ
As a result correlator for quantum operators is found to be: [61]
〈O(x1) . . .O(xn)〉EAdS = A(n)C2x−n∆+ (6.50)
So in this way any operator is in a one-to-one correspondence with a dual O(N)-invariant field whichis a statement of GKPY duality. In order to generalise the correspondence to dS space, we need to takethe analytical continuation. Using duality relation 6.40 correlation function for the bulk field in dS can bewritten as:
〈φ(τc,x1) . . . φ(τc,xn)〉dS = A(n)−C2
(−C2)n(−iτcx
)n∆+
. (6.51)
Similarly operators of corresponding Sp(N)-invariant QFT can be defined as:
O =
(1
−C2(−iτc)∆+
)φ
So boundary conformal field correlation functions then will be give by:[61]
〈O(x1) . . .O(xn)〉dS = A(n)−C2
x−n∆+ (6.52)
9In this quantum theory inner product is defined as 〈ψ′|ψ〉C ≡ 〈ψ′|C|ψ〉. [70]10This notation is chosen for convenience. As an example for a two-point function x2∆ = x2∆
12 ≡ |x1 − x2|2∆.
43
It can be seen that there is a one-to-one map between the correlators in the bulk HS gravity dS4 andcorrelation functions on the three-dimensional conformal theory:
〈O(x1) . . .O(xn)〉CFT3 ↔ 〈φ(τc,x1) . . . φ(τc,xn)〉dS4 (6.53)
44
Chapter 7
Cosmology
Inflation was a period of exponential expansion in a de Sitter background during the early universe beforethe hot big bang led into the radiation era. Inflation theory was successful in solving the old problemsabout flatness, horizon, and monopoles, and could predict large structure formation based on inhomogeneousquantum fluctuations. However inflation is not a complete theory and is unable to give any information aboutthe initial values of observables. In this section we study the application of dS/CFT duality in cosmology asthe main topic of this report, and try to show how one can define a holographic formalism for the universeduring the inflationary era. The ultimate goal is to introduce a quantum model that can provide quantitativepredictions that are compatible with latest observations.
In this chapter, at first a brief review of perturbation theory in cosmology is provided. At next stage, westart applying dS/CFT duality to the cosmological universe by explaining the correspondence between bulktime evolution and renormalisation group flow on the boundary, and address the issues regarding the numberof degrees of freedom. In the remainder of this chapter formulation of a holographic framework for cosmologyis review in details. Firstly we explain domain-wall cosmology correspondence, and employ the results offamiliar AdS/CFT connection to find the quantum field corresponding to the domain-wall. subsequentlyanalytical continuation N → −iN is used to map the quantities to the pseudo-QFT dual to the originalcosmology. Then it is shown how holographic formulae can be used to compute all inflationary observablesform predictions of the proposed holographic model. Finally a prototype system for phenomenological analysisis introduced, and predictions of this model is verified against latest observational data.
7.1 Inflationary Epoch
Theory of inflation was originally introduced so as to provide solutions to some cosmological puzzles includinghorizon, flatness and monopole problems. Later on it was realised that inflation can address another mysteri-ous problem in cosmological particle physics which was generation of inhomogeneity. Quantum perturbationson a homogeneous classical background are amplified during inflation and amount to formation of currentlarge structure of the universe. In this section we present an introductory review of selected topics aboutprimordial fluctuations generated by quantum fluctuations.
7.1.1 Metric Perturbation
In the context of general relativity, space-time is dynamical and will be perturbed as a result of fluctuationsin the matter. During inflationary period it is adequate to work with linear perturbation which involves smallinhomogeneities δgµν on the top of a homogeneous and isotropic background: 1 [18]
ds2 = gµν(t) + δgµν(t, x) dxµdxν , |δgµν | |gµν | (7.1)
It is natural to decompose this metric perturbation into three modes: scalars φ, ψ, ν and χ, vectors ωiand νi, and tensor γij . The vectors are transverse ∂iω
i = ∂iνi = 0, and the tensor is transverse traceless
1Linear approximation ceases to be valid at current time as inhomogeneities grew over time.
45
∂iγij = δijγij = 0. There are therefore six constraints, and the perturbed metric has ten independentcomponents. So linear perturbation of FRW metric can be written as: [74]
ds2 = − (1 + 2φ) dt2 + 2a2(t) (∂iν + νi) dtdxi + a2(t)
(δij − 2ψδij + 2∂i∂jχ+ 2∂(iωj) + γij
)dxidxj (7.2)
It turns out that vector perturbations decay over time so they can be ignored. So we centre on the scalarperturbations which affect structural formation, and tensor perturbation which generate gravitational waves.Since these three modes can be considered independently, from now onwards we set vector modes to zero. 2
[74]
7.1.2 gauge transformation
A general perturbation in metric can be defined as the difference between the physical perturbed metric g′µνand the unperturbed background metric δgµν(p′) = g′µν(p′)− gµν(p). But this comparison makes sense onlyif one defines a gauge choice which is a one-to-one map between the point p′ in the perturbed space-timeand the point p in the background. A change in this map or choice of coordinates is referred to as gaugetransformation. For instance consider the linear gauge transformation:
xµ → xµ + ξµ, ξµ = (ξ0, ∂iξ + ξi⊥) (7.3)
Then scalar fluctuations change like: [17]φ→ φ− ξ0
ν → ν + ξ0/a− aξχ→ χ− ξψ → ψ +Hξ0
(7.4)
However tensor perturbation is gauge invariant to linear order because there is no tensor in componentsof ξ. 3
There is no unique gauge choice and this freedom in selection of coordinates will give rise to fictitiousgauge. In general, there two ways to avoid spurious gauges:
1. Impose a gauge constraint to remove ambiguity. A familiar example is comoving gauge which requiresthat both comoving momentum δq and χ vanishes: 4 [18]
δq = 0, ξ = 0 (7.5)
2. Work with gauge invariant quantities which do not change under a coordinate transformation. Theexample we use in this chapter is curvature purterbation on hypersafes of uniform energy-density :
ζ = −ψ − H
ρδρ (7.6)
In addition to ζ there are other gauge invariant quantities obtained from metric perturbations: [74]
α = φ− δρ
ρ(7.7a)
β = ν − χ+δρ
a2ρ(7.7b)
Imposing Hamiltonian and momentum constraints on the above invariant parameters yields: [74]
α = − ζ
H, β =
ζ
a2H+εζ
q2(7.8)
where ε = −H/H2. 5 This means that the dynamics depend only on two invariant free parameters thatare curvature perturbation ζ, and graviton γij .
2This is the result of the translation invariance of the perturbed metric which means Fourier transformation of the modes donot interact. Se it is possible to study them separately and set unwanted ones to zero. [74]
3However γij is not invariant in quadratic order. [17]4Comoving gauge requires that slicing is comoving i.e. comoving momentum perturbation equals shift scalar δq = χ, and
that threading is also comoving i.e. momentum perturbation vanishes δq = 0 which gives the definition 7.5 above. [18]5This is the same ε as the slow-roll parameter; however,note that slow-roll condition is not used here.
46
Helicity
In Fourier space, it is convenient to expand a tensor like γij in helicity basis:
γij(t,q) =∑s=±2
γ(s)(t,q)ε(s)ij (q) (7.9)
where helicities satisfy orthogonality and normalisation conditions of: [17]
ε(s)ij (q) ε
(r)ij (−q) = 2δsr,
∑s=±2
ε(s)ij (q) ε
(s)kl (−q) = 2Pijkl
and Pijkl is traceless transverse projection operator. 6
7.1.3 Statistics of Perturbation
Consider a real scalar field like ζ. Due to homogeneity two-point correlation function should only depend onthe distance between two points: 〈ζ(x)ζ(y)〉 = f(|x−y|). Taking the Fourier transform it can be written as:
〈ζ(x)ζ(y)〉 =
∫〈ζqζq′〉e−i(qx+q′y)d3qd3q′
and symmetry constraints yield:[73]
〈ζqζq′〉 = (2π)3ps(q)δ(3)(q + q′) (7.10)
where q ≡ |q|, and p(q) is known as 3D power spectrum and has dimensions of volume.In an isotropic space we can do the angular Fourier transformation: [73]
〈ζ(x)ζ(x + r)〉 =
∫ps(q)e
−iq.r d3q
(2π)3
=
∫ ∞0
q2
2π2ps(q)
sin qr
qrdq
=
∫ ∞−∞
∆2s(q)
sin qr
qrd log q
where dimensionless power spectra ∆2(q) representing power per logarithmic intervals in momentum isdefined as:
∆2s(q) ≡
q3
2π2ps(q) (7.11)
In order to evaluate scale-dependence of power spectra it is convenient to define scalar spectral index as:[17]
ns − 1 ≡ d ln ∆2s
d ln q(7.12)
Another useful measure of primordial scalar fluctuations is running of the spectral index :
αs ≡dnsd ln q
(7.13)
For observational purposes, power spectrum is approximated by a power law form as: [17]
∆2s(q) = ∆2
s(q∗)(q
q∗
)ns(q∗)−1+ 12αs(q∗) ln(q/q∗)
(7.14)
6It is defined via Pijkl =(pikpjl + pilpjk − pijkl
)/2 where transverse projector is pij = δij − qiqj/q2. [17]
47
where q∗ is a chosen pilot scale.The same statistical quantities can be defined for a tensor like graviton γ, but it should be noted that
a tensor has two polarisation modes γ(s) = γ+2, γ−2. power spectrum for tensor perturbations is defined asthe sum of power spectra of two modes ∆2
T ≡ 2∆2γ .
∆2T (q) ≡ q3
2π2pT (q) 〈γqγq′〉 = (2π)3pT (q)δ(3)(q + q′) (7.15)
On historical grounds, tensorial spectral index is defined without the −1 term: [18]
nT ≡d ln ∆2
T
d ln q(7.16)
and one can run the tensor spectral index in a similar fashion:
αT ≡dnTd ln q
(7.17)
Power spectra is fitted by a power law of the form:
∆2T (q) = ∆2
T (q∗)(q
q∗
)nT (q∗)
(7.18)
and the ratio of the tensor to scalar power spectra is often shown as:
r ≡ ∆2T (q)
∆2s(q)
(7.19)
7.1.4 Dynamics
Action of a scalar field with potential V in a classical gravity background is given by: 7
S =1
2κ2
∫ √−g−R− gµν∂µΦ ∂νΦ + 2κ2V (Φ)
d4x (7.20)
We restrict to the case where Φ(t) is monotonic in time so that invert function t(Φ) is well-defined. Hubbleconstant can therefore be expressed in terms of Φ such that H = −(1/2)W (Φ). Then known equations ofmotion for inflation field on a classical background can be re-written as: [76, 17]
Φ = ∂φW, −2κ2V = (∂φW )2 − 3
2W 2 (7.21)
One can solve the Hamiltonian and momentum constraints 7.8 for free parameters ζ and γij , and insertthe results back into the action above. Then the action for perturbations at second order becomes: [18]
S =1
κ2
∫ (a3εζ2 − aε (∂ζ)2 +
1
8a3γ2
ij −1
8a∂kγ
2ij
)d4x
varying the action with respect to ζ and γij will result in the following equations of motion:
ζ +
(3H +
ε
ε
)ζ +
1
a2q2ζ = 0, γ(s) + 3Hγ(s) +
1
a2q2γ(s) = 0 (7.22)
Note that first equation implies that outside the Hubble horizon k/aH 1 on super-horizon scales scalarmode ζ is frozen in time. This is a general property of matter fields when they leave the horizon. It willremain constant until reentering the horizon after inflationary epoc. 8 [18]
7This is the minimally coupled case. More details about coupling of a scalar to gravity will be given in section 7.4.2.8More precisely this is true if the process is adiabatic and no entropy is produced. [17]
48
Linear Equations of Motion
When working to the first order, linearised wave equations facilitates the required amount of calculations.At first we need to define the conjugate momentum as:
Π = κ2 δLδζ
= 2εa3ζ, Πij = κ2 δLδγij
=1
4a3γij (7.23)
where for future convenience a factor of κ2 is included in the usual definition. Then linear responsefunctions are defined so as to relate the conjugate momenta to the perturbation fields:
Π(t,q) = Ω(t, q)ζ(t,q), Π(s)(t,q) = E(t, q)γ(s)(t,q) (7.24)
as mentioned Π(s) denotes helicity component of the linear response function Πij . Substituting back intothe equations 7.22, linear equations of motion can be written as:
Ω +1
2a3εΩ2 + 2aεq2 = 0, E +
4
a3E2 +
a
4q2 = 0 (7.25)
In most cases like calculation of power spectra, linear response functions suffice for derivation of cosmo-logical observables.
7.1.5 Cosmological Power Spectra
Having derived the classical equations of motion, standard prescription can be used to quantise the fields.Scalar and tensor quantum fields are now operators, and follow the equal time commutation relation:
[ζ(t,q), κ−2Π(t,q′)] = i(2π)3δ(3)(q + q′) (7.26a)
[γij(t,q), κ−2Πkl(t,q′)] = i(2π)3δ(3)(q + q′)Πijkl (7.26b)
The latter commutation can be rewritten in the helicity basis 7.9 as:
[γ(r)(t,q), κ−2Π(s)(t,q′)] =i
2(2π)3δ(3)(q + q′)δrs (7.27)
with other commutations vanishing. Note that κ−2Π and κ−2Πij are the actual canonical momenta.Operators can be expanded using creation and annihilation operators as follows:
ζ(t,q) = a(q)ζq(t) + a†(−q)ζ∗q (t), (7.28a)
γ(s)(t,q) = b(s)(q)γq(t) + b(s)†(−q)γ∗q (t) (7.28b)
and they follow the usual commutation relations:
[a(q), a†(q′)] = (2π)3δ(q− q′), [b(r)(q), b(s)†(q′)] = (2π)3δ(q− q′)δrs (7.29)
Using the mode decomposition 7.28 and the explicit expression for momentum one can reach to theWronskian relations: [76]
2εa3κ−2(ζq(t)ζ
∗q (t)− ζq(t)ζ∗q (t)
)= i, (7.30a)
1
4a3κ−2
(γq(t)γ
∗q (t)− γq(t)γ∗q (t)
)=i
2(7.30b)
Now, we can use the definition of linear response functions E and Ω in order to find the amplitude of thescalar and tensor perturbations. It follows that: [76]
|ζq(t)|2 = − κ2
2Im[Ω(t, q)], |γq(t)|2 = − κ2
4Im[E(t, q)](7.31)
Recalling the definition of power spectra, it is straightforward to define them in terms of constant latetime linear response functions:
49
∆2S(q) = − κ2q3
4π2Im[Ω(0)(q)], ∆2
T (q) = − κ2q3
2π2Im[E(0)(q)](7.32)
The relation above is the first step towards the holographic treatment of inflationary cosmology. In thenext sections we try to find a similar relation between domain wall two-point correlation function and linearresponses.
7.2 RG-Flow Universe
Recent observations confirms that universe had a de Sitter geometry far in the past during inflation and willbecome a de Sitter space again far in future. Value of cosmological constant raises by an order of about ahundred. [82, 83] During this time the metric describing the space-time is:
ds2 = −dt2 + e2Htdxidxi (7.33)
where Hubble constant is H =√
Λ/3. As discussed earlier isometry under t → t + τ , and xi → e−Hτxi
indicates that time evolution in the bulk corresponds to scaling of the conformal field on the boundary.writing the latter in momentum space as k → eHτk shows that temporal evolution is identified with energyscale of the gauge theory, hence to renormalisation group flow. Conformal invariance indicates that the fieldlives at a fixed point. This means that infra-red fixed point of the conformal field corresponds to the earlyinflationary times, whereas ultra-violet point is related to the late cosmological constant-dominated times.[56]
Between these two points, the universe is described by a flat FRW metric:
ds2 = −dt2 + a2(t)dxidxi (7.34)
The metric is no longer symmetric with respect to time evolution-scale transformation since Hubbleconstant H(t) = a(t)/a(t) is in general a function of time. Lack of symmetry in the bulk supposedlycorresponds to a boundary quantum field which is not conformal, and to a flow in the renormalisation group.So RG flow starts at the UV point and flows to the end IR fixed point. [58]
The universe is a RG flow from UV to IR fixed points of the dual Euclidean field theory.
Note that renormalisation group flows opposite direction to the time evolution. Consistent increase inthe value of Hubble constant shows that RG flow is irreversible. [75]
7.3 Holography for Cosmology
As discussed in previous chapter the gravitational theory describing the bulk space-time in dS/CFT cor-respondence is higher spin gravity, and it was shown that it corresponds to a fermionic Sp(N) invariantquantum field. Holographic renormalisation formalism for these theories is not yet very well understood. Inabsence of such a prescription we use the correspondence between cosmology and domain wall spaces to moveto anti-de Sitter space, and then apply the known AdS/CFT methodology to find the dual QFT.
Theses steps are schematically described in figure 7.1. The first step is to switch from cosmology todomain wall space time. Then the holographic renormalisation group flow machinery is employed to findthe quantum observables in terms of the corresponding domain wall variables. Finally we take an analyticalcontinuation to move to the QFT theory dual to the original cosmology. In this report, we refer to this gaugetheory as pseudo-QFT.
50
Figure 7.1: Schematic diagram illustrating methodology to find quantum field theory dual to the inflationarycosmology (reproduced from [77])
7.3.1 Domain-Wall / Cosmology Duality
Consider a scalar field Φ with a minimally coupled potential V (Φ) on a flat four dimensional background,the metric reads:
ds2 = σdz2 + a2(z)dxidxi, Φ = Φ(z) (7.35)
where i = 1 . . . 3, and σ = ± is a constant which allows for two different physical interpretation of themetric:
1. If σ = −1 then metric corresponds to the inflation field Φ on a flat FRW cosmology while z coordinatesinterpreted as time t.
2. If σ = 1, the metric describes a domain wall space-time, and z will be the radial r coordinates.
Without loss of generality, the domain wall metric shown here is Euclidean, but one can do a Wickrotation for one of xi coordinates to switch to Minkowski metric. Following similar procedures to section7.1.4 one can derive common classical equations of motion 7.21 for this metric:
Φ = ∂φW, 2σκ2V = (∂φW )2 − 3
2W 2 (7.36)
as well as the equations of motion for scalar and tensor fields 7.22:
ζ +
(3H +
ε
ε
)ζ − σ
a2q2ζ = 0, γ(s) + 3Hγ(s) − σ
a2q2γ(s) = 0 (7.37)
and relevant equations for linear responses 7.25:
Ω +1
2a3εΩ2 − 2σaεq2 = 0, E +
4
a3E2 − σa
4q2 = 0 (7.38)
To recognise similar quantities in two pictures, we show variables in the domain wall with a tilde on thetop. The claim is that domain wall space is related to the cosmology space-time via the analytic continuation:
κ2 → −κ2, q → −iq (7.39)
The correspondence applies to both homogeneous background and linear perturbation around that:
1. First we consider background solutions. Continuation κ2 → −κ2 is due to the change in the sign ofpotential in the background equation 7.36. The second map q → −iq comes from reversing the sign ofq2 in the equations 7.37. 9
9The choice of minus sign in −iq results from the fact that behaviour of Bunch-Davies vacuum ζ, γ(s) ∼ e−iqτ at early timesτ → −∞, should correspond to solution of the domain-wall as it decays ζ, γ(s) ∼ eqτ . [76]
51
2. As to linear perturbations, it can be noticed that if E(z,q) and Ω(z,q) are solutions to the cosmologicalequations 7.25; then the analytically continued responses E(z,−iq) and Ω(z,−iq) will solve the domainwall version of the equation i.e. 7.38.
Recall the discussion about Maldacena’s duality where it was shown κ2 ∼ N2. So keeping the cosmologicalconstant Λ fixed, continuation κ2 → −κ2 yields:
N → −iN, q → −iq (7.40)
Choice of sign in −iN will be discussed later in section 7.4.3.
7.3.2 Hamiltonian Formalism
As explained earlier in this paper holographic version of Hamiltonian methodology is identical to the usualHamiltonian-Jacobi theory with radial coordinate replacing time. This method features a number of ad-vantages: it admits an arbitrary potential for scalar fields, and also provides a universal formula for theexpectation value of energy-momentum: 10 [78]
〈Tmn 〉 =
(− 2√−g
Πmn
)(3)
(7.41)
with Πmn being radial canonical, and subscript three means components with conformal dimension three.
11
In an asymptotically AdS space radial momentum is given by: [76]
Πmn =
1
2κ2
√−g (Km
n −Kδmn ) (7.42)
where Kmn = 12∂zgmn is defined as extrinsic curvature on constant z hypersurfaces.
Finally by varying the Lagrangian, one can obtain Hamilton and momentum constraint equations 7.8.
7.3.3 CFT Two-Point Functions
In this section rules of holographic renormalisation already introduced in section 4.7 are employed to derivean expression for boundary quantum observables in terms of bulk domain wall linear response functions.This corresponds to the horizontal line on the top of schematic diagram 7.1. At first, holographic version ofHamiltonian formalism is reviewed so as to derive a useful expression for the stress two-point correlator. finallyrelation between domain wall response functions and two-point functions of the corresponding conformal fieldis introduced.
In previous parts of this chapter the expression for cosmological power spectra in terms of responsefunctions was derived. So at the end, we would be able to define the relation between cosmological observablesand pseudo quantum fields.
The ultimate goal is to show how cosmological observables are related to quantum correlation functions inorder to find a dual description of the universe. In general correlation functions are computed by solving theequations of motion for domain wall metric with asymptotic Dirichlet boundary conditions. In this section,we focus on energy-momentum correltor, and decompose it into a traceless part and a trace part, encodedby coefficients A and B respectively: [77]
〈Tmn(q)Trs(−q)〉 = A(q)Pmnrs +B(q)pmnprs (7.43)
10In this section Latin letter indices stating from i, . . . are used for spatial coordinate in the bulk while those starting fromm, . . . used for quantum operators on the boundary.
11This result is actually expected as in three dimensions energy-momentum has scaling dimension three. [76]
52
The aim is to find a relation between energy-momentum and domain-wall parameters, so we try to findan expression for coefficients A(q) and B(q) above in terms of domain wall response function.
Using metric perturbation formalism already discussed, variation of expectation value of energy-momentumis found to be (refer to appendix F for details):
δ〈T (s)(q)〉 =1
2A(q)γ
(s)(0)(q), δ〈T (q)〉 = −4B(q)ψ(0)(q) (7.44)
On the other hand after expanding 7.41 in momentum space, variation of stress tensor to linear orderwill be: (appendix F)
δ〈T (s)(q)〉 = − 2
κ2
[a−3E(q)γ(s)
](3), δ〈T (q)〉 =
1
κ2
(Ω(q)
a3− 2q
a2H
)ψ (7.45)
and comparing two equations above, one can work out transverse traceless A(q), and trace B(q) piecesas: [77]
A(q) = −4κ−2E(0)(q) B(q) =1
4κ−2Ω(0)(q) (7.46)
where the zero subscripts refers to the term in the response function that has zero scaling dimension orin other word E(0) and Ω(0) are independent of r. 12
To summerise the computations so far, at first in 7.32 we defined the cosmological power spectra in termsof imaginary parts of cosmological linear response functions. Then in 7.46 we expressed energy-momentumcorrelation function in terms of zero-scaling dimension pieces of domain-wall response functions. Now, allthe ingredients are in place. We only need to take the continuation 7.40 and find the imaginary parts ofthe response functions ImΩ0 and ImE0. The expression for dimensionless power spectrum ∆2
s(q) and ∆2T (q)
7.32 will amounts to our principal result:
∆2s(q) =
−q3
16π2Im B(−iq), ∆2
T (q) =−2q3
π2Im A(−iq)(7.47)
This equation shows that power spectrum in the cosmology can be obtained straight away from correlationfunctions in the dual quantum field. Note that the scalar power spectrum ∆2
s(q) is dual to the trace piece ofenergy-momentum two-point correlator while the tensor power spectrum ∆2
T (q) corresponds to the tracelesspiece.
7.3.4 Weak versus Strong Gravitational Coupling
In this section, It was explained that cosmological observables can be described in terms of correlationfunctions of a dual quantum field. In particular it should be emphasised that from holographic point ofview, inhomogeneities are not originating from perturbations around a classical FRW background metric asconventional inflation predicts, but form the fully quantum dynamics of the dual gauge field. This providesa new insight into the theory of inflation; however the proposed holographic model can be used to offer abroader scope beyond other current theories.
At early times before inflation universe was in a complicated strongly coupled quantum gravity phase withno clear description in low energy effective fields. However gauge/gravity relation is a strong-weak duality sothat weakly coupled gravity era can be mapped to weakly interacting low energy fields in the dual quantumfield theory. [37, 56]
12E(0) and Ω(0) can also be thought of as the leading term in expansion of the response functions E and Ω as r →∞.
53
It can be conjectured that early strongly-coupled quantum gravity phase smoothly transformed into theweakly interacting inflationary epoc. So it is possible to investigate the early time behaviour of the universeusing the proposed holographic model in a well-understood weakly coupled regime of QFT, and providequantitative predictions about the unknown features of quantum gravity phase. holographic cosmology cantherefore offer a new perspective on the problem of initial conditions which is beyond the scope of currentstandard inflation formalism. [76]
7.4 Holographic Phenomenology
The dual conformal field theory ideally should be derived from the quantum gravity theory in dS space;however, the map between the cosmological gravity and the dual field is not theoretically very well understood.Nevertheless we are still interested in a testable field theory that complies with recent observations. So inthis section, we leave the fundamental approach taken throughout this report, and set up a phenomenologicalplan to find a prototype model. An intelligent choice is to use quantum field models which their dual gravitytheory is known. In what follows, at first step a prototype CFT model on the AdS boundary dual to domain-wall gravity is defined. Then this model is used to compute all relevant QFT observables like two-pointcorrelation functions. At the end analytical continuation is performed to move to the conformal field theorydual to the original cosmology, and thus to deduce such cosmological observables as power spectrum andrunning of spectral index.
7.4.1 Prototype CFT Theory
Prototype theory is supposed to be a typical model emerging in AdS/CFT discussions. It turns out thatsuch a model needs to satisfy some restrictions: rank of gauge group N should be large, all the fields arerequired to be massless, coupling constant must be dimensionful, and scaling dimension of Lagrangian termsshould not equal three. [59] The proposed model is a three-dimensional SU(N ′) Yang-Mills theory with thefollowing action: [76]
S =1
g2YM
∫ 1
2F ImnF
Imn +1
2Dmφ
JDmφJ +1
2Dmχ
KDmχK + ψL /DψL
+ λM1M2M3M4ΦM1ΦM2ΦM3ΦM4 + µαβML1L2
ΦMψL1α ψL2
β
d3x (7.48)
where Fmn is field strength, φ denotes an ordinary scalar, but χ is a conformal scalar, Φ = (φ, χ) representsa pair of these scalars, and ψ is a spinor. Note dimensionful coupling gYM has dimension one 13 , so thetheory is super-renormalisable, whereas quartic λ, and Yukawa µ couplings are dimensionless. Here capitalletters are used to denote gauge group indices, Greek letters show fermionic indices.
7.4.2 Energy-Momentum Tensor
At this point we need to review some background information about quantum field theory in the curvedspace-time as energy-momentum tensor in a curved space is different than the one on a flat background.Lagrangian for a massless scalar field in presence of gravity is given by: [7]
L =1
2
√−ggmn∂mφ ∂nφ+ ξRφ2
(7.49)
where g is determinant of metric, and R is Ricci scalar. Note the last term ξRφ2 is the coupling betweenscalar field and gravity, and on dimensional grounds it is the only possible option. The dimensionless couplingconstant ξ determines the strength of interaction. If it vanishes then there is minimal coupling i.e. gravityand scalar are independent.
13For simplicity we normalised all the field so as to gYM appears in front of the action.
54
Figure 7.2: Feynman diagram for two-point function of energy momentum tensor 〈Trs(q)Tmn(−q)〉 at one-looporder. (reproduced from [76])
For an arbitrary field, the energy momentum tensor can be obtained through variation of the action withrespect to the metric: [2]
Tmn =2√−g
δS
δgmn(7.50)
Thus for a minimally coupled scalar energy-momentum is found to be: [8]
Tmn = ∂mφ ∂nφ−1
2gmng
rs∂rφ ∂sφ (7.51)
Another value of interest for coupling constant is ξ = (d − 2)/4(d − 1) which amounts to conformalcoupling. In this case metric is invariant under a conformal transformation 2.1. Energy-momentum tensorfor such a massless scalar χ will be given by: [7]
Tmn = (1− 2ξ)∂mχ ∂nχ+ (2ξ − 1
2)gmng
rs∂rχ ∂sχ− 2ξ(∂m∂nχ)χ
+2
dξgmn χχ− ξ
(Rmn −
1
2Rgmn + 2
d− 1
dξRgmn
)χ2 (7.52)
Furthermore energy-momentum of a spinor is found to be:
Tmn =1
2
[ψγ(m∂n)ψ − (∂(mψ)γn)ψ
](7.53)
and for the gauge field will be:
Tmn = 2F rmFrn −1
2gmnF
rsFrs (7.54)
Replacing for covariant derivative, and adding the Yukawa and quartic interaction terms the total energy-momentum tensor of the non-abelian model reads: [76]
Tmn =1
g2YM
tr
F Irm F Irn +Dmφ
JDnφJ +
3
4Dmχ
KDnχK − 1
4(DmDnχ
K)χK
− 1
8
(Rmn −
1
2Rgmn +
1
6Rgmn
)(χK)2 +
1
2
[ψLγ(mDn)ψ
L − (D(mψL)γn)ψ
L]
− gmn(F IrsF Irs +
1
2grsDrφ
j Dsφj +
1
4grsDrχ
K DsχK +
1
12χKχK
+ λM1M2M3M4ΦM1ΦM2ΦM3ΦM4 + µαβML1L2
ΦMψL1α ψL2
β
)(7.55)
7.4.3 Cosmological Observables
In the following sections the aim is to calculate such cosmological observables as power spectra and running ofthe spectral index, and make a comparison with current observations. Even though a different phenomenologyis used, compatible results would confirm validity of holographic model introduced here.
55
Figure 7.3: Feynman diagrams for two-point function of energy momentum tensor 〈Tmn(q)Trs(−q)〉 at two-loop order. (reproduced from [76])
In order to do so, first we need to find the coefficients A(q) and B(q) i.e. traceless and trace componentsof energy-momentum 2-point function 7.46. Then the result is analytically continued to the pseudo-QFT,and finally power spectra 7.47, and other observables could be readily computed. Also along the way, wecan check correctness of our assumptions regarding large N limit, and small effective interaction geff .
1-Loop Order
Summing over contribution from different fields in 7.48, 2-point function for energy-momentum could beobtained. In addition power counting can be used as a calculation shortcut. Because Tmn has dimensionthree, there should be a q3 contribution. 14 [76] Also due to conformal structure, and large N counting thegeneral format of coefficients will be given by:[77]
A(q) = a(0)N2q3, B(q) = b(0)N2q3 (7.56)
where the coefficient N2 stems from tracing over gauge indices. 15 Coefficients a(0) and b(0) are a functionof dimensionless effective coupling geff = g2
YM N/q, and they also depend on the field content: [76]
a(0) = (NA +Nφ +Nχ + 2Nψ) /256, b(0) = (NA +Nφ) /256 (7.57)
where NA indicates number of gauge fields A in the system etc. One-loop diagram for 〈Tmn(q)Trs(−q)〉is shown in figure 7.2. Applying the continuation N = −iN , and q = −iq, the cosmological power spectrumwill be:16 [76]
∆2s(q) =
1
16π2N2b(0), ∆2
T (q) =2
π2N2a(0)(7.58)
This clearly shows that power spectra is scale independent to the leading order which is consistent withobservational data.
The tensor to scalar amplitude ratio is found to be:
r =∆2s
∆2T
= 32b(0)
a(0)(7.59)
Maximum value of r will be constrained by observational data which in turn clarifies the field content7.56. For instance if r is decreased, the number of gauge fields and scalars need to decrease or fermions andconformal scalars must increase in number.
Using equation 7.58 above, one can verify our assumption about the large value of N . According toWAMP data [82], scalar power spectrum is ∆2
S(q∗) ≈ 10−9 which means N ≈ 104.
2-Loop Order
In order to find correction to one-loop order calculations, one needs to evaluate the energy-momentum tensorcorrelator at two-loop order. Feynman diagram for 〈Tmn(q)Trs(−q)〉 at two-loop order is shown in figure
14Note only comoving momentum q with dimension one counts because Yang-Mills coupling gYM does not contribute at oneloop order.
15Recall that there was a factor of κ2 in the expression for power spectrum 7.32 which corresponds to the factor of N2 hereon the CFT side. Also recall the relation κ2 ∼ N2.
16Dimensionless ’t Hooft coupling geff remains invariant under this continuation hence justifying minus sign in N = −iN .
56
Figure 7.4: Error bar plot for WAMP data from figure 4 in [82] which shows 68% and 98% CL constraintsat the pilot scale q∗ = 0.002Mpc−1. The straight line is QFT prototype predictions.(reproduced from [77])
7.3. Scale-free theories which are super-renormalisable have infra red divergences. [12] Power counting showsthat each of the diagrams has a quadratic IR divergence, and after regularisation it is evaluated as q2 ln q/q∗.There is also an overall factor of N3g2
YM . So in total contribution of each diagram to the stress correlationfunction will be found to be N3g2
YM q2 ln (q/q∗) = N2q3g2
eff ln (q/q∗). Therefore at this order, the tracelessand trace coefficients can be written as: [76]
A(q) = a(0)N2q3[1 + a(1)g2
eff ln (q/q∗)], (7.60a)
B(q) = b(0)N2q3[1 + b(1)g2
eff ln (q/q∗)]
(7.60b)
where coefficients a(1) and b(1) depend on the field content and, Yukawa and quartic coupling constants.Substituting this result back into the expression for 〈Tmn(q)Trs(−q)〉 in 7.47 we obtain: [77]
∆2S(q) = ∆2
S(q∗)[1− b(1)g2
eff ln(q/q∗)], (7.61a)
∆2T (q) = ∆2
T (q∗)[1− a(1)g2
eff ln(q/q∗)]
(7.61b)
where ∆2s(q∗) ≡ 1/(16π2N2b(0)) and ∆2
T (q∗) ≡ 1/(16π2N2a(0)).In order to make a comparison, we expand 7.14 and 7.18 to first order in (nS − 1) and nT which yields:
∆2s(q) = ∆2
S(q∗) [1 + (nS(q)− 1) ln(q/q∗)] , (7.62a)
∆2T (q) = ∆2
T (q∗) [1 + nT (q) ln(q/q∗)] (7.62b)
Comparing 7.61 and 7.62 indicates:
ns(q) = 1− b(1)g2eff + +O(g4
eff ) nT = −a(1)g2eff + +O(g4
eff ) (7.63)
It can be seen that at two-loop order there is a slight deviation from the original scale-invariance ofspectral index. To investigate scale dependence, one can run the spectral index, wich yields:
αS =dnSd ln q
= −(nS − 1) +O(g4eff ) (7.64)
However slow-roll approximation predicts that αS if of second order in (nS − 1). To make a betterjudgment, error bar plot for WAMP data showing spectral tilt nS versus αS is illustrated in figure 7.4. It isclear that predictions of the model studied lies close to the allowed region which means that the results arein agreement with observational data. [82]
Depending on the sign of a(1) (b(1)) red or blue power spectrum is favored. Power spectrum for positiveand negative values of the coefficient b(1) is plotted in figure 7.5. It can be seen that for large momenta,
57
Figure 7.5: Relative power spectrum in momentum scale. The lower graph illustrates negative value of b(1),and upper one corresponds to positive values.(reproduced from [79])
spectrum is scale independent, and positive value of b(1) corresponds to a slightly blue index while negativevalue amounts to a slightly red tilt.
Infra red divergences causes a pick at smaller values of momentum at ln |q/q| = 1. Nevertheless note thatthe perturbative treatment used in analysing the energy-momentum is no longer valid for small momenta.So the results around the IR peak could be unreliable. [79]
We also need to check the value of effective coupling geff to verify validity of perturbation method usedhere. According to WAMP data at q = 0.002Mpc−1 spectral index reads (nS − 1) ≈ O(10−2) which givesthe small value of geff ≈ O(10−2). [82]
Once values of N , gYM , and the field content are determined, one can compute any other cosmologicalobservables using this model. As discussed above, in conclusion, the phenomenological approach is able tosatisfy constraints given by current observations.
58
Chapter 8
Conclusion
In this report, gauge/gravity correspondence for de Sitter space was studied and different properties wereinvestigated. In the first chapters basic materials necessary to understand the duality was reviewed whichincluded conformal field theory, and geometry of constant curvature spaces specially anti-de Sitter and deSitter. The formulation of dS/CFT duality has been guided by the anti-de Sitter counterpart correspondence.As a result, at first a review of AdS/CFT duality was provided, and different aspects of the correspondenceincluding concrete example of AdS5 × S5/N = 4, SYM relation were studied.
However since in the dS/CFT correspondence time direction in the gravitational theory is holographi-cally reconstructed, this duality has some unique features. It was shown that corresponding conformal fieldlives in the asymptotic future of the de Siter, and there exist a renormalisation group which flows back tothe asymptotic past in opposite direction to the time. It was also explained that the fundamental theorydescribing the gravity in this space is higher spin theory which corresponds to a fermionic Sp(N) invariantconformal field.
The application of dS/CFT duality to cosmology as the main topic of this dissertation was studiedin details, it was explained that how holographic renormalisation group could be formulated for universeduring the inflationary period. The exact correspondence between cosmology and the gauge theory is notyet well understood. Using the correspondence between cosmology and domain wall spaces, nevertheless itwas shown that there is a methodology which makes this connection through the well established anti-deSitter and conformal field AdS4/CFT3 duality. The main point was to relate the cosmological observablesto correlation functions of the quantum field on the other side.
A proposed prototype quantum field model was reviewed in details and values of cosmological observablesin particular power spectra was computed using this model. It was explained that predictions of the prototypequantum field is consistent with latest observational data available. We also argued that holographic modelis able to make predictions where current inflation theories do not apply.
The gauge-gravity connection is a kind of weak-strong duality which means when the gravity is stronglycoupled the corresponding quantum field has weak interactions. It was discussed that the dual QFT ispotentially able to provide predictions on the strongly-interacting era which is beyond the scope of currentstandard inflationary theories. This means that the holographic model has the potential to solve majorproblems in the contemporary cosmology about the initial conditions of the early universe.
8.1 Futer Works
In this section some suggestions for possible future works on applications of the gauge/gravity duality in cos-mology are provided. They range from computational simulations to analytical studies and more fundamentalideas at early conceptual stages.
More recent results of the Planck space observatory have been released providing a more detailed imageof cosmic microwave background. It would be interesting to investigate the effect of new information onthe predictions of the prototype quantum model, and to examine whether predictions of the model are stillcompatible with resent observations. In addition over the coming years, large amount of observational datawill impose more constraints on the values of cosmological observables. If new results confirm the predictions
59
of the dual QFT model, it could be regarded as the first evidence for the holographic description of ouruniverse.
In principal holographic formalism is applicable to any cosmology with asymptotic de Sitter behaviournot only the inflationary epoc. In particular cosmological holography explained in this paper could be appliedto the far late cosmological constant dominated universe. At first step we need to provide a clear definitionof cosmological observables for the late universe. Then we need to try to derive an RG flow formulation forthat era along the lines of what was accomplished for the inflationary period.
Before the inflationary epoc, universe was in an unknown intractable state where gravity was stronglycoupled when current standard theories of inflation can not provide any explanation. As discussed earlier,dS/CFT prescription is able to map this system of strongly interacting gravity to a theory of quantumfield with weak coupling. The proposed holographic model of inflation can provide a complete descriptionincluding the early behavior of the system. It would be desirable to find an explicit definition of the dualquantum model. Such a model will be able to give quantitative predictions about pre-inflationary state andoffer some solutions for the problem of initial conditions.
It was discussed that loop corrections of the correlators in cosmology correspond to 1/√N corrections
to the quantum system. In the limit where N is large, the analytic continuation is still valid, and thismodel is able to predict all cosmological observables. The next step is to move beyond the perturbativelimits and find a more general definition of the holographic model. The gravitational theory supporting thedS/CFT correspondence is the higher spin formalism of gravity. A longer-term goal would be to find aholographic framework in the context of correspondence between HS gravity and Sp(N)-invariant fermionictheory of quantum field. In absence a clear relation between the mentioned theories one can try an analogousmethodology to what proposed in this report. Using analytic continuation it is possible to map de Sitter toa Euclidean AdS space, and then use the known GKPY duality to provide a definition of the correspondingQFT with a familiar O(N) gauge group. All the computations can be carried out in this regular quantumtheory, and finally results are analytically mapped to the pseudo-QFT dual to the higher spin gravity.
8.2 Acknowledgment
I would like to acknowledge Dr Toby Wiseman supervisor of this project, for his instructive comments anddiscussions over the course of the project.
I would also like to acknowledge Prof. Carlo Contaldi, and Dr Paul McFadden for their contributions tothis project.
60
Appendix A
Anti de Sitter Equation of Motion
The equation of motion for a scalar field in AdSd is given by:
(−m2) φ = 0
where Laplacian can be computed as:
φ = gµνφ,µ;ν =1√g∂µ (√ggµν∂νφ)
=zd
ld
[∂z
(ld
zdz2
l2∂z
)+ld
zdz2
l2∂2x
]=z2
l2[zd−2∂z
(z2−d∂z
)+ ∂2
x
](A.1a)
=z2
l2[∂2z − (d− 2)z−1∂z + ∂2
x
](A.1b)
Separating the variables via φ(z,x) = f(z)Φ(x), and using relation A.1a, equation of motion will bewritten as:
−z2
l2[zd∂z
(z2−df ′
)Φ + f∂2Φ
]+m2fΦ = 0
where ∂2 denotes Laplacian in Minkowski space-time. Dividing both sides by fΦ we reach to two inde-pendent equations:
−zd−2
f∂z(z2−df ′
)+m2l2
z2=∂2Φ
Φ= −k2
where kµ turns out to be momentum with modulus square k2. Treating both of these equations indepen-dently, one reaches to the set of equations of motion in 3.24:
(∂2x − k2)Φ(x) = 0 (A.2)[
−zd∂z(z2−d∂z) +m2l2 + k2z2]f(z) = 0 (A.3)
and the complete solution is given by the superposition of the form φ(z,x) =∫
Φk(x)fk(z)dd−1k. Theequation of motion for Φ is the known Klein-Gorden equation with plane wave solutions Φ(x) = eikx/(2π)d−1.So superposition of the results gives rise to 3.25:
φ(z,x) =
∫fk(z) eikx
dd−1x
(2π)d
In addition, equation of motion for f can be rewritten as:
z2f ′′k − (d− 2)zf ′k −(m2l2 + k2z2
)fk = 0
a
change of variable gk = z(1−d)/2fk gives:
z2g′′k + zg′k −(
(d− 1)2
4+ k2z2 +m2l2
)gk = 0
Writing the above equation in terms of kz rather than z, we end up with modified Bessel equation: [38]
(kz)2g′′k + (kz)g′k −(
(d− 1)2
4+m2l2 + k2z2
)gk = 0 (A.4)
with the solution:gk = akKδ(kz) + bkIδ(kz)
where δ =√
(d− 1)2/4 +m2l2, and Iδ and Kδ are modified Bessel functions of first and second type
respectively. Rewriting in terms of fk, the solution to the wave equation will be found as:
fk(z) = ak(kz)(d−1)/2Kδ(kz) + bk(kz)(d−1)/2Iδ(kz)
The solution has to be regular throughout the interior, and particularly when approaching the deepinterior z →∞. Modified Bessel equations in the limit of infinity behave like: [23]
limz→∞
Iδ(z) = ez/√z, lim
z→∞Kδ(z) = e−z/
√z (A.5)
So first Bessel function Iδ diverges unless bk = 0, which reduces the solution to:
fk(z) = ak(kz)(d−1)/2Kδ(kz)
On the other side, when approaching the origin, modified Bessel functions behave like:
limz→0
Iδ(z) =1
Γ(δ + 1)
(z2
)δ, lim
z→0Kδ(z) =
Γ(δ)
2
(2
z
)δ(A.6)
So near the boundary solution z → 0 will be:
fk(z) ≈ ak(kz)(d−1)/2
[Γ(δ)
2
(2
kz
)δ+
Γ(−δ)2
(kz
2
)δ]
Defining φ− ≡ ak2δ−1Γ(δ)k∆− , and φ+ ≡ ak2−(δ+1)Γ(−δ)k∆+ one reaches to the asymptotic solution inthe momentum space 3.28:
fk ≈ φ−(k)z∆− + φ+(k)z∆+
with usual scalar mass-dimension relation ∆± = (d−1)2 ± δ = (d−1)
2 ±√
(d−1)2
4 +m2l2.
Finally applying the reverse Fourier transformation we find the familiar scalar solution in the anti-deSitter space, 3.30:
limz→0
φ(z,x) = z∆−φ−(x) + z∆+φ+(x) (A.7)
b
Appendix B
Anti-de Sitter Bulk-to-BoundaryPropagator
Bulk-to-Boundary Propagator in AdSd is given by 4.21
K(z,x;y) = C∆+
(z
z2 + (x− y)2
)∆+
(B.1)
In this section we show that near the boundary z−∆−K behaves like a delta function, and fix the nor-malisation constant C∆+ . So it is required to integral over x and expect to reach to a constant value. Dueto translation invariance we can set y = 0.∫
K(z,x)dd−1x = C∆+z∆+
∫1
(z2 + x2)∆+dd−1x
= C∆+z∆+Ωd−2
∫ ∞0
rd−2
(z2 + r2)∆+dr
changing the variable t = r/z:
= C∆+zd−1−∆+Ωd−2
∫ ∞0
td−2
(1 + t2)∆+dt
Again change of variable x = t2:
=1
2C∆+
zd−1−∆+Ωd−2
∫ ∞0
x(d−3)/2
(1 + x)∆+dx
=1
2C∆+
zd−1−∆+Ωd−2 B(d− 1
2,∆+ −
d− 1
2)
=1
2C∆+z
d−1−∆+2π(d−1)/2
Γ(d−1
2
) Γ(d−1
2
)Γ(∆+ − d−1
2
)Γ(∆+)
where Ωd−2 = 2π(d−1)/2/Γ(d−1
2
), and beta function is defined as: ??
B(x, y) =
∫ ∞0
tx−1
tx+ydt =
B(x)B(y)
B(x+ y)(B.2)
So normalisation constant C∆+is fixed to:
C∆+=
Γ(∆+)
π(d−1)/2 Γ(∆+ − d−12 )
(B.3)
c
such that: ∫K(z,x)dd−1x = z−∆− (B.4)
This shows that K does not behave like a delta function but z−∆−K does:
limz→0
z−∆−K(z,x;y) = δ(d−1)(x− y) (B.5)
d
Appendix C
Anti de Sitter Two-Point Function
It was shown in section 4.4.1 that action of interest for a two-point correlator in the AdSd is given by:
S =1
2
∫ √−ggµν∂µφ∂νφ−m2φ2
dd−1xdz
For the first term, integration by part gives:√−ggµν∂µφ∂νφ = ∂µ
(√−ggµνφ ∂νφ
)− φ√−g φ
with usual definition of Laplacian in 3.22. Imposing the on-shell condition (−m2)φ = 0, the actionreduces to:
S =1
2
∫ √−g∂µ (gµνφ ∂νφ) dd−1xdz − 1
2
∫ √−gφ(−m2)φ dd−1xdz
=1
2
∫ √−g∂µ (gµνφ ∂νφ) dd−1xdz
Using Stoke’s theorem this can be integrated over the boundary: [43]
S =1
2
∫∂AdS
√−gφ (n.∇φ) dd−1x (C.1)
Without loss of generality one can choose the normal vector n parallel to z axis. Working in Poincarecoordinate
√−g = z−d and gzz = z2, when approaching the boundary the action behaves like:
limz→0
S = −1
2
∫∂AdS
√−g gzzφ ∂zφ dd−1x
= −1
2
∫∂AdS
z2−dφ ∂zφ dd−1x (C.2)
Now, recall that field is given by the convolution of the bulk-to-boundary green function with the source4.18:
∂zφ(z,x) = ∂z
∫K(z,x;y)φ0(y)dd−1y
=1
C∆+
∫∂z
[z∆+
(z2 + |x− y|2)∆+
]φ0(y)dd−1y
=∆+
C∆+
∫ [z∆++1 + z∆+−1|x− y|2 − 2z∆++1
(z2 + |x− y|2)∆++1
]φ0(y)dd−1y
When approaching the boundary this will behave like:
limz→0
∂zφ(z,x) =∆+
C∆+
∫z∆++1
|x− y|2∆+dd−1y (C.3)
e
In addition, recall that near the boundary the field behaves like φ ≈ z∆−φ0. Inserting these results intoC.2 powers of z cancel out, and asymptotic action is simplified to:
S =1
2
∆+
C∆+
∫φ0(x)φ0(y)
|x− y|2∆+dd−1xdd−1y (C.4)
Finally two-point correlation function can be computed by taking the derivative with respect to the source:
〈O(x)O(y)〉 =−iδ
δφ0(x)
−iδδφ0(y)
S∣∣∣∣φ0=0
=
(− δ
δφ0(y)
)[∆+
2C∆+
(φ0(x)δ(d−1)(x− x′)
|x′ − y′|2∆++φ0(x)δ(d−1)(x− y′)
|x′ − y′|2∆+
)S] ∣∣∣∣φ0=0
= − ∆+
2C∆+
(δ(d−1)(x− x′)δ(d−1)(y− y′) + δ(d−1)(x− y′)δ(d−1)(y− x′)
|x′ − y′|2∆+
)which gives the expression 4.29 in the text:
〈O(x1)O(x2)〉 = − ∆+
2C∆+
1
|x− y|2∆+(C.5)
f
Appendix D
De Sitter Green Function
As explained in section 6.5 green function in the de Sitter space is the solution of Laplace equation:(−m2
)G(x− y) = δ(d)(x− y)
which leads into the hypergeometric equation 6.13, and the solution will be a multiple of hypergeometricfunction F :
G(P ) = C∆F
(∆+,∆−,
d
2;P + 1
2
)here we need to fix the constant C∆. The exact behavior of the hypergeometric function F (a, b, c; z) near
the singularity z = 1 is given by: [23]
F
(∆+,∆−,
d
2; z
)≈(D2
4
)1− d2 Γ(d2
)Γ(d2 − 1
)Γ (∆+) Γ (∆−)
(D.1)
where D is geodesics separation defined in section 5.3. A short distance singularity in de Sitter behaveslike: [55]
Γ(d2 − 1
)2(d− 2)πd/2
D2−d (D.2)
Value of coefficient C∆ can be obtained by equating D.1 and D.1 so that:
C∆ = 41−d/2 Γ (∆+) Γ (∆−)
Γ(d2
)Γ(d2 − 1
) × Γ(d2
)2(d− 2)πd/2
=Γ (∆+) Γ (∆−)
(4π)d/2Γ(d2
) (D.3)
So the green function in de Sitter space is given by:
G(P ) =Γ (∆+) Γ (∆−)
(4π)d/2Γ(d2
) F (∆+,∆−,d
2;P + 1
2
)(D.4)
D.1 Asymptotic behaviour
Near the I− boundary, invariant distance between two points P diverges because recalling 5.18:
limt,t′→−∞
P (t,x; t′,y) = −1
2e−(t+t′)|x− y| (D.5)
g
So the last argument in D.4 above, diverges. There are transformation formulas for hypergeometricfunction which relate the functions with different values in z, the last argument: [23]
F
(∆+,∆−,
d
2; z
)=
Γ(d2
)Γ (∆− −∆+)
Γ (∆−) Γ(d2 −∆+
) (−z)−∆+F
(∆+,∆+ −
d− 1
2,∆+ −∆− + d− 2;
1
z
)+ (∆+ ↔ ∆−)
(D.6)Also note:
F (a, b, c; 0) = 1
So asymptotic behaviour of green function near the I− boundary, will be:
limt,t′→−∞
G(t,x; t′,y) =(d+ 1)∆+Γ
(d2
)Γ (∆− −∆+)
Γ (∆−) Γ(d2 −∆+
) e∆+(t+t′)
|x− y|2∆++ (∆+ ↔ ∆−) (D.7)
h
Appendix E
De Sitter Two-Point Function
As explained earlier, the green function in de Sitter space dSd is defined as 6.9:(−m2
)G(x− y) = δ(d)(x− y)
Therefore the field can be expressed:
φ(t,x) =
∫ √−gφ(t′,y)(−m2)G(t,x; t′,y)dt dd−1y (E.1)
Integrating the first term twice by parts yields:
φ G = ∂µ(gµνφ ∂νG)− gµν∂µφ ∂νG= ∂µ(gµνφ ∂νG) + ∂µ(gµν∂νφ G) + φ G
So E.1 becomes:
φ(t,x) =
∫ √−g∂µ(gµνφ ∂νG+ gµν∂νφ G) + (−m2)φ G
dt dd−1y
=
∫ √−g∂µ
(gµνG
↔∂µφ
)dt dd−1y
where in the second line on-shell constraint ( −m2)φ = 0 is imposed. Using Stoke’s theorem this canbe rewritten as:
φ(t,x) =
∫∂dS
√−gG(t,x; t′,y)n
↔.∇φ(t′,y)dd−1y
Without loss of generality, we can take the normal vector n to be in parallel with time axis. Using planarcoordinates 5.13 have
√−g = e−(d−1)t/l, and gtt = −1. So near the boundary field goes like:
limt→−∞
φ(t,x) = −∫I−e−(d−1)t/lG(t,x; t′,y)
↔∂t′φ(t′,y)dd−1y (E.2)
On the other hand, it was discussed in section 6.6 that required gravity action is given by:
S =1
2
∫ √−ggµν∂µφ∂νφ−m2φ2
dt dd−1x
Following the same steps as before, by integration by part, and applying the Stoke’s theorem on the pastnull infinity I−, the on-shell action behaves like:
limt→−∞
S =
∫I−e−(d−1)t/lφ(t,x)
↔∂tφ(t,x) dtdd−1x (E.3)
Now, inserting expression for the field E.2 into the above integral gives the following relation for thegravity on-shell action:
i
limt,t′→−∞
S = −∫I−e−(d−1)(t+t′)/lφ(t,x)
↔∂tG(t,x; t′,y)
↔∂t′φ(t′,y) dd−1ydd−1x (E.4)
which is the same equation as 6.21 in the text.
E.1 Asymptotic Behaviour
Referring to D.7, near the past boundary I−, green function behaves like:
limt,t′→−∞
G(t, xi; t′, yi) = C∆+
e∆+(t+t′)
|x− y|2∆++ C∆−
e∆−(t+t′)
|x− y|2∆−(E.5)
where C∆± are constants fixed in the appendix D. Inserting asymptotic green function above E.5 andthe field on the boundary 6.18 in the action E.4, we find:
S ≈∫φ−(x)φ−(y)
|x− y|2∆+dd−1xdd−1y (E.6)
Taking the derivative of action with respect the source, two-point correlation function will be given by:
〈O(x)O(y)〉 =δ
δφ−(x)
δ
δφ−(y)S∣∣∣∣φ−=0
=
(δ
δφ−(y)
)[∆+
2C∆
(φ−(x)δ(d−1)(x− x′)
|x′ − y′|2∆++φ−(x)δ(d−1)(x− y′)
|x′ − y′|2∆+
)S] ∣∣∣∣φ−=0
=∆+
2C∆
(δ(d−1)(x− x′)δ(d−1)(y− y′) + δ(d−1)(x− y′)δ(d−1)(y− x′)
|x′ − y′|2∆+
)This gives the relation 6.24 in the main text:
〈O(x)O(y)〉 =∆+
2C∆
1
|x− y|2∆+(E.7)
j
Appendix F
Holography for Cosmology
In this appendix it will be demonstrated how energy-momentum tensor two-point functions can be expressedin terms of domain wall linear response functions. Hamiltonian formalism already introduced in section 4.7is used to perform the computations.
Before dong so consider quantum field theory on the boundary which is a flat metric in absence of anysources. Ward identity indicates that energy-momentum 2-point function must be transverse:
qm〈Tmn(q)Trs(−q)〉 = 0
Considering symmetry constraints, there are only two ways to construct transverse tensors which impliestow-point stress correlator can be decomposed as follows: (equation 7.43)
〈Tmn(q)Trs(−q)〉 = A(q)Pmnrs +B(q)pmnprs (F.1)
As explained in the text, it is desirable to express transverse traceless A(q), and trace B(q) coefficientsof energy-momentum tensor in terms of domain-wall response functions E(q) and Ω(q).
Note that in general variation of a two-point function like 〈Tmn(q)Trs(−q)〉 is given by variation of theone-point function with respect to the source, in this case g(0)ab. In the flat boundary metric, perturbation7.1 can be written as gmn = δmn + δgmn, so :
δ〈Tmn 〉 = δmrδ〈Trn〉 = −1
2
∫δmr〈TrnTst〉 δgst(0)d
3x
Taking the Fourier transformation, in momentum space:
δ〈Tmn (q)〉 = −1
2δmr〈Trn(q)Tst(−q)〉 δgst(0)(q) (F.2)
Using the the stress two-point function decomposition F.1, and metric perturbation formalism 7.2 wereach to the equation 7.44:
δ〈T (s)(q)〉 =1
2A(q)γ
(s)(0)(q), δ〈T (q)〉 = −4B(q)ψ(0)(q) (F.3)
On the other side expanding canonical momentum 7.41 to linear order gives:
δ〈Tmn 〉 =1
κ2(δK δmn − δKm
n )(3)
and imposing Hamiltonian and momentum constraints 7.8:
δ〈Tmn 〉 = − 1
κ2
(2ψδmn + q2χ pmn +
1
2γmn
)(3)
(F.4)
k
Then using the gauge φ = ν = 0, one finds that: [76]
2ψ = ρ δρ, (F.5a)
q2χ =2q2ψ
a2H− εζ =
2q2ψ
a2H− κ2Ωζ
a3
=
(2q2
a2H+κ2Ω
2a3
)ψ − κ2ΩH
a3ρδρ (F.5b)
1
2γmn = 2
E
a3γmn (F.5c)
But terms proportional to δρ contribute to the dual operator 〈O〉, not to the transverse or traceless parts.So these terms can be omitted and we end up with equation 7.45:
δ〈T (s)(q)〉 = − 2
κ2
[a3
E(q)γ(s)
](3)
, δ〈T (q)〉 =1
κ2
(Ω(q)
a3− 2q
a2H
)ψ (F.6)
and finally following comparison with F.3, coefficients can be written in terms of response functions:(equation 7.46)
A(q) = −4κ−2E(0)(q) B(q) = − κ−2
4Ω(0)(q) (F.7)
l
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