Master Thesis Domain Wall Solutions in the Ads/CFT Correspondence Author

Master thesis

Domain wall solutions in the AdS/CFT correspondence

Author: Eduardo Mateos González Supervisor: Giuseppe Dibitetto 2 Acknowledgments

Firstly, I would like to express my sincere gratitude to my supervisor, Professor Giuseppe Dibitetto, for his continuous support during the elaboration of my Master thesis, for his patience, motivation, and immense knowledge. I also want to thank Professor Luis Miguel Nieto Calzada, who helped me initiate in the world of research and gave me his advice during my Bachelor studies. I am especially grateful to my parents, who have given me the opportunity to study what I enjoy and pursue a career in research, and they have always fully supported me, to my girlfriend, Elisa, who I can always count on, through thick and thin, to my grandparents and my cousin Samuel, for always being there when I needed to call them, and also to the rest of my family. Additionally I want to thank my fellow classmates for all the stimulating discussions, for the sleepless nights we were working together before deadlines, and for all the fun we have had these years. To all the friends who have helped me by reading parts of the thesis and proposing corrections, especially Jose, Maor, Souvanik and Carlos. Finally, to all the professors in the Theoretical Physics department in Uppsala University that helped me whenever I had a question, thank you.

i ii ACKNOWLEDGMENTS Contents

Acknowledgments i

Abstract vii

1 Introduction 1 1.1 Current goals in Theoretical Physics ...... 1 1.2 Tools to study strongly-coupled systems ...... 3

1.3 AdS7, SCFT6 ...... 6

2 Theoretical background 7 2.1 Quantum Field Theory (QFT) ...... 7 2.1.1 Quantum mechanics ...... 7 2.1.2 Special Relativity ...... 14 2.1.3 General Relativity ...... 19 2.1.4 Quantum Field Theory ...... 25 2.2 Supersymmetry (SUSY) ...... 32 2.3 Conformal Field Theory (CFT) ...... 35 2.3.1 Superconformal Field Theory (SCFT) ...... 40 2.4 String theory ...... 41 2.4.1 Superstring theory ...... 48 2.5 Supergravity (SUGRA) ...... 50 2.6 Anti de-Sitter spacetime (AdS) ...... 52 2.7 Anti de-Sitter/Conformal Field Theory correspondence ...... 61

3 Calculations 67

3.1 Supersymmetric domain wall in AdS7 ...... 68 3.1.1 Effective action for the domain wall ...... 75

3.2 Non-conformal deformation of the CFT6 dual ...... 79 3.3 Equivalent Quantum Field Theory ...... 82

Bibliography 87

iii iv CONTENTS List of Figures

1.1 Standard Model of particle physics. Credit: wikimedia.org ...... 1 1.2 Artist’s conception of a realization of the AdS/CFT correspondence. Credit: researchgate.net ...... 4

2.1 Schematic description of a quantum measurement. Credit: csee.umbc. edu/~lomonaco/graphics/Lomonaco-Quantum-Measurement.jpg .... 8 2.2 Visual representation of the double cover of SO(3), the special unitary group SU(2). Credit: wikimedia.org ...... 14 2.3 Causal cone of an event in spacetime. Credit: researchgate.net .... 18 2.4 Example of mass being transformed into energy. Credit: wikimedia.org 19 2.5 Artist’s conception of the spacetime curvature. Credit: sciencenews.org 23 2.6 Visual aid for lattice QFT represented as harmonic oscillators in a lattice. Credit: ribbonfarm.com ...... 27 2.7 . Credit: slimy.com ...... 30 2.8 An example of a Feynman diagram. Credit: Modified from Peskin and Schröder ...... 31 2.9 Minimal Supersymmetric Standard Model (MSSM). Credit: arstechnica. org ...... 33 2.10 Scale invariant system at four different distance scales. Credit: Douglas Ashton ...... 36 2.11 Pictorial representation of a string attached to a D-brane. Credit: David Tong...... 46 2.12 Pictorial representation of two closed strings interacting. Credit: slimy.com 47 2.13 String theory dualities. Modified from originals found in quantamagazine. org and physics.stackexchange.com ...... 49 2.14 An example of a Calabi-Yau manifold. Credit: wikimedia.org ...... 52

2.15 Penrose diagram for AdS2. Credit: researchgate.com ...... 55

2.16 Diagramatic representation of the Poincaré Patch for AdS3, corresponding with the shaded area in the figure. Credit: Freedman and Van Proeyen (2012, p. 494) ...... 56

v vi LIST OF FIGURES

2.17 AdS/CFT correspondence. Credit: philosophy-of-cosmology.ox.ac.uk 62 2.18 Relationship between the mass of a scalar field in AdS and the scaling dimension of its dual for unitary operators. Credit: Benini (2018, p. 34) . 65 2.19 Examples of Witten diagrams. Credit: Freedman and Van Proeyen (2012, p. 542) ...... 66

3.1 Brane. Credit: Bobev et al. (2017, p. 5) ...... 70 3.2 Scalar potential plotted for h = g = 1 ...... 71 3.3 Plot of the coefficients of the metric defined in (3.13) for h = g = 1 √ showing the divergence at r = 10 8 ...... 74 3.4 Plot of the c function for h = g = 1 ...... 84 3.5 Plot of the β function for h = g = 1 ...... 85 Abstract

In this thesis we study a particular realization of the Domain wall/Quantum Field Theory correspondence, a modification of the Anti de-Sitter/Conformal Field Theory correspondence that is used to study deformations of a Conformal Field Theory. In the Quantum gravity side of the duality we analyze a N = 1 gauged supergravity theory in 7 dimensions which presents two different Anti de-Sitter vacua, one of which preserves the full supersymmetry and one that breaks half of the supercharges. We will find a scalar 1/2-BPS solitonic solution describing a domain wall in an asymptotically Anti de-Sitter spacetime which interpolates between the supersymmetric AdS vacuum and a divergent AdS space situated at infinity, and we will calculate its tension and the effective mass of the scalar field when evaluated at the AdS vacuum. The dual theory of ourgauged supergravity is the 6-dimensional N = (1, 0) Superconformal Field Theory, and the scalar 1/2-BPS field is dual to two relevant operators that induce a relevant deformation of the SCFT which can be identified with a renormalization group-flow. Here we will first compute the scaling dimension and the one-point functions of these operators in the SCFT, as well as indicating how to compute the two-point and three-point functions, and then we will study the c-function along the renormalization group-flow they induce and the beta function that characterizes this flow in order to derive some properties of the resulting Quantum Field Theory.

vii viii ABSTRACT Chapter 1

Introduction

1.1 Current goals in Theoretical Physics

Theoretical physicists are trying to find a theory that accommodates all the observed phenomena in our world in a single framework, usually referred to as a Theory of Everything, in order to advance our understanding of the universe and discover new results. A theory like this must at least be able to describe the four fundamental forces that we know of so far: Gravity, Electromagnetism, Strong interaction and Weak interaction. The last three have a unified description in terms a Quantum Field Theory known as the Standard Model, while the first one receives a satisfactory treatment using the geometrical viewpoint of General Relativity; however, we find difficulties when we try to reconcile both frameworks in a theory of quantum gravity.

Figure 1.1: Standard Model of particle physics. Credit: wikimedia.org

1 2 CHAPTER 1. INTRODUCTION

String theory has received a lot of attention from the scientific community for the last five decades as a possible candidate for a quantum gravity theory and as a consequence there has been a great deal of theoretical results, especially in the perturbative regime of the different string theories. Nevertheless, String theory is not a unique description of a system, but a broad framework that encompasses a wide variety of models, so it is possible to generate a very broad range of outputs by adjusting the initial conditions of the theory; as a result, String theories have low predictivity at scales much larger than the strings themselves, and the energies needed to test whether the predicted effects beyond the Standard Model are correct still remain far away from our reach. This situation contrasts strongly with the case of Quantum Field Theories (QFTs): the QFTs that we have built so far cannot describe all known physics, as we do not know how to accommodate a quantum description of gravity within our current formalism, but these theories have managed to predict with astonishing precision many phenomena in the energy regimes accessible with our prevailing technology, consistently matching experimental results. A particularly impressive accomplishment is the prediction for the anomalous magnetic dipole moment of the electron that can be computed within Quantum Electrodynamics (QED), whose theoretically calculated value matches the experimental result up to the ninth significant digit, an achievement not matched in any other realm of science. The Standard Model of particle physics proves to be our best tool to understand the nature of fundamental particles, and with only a handful of input parameters (19 in the original Standard Model, 26 when including masses for the neutrinos (Thomson, 2013, p. 499)) it offers testable predictions that have been verified using our best particle ac- celerators, and the solutions we obtain are in better agreement with current experiments than those from any alternative theory or any proposed extension of new physics beyond the Standard Model. However, despite its impressive achievements, we are presented with two shortcomings in its role as a fundamental description of the universe: first, it is not a Theory of Everything. Not only we do not know how to accommodate a quantum description of the gravitational interaction with our current QFT formalism, but also it is likely that there are other phenomena in Nature that we are not aware of yet, for example there could be additional fundamental interactions or extra dimensions that are not apparent at our energy scale. The second problem is that it is very computationally challenging to obtain exact analytic solutions for the complete Standard Model. So far we have only been able to completely solve analytically a handful of QFT theories, generally those with a great deal of symmetry, but most prominently the non-interacting cases in which the degrees of freedom of the solutions are independent of each other. The solutions for these non-interacting models are very useful, as they can be used as a starting point when studying weakly interacting theories, but they do not constitute good models for our Universe. For a general QFT we have to resort to approximations 1.2. TOOLS TO STUDY STRONGLY-COUPLED SYSTEMS 3

if we want to study the behavior of its solutions, and even then our best tools are only effective when dealing with linear interactions and small corrections of known theories, but they stop producing reliable results the further we strive from the starting theory. Most calculations that we rely on today can only be carried out for weakly interacting QFTs, and they are usually obtained as a perturbative expansion of a similar non- interacting theory (that is, a theory that we can solve exactly so that we can evolve all relevant states independently, like free particles which do not interact with each other); as a consequence these solutions are only valid when the degrees of freedom of the theory that we want to study do not differ much from their free-particle counterparts. Nevertheless, for most Quantum Field Theories the value of the interaction couplings between these quasi-free states are not fixed: the interaction strength changes with the energy scale due to quantum corrections (these parameters are often referred to as ‘running couplings’), which means that there are energies for which they will become strong enough to make our perturbative approach unreliable. This is evident for the Quantum Field Theory describing the strong force (QCD), whose coupling is strongest at low energies and as a result the particles that we find in everyday experiments look nothing like small perturbations of the fundamental degrees of freedom (the quarks). Instead, at low energies these quarks group into strongly interacting states, called hadrons, which are strongly non-linear solutions of QCD. The current goals in physics must then be two-fold: finding new theories that accommodate all known phenomena, and at the same time developing new tools that allow us to study strongly-coupled systems.

1.2 Tools to study strongly-coupled systems

In order to study strongly-interacting theories with a degree of accuracy comparable to that achieved in their perturbative regime we can think of several possible strategies: we could, for example, try to find new powerful tools that allow us to study non-linear solutions analytically, making it possible to solve the theory completely for any regime; a different approach could consist on developing new perturbation techniques that provide a better approximation with fewer higher-order terms, improving the range in which our perturbative solutions make accurate predictions; or finally, we could search exact non-perturbative solutions as close as possible to the regime that we want to study and use those solutions as the approximation that we will build on using perturbation theory. The first approach is clearly the most ambitious but also the most complicated one, while the last one just needs us to find explicit examples of exact solutions within the non-perturbative regime we are interested in. For the last case the Anti-de Sitter/Conformal Field Theory (AdS/CFT) corre- 4 CHAPTER 1. INTRODUCTION spondence can be used as an extraordinarily useful tool that provides us with some non-perturbative solutions for some special cases of Quantum Field Theories and String Theories due to the fact that it is conjectured that the dual of a strongly-coupled Quan- tum gravity theory in an AdS space is a weakly-coupled CFT (which we can solve perturbatively) and vice versa. By virtue of this duality we can investigate the strong- coupling regime of a theory using our linear perturbative techniques in the weak regime of its dual, and the accuracy of the result will be as good as that of the perturbative analysis performed in the weak coupling limit, in which we know we can obtain a high degree of accuracy as we have already been able to test its predictions against Nature with the Standard Model, for example.

Figure 1.2: Artist’s conception of a realization of the AdS/CFT correspondence. Credit: researchgate.net

The duality is a special case of the so-called holographic principle that is expected to be present in a complete theory of Quantum gravity, according to which when we study a Quantum gravity theory on an AdS background it will produce a local CFT in the boundary of the spacetime, and both the theory in the bulk and that in the boundary will share the same global symmetries, as we will see in section 2.7. String theory is a candidate for a theory of quantum gravity, so it can be used to formulate the AdS side of the correspondence; due to the conservation of global symmetries, the supersymmetries present in Superstring theories will still be present in their duals, which we then refer to as Superconformal Field Theories (SCFT). Looking at the String theory side of the correspondence we can distinguish three limits in the parameter space in which we could study this duality depending on the value of the string coupling (denoted by gs), and the dimensionless ratio of the curvature of the AdS space divided by the string length, L/ls. For general arbitrary values of these 1.2. TOOLS TO STUDY STRONGLY-COUPLED SYSTEMS 5 parameters we need to do our calculations in the frame of a Quantum String theory, whose solutions are usually only known in the perturbative regime; if we restrict the value of gs to be close to zero instead, we can work with Classical String theory and disregard the quantum corrections that arise from higher-order terms; finally, if gs is small and the string length is negligible compared with the characteristic length of the AdS space,

L ls, we can forego the stringy description and work with an Effective Field Theory for gravity instead, supergravity, a QFT for point particles with gauged supersymmetry that approximates String theory at low energies. When performing our calculations we will focus in the supergravity limit, and in turn the CFT side will be described in this 2 range by a large N expansion in which the ’t Hooft coefficient λ = NgYM is large; that is, the CFT will be in the strong-coupling regime. There is still one last caveat, however, if we want to use the AdS/CFT correspondence to obtain a QFT theory that is suitable for describing particle physics: CFT theories are, by definition, Quantum Field Theories that are invariant under conformal transformations and its solutions are scale-invariant states, which in general are forced to be massless.1 But the particles that we have detected in Nature are not only massive, but they also span a wide range of mass values; from the neutrinos, whose mass must be smaller than 0.1eV/c2, to the top quark, with a mass of 173GeV/c2, there are 12 orders of magnitude. If we assumed all the particles to have the same mass (zero) the resulting calculations would be off by a large margin. Nevertheless, despite not being suitable models for particle physics, CFTs are still useful when we are interested in physical models based on Quantum Field Theories. Any well-defined QFT that strives to explain fundamental phenomena has to be well-behaved when we study it at arbitrarily small distances (we say in that case that the theory has a ultraviolet (UV) completion); for a theory to be UV-complete its renormalization group flow has to be a relevant deformation of a UV-fixed point, and theories atfixed points show conformal invariance. Thus, for a Quantum Field Theory to be applicable at arbitrarily large energies it must be defined as a small perturbation of a Conformal Field Theory! Additionally, QFTs that do not need a cut-off neither for short nor for long distances can be defined as a renormalization group flow between a UVCFTand an IR CFT. According to the AdS/CFT correspondence, when the quantum gravity theory is only asymptotically AdS the theory that appears in its boundary is an approximate CFT. The states that break the isometries of the maximally symmetrical AdS side are related to the non-conformal operators that take us away from the renormalization group-invariant point in its CFT dual, so that different operators will define different RG-flows, and as such, different QFTs. An example of an asymptotically AdS solution is a solitonic

1We will clarify these assumptions in more detail in section 2.3 6 CHAPTER 1. INTRODUCTION state that interpolates between two different AdS vacua, usually called a domain wall, for which its dual will be a renormalization group flow interpolating between a UV CFT and an IR CFT. Different points of the RG-flow can be interpreted as a QFT evaluated at a certain energy scale. It is also possible for the UV or IR CFTs not to exist, in which case the associated QFT will not have a well-defined behavior at small scales or large scales respectively.

1.3 AdS7, SCFT6

In this thesis we are going to apply a deformation of the Anti de-Sitter/Conformal Field Theory correspondence (sometimes referred to as the Domain wall/QFT correspondence) to a particular case, focusing on the supergravity limit. For the AdS part of the correspondence we will choose a half-maximal gauged supergravity theory in 7 dimensions for which we know from the literature that it presents two different AdS vacua (for example from the analysis in Dibitetto and Petri (2017)), and whose dual is known to be a 6-dimensional SCFT without a Lagrangian formulation. We will follow the procedure first developed in the pioneering paper by Boonstra et al. (1998), which starts by studying a BPS-flow in the supergravity side corresponding to a supersymetric domain wall in AdS7, and by means of the AdS/CFT correspondence we will find the associated non-conformal operators living in the 6-dimensional SCFT, thanks to which it will be possible to study the renormalization group flow that they induce on the Conformal Field Theory, and we will use it to characterize the Quantum Field Theory associated with this flow. The explicit shape of the scalar domain wall will be calculated in section 3.1 after reviewing the properties of the gauged supergravity theory it is derived from, obtaining also the effective mass and the tension of the BPS scalar. In section 3.2 wewilluse the dictionary that we will derive for the AdS/CFT correspondence in section 2.7 to identify the relevant operators (also called defects) dual to the AdS scalar field, for which we will compute their scaling dimension and their one-point, and we will indicate how to compute their two-point and three-point functions. Finally, in section 3.3 we will study the beta function associated with the renormalization group-flow induced by the non-conformal operators, as well as the c-function along the flow, which will allow usto derive important properties of the associated QFT. Before proceeding with the calculations, we will present the main results in all the areas that we have named throughout this introduction and that will be necessary for the calculations present in this thesis. We will first describe the physical content of each theory and then we will proceed to state the main equations and tools that we will need to use later on in order to apply the AdS/CFT correspondence. Chapter 2

Theoretical background

2.1 Quantum Field Theory (QFT)

A Quantum Field Theory is, strictly speaking, any quantum theory that deals with extended objects called fields, but more often than not we will use this term to refer to the subset of QFTs that are invariant under the Poincaré symmetry. The general consensus nowadays is that every Quantum Field Theory must be understood as an Effective Field Theory (EFT), which is not defined throughout the complete parameter space of the theory and instead describes the relevant physics only up to a given energy scale at which new degrees of freedom will appear. This maximum energy is dubbed the cut-off and denoted by Λ, and the predictions of the EFTs become less reliable when the scale of interest becomes comparable with Λ. As we mentioned in the introduction, Quantum Field Theories are the most accurate tools that we have today to investigate the fundamental properties of Nature. In spite of this, a mathematically rigorous approach to QFT has proven hard to pin down, and every attempt we have made since the 1950s has had its share of short-comings. To understand what exactly is a Quantum Field Theory and what problems appear when we try to axiomatize it, let us look at the two fundamental paradigms that we try to reconcile through QFT: Quantum Mechanics and Special Relativity.

2.1.1 Quantum mechanics

The difference between Quantum mechanics and Classical mechanics resides on their description of how a system behaves under a measurement. Measuring is fundamental in order to describe the world around us; according to the Relational Quantum mechanics interpretation we can consider any type of interaction between two systems as an information exchange or, equivalently, a measurement. Quantum mechanics is a broad framework that describes physical systems in conjunc-

7 8 CHAPTER 2. THEORETICAL BACKGROUND tion with the measurements that we can perform on them and how they modify the state being measured. In order to study this theory we will first have to set some notational conventions to understand its core principles and its differences with Classical physics; for the moment we will denote the state of a system by a Greek letter, as could be ψ, and we will be interested in studying the properties of such states (like its position or its electric charge) that we will generically write with capital Latin letters, for example A. To obtain information about a system ψ, we will make it interact with a second system, φ, whose initial state we know and for which we have tabulated and calibrated all its possible final states to correspond with particular quantities of the magnitude relevant for the interaction between the systems, A; for example, if after the measurement of ψ the calibrated system changes from its initial state φ0 to some other state φa we will say that the quantity which mediates the interaction, A has been measured for ψ and it has been found out to be equal to a. We can represent this process by a function fA which describes the measurement procedure; when this function acts on a particular initial state of ψ it returns a numerical value and it drives the system into a (possibly 0 different) final state, so we can write this process schematically as fA(φ0, ψ) = (φa, ψ ), where a is the numerical outcome obtained.

Figure 2.1: Schematic description of a quantum measurement. Credit: csee.umbc.edu/ ~lomonaco/graphics/Lomonaco-Quantum-Measurement.jpg

In Classical physics it is implicitly assumed that for any given state ψ and any property A, there exist one and only one real number a associated with that state, and as a consequence the value that we will find when we try to measure A will be independent of the particular experimental procedure used (after we take into account all possible error sources). In other words, in Classical mechanics we can have several functions that measure the same property and all of them will return, within the available 2.1. QUANTUM FIELD THEORY (QFT) 9 accuracy, the same outcome. Regarding the final state obtained after the measurement it will in general be different from the initial state and, unlike the numerical outcome, it can depend on the particular measurement process realized; due to this fact we can have more invasive and less invasive measurement processes depending on how different the initial state and the final state are. In Quantum mechanics, on the other hand, the possible results of a measurement are no longer assumed to be independent of the experimental procedure. The starting point of the theory is instead that all the possible outcomes (both the numerical values and the final states) that we can obtain from an experiment described by the function fA are intrinsic to that particular function and uniquely determined beforehand. Regardless of the initial state that we are measuring! That is, we could elaborate a list of possible numerical outcomes and their corresponding final states for fA, which we can write for example as {(ai, ψi)}, and when measuring fA(φ0, ψ) we will find that regardless of the properties of ψ it will return one of the pairs from the aforementioned list. Each of the possible numerical values ai of the function fA are referred to as eigenvalues, and the allowed final states are called eigenstates. Data from experiments indicates that measuring an eigenstate will keep the system unchanged, while the possible outcomes after measuring any other possible state are probabilistic in nature and transform linearly under small perturbations of the state to be measured. To compile all of these properties into a mathematical framework we will require that states are described by linear objects that can be decomposed as a sum of other states (that is, they must be vector states), and the framework must also include linear functions to describe observables and an inner product between states. All these properties naturally lead us to choose a Hilbert space as the natural mathematical framework for Quantum mechanics, defined rigorously through the Dirac- von Neumann axioms, enumerated for example in Cohen-Tannoudji et al. (1978, p. 215) and Galindo Tixaire and Pascual de Sans (1990, p. 37). These axioms define Quantum mechanics as the study of the expectation values of unitary operators acting on a projective Hilbert space of countable infinite dimension, P (H). A Hilbert space, H, is a vector space with an internal product, h·, ·i : H × H → C that must be complete with respect to the distance function induced by the inner product. Its projective space, P (H), is the set of equivalence classes that identify v, w ∈ H \ {0} if v = λw for some non-zero complex number λ; we call these equivalence classes rays. In Quantum mechanics it is conventional to use the so-called bra-ket notation, where vectors in H are represented as |φi and go by the name of kets, and vectors in the dual space, H∗, are called bras and we write them as hψ|. If we choose the dual of a vector |φi to be the linear functional that acts on some other arbitrary vector |φ0i and returns the inner product of the two, |φi∗(|φ0i) = hφ, φ0i, then we can rewrite the definition of the inner product as the result of multiplying a bra and a ket, hφ|φ0i = hφ, φ0i ∈ C. In 10 CHAPTER 2. THEORETICAL BACKGROUND addition, the internal product naturally defines a norm for the vector space, which in p the bra-ket notation is written as kφk = hφ|φi ∈ R+. Two states are said to be orthogonal if hφ|φ0i = 0. The norm of a well-defined state must be exactly 1, as expected from probability theory; because all operators in Quantum mechanics are unitary, the norm of a given state will always remain constant. The inner product can be understood from a physical point of view as the amount of overlap between two vector states, that is, the probability that if we measure a state that is in an initial state |φ initiali we will find it to be in the state |φ finali after the 2 measurement is equal to P( φ final | φ initial) = |hφ final|φ initiali| . Regarding the process of measuring, we say that we are taking a measurement of a system when we subject its state to a transformation in order to gain information about it (Cohen-Tannoudji et al., 1978, p. 226). As we saw, in a measurement we are applying 0 a given operator onto the state and recording a numerical outcome (fA(ψ) = (a, ψ )), and because every information exchange can be interpreted as a measurement of the states involved, the only quantity with physical significance is the expected value of the operator defining the transformation associated with a given measurement procedure fA. An operator that describes a measuring transformation is called an observable, and one of the Dirac-von Neumann axioms states that any observable must be self-adjoint (Galindo Tixaire and Pascual de Sans, 1990, p. 41); as a result, every observable must be a diagonalizable operator, which implies that its eigenvectors can be chosen to form an orthonormal basis of the Hilbert space. All self-adjoint operators are Hermitian, but the converse is only true in the case of bounded operators. The expectation value of an observable A for a system in the state |φi is given by hAiφ = hφ|A|φi = hφ|Aφi, which must be a real number (Cohen-Tannoudji et al., 1978, p. 232). If the dynamics of the system do not depend on time explicitly, that is, if we have a symmetry under translations in time, then one of the observables of the theory will be the operator that generates time translations, which turns out to be the Hamiltonian, referred to as H. In order to study a quantum mechanical system we will be interested in selecting a state and then following its time evolution, so H will play a central role in the theory. In the Schrödinger picture (where the states evolve with time and the operators do not depend explicitly on it), the operator that governs the transformation of the states as a function of time is the evolution operator (Cohen-Tannoudji et al., 1978, p. 222), which is defined as:

i − H(t−t0) |φ(t)i = e ~ |φ(t0)i (2.1)

This is a unitary operator, so the norm of the states are always preserved. From this definition it is easy to derive Schrödinger equation as the infinitesimal change ofastate 2.1. QUANTUM FIELD THEORY (QFT) 11 under time evolution

d i |φ(t)i = H|φ(t)i (2.2) ~dt Both (2.1) and (2.2) are operator equations that depend on the Hamiltonian of the theory, H, and from the Schrödinger equation it is clear that different eigenstates of the Hamiltonian will evolve without mixing. As the Hamiltonian is an observable in Quantum mechanics, its eigenvectors can be chosen to span an orthonormal basis of the whole Hilbert space, and thus it is enough to study them to automatically solve the complete theory. When the spectrum of H is non-degenerate, we can unequivocally label all basis states just using their eigenvalues (the energy of the state); if several eigenvectors have the same energy, however, the basis formed from the Hamiltonian is not unique and we will need to use additional labels in order to distinguish energy-degenerate states from one another. We can either make an arbitrary choice of orthonormal vectors for each energy-degenerate subset, assigning them labels arbitrarily (which will necessarily require us to explicitly state all their components, of which there can be infinitely many) or else we need to find other physical labels to distinguish between these states. To produce distinguishable labels for the Hamiltonian eigenstates we want to find other operators with the same eigenvectors as the Hamiltonian but different eigenvalues. It is easy to prove that two operators have the same eigenvectors if and only if they commute (Cohen-Tannoudji et al., 1978, p. 139), so we need to find enough operators, all commuting with each other, until we have unique labels for all basis states. As the eigenvectors of all these operators span a basis of the state, they will necessarily have to be self-adjoint, automatically turning them into observables. The set of these operators is called Complete Set of Commuting Observables (CSCO), and the eigenvalues of a given state with respect to these operators are usually referred to as its quantum numbers (p. 139 Cohen-Tannoudji et al., 1978; Galindo Tixaire and Pascual de Sans, 1990, p. 55). Any operator that commutes with the Hamiltonian defines a transformation that leaves the theory invariant; we call these transformations symmetries and their associated operator generator of the symmetry. We can rewrite the Schrödinger equation as a condition on the time evolution of operators to obtain (in the Scrödinger picture, where the operators do not have explicit time dependence)

dA i = [H,A] (2.3) ~ dt where we immediately see that all operators in the CSCO must be independent of time, and in turn this fact can be used to prove that their eigenvalues cannot depend on time either; thus, the quantum numbers of the vector basis must then remain constant through the state’s evolution. Two states with the same quantum numbers are indistinguishable from each other, 12 CHAPTER 2. THEORETICAL BACKGROUND and even if we try to label them at some initial point when they occupy different regions of the space and their wavepackets are completely non-overlapping, whenever they get close enough (the wavelength of the packet becomes comparable to the distance separating the particles), the uncertainty associated to the measurement process makes it impossible to know which state corresponds to which initial particle (Cohen-Tannoudji et al., 1991, p. 1374). This result has profound consequences on the behavior of quantum systems, and it forces the wavefunctions of two indistinguishable quantum particles to either commute or anticommute when we exchange them (depending on their statistics), resulting in bosons and fermions respectively. Following our derivation it is clear how the time coordinate is treated on a very different footing compared to the spatial ones, and additionally the dynamical equation giving the time evolution of the system is not necessarily invariant under a Lorentzian boost; this means that, in general, Quantum mechanics is a non-relativistic framework. Instead, most of the theories studied using non-relativistic Quantum mechanics are usually assumed to be invariant under Galilean transformations, a symmetry group which contains rotations around the three spatial axis (described by the symmetry group SO(3)), Galilean boosts and space and time translations. We should clarify that every time we mention a vector basis we are actually referring to a projective frame, also known as a simplex, due to the projective nature of the Hilbert space of the theory. It is easy to understand why Quantum Mechanics must be defined in a projective Hilbert space instead of a linear one if we remember that the quantities with physical significance in the theory are the expectation values of the observables and not the states themselves. We can see that due to the properties of the internal product, multiplying a state by a complex number will not change the expectation value of any linear operator. If we take λ = a + ib and its conjugate λ∗ = a − ib, we have:

hφ˜|A|φ˜i hλ φ|A|λ φi hφ|A| φi |φi −→ |φ˜i = |λ φi, = = (2.4) hφ˜|φ˜i hλ φ|λ φi hφ|φi If the correct framework for Quantum mechanics was a non-projective complex linear vector space, two states differing in a complex phase would describe different objects. However, due to equation (2.4) it is physically impossible to distinguish between two such states, and thus the correct description of the theory must treat them as identical object. Indeed, by working on a projective space we are only considering physically distinct solutions as inequivalent states, which we call rays. This projective nature must be taken into account when we look for the eigenstates of our theory. If the symmetry group we are interested in has a matrix representation, it can be shown that all the eigenstates and eigenvalues of the linear representations of an operator O can be calculated using an eigenvalue equation, AO|vi = λ|vi, where AO is a matrix in a particular linear representation of the group. In a projective space, though, 2.1. QUANTUM FIELD THEORY (QFT) 13 we have additional eigenstates beyond the linear representations; these additional states cannot be obtained from a linear equation like the eigenvalue equation, and they may have different eigenvalues that do not appear in the linear representations. Working with projective representations is in general not easy and involves dealing with bigger symmetries with so-called superselection rules, but for most of the Lie groups of interest for physics we can obtain the projective states of a given symmetry group by studying instead the linear representations of its universal cover group. The universal cover group of a connected group G is a simply-connected group C such that C = G/H, where H is an invariant subgroup of C. In some circumstances both C and G can be the same group, and in that case all the eigenstates will be linear representations of the original group and our job is then reduced to studying its eigenvalue equation. However, C can enlarge G in two possible ways: if the Lie algebra of G admits a central charge (a new algebraic term that appears in the Lie bracket of the generators of the group that is proportional to the identity element and cannot be reabsorbed by a redefinition of the group generators), C is said to be a central extension of G and it will include new representations; it is never possible to add a central charge to a semi-simple group. The second possible enlargement arises if G is connected but not simply-connected; as C always has to be simply-connected the topology of this group will be different from that of G, namely, two or more elements of C will be associated with a single element in G. The number of elements that are put on correspondence depends on the homotopy group of the manifold of G, where for example if the original group has two simply-connected components there will be two elements of C associated with each element in G, and we will say that G is doubly-covered by its universal covering group. It can be proven that a projective representation of G, which requires superselection rules if we want to obtain them from the linear equations for G, will always be equivalent to a linear representation of C without superselection rules (Weinberg, 1995, p. 90). For clarity we can look at a theory with rotational symmetry under the Galilean group in 3 spatial dimensions to see how the projective representations of the symmetry group are realized. Rotations are generated by the angular momentum operators, Li (because they are generators of a symmetry of the theory, t hese operators have to commute with the Hamiltonian) and their Lie group is SO(3). The linear irreducible representations of the rotation group are scalar states, vector states and tensor states of arbitrary order, n. To study the projective representations we turn to the universal cover of SO(3), which is the simply-connected Lie group SU(2); the special unitary group also has scalar, vector and tensor irreducible representations, but it also admits half-integer spin representations, usually called fermionic states. This example also demonstrates how fermions arise naturally due to the projective nature of Quantum mechanics when combined with a rotationally symmetric theory, explaining why fermions often appear 14 CHAPTER 2. THEORETICAL BACKGROUND when we study non-relativistic quantum systems while they do not play an important role in General Relativity, for example.

Figure 2.2: Visual representation of the double cover of SO(3), the special unitary group SU(2). Credit: wikimedia.org

To specify a particular Quantum mechanical theory we only need to write its explicit evolution equation, as given in (2.2); thus, given the Hamiltonian operator the theory is completely determined, as we can obtain the time-evolution of any state and the symmetries of the theory by studying the Hamiltonian and its eigenstates (nevertheless, it may not be easy to find the eigenstates or to diagonalize H even if we know its explicit expression). However, if we already know the behavior of a system in the classical limit and we want to build a quantum theory that recovers that limit by quantizing the classical theory instead of directly guessing the appropriate Hamiltonian operator for the quantum regime, the procedure is not straightforward. To summarize, Quantum mechanics is a physical framework in which the relevant objects in a physical system are not the states per se, but the expected values of the observable operators acting on those states. The space of states is a projective Hilbert space for which we can find a basis by studying the eigenvectors of a group of operators, a CSCO (which includes the Hamiltonian operator, that define the symmetries of the system). To completely solve a theory we only have to find all the vectors in the basis and time-evolve them using the evolution operator or an equivalent description using Schrödinger’s equation. If we want to use the symmetries of the problem to our advantage to simplify its resolution, we must remember to pay attention to the projective representations of the symmetry group and not just the linear representations.

2.1.2 Special Relativity

The theory of Special Relativity is a physical framework that constraints the possible behavior of a system under certain coordinate transformations, those given by the 2.1. QUANTUM FIELD THEORY (QFT) 15

Poincaré group. Theories in agreement with Special Relativity are those for which the dynamic equation describing the evolution of the system is invariant under Poincaré transformations; this requirement places very strong constraints on the theory, but this symmetry has been observed to hold in Nature to a very high degree of accuracy, with no apparent violation detected in any particle physics experience or in the cosmological experimental data.1 The Lie group for the Poincaré symmetry in d spacetime dimensions with a single time-like coordinate is ISO(d−1,1), which is the maximal group of isometries of a metric space whose metric has signature (d − 1,1); this group can be decomposed into a local component and a translation component, as ISO(d−1,1) is the affine group of O(d−1,1), the Lorentz group, so it can be rewritten as a semidirect product: ISO(d−1,1)= Rd−1,1o O(d − 1,1). We can further decompose the Lorentz group into its proper orthocronous component, SO(d − 1,1) (which is the component connected to the identity element), together with the parity and time reversal transformations. Special Relativity poses constraints on how all physical magnitudes vary under an affine transformation, x → x0 = Rx + s, and the most adequate mathematical frame to describe these coordinate transformations is an affine space. Additionally, just as it was the case with the Poincaré group, a general affine transformation can be written asa composition of a linear transformation with a translation; this fact means that we do not need to work with the whole affine group, and instead we can study separately each of its subgroups, that of linear transformations and that of translations. Linear coordinate transformations can be studied using the mathematical framework that describes vector spaces with an inner product (though, as we will see, the properties of the inner product defined in Special Relativity are slightly different from the canonical examples that one first encounters when studying vector spaces in Mathematics, so sometimes it is called instead a pseudo-inner product to mark the differences). This vector space must be invariant under Lorentz transformations, which are the Poincaré transformations that are local, and is usually referred to as a Minkowski space and denoted by some authors as Mkw; for example, Mkw4 would be a spacetime with 4 dimensions. By defining an independent vector space for each point of the spacetime and constructing translation transformations that take us from each particular point to any other point of the space we are able to describe the relevant affine space in its entirety. The vector space Mkw is equipped with a non-degenerate indefinite (that is, with non-definite signature) bilinear that lets us define a pseudo-inner product between two arbitrary vectors of the space, h·, ·i : Mkw × Mkw → R. This bilinear does not hold one of the properties of the usual inner product, namely it does not require that hx, xi > 0

1Anisotropies in the speed of light can be excluded with a confidence greater than 1 part in 1017 16 CHAPTER 2. THEORETICAL BACKGROUND for all elements different from the null one. In the same way we can write aninduced pseudo-norm for the vector space in terms of the inner product as kxk2 = hx, xi, which will return results in R instead of just R+. We will say that a vector is space-like if kxk2 > 0, light-like if kxk2 = 0 and time-like if kxk2 < 0 (Carroll, 2019, p. 23). Finally, the norm can be used to induce a distance function between two points of the vector p space defined as d(x, y) = kx − yk = |hx − y|x − yi|. This function is non-negative, but it can be zero for distinct x and y points (those whose relative position vector is light-like), violating an axiom usually required for distance functions. By focusing now on the specific properties imposed by the Poincaré invariance present in theories compatible with Special Relativity we are able to derive additional results that these theories must hold, increasing the predictive power of the formalism. The vectors of Mkwd will have exactly d components, corresponding to each the spacetime coordinates (Carroll, 2019, p. 17), and the bilinear can be written in matrix form as a d × d object, which in physics is denoted as ηµν and dubbed the metric of the µ ν spacetime. In this notation the inner product is written as hx, yi = x ηµν y . Spacetime symmetries appear reflected as symmetries of the metric tensor; in particular, Lorentz transformations (which are usually represented as matrices that we will write as Λ) must be symmetries of the theory and thus they must keep the internal product invariant. That is, after a coordinate transformation x0 = Λx we have to find that hΛx, Λyi = hx, yi. By writing this equation with the vector indexes explicitly we obtain the constraint that any metric η has to hold in order for it to be Lorentz invariant (Carroll, 2019, p. 12)

α β Λµ ηαβ Λν = ηµν (2.5)

As we know the properties that characterize Poincaré transformations, we can use them to derive the constraints that the spacetime metric must hold. In particular, all d − 1 spatial coordinates must be treated in equal footing (as a consequence of spatial rotations being a symmetry of the theory), while the time coordinate has to appear in the metric with opposite sign to that of the spatial ones, as a boost is a linear transformation of the coordinates in which the time and the spatial coordinates appear together with different signs; as a result the metric must be non-positive definite, and in particular it must have d − 1 signature. In this thesis we will take the metric with mostly plus signature, but this is a matter of convention. Lastly, by imposing translation invariance we conclude that the metric cannot depend explicitly on the spacetime point in which it is evaluated, and thus we arrive at the conclusion that we can always choose our coordinates such that ηµν is a diagonal matrix with all its entries equal to 1 (in natural units) and signature d − 1, commonly written schematically as (− + ... +); this coordinate system is called Cartesian coordinates. We can take a further simplification by incorporating a principle from group theory: 2.1. QUANTUM FIELD THEORY (QFT) 17 all states with Poincaré invariance can be written in terms of the irreducible representations of its associated Lie group, where each of the representations has a well defined transformation law under spacetime coordinates transformations (Peskin and Schroeder, 1995, p. 38). We can find scalars, vectors and tensors of arbitrary order, but they canall be obtained as direct products of states in the vector representation, namely any tensor of order n comes from the product of n vectors while a scalar can be seen as an empty product. A very powerful and useful tool that Special Relativity provides us with is the so- called covariant notation, which allow us to write any equation in a Poincaré-invariant fashion. All the terms in a covariant equation must have the same index structure, and the quantity and location of the indexes characterize what must be the transformation rule of that object under a Poincaré transformation. A major consequence of using this notation is that any covariant expression that is true in general for a given inertial reference frame will automatically hold true in any other possible inertial frame; this simplifies many calculations, as we can always choose a frame where the equations that we want to solve are easier.

According to its index structure, the metric ηµν has to be a (0,2)-tensor (Carroll, 2019, p. 31). The norm of a given vector can then be reinterpreted as the contraction of 2 µ ν µ the vector it with its associated covector, kxk = ηµνx x = x xµ. It is straight-forward to generalize this idea to objects with arbitrary many indexes, so that multiplying a tensor with the metric will result in an equivalent tensor with one of its indexes lowered, and multiplying it with the inverse metric will raise an index; we can contract together an upstairs and a downstairs index in a tensor by summing over them. One of the most ground-shattering consequences of Special Relativity is that the Poincaré spacetime transformations, and in particular the boosts, cause the time coordinate to mix with the spatial ones, blurring the concepts of simultaneity and causality that we had in Classical mechanics. In classical theories we have an absolute time for all observers, and ‘previous’ events can affect ‘future’ ones, but not the other way around; on the other hand, in a relativistic theory the time coordinate assigned to two different events far away from us and from each other is frame-dependent, and thus ill-defined. The apparent loss of causality is worrying, as in our experience we have always seen that causes precede their effects; nevertheless, Poincaré symmetry naturally produces a different kind of causal evolution: instead of relying on an absolute time coordinate to identify precedence, we will impose that the speed of information must always be finite, so that a given event can only affect or be affected by other events inside its ‘causal-cone’.

In order to have a finite speed of information, say v1, physical particles cannot travel faster than a maximum speed set equal or lower to the maximum speed for information, for example v2, otherwise we could transmit data using particles going faster than the theoretical limit for information, which is a contradiction. Because 18 CHAPTER 2. THEORETICAL BACKGROUND

Figure 2.3: Causal cone of an event in spacetime. Credit: researchgate.net

Poincaré transformations leave one and only one speed constant in all reference frames (namely the constant usually denoted c and called the ‘speed of light’, but that is more precisely defined as a fundamental constant of the spacetime), if we want the maximum speed to be frame-invariant we must set v1 = v2 = c. As a consequence, Poincaré invariance naturally introduces a maximum speed that particles cannot surpass, and that coincides with the maximum speed at which information can travel. But even though a causal theory limits the speed of information, it does not necessarily enforce how this information has to be transmitted: we can have theories with an ‘action at a distance’ interaction while still respecting causality. Nevertheless we can eliminate action at a distance from our theory by imposing an additional constraint: locality. If two states must be in the same spacetime point in order to interact, we can forego any ‘action at a distance’ forces and all information in our theory is thus automatically carried by physical particles. To clarify, while causality is required by Special Relativity, locality is an ad hoc condition that is not necessary in principle, although it has so far always been observed in our universe. Furthermore, from a metaphysical perspective, it would always be possible to reinterpret any causal ‘action at a distance’ as a particle, justifying in this way its adoption as a core tenant of the theory. On a separate note, imposing locality alone does not guarantee causality in the relativistic sense because information can travel at arbitrarily large speeds in some local theories. Finally, we must highlight the so-called mass-energy equivalence that appears as a consequence of Poincaré invariance, which states that mass is another manifestation of energy. This relation appears due to the non-conservation of the apparent mass of a 2.1. QUANTUM FIELD THEORY (QFT) 19 body under boosts, which can be seen by studying the behavior of the linear momentum µ dxµ of the particle, p = m dτ (Carroll, 2019, p. 14). In particular, the µ = 0 component of this expression gives us Einstein’s famous formula E = mc2 (Carroll, 2019, p. 32).

Figure 2.4: Example of mass being transformed into energy. Credit: wikimedia.org

The take-away points of Special Relativity are that it is a physical framework that describes the properties that theories invariant under Poincaré symmetries must hold. We can guarantee that a theory is Poincaré invariant by construction by writing its equations in a covariant form; a causality principle arises naturally when we impose a finite maximum speed, and it can be further enhanced by also imposing locality.

2.1.3 General Relativity

General Relativity describes the movement of particles in spacetime in the presence of a gravitational source, and it describes how energy density modifies the curvature of the locally-flat spacetime. The theory can be interpreted geometrically thanks to the property of universality of the gravitational interaction, stated formally through the equivalence principle (Carroll, 2019, p. 48). This principle hypothesizes that the relevant charge for gravity, the gravitational mass, is exactly equal to the inertial mass that determines a body’s resistence to be accelerated, that is, mg = mI . In spite of the fact that scholars knew since antiquity that different bodies tend to fall at nearly the same speed, this principle was first tested rigorously in pioneering work made by Galileo in 1610. No appreciable difference has ever been found between two quantities regardless of their composition, as shown by Eötvös in ground-breaking experiments in 1908, when he set the bound for a possible 20 CHAPTER 2. THEORETICAL BACKGROUND difference as low as one part in 109. Current experiments have pushed this limit to less than one part in 1015 (Asmodelle, 2017, p. 23). Such an astonishing accord hints at a very fundamental relationship instead of just a mere coincidence, and General Relativity arises from elevating this hypothesis to a core principle of the theory. Nevertheless, there are different formulations of the principle with varying degrees of restrictiveness (Carroll, 2019, p. 50).

• The Weak Equivalence Principle, also known as the Galilean principle, states that objects in free-fall have to follow the same trajectory regardless of their composition and their structure, but it does not rule out the possibility to distinguish if an observer is free-falling or if it is in an accelerating frame of reference by performing other non-free-fall experiments. For example, the laws of electromagnetism could be different in each of these frames, so that electromagnetic binding energies could contribute differently to the gravitational mass of a body and to its inertial mass. Another possibility could be that an observer in a gravitational field could start to rotate as it falls, following the same trajectory as in an accelerated frame but also allowing us to distinguish between the two situations.

• A stronger constraint is given by the Einstein Equivalence Principle, which automatically implies the Weak Equivalence Principle, but in addition it imposes that local non-gravitational experiments cannot be used to distinguish between free-falling observers and accelerated ones. These two observers could still be distinguished by gravitational effects however, by finding discrepancies arising due to gravitational self-interaction; for example, two particles could feel a stronger gravitational attraction between them when they are falling in a third body’s gravitational field than they would if they were in an accelerated frame.

• The most restrictive version is the Strong Equivalence Principle, which completely rules out the possibility to distinguish both scenarios through local experiments. The experiments testing this principle have ‘only’ ruled out a possible discrepancy to up to one part in 104, as opposed to one part in 107 for the Einstein Equivalence Principle and one in 1015 for the weak one (Asmodelle, 2017, p. 23). General Relativity is believed to be one of the only theories that satisfy the Strong Equivalence Principle, thus providing a very strong falsifiable prediction that rules out most other theories of gravity, at least up to the tested scales.

Due to the equivalence principle all particles must have the same gravitational charge per unit mass regardless of their composition, and as a result the effect of the gravitational 2.1. QUANTUM FIELD THEORY (QFT) 21 interaction can be reinterpreted in terms of a geometrical description on the grounds that the curvature is also a property of space that all objects will feel with the same intensity irrespective of their nature. In the absence of gravitational interaction our experiments show that Einstein’s Special Relativity gives accurate predictions, so we are going to hypothesize that the geometry and dynamics of spacetime when gravity is turned off must be well described by a Minkowky spacetime, a space invariant under Poincaré transformations and described by a flat metric, ηµν. The Strong Equivalence Principle can then be physically interpreted in the following way: the local description of the world given by an observer in free-fall in a gravitational field must be identical to that of an accelerated observer not subject to gravity. In particular, as accelerated observers in the absence of a gravitational field must be described using Special Relativity, the geometry of spacetime seenbya free-falling observer in a gravitational field must also be locally flat. Additionally, as all spacetime points are assumed to be equivalent, we arrive to the conclusion that the geometry of spacetime predicted by General Relativity is locally Minkowski everywhere. The appropriate mathematical concept to describe a spacetime with this characteristics is that of differential manifolds, geometrical surfaces that are locally isomorphic to Rn everywhere. The curvature of these manifolds is given by the Riemann tensor, a geometrical object that in General Relativity satisfies a differential equation relating it to the energy density present in the manifold. The theory has thus a solid mathematical formulation in terms of differential geometry. Formally we will define a Lorentzian manifold, denoted as M, in conjunction with a metric tensor g, which holds that for any point p ∈ M there exists a coordinate neighborhood U ⊆ M such that the metric g can be written in Minkowski form with λ vanishing Levi-Civita connection (given by the Christoffel symbols Γµν) at the point p, but not necessarily in the punctured neighborhood U \{p}, and the Riemann curvature ρ tensor Rσµν does not necessarily vanish at p. The frames in which the first derivatives vanish are called local inertial frames; all the local inertial frames at a given point are related by Lorentz transformations (Carroll, 2019, p. 74). Conversely, the second derivatives of the metric with respect to the coordinates are related to the Riemann tensor, which as we said cannot be made zero in general because curvature is a property of the underlying geometry of the space and not of the coordinate system chosen, so the second derivatives cannot all be made to vanish at a point in which the Riemann is non-zero. When working with objects defined in an open set of a non-flat manifold, U ⊆ M, curvature forces us to substitute the flat-space derivatives ∂µ with covariant derivatives

∇µ. These new objects take into account the curvature of the space and how it affects different tensorial objects on which the operator is acting; for example, for each vector index that the object possesses, the covariant derivative will include a connection term 22 CHAPTER 2. THEORETICAL BACKGROUND

λ given by the Christoffel symbols Γµν. These symbols can be calculated from the metric as (Carroll, 2019, p. 97):

λ 1 λσ Γ = g (∂µgνσ + ∂νgσµ − ∂σgµν) (2.6) µν 2 However, the Christoffel symbols do not directly contain all the information about curvature because, as we have seen, they can be made equally zero in a local inertial frame. Instead we can calculate the Riemann tensor (Carroll, 2019, p. 126), a (1,3) tensor related to the derivatives of the connection, or equivalently the second derivatives of the metric, that measures the extent to which the metric is not locally isometric to that of the Minkowsky space

ρ ρ ρ ρ λ ρ λ Rσµν = ∂µΓνσ − ∂νΓµσ + ΓµλΓνσ − ΓνλΓµσ (2.7)

The physics described with this formalism will always be invariant under coordinate transformations that do not alter the geometry of the spacetime itself. These transformations are called the diffeormorphisms of the manifold, and they are the invertible functions that map one differentiable manifold into another while guaranteeing that both the coordinate change and its inverse are smooth (Carroll, 2019, p. 59). Beyond coordinate symmetries, the underlying manifold can also be invariant under some transformations that change the geometry of the spacetime, in which case we refer to these transformations as spacetime symmetries. Continuous symmetries are associated with conserved Killing vectors, denoted Kµ, and it is easy to prove that if a manifold has a Killing vector that is conserved then the metric of the spacetime cannot explicitly depend on the coordinate defined by that vector (Carroll, 2019, p. 134). We call maximally symmetric spaces to those manifolds that are invariant under all isometries, and thus they contain the maximum number of linearly independent Killing 1 vectors. In d dimensions there can be at most 2 d(d + 1) of these vectors. General Relativity must give closed formulas not only for how objects propagate through a background spacetime manifold, but also for how exactly those very particles deform the same space due to the gravitational interaction as they move through it. Regarding the movement through the manifold, all particles tend to propagate through the least resistance path through the curved background, which is the one for which any two points of the curve are joined by the shortest possible path along the manifold. In flat space this would be a straight line, but for a general differential manifold these curves are the geodesic curves of the manifold (Carroll, 2019, p. 106), and they can be obtained from the parallel-transport equation for the velocity vector of the dxµ particle, dλ . The equation can be written for a general manifold using the Levi-Civita connection, and it reads 2.1. QUANTUM FIELD THEORY (QFT) 23

d2xµ dxρ dxσ + Γµ = 0 (2.8) dλ2 ρσ dλ dλ From the role that the connection plays in equation (2.8) it is clear than in a local inertial frame all the particles will follow the same trajectories as they would in a flat d2xµ manifold, dλ2 = 0. Regarding how the spacetime bends under gravity, the shape of a given manifold will be a result of the gravitational field produced by all the objects it contains. In order to find out the precise relationship we need to find what physical quantities makethe space bend and how exactly they are related to the curvature at each point.

Figure 2.5: Artist’s conception of the spacetime curvature. Credit: sciencenews.org

In Newtonian mechanics mass is the sole source of gravitation, but we have already seen in section 2.1.2 that mass is just a particular manifestation of energy, and that the apparent mass of an object will not be the same when observed from two different inertial frames in relative motion. It is only natural then for General Relativity to admit the energy-momentum tensor T µν as the appropriate generalization for the source of the gravitational interaction in General Relativity. The reason we need to work with the energy-momentum tensor instead of the energy-momentum vector P µ can be understood mathematically from the fact that in order to write an equation that involves the Riemann curvature tensor and its contractions we will always need to have an even number of indexes, and it turns out that T µν is the appropriate object. From a physical perspective, as the gravitational interaction carries energy and thus it must source itself, which means it must behave like a field; as fields need to be defined by a 2-tensor, just like a fluid does, that explains the appearance of T µν. The relationship between energy-momentum and spacetime curvature is then given by Einstein’s equation (Carroll, 2019, p. 156).

1 8πG Rµν − Rgµν = Tµν − Λgµν (2.9) 2 c4 24 CHAPTER 2. THEORETICAL BACKGROUND

The Field Equations for General Relativity also accept a Lagrangian formalism, and thus they can be derived as the Euler-Lagrange equations for the Einstein-Hilbert action,

1 Z √ S = R −gd4x (2.10) 16πG

where R is the Ricci scalar and g is the determinant of the metric gµν. This action can be modified to include a cosmological constant and the propagation of matter fields in a gravitational fields by supplementing (2.10) with additional terms

Z 1 √ 4 S = (R − 2Λ) + LM −gd x (2.11) 16πG

Contrary to other physical theories for which energy cannot be measured absolutely and only energy differences are observable, in General Relativity the total energyis directly related with the curvature of spacetime. Furthermore, due to the general coordinate invariance of the theory the energy is not well defined in the bulk, and instead it must be defined as a surface integral in the boundary of the space, as describedin Regge and Teitelboim (1974). As the energy is given by the Hamiltonian function, which governs the dynamics of the theory, the fact that this object can only be well defined in the boundary means that General Relativity presents holography, the principle according to which all the information contained in a region of spacetime can be stored in a lower-dimensional space. A striking fact of General Relativity is that it predicts solutions to Einstein’s equations where the curvature can get arbitrarily big at a given point despite the energy remaining finite; these points are called singularities, and they are the source of many unexpected results and complications in the theory. For example, the no-hair theorem states that two black holes with the same mass, charge and spin must be indistinguishable, which can makes us wonder what happens with the information carried by the objects that fell into it. The situation seems even more paradoxical when we study a black hole in a curved-background QFT, as in that case a black hole can evaporate through Hawking radiation; if the information of the states were stored locally, it would be lost, violating unitarity. This problem is called the information paradox. However, holography can be used to restore unitarity for a prospective quantum gravity theory, as all the relevant information is stored on the boundary even when there are singularities present. To summarize, General Relativity provides a geometrical interpretation of gravity in terms of differentiable manifolds, where particles follow geodesics and the geometry of spacetime is modified by the energy-momentum tensor, Tµν of the objects in the manifold. 2.1. QUANTUM FIELD THEORY (QFT) 25

2.1.4 Quantum Field Theory

Quantum Field Theory is a physical framework, and as such it serves as a foundational base of keystone principles that accommodate many particular theories. QFTs are a special subset of quantum systems in which the Hamiltonian is invariant under Poincaré transformations; the states in these theories must necessarily be extended configurations throughout the spacetime instead of point particles, and they are usually referred to as fields. The properties of these states will be based on those of classical fields as studied by Classical Field Theory, while also incorporating the basic properties of both the Quantum Mechanics and the Special Relativity frameworks that we have reviewed in this chapter. These properties include: the observability of expectation values of operators and unobservability of the states themselves; the unitary evolution and linearity of the physical states; the Poincaré symmetry of spacetime; causality, defined as a finite speed of information (usually in conjunction with locality); and the nature of mass as another realization of energy. With just those principles we can already construct the general framework and even make some predictions that have to hold for any QFT, while the particular theories of interest that will try to explain experimental results will be built by adding additional interactions and constraints to the general framework. In order to study Quantum Field Theory we would ideally write down some axioms and derive from them all the resulting properties. One possibility is to use Wightman axioms, derived in the 1950s by Lars Gårding and Arthur Wightman, which consist on four axioms that include all the core principles that we stated in the previous paragraph. From these starting axioms it is possible to derive a number of results, the most important of which is probably the CPT theorem, which states that all Lorentz invariant states must remain invariant under a simultaneous change of parity, particle-antiparticle reversal and time inversion. Another relevant result is the spin-statistic connection, which asserts that the algebra involving operators of fields with integer spin has to be defined in terms of commutators, while that of half-integer spin operators must be stated using anticommutators. It is also expected that many QFTs (especially those with non-Abelian symmetries) develop a mass gap between their ground states and the first excited states, but this result has not been mathematically proven yet for the more general case. Wightman axioms are the first example of the so-called Axiomatic (or Algebraic) Quantum Field Theory (AQFT), an approach which aims to infuse mathematical rigor into some of the tried and tested techniques that are used in theoretical and particle physics. However, the AQFT framework is too broad to describe many QFTs of interest, and when we add additional constraints in order to replicate some of the concrete calculations that can be successfully obtained through less rigorous approaches we often find roadblocks. Other mathematical frameworks include Vertex Operator Algebras and also Topological QFT, developed in the late 1980s, but both of them are too restrictive 26 CHAPTER 2. THEORETICAL BACKGROUND to accommodate many of the interacting theories that we observe in Nature. Being a quantum theory, the important objects in QFT are the observables: unitary self-adjoint operators that act linearly on states. Nevertheless, the vector space for an interacting Quantum Field Theory cannot be defined in terms of single-particle Hilbert spaces, as it is the case for non-interacting theories. We can see that in those cases, when we impose that the theory must be frame independence as required by Special Relativity (and more specifically due to the energy-mass equivalence), the eigenvalues of the particle number operator stop being good quantum numbers. The physical meaning of this statement is that a state in QFT can vary its number of particles (that is, particles can be generated or annihilated) while still evolving unitarily. In a non-interacting theory the vector space can be written as a sum of Hilbert spaces with different but fixed number of particles (this construction is called a Fock space); however, for an interacting QFT a generalization of Haags theorem states that the vector space cannot be a Fock space. Wavepackets describing a single particle stop being good descriptions of the underlying physics if new indistinguishable states can be created at any point in space and any moment in time. The simplest modification to accommodate this possibility isto work with fields extending throughout the spacetime and consider each spacetime point as a separate variable where quanta of the field can be created or destroyed. Inthe non-relativistic theory we have operators associated with the space coordinates whose eigenstates have a perfectly defined position coordinate, as it is characteristic of a point particle. In Quantum Field Theory, on the other hand, particles can be created and destroyed, and this process can even happen when we attempt to perform a measurement, so given a field state it no longer makes sense to wonder ‘where the particle is’, but instead how many quanta exist at a given spacetime point. As a result, spacetime coordinates stop being quantum operators that can be directly applied to a given state, becoming instead labels on every other operator that characterize the field at that point. We can look at a free field on a spacetime lattice for concreteness: for this system each point behaves as a quantum harmonic oscillator, which has a discrete spectrum and evenly-spaced energy states. The harmonic state with the nth lowest energy can be interpreted as a field composed of n (non-interacting) quanta at that spacetime point. A visual representation of this situation can be found in figure 2.6. For interacting theories the harmonic oscillators are not longer free, but coupled to each other, a fact that complicates calculations. As a Field theory, a QFT in a continuous spacetime possesses infinitely many degrees of freedom, due to the fact that regardless of their matter content these theories must have at least one free variable defined in every spacetime point. This fact greatly increases the difficulties that we encounter when performing both analytic and numerical calculations, and it is responsible for many of the divergences that appear in naive calculations. As a quantum theory, QFT states are invariant under global phase shifts, which 2.1. QUANTUM FIELD THEORY (QFT) 27

Figure 2.6: Visual aid for lattice QFT represented as harmonic oscillators in a lattice. Credit: ribbonfarm.com as we saw in section 2.1.1 implies the mathematical framework of the theory must be a projective vector space, just as the Hilbert space was in the non-relativistic case, and the eigenstates of the operators are realized as projective representations of the corresponding symmetry groups. As a relativistic theory, the causality constraint imposed by Special Relativity that in classical theories forbids the superluminal movement of particles must be reinterpreted in a quantum theory in terms of operators and not states, as due to the unobservability of states these objects can be non-local and non-causal as long as all the operators’ expected values retain causality. The result is that we must impose that the commutators of operators have to be identically zero outside of the light cone (the physical interpretation is that a given measurement in a spacetime point cannot affect another measurement if it is outside of its causal cone). When combining Special Relativity and Quantum Mechanics we obtain that the spacetime symmetry group of a QFT has to be the universal cover of the orthocronous connected component of the Lorentz group SO(3,1), usually denoted as SO+(1, 3). This universal cover is SL2(C), the special linear group of transformations in the complex plane, which together with the bosonic particles also contains relativistic spinors. The main object in a quantum theory, the Hamiltonian operator H, responsible for the time-evolution of the states through equation (2.1), is automatically determined in QFT as the time translation operator, which is contained in the Poincaré group. However, as time is an observer-dependent coordinate, in order to define a Hamiltonian we need to explicitly choose a particular reference frame what immediately breaks the explicit Poincaré symmetry of the theory. It can still be used to completely determine the evolution of all the states of the theory from the point of view of an observer in the reference frame that we choose, but the expressions obtained will not be explicitly Poincaré invariant, and thus we will lose the powerful mathematical results that Special Relativity provides us with. As in a relativistic theory all inertial observers must give equivalent descriptions of 28 CHAPTER 2. THEORETICAL BACKGROUND the universe, we must enlarge our concept of symmetries of the theory to include not only those transformations that leave the Hamiltonian of all possible inertial observers invariant, but also those transformations that change the initial Hamiltonian into any expression valid for an inertial observer. As an object that describes the time evolution as seen from a particular reference frame, the Hamiltonian will always commute with the generators of transformations that leave the theory invariant from the point of view of that particular observer, but not with those that modify the time coordinate. A very clear example is given by boosts, which we know are a symmetry of the theory as they belong to the Poincaré symmetry group, while from their expression we see that [H,Ki] = P i and thus not zero. By taking advantage of the covariant formalism developed in section 2.1.2 to study relativistic theories we can see that the Hamiltonian does not transform as a scalar under Poincaré transformations, but instead it behaves as a vector whose spatial components are given by P i. Just as we did for their non-operator-valued-counterparts, we can then define a new operator vector P µ, where H = P 0. Nonetheless, vector transformation laws are rather cumbersome and require us to know all the vector components in order to find its expression in another frame even if we are only interested in one of the components, the Hamiltonian. The study of the theory would become easier if we could obtain all the dynamic quantities of interest from a Lorentz scalar instead, as the transformation law for this objects is trivial. It turns out that for theories that can be derived from a variational principle there is one such object: S, the action. S is a non-local functional, but for a local theory it is possible to write it as the integral of a local scalar density, the Lagrangian density, as S = R Ld4x. There are however theories that cannot be written in terms of a variational principle, and thus they do not have an action or a Lagrangian. Some examples of this are given by systems with non-holonomic constraints; in spite of the fact that it is possible to incorporate these constraints in a Lagrangian formulation by using Lagrange multipliers, doing so would render the multiplier a dynamical variable and thus it would modify the field content of the original theory. Once we have the action of the theory it is possible to derive the Hamiltonian for a given observer by means of the Legendre transformation in the cases when this transformation is not singular, and then derive all the quantities of interest for the theory just like we do in the non-relativistic theories (including the same complications, like the difficulty to calculate the eigenstates for certain Hamiltonians despite knowing its functional form). However, it is possible to compute all the quantities of interest directly from the action by means of the path integral formalism (which naturally quantizes a classical theory) by computing the so-called correlation functions, which are a summation over all possible field configurations with given initial and final states where every configuration is ponderated by the exponential of its action. 2.1. QUANTUM FIELD THEORY (QFT) 29

In order to study scattering events we can introduce the so-called asymptotic states, which are approximate solutions of the theory under the assumption that all other fields in a sufficiently large neighborhood around them are in their vacuum states. Wewill make use of two different vector basis of the QFT, one that we will use for initial states, also called in-states and denoted by |αii, and one final states or out-states which we will write as |βii. When working in a flat minkowskian spacetime, in any given scattering event between multiple states in which no two particles have exactly the same momentum it is possible to follow each individual state back in time until they are arbitrarily far away from each other and their interaction with all other states is negligible, thus being well approximated by asymptotic states. Consequently any interaction can be described in terms of the time evolution of an asymptotic in-state defined in the infinite past, and after the scattering all the resulting states evolve towards asymptotic out-states in the infinite future (Weinberg, 1995, p. 109). The transition amplitudes between all possible combinations of in and out-states (the combinations that have non-zero amplitude are called scattering channels) can be arranged into an object called the S-matrix, which contains all the information of the theory; in components this matrix is defined as Sij = hβi|αji. The unitarity of the QFT is directly translated into an unitary S-matrix, and the poles in the complex-energy plane are identified with bound states, virtual states or resonances; branch cuts correspond with the opening of a new scattering channel.

The elements Sij can be regarded as the components in either the base {|αii} or in

{|βii} of an operator S that relates in-states with out-states so that S|βii = |αii and † hαi|S = hβi|, or equivalently S |αii = |βii, thus obtaining:

Sij = hβi|αji = hαi|S|αji = hβi|S|βji (2.12)

Thanks to this formula it is possible to work with only one of the two bases for all our calculations, simplifying the calculations.

If we know the Hamiltonian of the theory, the elements of the S-matrix (Sij) can be calculated as the time ordered exponential of the interaction Hamiltonian,

Hint, which assuming that the vacuum state is invariant under time evolution reads R S = T exp −i Hint(t) dt . The series expansion of this formula produces the so-called Dyson series, from which the S-matrix can be calculated perturbatively

∞ X −i ZZ Z Z S = ... T Hint(t1) Hint(t2) ...Hint(tn) dt1 dt2 ... dtn (2.13) n! n=0

It is also possible to calculate the S-matrix directly from the Lagrangian approach by 30 CHAPTER 2. THEORETICAL BACKGROUND means of the path integral and the LSZ (Lehmann-Symanzik-Zimmermann) reduction formula, which relates the amplitudes associated with the Sij elements to the so-called correlation functions that describe the propagation of a particle between two given spacetime points. The n-point correlation function can be defined in terms of the path integral of n given QFT field operators as

R −S[φ] φ(x1)φ(x2) . . . φ(xn) e Dφ hφ(x1)φ(x2) . . . φ(xn)i = (2.14) R e−S[φ] Dφ As the path integral is hard to calculate exactly in most cases, it is usually expanded in series as a small perturbation of a simpler action for which we know how to perform this integral (which most always are those that are gaussian with respect to the field variables). Each of the terms obtained in this perturbative approach can be interpreted as a separate process where interactions are mediated through the exchange of virtual particles, and the total amplitude is then the sum of all these possible processes.An example of one a process contributing to the scattering amplitude between two particles can be seen in figure 2.7.

Figure 2.7: . Credit: slimy.com

The pictorial representation to the right of figure 2.7 is called a Feynman diagram, and they are used as computational aids in order to find all non-trivial terms necessary to calculate any scattering event. For each theory we will need to find the drawing rules that every Feynman diagram must respect, which will change depending on the type of interactions that appear in the theory. Even though these diagrams are usually easier to obtain than the series expansion of the path integral, higher order terms can be very complicated, as can be seen in figure 2.8. Although the LSZ reduction formula cannot be directly used to compute bound states, massless particles or topological solutions, it is possible to generalize it by including composite fields to the action in the path integral. Even though these fieldsmaybe non-local, their behavior still respect all the constraints of the theory. However, if the 2.1. QUANTUM FIELD THEORY (QFT) 31

Figure 2.8: An example of a Feynman diagram. Credit: Modified from Peskin and Schröder

theory is confining as is the case with QCD, the Hamiltonian formalism is usually easier to work with than the Lagrangian approach.

It is easy to generalize a Quantum Field Theory defined in a flat background in order for it to admit a static curved spacetime background; this process constitutes the first step towards including gravity in our framework. It is enough to first substitute the

flat-space Minkowski metric, ηµν, with its curved equivalent, gµν, then modify the flat

derivatives ∂µ to obtain covariant derivatives ∇µ that include the connection induced by √ the curved spacetime, and finally change the integration measure from ddx to −g ddx. Most results from flat space are easily generalizable to a curved background, though there are some new interesting results that do not have a flat-space equivalent, like the Unruh effect. Nevertheless, having a description for particle propagation on a static space is not enough to obtain a quantum description of gravity: we need an operator equation that plays the same role as Einstein’s field equations to determine what is the source of curvature and how spacetime changes under in, and it is here where we start running into multiple complications. For example, the notion of spacetime itself is not well defined at scales of the order of Planck’s length, where all Effective Quantum Gravity theories that we have built so far break down due to the non-renormalizable terms introduced by gravity. This is because the extremely small (UV) effects are not well described by the Effective Field Theory we have, we will need better knowledge of the physicsat very short distances in order to complete our theory and turn it renormalizable. For the moment however we have to work with Effective theories with a physical cutoff at the Planck length, where we expect new degrees of freedom to appear, and limit our predictions to energies well below that scale. 32 CHAPTER 2. THEORETICAL BACKGROUND 2.2 Supersymmetry (SUSY)

A supersymmetrical theory is a Quantum Field Theory that is invariant under some mixing of the bosonic and fermionic states in the theory. As bosons are described by even fields while fermions need odd ones (that is, the algebra of its operators iswritten using commutators and anticommutators respectively), the algebra of a Supersymmetric theory must be a Z2-graded algebra, also called a superalgebra. The (anti-)commutation relations between operators can be written following the treatment by Ammon and Erdmenger (2015, p. 121) as

[A, B} = AB + (−1)1+gr(A) gr(B)BA (2.15)

where gr(O), the grade of an operator, is an integer defined modulo 2 which is equal to 0 for bosons and 1 for fermions (we can also take it equal to −1 for fermions, as 1 mod 2 = −1 mod 2). From this definition we can derive the expression for the opposite object, as described in Dibitetto (2012, p. 29)

{B,A] = (−1)1+gr(A) gr(B) [A, B} (2.16)

The (anti-)commutator is bilinear and additive with respect to the grade function, which can be written as

gr([A, B}) = gr(A) + gr(B) (2.17)

. Additionally, it has to satisfy the super-Jacobi identities which, by using the short- hand notation gO = gr(O), can be written as

(−1)gA gC {{A, B] ,C] + (−1)gB gA {{B,C] ,A] + (−1)gC gB {{C,A] ,B] = 0 (2.18)

The supergroup associated with this graded algebra enlarges the Poincaré group by adding fermionic operators. By virtue of (2.17), the anticommutator of two fermionic operators must be equal to a sum of non-zero bosonic symmetry generators (Weinberg, 2000, p. 32). In the case of an extended supersymmetry, those that have several independent fermionic generators, the anticommutators between these operators introduce bosonic scalar operators that we call central charges, as they commute with all other generators of the super-Poincaré group. The supersymmetry algebra is invariant under a global non-Abelian symmetry called the R-symmetry. The most general superalgebra structure that includes the Poincaré group is found in Wess and Bagger (1992, p. 8). 2.2. SUPERSYMMETRY (SUSY) 33

A superalgebra is, under certain hypothesis, the only way to combine the Poincaré and internal symmetry groups into a larger group that cannot be written as a direct product. According to the Coleman-Mandula no-go theorem, first stated in Coleman and Mandula (1967), for a local 4-dimensional QFT theory that contains only a finite number of different particles of any given mass and that possesses an energy gap between the vaccum and the one-particle states, the Poincaré algebra can only be enlarged by scalar operators. However, in Haag et al. (1975) it was concluded that it is possible to relax some of the hypothesis of this theorem, in which we find that both conformal transformations and graded Lie algebras constitute non-trivial extensions of the Poincaré algebra; supersymmetry takes the second path, relaxing the assumption that the mathematical structure must be that of a group algebra and instead using a superalgebra.

Figure 2.9: Minimal Supersymmetric Standard Model (MSSM). Credit: arstechnica. org

The super-Poincaré algebra is composed of even an odd operators, and we will assume that for any fermionic operator in the algebra, Q, its hermitian conjugate Q† must also be present. The minimal supersymmetric theories have two fermionic operators, conjugate to each other, while we can define extended supersymmetries that have N pairs of fermionic operators. It can be proven, as it is shown in Wess and Bagger (1992, p. 4) that if we want fermionic operators to act correctly on a Hilbert space with positive definite metric (as is the case in a QFT), they must be necessarily spin-1/2 objects. Schematically A we write them as Qα , where A counts the number of supersymmetries and it ranges from 1 to N , while α is the Lorentz index for the irreducible spin-1/2 representation in a spacetime with d dimensions. A spin-1/2 representation has 2bd/2c+1 real components, but in certain dimensions not all of them may transform among themselves, and thus the irreducible representations will have a smaller number of components depending on whether the irreducible spinors are Dirac, Majorana or Majorana-Weyl in that dimension; a table summarizing the results for each value of d can be found in Dibitetto (2012, 34 CHAPTER 2. THEORETICAL BACKGROUND p. 28). For a theory in a dimension where the irreducible spinor has C real components, the Lorentz index α ranges by convention from 0 to C − 1, and we thus have N·C independent supercharges in an extended supersymmetry. For massless particles, defined by their 4-momentum and their helicity, |pµ, λ}, the action of half of the supercharges on them is identically zero, while the other half act as raising and lowering operators respectively. Thus, the difference between the minimum and the maximum helicity of the massless fields present in a supersymmetric theory with a single multiplet will be equal to the amount of non-zero raising supercharges needed to go from the lowest helicity state to the maximum helicity in half-integer increments, that is

N·C λmax − λmin = (2.19) 8 The Weinberg-Witten theorem stated in Weinberg and Witten (1980) states two separate results:

• A 4-dimensional Quantum Field Theory with a conserved 4-vector current J µ which is Poincaré covariant (and also gauge invariant if the theory also possesess gauge symmetries) does not admit massless particles with helicity |λ| > 1/2 if they also have nonzero charges associated with the conserved current.

• A 4-dimensional Quantum Field Theory with a non-zero conserved stress–energy tensor T µν that is Poincaré covariant (and gauge invariant if it applies) does not admit massless particles with helicity |h| > 1.

This theorem can be interpreted as the fact that massless fundamental degrees of freedom with helicity |λ| > 1/2 has to be associated with a conserved current for the QFT theory to be consistent: A conserved vector, |λ| = 1, implies a gauge symmetry; spin 3/2 currents can be identified as supersymmetric currents, of which there must be exactly N ; and the only conserved 2-tensor is the energy momentum tensor, which implies that there can only be one graviton. Any further conservation laws, especially for higher spins, will result in trivial scattering amplitudes. The only workaround in order to include higher-dimensional spins is to use towers of infinitely particles with arbitrarily high spin that move in sync with each otherin order to cancel out divergences, as is the case in String theory, for example. In any other situation we would like to avoid including massless particles with spin greater than 2 in our theories, and so λmax − λmin ≤ 4. If we are only interested in renormalizable

(non-gravitational) theories then we will take λmax − λmin ≤ 2 instead. In a spacetime of given dimension d, equation (2.19) imposes a superior bound on the number of supersymmetries that any theory can have, N C ≤ 32. In 4 dimensions, 2.3. CONFORMAL FIELD THEORY (CFT) 35 as C4 = 4, this inequality means that N ≤ 8, while we find the surprising result that there is a maximum spacetime dimension for which we can have supersymmetry without introucing high spin particles, and that is d = 11, for which C = 32 (Dibitetto, 2012, p. 34). For non-gravitational theories we require N ≤ 4 in 4 dimensions and d = 10 is the maximum dimension in which supersymmetry is still consistent. For an unbroken supersymmetry, all particles comprising an irreducible representation must have the same mass, as the fermionic operators commute with the mass Casimir, P 2 (Wess and Bagger, 1992, p. 11). Also, in the case of finite-dimensional representations there has to be the same number of fermionic and bosonic degrees of freedom in any given multiplet (Wess and Bagger, 1992, p. 12). Spin is no longer a good quantum number when we want to describe a state that is invariant under super-Poincaré transformations, as it can be changed by acting on the state with a fermionic operator. However, as we have not found any superpartners of the known particles of the Standard Model, if Supersymmetry is a theory of the Universe it must be broken, pushing all the superpartners to an energy scale much bigger than that of the Standard Model. As a result, for energies well below this scale spin can be considered a good approximation for a quantum number, and so we can decompose irreducible supersymmetric representations of the super-Poincaré group into subgroups that are almost invariant under the supergroup transformations at those energies, returning us to our usual single-spin irreducible representations.

2.3 Conformal Field Theory (CFT)

Conformal Field Theories is the name with which we refer to QFTs whose symmetry group includes the conformal group. These are the transformations under which the metric is invariant up to a multiplicative factor Ω2(x) ∈ R+, that can be a function of the coordinates, and thus it depends on each particular spacetime point at which the metric is evaluated. If we perform a conformal transformation from a coordinate system denoted by xµ to another system given by the coordinates xµ0, the metric in the new system will be

α β 0 ∂x ∂x −2 η (x) = ηαβ(x) = Ω (x) ηµν(x) (2.20) µν ∂xµ0 ∂xν0

In particular, Poincaré transformations form a subgroup of the conformal group, namely they are the coordinate changes for which Ω2(x) = 1. For a conformal theory with euclidean signature, an alternative definition is that these are the most general transformations that preserve angles, while for lorentzian signature they are instead the most general transformations that locally preserve the causal structure of the spacetime. 36 CHAPTER 2. THEORETICAL BACKGROUND

For a theory in a d-dimensional spacetime (for d > 2) with lorentzian signature, the Lie group associated with conformal symmetry is SO(d,2), which expands the d-dimensional isometry group (that is, the Poincaré group ISO(d − 1,1) = Rd−1,1o O(d − 1,1), as seen in section 2.1.2), by enlarging it with non-isometric transformations. If in addition to the continuous conformal transformations we also add the inversion, that is, the discrete transformation that takes the point at xµ into xµ/x2, we enlarge the symmetry group to O(d,2) (Zaffaroni, 2000, p. 6). Because a CFT can be interpreted as a Poincaré-invariant QFT with additional constraints, all of the basic properties reviewed in section 2.1.4 still apply for conformal theories. Conformal symmetry is a stronger constraint on a system than scale invariance is; however, if we want to quantize a local CFT and we do not want quantum corrections to break the conformal symmetry (that is, we want the conformal current to be conserved in the quantum theory), then under the reasonable physical assumptions of locality and unitarity we can prove that for a CFT scale invariance implies also conformal invariance (for most cases this implication can be further proven to be a sufficient and necessary condition) (Zaffaroni, 2000, p. 7). For this reason, the terms ‘conformal invariance’ and ‘scale invariance’ are often used interchangeably in the literature regarding CFTs, despite the caveats.

Figure 2.10: Scale invariant system at four different distance scales. Credit: Douglas Ashton

A necessary condition to have scale invariance is that the integral of the trace of the energy-momentum tensor is equal to zero, while a sufficient condition is for the tensor µ to be traceless, Tµ = 0 (p. 118 Ammon and Erdmenger, 2015; Benini, 2018, p. 11). A traceless energy-momentum tensor represents massless particles, so in the theories for 2.3. CONFORMAL FIELD THEORY (CFT) 37

µ which Tµ = 0 all states have to be massless; on the other hand, if in a given CFT a representation of the conformal group were to contain a state with a given non-zero mass, it must also contain states with all the masses ranging between 0 and infinity that can be obtained by acting on the first state with the dilation operator, D, in order for the integral of the trace to cancel out (Zaffaroni, 2000, p. 10). The conformal algebra exhibits a very different behavior in two dimensions than it does in higher dimensions. For d = 2 there are infinitely many conformal transformations, so the possible solutions are very constrained and it is relatively easy to find complete solutions to models in CFT2 using algebraic techniques (Kaplan, 2018, p. 59). When we quantize a 2-dimensional CFT the conformal group is replaced by its universal cover group, defined through the Virasoro algebra; the commutators of the operators receivea central extension involving a new number, the central charge c (Ammon and Erdmenger, 2015, p. 107). In dimensions d > 2, on the other hand, the number of conformal transformations is finite and its algebra is locally isomorphic toSO(d,2) in Minkowky signature. The irreducible representations of this group are usually classified into primary states and descendant states. As all other unitary representations of the conformal group can be decomposed as linear combinations of irreducible representations, we can restrict our study to that of the properties of primary and descendant states. Primary states are those of lowest scaling dimension in a given representation, and thus they are composed by the states that are annihilated by the generator of special conformal transformations, Kµ|φi = 0 (Kaplan, 2018, p. 49). Descendant states can be obtained from primary states by repeatedly acting on them with Pµ (the translation gen- µ n erators), and thus can be represented schematically as |φn,li = (P Pµ) Pµ1 Pµ2 ··· Pµl |φ0i, which must be normalized and symmetrized appropriately (Kaplan, 2018, p. 26). The set of a primary field and all its possible descendants is called a conformal family; these families can be classified with respect to its behavior the dilation operator D, whose eigenvalues are denoted by ∆, with the primary field having the lowest value in any given family. Through the commutation relationship between P and D we can see that the translation generators act as raising operators with respect to D, while the special conformal transformations K act as lowering operators. When we impose unitarity for the CFT we obtain the so-called unitarity bounds that relate the scaling dimension of a field with its Lorentz quantum numbers, as seen in Ammon and Erdmenger (2015, p. 107) for the 4-dimensional case and in Benini (2018, p. 14) in general dimension. In particular, scalars must have conformal dimension d−2 ∆ ≥ 2 , where the equality holds for free fields, and for vectors we have ∆ ≥ d − 1, the saturation bound corresponding to conserved currents. Upon quantization the conformal symmetry of the theory can become anomalous due to the renormalization group flow, which in general can introduce a dimensionful 38 CHAPTER 2. THEORETICAL BACKGROUND parameter in the theory (the dependency of the coupling with the energy scale). To prevent this, a CFT must sit on a fixed point of the renormalization group, where the beta function of the coupling parameters is identically zero. Otherwise it is also possible to obtain a consistent CFT from finite theories, that is, systems with a great deal of symmetry that present no UV or IR divergences upon quantization; in that case, the beta function is always zero for any value of the interaction parameters and thus the renormalization group flow presents a line of fixed points instead of isolated ones. Nevertheless, even if the renormalization group flow is equal to zero the classical dimension of the fields in the theory, ∆0, will be corrected after quantization by a so- called anomalous dimension, γ, so that ∆ = ∆0 + γ. The anomalous dimension can be calculated in terms of the variation of the field strength renormalization counterterm, Z, 1 d ln Z with respect to the energy scale, µ, as γ = 2 µ dµ evaluated at the fixed point (Zaffaroni, 2000, p. 8). Operators in a CFT must transform homogeneously under dilations, generated by the operator D which induces the coordinate transformation x0 = λx, meaning that we can write O0 = λ∆O where ∆ is referred to as the scaling dimension of the operator. We are interested in studying correlation functions, which measure the ammount of interdependence between different fields at different spacetime points; these objects possess a great deal of symmetry in a CFT, and the strict conditions that are required to maintain conformal invariance is in many cases sufficient to determine the behavior of the objects in the theory. In particular, correlation functions between operators have to be conformally invariant too, and as a consequence they show a very simple dependency on spacetime coordinates. Correlation functions of two scale invariant operators in the same conformal family

Oi, Oj with scaling dimension ∆i and ∆j respectively, must scale as hOi(λx)Oj(λy)i =

∆i+∆j λ hOi(x)Oj(y)i as proven in Ammon and Erdmenger (2015, p. 110). This means that the two-point correlator can only depend on the difference between the two spacetime points as |x − y|∆i+∆j , so for any two fields the 2-point function has to be exactly equal to

cij hOi(x)Oj(y)i = (2.21) |x − y|∆i+∆j

where cij are real and symmetric numbers depending on the particular fields in question, but for a CFT the primary fields are usually chosen so that these factors are normalized and diagonalized, in which case cij = δij (Ammon and Erdmenger, 2015, p. 111). The 2-point functions of descendant operators are obtained by taking derivatives of the 2-point functions for their corresponding primary fields. The correlators of three invariant operators, also known as 3-point functions, can also be shown to be greatly constrained by scale symmetry so that its form is fixed up 2.3. CONFORMAL FIELD THEORY (CFT) 39 to a constant depending on the fields

cijk hOi(xi)Oj(xj)Ok(xk)i = ∆ +∆ −∆ −∆ +∆ +∆ ∆ −∆ +∆ |xi − xj| i j k + |xj − xk| i j k + |xk − xi| i j k (2.22)

The cijk for canonically normalized primary fields are called the structure constants of the theory, and they play a critical role in the study of a CFT (Benini, 2018, p. 14). Higher point correlation functions can depend on dimensionless cross-ratios of distances between points, complicating their structure. However, all correlators are completely determined once we know the spectrum of primary fields and their three point functions due to another great simplification present in CFTs: the possibility to rewrite non-local products of fields in terms of sums of local fields using the operator product expansion (OPE). For the product of two operators at two different points, Oi(x) and

Oj(y), we can write the OPE in terms of an infinite polynomial whose coefficients depend on the distance between them, |x − y|, multiplying a series of operators evaluated at one point only (Benini, 2018, p. 15).

∞ X (n) Oi(x)Oj(y) = cij (x − y) On(y) (2.23) n=1 where conformal symmetry requires that

(n) cijn cij (x − y) = (2.24) |x − y|∆i+∆j −∆n We can write this operator expansion in any local QFT, but the radius of convergence of the infinite series may be zero. However, in any CFT we are guaranteed that the series is convergent until the circle around y hits another operator; these convergence properties turn OPEs into very useful tools to study conformal theories. When the two fields in the left-hand side of equation (2.23) are canonically normalized primary operators, the summation in the right-hand side will only include primary operators, and the coefficients cijn will be the 3-point structure constants of the three operators. From this result it is straight-forward to derive the 3-point functions involving descendant operators as the derivatives of its primary counterparts. Finally, any n-point function can be written in terms of 3-point functions by repeatedly using the OPE; the result is called a conformal partial wave that is completely determined by conformal symmetry, and the contribution from each irreducible representation is called a conformal block (Kaplan, 2018, p. 67). As a result of the simplifying properties introduced by the OPE, we can completely solve a CFT by writing down the correlation functions we are interested on and then decomposing those functions in terms of structure constants and conformal blocks, a process usually called the conformal 40 CHAPTER 2. THEORETICAL BACKGROUND bootstrap principle.

2.3.1 Superconformal Field Theory (SCFT)

When enlarging the conformal group into a supergroup, as explanied in section 2.2, we realize that in addition to the fermionic generators, denoted by Q, we must also include a new set of fermionic superchargers, the superconformal operators S. Just like the a µ supersymmetry generators Qα are the superpartners of the translation generators, P , a the superconformal operators Sα are the fermionic superpartners of the special conformal transformations Kµ (Ammon and Erdmenger, 2015, p. 139). There must be as many superconformal generators as there are supersymmetry ones because the S appear in the right-hand side of the commutation relations between Q and K (Benini, 2018, p. 17); the algebra of the fermionic operators is given in Ammon and Erdmenger (2015, p. 139). The S operators can be shown to lower the scalar dimension of states. Thus, the new operators of lowest scaling dimension, usually called superconformal primary operators, must be annihilated by all the superconformal generators. From these primaries it is possible to construct all the descendants in the family by acting on them with each of the operators in the superconformal algebra. A particularly important set of descendants is that of the superdescendants, obtained by acting on them with Q, which increases the scaling dimension of the new operator by 1/2; it can be proven that these superdescendant operators are conformal primary operators, that is, they are annihilated by all the Kµ. When counting the amount of supercharges in a SCFT we have to consider both Q and S operators, and thus these theories have twice as many supercharges as the equivalent CFT. Another interesting set of operators are those superconformal primaries that are a annihilated by at least one of the Qα, called the chiral primary operators. By definition these objects must preserve some of the supersymmetry of the theory, which is also the definition of a Bogomol’nyi-Prasad-Sommerfield (BPS) operator (Ammon and Erdmenger, 2015, p. 140). An important property of the chiral primaries is that their scaling dimension does not receive any correction in the quantum theory, as due to supersymmetry this number is a fixed function of the spin and the R-symmetry quantum number of the operator, and thus it is a constant number. In a Superconformal Field Theory the R-symmetry is automatically determined as part of the algebra and not as an outer automorphism, as described in Benini (2018, p. 17). As a consequence, SCFTs greatly limit the total symmetry group of the theory more than CFTs or supersymmetric field theories do, but in exchange these theories are considered more natural and they have a higher predictive power. The limitations imposed on graded algebras that contain both the conformal group and supersymmetry limit the possible SCFTs to only a handful of theories, all of them in 6 dimensions or 2.4. STRING THEORY 41 less, as proven in the classification elaborated by Nahm (1978). A SCFT often used to study the AdS/CFT correspondence is the SU(2, 2|N ) Super- conformal Field Theory in 4 dimensions whose algebra we can find written explicitly in Ammon and Erdmenger (2015, p. 532).

2.4 String theory

String theory is a quantum theory for fundamental 1-dimensional objects that we call strings. These objects propagate in a background spacetime of (a priori) arbitrary dimension d, generally with a single time-like coordinate. An axiomatic non-perturbative background-independent formulation of string theory is not known yet. We want to study the dynamics of a string by tracking its position in spacetime as it moves, but being an extended object (a 1-dimensional line) we cannot write its movement as a function of a single parameter, the same way that we usually do for a point-like particle when we write its position in terms of its proper time, τ. Instead, to describe the position of the complete string we need to add a second parameter, represented by σ, which determines ‘which point in the line’ we are referring to. For closed strings we will take σ ∈ [0, 2π), while for open strings it is customary to choose σ ∈ [0, π] (p. 146 Ammon and Erdmenger, 2015; Tong, 2009, p. 14). The two parameters τ and σ, usually referred in combination as σa with a = 0, 1, define a worldsheet, Xµ(τ, σ), that the string sweeps as it propagates. However, as string are hypothesized to be fundamental objects (and thus indivisible) they cannot have subdivisions, and as a consequence a given point of the string will not be physically meaningful entity, and it will not be possible to follow the evolution of individual points. Due to this fact we cannot single out a privileged reference frame in the string worldsheet, as this would require comoving with a chosen point; thus all possible parametrization of the worldsheet must be equivalent and the underlying physics has to be invariant under reparametrizations of the coordinates σa (this property is also referred to as diffeomorphism invariance). The d-dimensional space in which the string moves is called the target space, and when studying perturbative String theory we will assume by hypothesis that the initial background is a flat manifold with lorentzian signature which is invariant under Poincaré symmetries, also known as a Minkowsky space. A complete non-perturbative String theory is expected to be background-independent, but so far all our constructions require us to define an initial spacetime. The physics described by String theory mustonly depend on the embedding of the worldsheet into the target spacetime, given by Xµ, and not on the parametrization chosen for the worldsheet or for the target space (Ammon and Erdmenger, 2015, p. 146). 42 CHAPTER 2. THEORETICAL BACKGROUND

In order to study the propagation of a quantum string we are going to find a Lorentz- invariant action for a classical string from which we will then derive the equations of motion and finally we will quantize them. The classical action is proportional toa coordinate invariant associated with the string, the proper area of its worldsheet, which is the natural generalization of the invariant that appears in the action of a point particle, the proper time (Tong, 2009, p. 15); the proportionality constant that accompanies the string action is a parameter T usually called the string’s tension. The proper area can √ be defined as dA = −γ d2σ, where γ is the pull-back of the Minkowski space metric onto the worldsheet. With these two ingredients we obtain the Nambu-Goto action for a string, which reads

Z Z √ S = −T dA = −T −γ d2σ (2.25)

After fixing the gauge for the worldsheet by selecting specific coordinates σ1 and σ2, µ ν the pull-back metric can be calculated as γab(X) = gµν(X) ∂aX ∂bX ; by calculating its transformation under coordinate reparametrizations we find that its determinant (and √ by extension also −γ) is independent of the choice of coordinates, as expected. The tension T is the only free parameter in the action, and sometimes it is given in terms of 0 1 the so-called universal Regge slope, which is defined as α = 2πT ; from this constant it is 2 0 also possible to obtain a length scale associated with the string, ls = α . The Nambu-Goto action that we wrote in equation (2.25) is easier to study and quantize if we are able to extract the fields from the square root. It is possible totake µ the coordinates and the metric of the target space, X and gµν respectively, out of the square root if we introduce an additional field: the metric of the worldsheet, hab (p. 147 Ammon and Erdmenger, 2015; Tong, 2009, p. 18). The resulting action is called the Polyakov action

Z √ T ab µ ν 2 S = − −h h gµν(X) ∂aX ∂bX d σ (2.26) 2 It can be easily checked that this action produces the same equations of motion as the Nambu-Goto action for the scalar fields Xµ that describe the position of the string.

For the new field hab not to introduce additional degrees of freedom that were not present in the Nambu-Goto action it has to be possible to gauge away all of its free parameters. A square matrix in two dimensions has four elements, but the signature of the metric must necessarily be (-1,1) due to the propagation of the string through one temporal and one spatial dimension, so after fixing this constraint we conclude that this object has three free parameters. Two gauge symmetries are automatically obtained as a result of imposing reparametrization invariance of the σ1 and σ2 coordinates, as the diffeomorphism group in two dimensions has two parameters (Tong, 2009, p. 22). 2.4. STRING THEORY 43

The third degree of freedom of hab can be eliminated by means of Weyl invariance, a transformation that leaves the coordinates invariant while rescaling the metric by an arbitrary function of the parameters; it is worth noticing that a Weyl rescaling is not a coordinate transformation and thus it is different from a conformal transformation, but in 2 dimensions when combining Weyl invariance with diffeomorphism invariance we obtain that the worldsheet must have conformal symmetry when the background metric is fixed (Tong, 2009, p. 62). As required, the equations of motion for Xµ that we obtain from the action in (2.26) are invariant under diffeomorphisms and Weyl transformations of hab, so we have proven that the additional parameters present are unphysical (they cannot be observed) and can be gauged away, and consequently the Polyakov action describes the same physics as the Nambu-Goto one.

Because all possible choices for hab have to be equivalent we can choose to fix the gauge in order to simplify our calculations; we will choose the worldsheet metric to be that of 2-dimensional flat spacetime, ηab, as it is the easiest one to work with. If we start with any other metric we can change to a worldsheet coordinate system, σa, in which the metric is flat by first using the diffeomorphism invariance to the coordinates inwhich 2φ hab = e ηab (a coordinate choice which is known as the conformal gauge, as defined in

Tong (2009, p. 22)). Finally by using a Weyl transformation we can set hab = ηab. In this gauge the Polyakov action reduces to Z T µ 2 S = − ∂aX ∂bXµ d σ (2.27) 2

with the additional condition that hab has to satisfy the equations of motion imposed by the action in (2.26), namely:

µ ν 1 cd µ ν gµν(X) ∂aX ∂bX − habh gµν(X) ∂cX ∂dX = 0 (2.28) 2 which has to hold for a, b = 0, 1. We can check that the left-hand side of these equations are proportional to the components of the energy-momentum tensor of the worldsheet, which is defined as:

2 1 ∂S Tab = − √ (2.29) T −h ∂hab

By comparing equation (2.29) with (2.28) we obtain the condition that Tab = 0, that is, all four components of the energy-momentum tensor of the worldsheet have to vanish in order for the equations of motion of hab to be satisfied. The physical interpretation of this surprising fact is that the string cannot store energy or resist external forces, it must automatically deform under a perturbation. An analogous explanation is that all physical modes of the string are transverse oscillations, there cannot be any longitudinal 44 CHAPTER 2. THEORETICAL BACKGROUND modes (Tong, 2009, p. 24). If we now vary the action in (2.27) with respect to the Xµ coordinates we will the equations of motion for each of the d parameters, which under target space diffeomorphisms transform as scalars. The equations of motion are

a µ ∂a∂ X = 0 (2.30)

In the lightcone gauge, that is, the coordinate choice for the worldsheet given by σ± = τ ± σ, the solution can be decomposed into left-moving and right-moving modes (Dibitetto, 2012, p. 15). These equations of motion have to be supplemented by the additional boundary conditions Tab = 0, whose components read

µ ν T01 = gµν(X) ∂τ X ∂σX = 0 (2.31)

1 µ ν µ ν T00 = T11 = gµν(X)(∂τ X ∂τ X + ∂σX ∂σX ) = 0 (2.32) 2

Equivalently we can obtain the boundary conditions by making the variation of the action in (2.27) equal to zero, as explained in Dibitetto (2012, p. 14) and Tong (2009, p. 50) Z µ µ δS = −T [∂σX δXµ|σ=¯σ − ∂σX δXµ|σ=0] dτ = 0 (2.33)

µ The integrand of (2.33) will be identically zero if and only if the term ∂σX δXµ|σ takes the same value when evaluated at σ = 0 than it does when evaluated at σ =σ ¯ (we take σ¯ = 2π for closed strings and σ¯ = π for open strings). These conditions have to hold for arbitrary values of Xµ, so there are only two possible ways for this expression to always cancel out for any possible configuration of the string:

• If the Xµ always take the same values at σ = 0 and at σ = 2π (that is, both ends of the string have the same coordinates, or equivalently, the string is a closed curve), the two terms in the integrand are necessarily identical. We call strings that hold this condition closed strings.

• For strings with non-coincident endpoints, known as open strings, we require that µ ∂σX δXµ = 0 at both σ = 0 and σ = π. At each endpoint we have a sum of d terms that in principle could cancel each other separately, but it is easy to prove that because the position coordinates are independent of each other we need d independent constraints in order to cancel the integrand in (2.33). As the string has two endpoints we need to impose 2d independent constraints simultaneously, 2.4. STRING THEORY 45

which for simplicity we will take them all equal to zero as it is always boost to the reference frame where all of the terms in one of the extrema of the string vanish. Consequently there are two possibilities for each of the 2d terms to independently i i cancel: either ∂σX = 0 or δX = 0.

i If we take ∂σX = 0, that is, if we impose that the speed at the end of the string has to be zero, we say that we are imposing Neumann boundary conditions for the i coordinate labeled by i at that endpoint, while choosing δσX = 0 amounts to fixing the string in place in the direction of that coordinate and the choice is referred to as a Dirichlet boundary condition. For the time coordinate (i = 0) we will usually choose Neumann boundary conditions, as otherwise we would be fixing the string to a specific point in time; however, there are some non-perturturbative solutions of String theory referred to as instantons for which we will choose Dirichlet boundary conditions in time (Ammon and Erdmenger, 2015, p. 147).

i When we choose δσX = 0 the energy-momentum tensor will not be automatically i conserved at the end of the string, as the terms of the form ∂σX in equations (2.32) and (2.31) for those directions will not necessarily vanish. In order to make the equations of motion of hab hold for this case we will need to introduce additional objects to the theory so that the string couples to them and transfers the excess momenta while keeping the string endpoint fixed in place, as required by Dirichlet boundary conditions (Tong, 2009, p. 51). Because it is always possible to choose Neumann boundary conditions for some spacetime directions and Dirichlet for others we conclude that the new objects that the string couple to have to be extended instead of point-like, and they must allow the string endpoint to move freely along their surface. Thus, if we choose Dirichlet boundary conditions in n of the d spacetime directions we will need to attach the string to a (d − n)-dimensional extended object that we will call D-branes, where the D stands for Dirichlet, or sometimes Dp-branes, with p being the spatial dimension of the brane: p = d − n − 1.A D0-brane is a particle, a D1-brane is a massive string or D-string, a D2-brane is a membrane and so on, with the surprising addition of the D(−1)-brane, which is an instanton. A p-dimensional brane explicitly breaks the Poincaré symmetry of the target space, assumed to be ISO(d−1,1), down to ISO(p,1 )× ISO(d−p−1). If a theory contains both open and closed strings then the branes must be dynamical objects, because as we will see these descriptions automatically include gravity, and in a theory of gravity we cannot have rigid objects (Tong, 2009, p. 57). Thus we see that String theories automatically generate extended dynamical objects of higher dimensions, resulting in more complex dynamics than what their action initially suggested; these objects are non-linear and they do not appear when studying perturbation theory. Once we have found the classical solutions we can proceed to quantize them by 46 CHAPTER 2. THEORETICAL BACKGROUND

Figure 2.11: Pictorial representation of a string attached to a D-brane. Credit: David Tong promoting all string parameters to operators (it would have also been possible to directly start from a quantum system and then solving it, a method usually referred to as covariant quantization). When we quantize a string we obtain that the vibration modes can be associated with particles with particular quantum numbers; some of these numbers are the momenta of the string, from which we obtain the mass for the particle state associated with that vibration mode. For the bosonic string we find the surprising result that the ground state has negative mass when the dimension of the target space is larger than 2, both for open and for closed strings, and as a consequence it is what is usually referred to as a tachyonic particle. These type of states appear to violate causality when we let them interact with other non-tachyonic states, but it is possible to find a more physical interpretation instead, as described in Tong (2009, p. 41). There it is shown that tachyons naturally appear when we expand a theory around an unstable vacuum for which the second derivative of the potential energy is negative (as would happen if we expand around a maximum of the potential instead of a minimum, for example). We can then reach the conclusion that we are trying to expand bosonic string theory around an unstable point; at the moment it is not known if these theories have another vacuum that is stable. The excited states of the theory, corresponding to higher vibration modes of the string, increase in mass linearly with the mode number for a configuration with a given number of generation operators. The first excited states fit into a representation of SO(d − 2), which is the little group of the Poincaré space SO(d − 1, 1); for the theory to be consistent we must then require that they are massless, which requires the target space to have 26 dimensions Tong (2009, p. 41). This choice for the dimension of spacetime is consistent for all states of the theory because all other excited solutions fit into some representation of SO(25), as massive states should. The first excited state of closed strings is a massless spin 2 particle. As we have seen, the Polyakov action for String theory naturally describes objects with arbitrary worldsheets, more complicated in general than those of the fundamental 2.4. STRING THEORY 47 open and closed strings, whose worldsheets are homeomorphic to a strip and a cylinder respectively; the objects with more complicated worldsheets than these can be interpreted as a group of fundamental strings interacting.

Figure 2.12: Pictorial representation of two closed strings interacting. Credit: slimy.com

The easiest approach to study these solutions to String theory is to write the path integral of the theory and integrate over all possible worldsheet geometries allowed in the theory, which are denoted by Σ. We can obtain different String theories depending on which worldsheet geometries we sum over, obtaining four combinations that differ on whether they have boundaries or not and whether they are orientable (Polchinski, 1998a, p. 80). The total space of possible geometries, Σ, can be further decomposed into a sum of integrals over worldsheets with the same genus g, which we will write as Σg. The partition function of the theory in question will then be

Z ∞ Z iS µ X iS µ Z = e Dhab DX = e Dhab DX (2.34) Σ g=0 Σg

where S is the Polyakov action in (2.27). The complex exponential makes the calculations more difficult, but it is possible to Wick-rotate the time direction ofthe target spacetime (and as a consequence, also the time direction of the worldsheet) to obtain an equivalent Euclidean theory for which the integral over the metrics is better behaved. The minkowskian action transforms under Wick rotations as iS → −SE, where

SE = S + λχ, λ being a free parameter of the theory and the constant χ is the Euler characteristic of the worldsheet manifold being summed over, which is defined as an integral involving the Ricci scalar and the geodesic curvature of the manifold. The Euler characteristic is related to the genus of the worldsheet, as χ = 2 − 2g, so we can rewrite the partition function in (2.34) as a ponderated sum where manifolds with higher genus contribute smaller amounts 48 CHAPTER 2. THEORETICAL BACKGROUND

∞ Z X −λ(2−2g) −S µ Z = e e Dhab DX (2.35) g=0 Σg

The worldsheets in Σ0 are referred to as planar, and they play an important role in the AdS/CFT correspondence. Adding a handle to a given worldsheet increases its genus by 1, reducing its weight in the series expansion, and can be interpreted as adding to a diagram the emission and reabsortion of a closed string. This interaction process involving a closed string adds a constant equal to e2λ to the original term in the partition function, and thus we say λ that the coupling to closed strings is given by gs = e . Interacting theories of open strings necessarily contain closed strings, but the reverse is not true (Polchinski, 1998a, p. 81). When studying a Quantum String theory perturbatively there are two perturbation parameters around which we can expand: the interaction coupling gs and the string 0 length ls, or equivalently the parameter α (Dibitetto, 2012, p. 20). The expansions on α0 are purely string-like and they have no analogue in QFT.

2.4.1 Superstring theory

Introducing supersymmetry in the worldsheet eliminates the tachyonic states from the spectrum of the string. We formulate this theory by supplementing the action in equation (2.26) with extra fermionic worldsheet degrees of freedom such that the complete action is invariant under some mixing of fermions and bosons. The new supersymmetric action can be found in reference Dibitetto (2012, p. 17) and it reads

Z √ T ab µ ¯ µ a 2 S = − −h h ∂aX ∂bXµ − i ψ ρ ∂aψµ d σ (2.36) 2 When we quantize this action the Weyl gauge symmetry of the worldsheet may become anomalous, rendering the theory inconsistent. A possible solution is to take the target space to have 10 dimension, in which case the conformal trace anomaly vanishes; superstring theories in this dimension are called critical string theories (Tong, 2009, p. 48). It is possible to study string theories in other dimensions if we manage to cancel the anomaly by other means, an approach referred to as non-critical string theory. Even in 10-dimensional string theories there are still tachyonic solutions in the string spectrum, but in this case it is possible to project out these solutions consistently by using the so-called GSO projection (for Gliozzi, Scherk and Olive), which retains only a subset of the operators of the CFT while guaranteeing that this set is closed under their Operator Product Expansion and are mutually local and respect modular invariance (Dibitetto, 2012, p. 19). This process also guarantees that there are the same number of 2.4. STRING THEORY 49 fermionic and bosonic states in each mass level (Ammon and Erdmenger, 2015, p. 159). There are only five consistent superstring theories that are inequivalent: (Tong, 2009, p. 49) • Type I: It contains open and closed strings and is N = 1 supersymmetric. Its vector supermultiplet has a SO(32) gauge symmetry group.

• Type II: It has both left and right-moving worldsheet fermions, N = 2 supersymmetry and SO(8) gauge group. There are two subtypes, IIA in which both fermion groups have the same chirality and IIB in which they have different chiralities.

• Heterotic: It combines left-moving bosons with right-moving fermions, and it has N = 1 supersymmetry. There are two gauge groups that are anomaly free, SO(8)

and E8 × E8. It has been proven that the 5 theories are related by suitable transformations of their degrees of freedom called dualities, some examples of which are the S, the T and the U dualities. The five theories fit together as different limits of a non-perturbative 11-dimensional framework called the M-theory.

Figure 2.13: String theory dualities. Modified from originals found in quantamagazine. org and physics.stackexchange.com

As we have seen, superstring theories are usually studied in 10 dimensions, but in order to use them to describe phenomena in our universe, where only 4 spacetime dimensions are apparent, we need some procedure to reduce or compactify the extra 6. 50 CHAPTER 2. THEORETICAL BACKGROUND 2.5 Supergravity (SUGRA)

Supergravities are Quantum Field Theories with gauged (also called local) supersymmetry. These theories automatically include a massless particle of spin 2, the so-called graviton, and therefore they must also describe gravity (Nastase, 2007, p. 34). As mentioned in section 2.2, theories with massless particles with spin greater than two automatically need to also contain superpartners of arbitrarily high spin, or else they become inconsistent. We are interested in theories with only a finite number of fundamental fields in order to keep our calculations tractable, in which case wemust require that the number of supercharges in such a supergravity theory to be lower or equal to 32. In combination with the number of spinorial components of the Dirac group in different dimensions, this condition implies that the highest dimensional spacetime with Lorentzian signature in which we can construct a consistent supergravity theory has 10 space dimensions and a time one. To study supergravity we can use one of two formalisms: in the second order formalism the spin connection of the theory, ω, is taken to be a non-dynamical object that is subject to the condition that the torsion of the theory has to be equal to zero; the equations of motion obtained from this formalism are second order partial differential equations, hence the name. Instead, by promoting ω to be an independent gauge field we need to modify the action with extra terms, but in exchange we can obtain first order differential equations of motion, and so this alternative approach is called the first order formalism of supergravity. A more in depth description can be found in Nastase (2007, p. 36). Supergravities find their main application as the massless, tree-level approximation of String theories, where all the dynamical objects found in the latter are also present in supergravity. In fact, whenever we try to approximate strings by point particles the resulting QFT will necessarily be a supergravity theory (Ammon and Erdmenger, 2015, p. 160). Nevertheless, it is possible that supergravities may accept UV-completions different from those offered by String theory, so studying supergravities can beapath towards a Quantum gravity theory regardless of the validity of String theory. As the low energy limit of String theory, supergravities inherit the universality principle hypothesized to be true in String theory. The string universality conjecture claims that all consistent Quantum Field Theories coupled to gravity must be realized as solutions to String theory; so far the conjecture has been found to be true for theories in 10 and 11 dimensions when considering only the matter content and the symmetries of the theories (Taylor, 2011, p. 4). If this principle turned out to be true for all possible dimensions, an immediate consequence would be that all supergravity theories in any dimension will necessarily have their origin in String theory. Some supergravities can also be obtained as a compactification of the unique 11-dimensional supergravity 2.5. SUPERGRAVITY (SUGRA) 51 on an appropriate background, possibly in the neighborhood space-filling branes or orbifolds, but it is suspected that some lower-dimensional supergravities may contain more information on the winding modes present in String theory than the 11-dimensional theory does, and thus they could not be obtained as compactifications of it. Beyond the easier examples of supergravity theories, in which the gravitinos are not charged with respect to any other interactions, we also have the so-called gauged supergravities for which the R-symmetry is gauged and the gravitinos become charged with respect to the gauge fields. In order to preserve supersymmetry most gauged supergravities require a potential for the scalar fields present in the theory or a potential involving the cosmological constant if the supergravity in question does not contain any scalar degrees of freedom. A gauged supergravity can be obtained from its ungauged counterpart by promoting a subgroup of its bosonic symmetry group to a local symmetry, and sometimes we will also need to add masses for the fermions, topological terms and a scalar potential not present in the ungauged theory in order to preserve supersymmetry (Dibitetto, 2012, p. 43). There is a precise correspondence between gaugings and generalized fluxes through the embedding tensor formalism, as reviewed in Dibitetto (2012), so a gauged supergravity theory automatically determines the background flux and the topology of the compactified dimensions. Besides gauging, we can also obtain gauged supergravities by compactifying a higher-dimensional supergravity theory on the appropriate background. The background spacetime in which we compactify the theory in order to dimensionally reduce it plays a big role in the dynamics of the subsequent lower-dimensional theory. The characteristics of the internal manifold determine if the symmetries present in the original theory are conserved or broken, and they also determine the field content of the new theory and their dynamics. Most compact manifolds have positive spacetime curvature, which means that if our original theory was formulated in flat space the dimensionally reduced model will have negative spacetime curvature; in the case of maximally symmetric spaces we will thus find Anti de-Sitter vacua solutions inthe non-compact dimensions. If we want the lower dimensional supergravity to conserve maximal supersymmetry it is necessary that the holonomy of the connection of the internal manifold, which is a geometrical property that measures how much geometrical information is lost when we parallel transport an object around a closed loop, admit covariantly constant spinors. Examples of manifolds with maximal honolomy are tori, for which we can compatify an ungauged maximal supergravity to produce a lower-dimensional maximally symmetric ungauged supergravity theory, and also coset reduction manifolds, of which a particular example are spheres, which instead give us maximally symmetric gauged supergravities with semisimple gauge groups (Dibitetto, 2012, p. 51). It is also possible to include fluxes in the compactification in order to generate different symmetry groups and possibly 52 CHAPTER 2. THEORETICAL BACKGROUND modify the resulting curvature of the non-compact dimensions. Manifolds with less than maximal holonomy will spontaneously break some of the supersymmetry. An important example of an internal manifold that preserves only some supersymmetry are the Calabi-Yau spaces, which are compact structures that are Ricci flat and admit a complex Kähler metric; squashed spheres also breaksome supersymmetry when we compactify on them, and it is also possible to obtain theories with explicit supersymmetry breaking by adding space-filling branes or O-planes before compactifying. Theories with these internal manifolds admit effective descriptions in terms of gauged supergravities with less than maximal supersymmetry.

Figure 2.14: An example of a Calabi-Yau manifold. Credit: wikimedia.org

Due to the strict constraints imposed on the theory by supersymmetry, it is possible to obtain the equations of motion of its matter fields just by imposing the conservation of supersymmetry (at least those solutions which keep the symmetry unbroken). As the differential equations that we need to solve in this case only involve one derivative itis simpler to find analytic solutions with this method.

2.6 Anti de-Sitter spacetime (AdS)

An anti de-Sitter space is a pseudo-Riemannian spacetime manifold with lorentzian signature and constant negative scalar curvature that is invariant under isometries, and 2.6. ANTI DE-SITTER SPACETIME (ADS) 53 in d-dimensions its symmetry group is given by SO(d−2,2). By including the inversion, a discrete transformation, the group get enlarged to O(d−2,2) (Zaffaroni, 2000, p. 6). It is also possible to study its properties by studying its embedding into a higher dimensional space, in particular AdS can be realized as an hyperboloid embedded in Rd,2.

As described in section 2.1.3, when we study a spacetime the properties that we are interested in the most are how this space behaves under a change of coordinates, which can be rigorously defined as a local diffeomrophism. An isometry is a diffeomorphism that leaves the metric tensor invariant at every point of spacetime, and those manifolds that are invariant under all possible isometries are referred to as maximally symmetric manifolds.

In order to further classify these spaces we must look at the scalar curvature of the space, given by the Ricci scalar, R; it can be proven that for a spacetime to be maximally symmetric this scalar must be the same in every point of the manifold. Depending on the sign of R we have three possible families of maximally symmetric spacetimes: de- Sitter (positive curvature spaces), flat (R = 0) and anti de-Sitter (negative curvature spaces). Within the negative curvature family we can classify all possible AdS manifolds q d(d−1) according to their characteristic length, L, given by L = |R| (note that L > 0). In gravitational theories that reduce to Einstein’s General Relativity at low energies we can assume that the vacuum solutions will be well characterized by Einstein’s Field equations, as the dynamics of spacetime itself become less important when we are situated at the minimum energy point. From equation (2.9) we can see that in the absence of matter, an anti de-Sitter space is obtained when the cosmological constant, Λ, is negative (this is because in a vacuum the Ricci scalar only depends on the value of d the cosmological constant, R = d−2 Λ). Many supergravity theories produce AdS vacua when compactified in a internal manifold with positive curvature, as in those caseswe obtain an effective negative cosmological constant originating from a scalar potential term that takes negative values when the theory is in its lowest energy state.

There are many different functional forms for a metric with constant negative scalar curvature, all of which must necessarily be related to each other by a diffeomorphism; the most common choices of coordinates used to parametrize an AdSd+1 spacetime are the hyper-polar coordinates, the global coordinates and the Poincaré patch coordinates; in the discussion of these metrics we will use the same conventions as in Benini (2018, p. 22). It is worth noting that starting with a theory with lorentzian signature it is possible to obtain an associated theory with euclidean signature by performing a Wick rotation such that t → iτ, and viceversa; this fact allows us to obtain some results for a theory in a given signature by analytic continuation of its counterpart (p. 14 Zaffaroni, 2000; Kaplan, 2018, p. 15). 54 CHAPTER 2. THEORETICAL BACKGROUND

Global coordinates

For an AdSd+1 space with lorentzian signature, the global coordinates are denoted by

(t, ρ, Ωi), with i ranging from 1 to d − 1, ρ ≥ 0 and t ∈ [0, 2π). We can further unwrap the time coordinate, t, by taking the universal cover space instead, in which t ∈ R. The metric written in these coordinates be written following the convention in Freedman and Van Proeyen (2012, p. 492) as

ρ ρ ds2 = − cosh2 dt2 + dρ2 + sinh2 (L2dΩ2) (2.37) L L where L is the characteristic length and L2dΩ2 is the metric of a d − 1-dimensional sphere of radius L that we can write in terms of d − 1 angular coordinates θi as

d−1 i−1 2 X Y 2 2 2 2 2 2 2 2 dΩ = ( sin θj)dθi = dθ1 + sin θ1 dθ2 + ... + sin θ1 · ... · sin θd−2 dθd−1 i=1 j=1

It is often necessary to refer to an individual summand in this expression, so it will simplify our work to define d − 1 new differential forms related to the angular 2 Qi−1 2 2 coordinates θi by dΩi = j=1 sin θj dθi , in terms of which the spherical metric is 2 Pd−1 2 simply dΩ = i=1 dΩi . This formalism is especially useful when embedding an AdS space in a higher-dimensional ambient space, as in that case it comes naturally to Pd 2 introduce an extra coordinate, Ωd, subject to the constraint i=1 Ωi = 1. The boundary of the AdS space in this coordinate system is situated at ρ → ∞, and the universal cover of the global coordinates (when we allow t ∈ R) covers the entire space. We can also find a different but equivalent description of AdS in global coordinates by defining a new coordinate, α, such that L tan α = sinh ρ/L; this coordinate results more useful than ρ when we want to study the boundary of the spacetime, as it has a finite range, α ∈ [0, π/2) and the boundary of the space is now situated at α = π/2, a finite value. In these alternative coordinates, the metric is realized as:

L2 ds2 = (−dt2 + dα2 + sin2 α dΩ2) (2.38) cos2 α

In this coordinate system we can obtain a Penrose diagram for AdSd+1 by dropping the global factor, obtaining thus a conformal representation with a finite range in θ (although it is not possible to also limit t to a finite range while maintaining the conformal structure of the space). As an example, the Penrose diagram for AdS2 can be seen in figure 2.15. In global coordinates a SO(d)×SO(2) subgroup of SO(d, 2) is realized explicitly; this group product is related to the quantization of a d-dimensional CFT on Sd−1 (Benini, 2.6. ANTI DE-SITTER SPACETIME (ADS) 55

Figure 2.15: Penrose diagram for AdS2. Credit: researchgate.com

2018, p. 22).

Hyper-polar coordinates

Another choice of global coordinates is given by the hyper-polar coordinates (t, ρ, Ωi), which span the whole AdS space (Natsuume, 2014, p. 96). They can be obtained from those in equation (2.37) by taking r/L = sinh ρ/L, so that the metric becomes

r2 1 ds2 = − 1 + dt2 + dr2 + r2dΩ2 (2.39) L2 1 + r2/L2 and the boundary of the AdS space is found at r → ∞, as explained in Freedman and Van Proeyen (2012, p. 492). This metric is specially easy to modify to describe asymptotically AdS spacetimes that include singularities.

Poincaré patch

The group of coordinate systems that presents an explicit d-dimensional Minkowsky submanifold are referred to as Poincaré patch coordinates. In d + 1 dimensions we can group these coordinates as (xµ, z), where z is referred to as the radial coordinate, which can be chosen in different but equivalent ways, and the parameter µ ranges from 0 to µ 0 0 µ d − 1, such that x = (x , ~x) with x = t. These x coordinates parametrize the Mkwd subspace, and we can simplify our AdS metric expression by grouping them together as 56 CHAPTER 2. THEORETICAL BACKGROUND ds2 = dxµdx = −dt2 + d~x2 z Mkwd µ . We will start choosing the radial coordinate so that z > 0, obtaining the following metric:

dz2 + ds2 ds2 = Mkwd (2.40) (z/L)2 In a spacetime with lorentzian signature, the Poincaré patch coordinates do not cover the whole space, as can be seen by putting them in correspondence with global coordinates; a graphic example for the case of AdS3 can be seen in figure 2.16. The boundary of the space is situated at the point in the radial coordinate at which the size of the d dimensional Minkowsky slice becomes infinitely large, which happens at z = 0; however, the metric in (2.40) also presents a horizon in z → ∞, where the Minkowsky slice grows infinitely large.

Figure 2.16: Diagramatic representation of the Poincaré Patch for AdS3, corresponding with the shaded area in the figure. Credit: Freedman and Van Proeyen (2012, p. 494)

Other representations of AdSd+1 in Poincaré patch coordinates can be obtained by performing a coordinate change for the radial coordinate z while leaving the xµ unchanged, for example by taking z/L = L/u (Ammon and Erdmenger, 2015, p. 535), with u > 0, for which the metric reads

du2 ds2 = + (u/L)2 ds2 (2.41) (u/L)2 Mkwd 2.6. ANTI DE-SITTER SPACETIME (ADS) 57

and now the boundary is situated at u → ∞ and the horizon is at u = 0. Another possibility is to take r/L = − log z/L or equivalently z/L = e−r/L, with −∞ < r < ∞. In this coordinate system the boundary is at r → ∞ while the horizon is situated at r → −∞, and the metric can be written as:

ds2 = dr2 + e2r/Lds2 Mkwd (2.42) The metric expressed in the Poincaré patch coordinates has an explicit SO(d − 1, 1)× SO(1, 1) symmetry group, a proper subgroup of SO(d, 2). This symmetry structure is equivalent to the one obtained after radial quantization of a d-dimensional CFT, where SO(1, 1) is related to the dilations of the CFT (Benini, 2018, p. 22).

Classical solutions for a free theory on an Anti de-Sitter vacuum

It is interesting to study free solutions in an AdS background, as many calculations in effective quantum gravity theories rely on them as a first order approximation. Fur- thermore, when studying the AdS/CFT correspondence we will see that free theory calculations in AdS completely define 2-point and 3-point functions of the equivalent CFT. In a non-quantum theory in an AdS space, we can write the action for a classical particle in terms of its geodesic equation of motion parametrized by a parameter λ (which for a massive particle will be its proper time, τ) r Z Z dXµ(λ) dXν(λ) S = m dτ = m gµν(X(λ)) dλ (2.43) dλ dλ where for the cases we are interested in we will restrict the action to an AdS space by choosing an appropriate functional form for gµν. Due to the square root some of the calculations that we are interested in are difficult to perform, and furthermore we would also encounter problems if we want to quantize this action. As an alternative to this formulation, we can rewrite the action principle in terms of a non-dynamical Lagrange multiplier, α, whose equations of motion are satisfied trivially and take us back to our original action. Another additional benefit is that this alternative action is simpler to use in the case of massless particles. The functional form of this alternative expression is

Z µ ν 1 dX (λ) dX (λ) α 2 S = gµν(X(λ)) + m dλ (2.44) 2α dλ dλ 2 The classical solutions for this action can be calculated and written in closed form, as for example in Kaplan (2018, p. 19). After fixing our origin of coordinates wefind that all solutions describe periodic trajectories around the origin, all of the orbits having 58 CHAPTER 2. THEORETICAL BACKGROUND exactly the same period. This does not contravene the fact that AdS is a maximally symmetric space, and thus the description of the solutions must the same regardless of the origin of coordinates that we choose, as studying the transformation of the solutions under a diffeomorphism we would find that the trajectories will always orbit aroundthe origin of coordinates that we choose, regardless of the particular choice (Kaplan, 2018, p. 20). Furthermore, and opposite to what happens in flat space, the solutions for some particles with negative mass are not tachyonic, and instead can have positive total energy as a result of the positive contribution from the metric connection (sometimes this effect is referred to as a gravitational potential). The minimal negative massvalue, m, for which it is still possible to obtain positive-energy solutions can be obtained as a function of the curvature of the space and it is referred to as the Breitenlohner-Freedman d2 2 2 bound. For scalar particles in AdSd+1 this bound is given by − 4 ≤ m L ≤ 0 (Zaffaroni, 2000, p. 28).

Quantum Field Theory in an Anti de-Sitter background

In a Field Theory in an static AdSd+1 background we can promote the coordinates of worldline of the particle that appear in (2.44) to field-valued parameters defined throughout the whole spacetime, φ = φ(x), obtaining the following action for a scalar field Z 1 µν 1 2 2 √ d+1 S = g ∇µφ∇νφ − m φ −g d x (2.45) 2 2 If we take gµν to be a non-dynamical object that describes a static AdS background, we can calculate the field solutions for φ by solving the Euler-Lagrange equations. The general solution for the scalar field φ can be written as a function of three quantum P numbers, n, ` and J such that φ = n,`,J φ n,`,J ; we will write the explicit coordinate- dependent solutions in the global coordinates defined in (2.38), for which following the process in Kaplan (2018, p. 25) we obtain

Y (Ω) d `,J −iEn,`t ` ∆ 2 φ n,`,J (t, ρ, Ω) = e sin ρ cos ρ 2F1(−n, ∆ + ` + n, ` + ; sin ρ) (2.46) N∆,n,` 2

2 2 In this equation ∆ is calculated using the relation L m = ∆(∆ − d); Y`,J (Ω) are the spherical harmonics in d − 1 dimensions, which codify the angular dependence of the solution; the energy, E, is independent of the angular parameters codified by J, and thus it is given by En,` = ∆ + 2n + `; the function 2F1(a, b, c; z) is the ordinary hypergeometric function, and finally N∆,n,` is a normalization factor that will simplify the interpretation of the field as a probability amplitude. 2.6. ANTI DE-SITTER SPACETIME (ADS) 59

The ground state has quantum numbers n = 0, ` = 0 and J = 0, for which we find 2 En=0,`=0 = ∆ and 2F1(0, 0, 0; sin α) = 1, so following Kaplan (2018, p. 26), the state can be simplified to

i∆t ∆ φ 0(t, α, Ω) = e cos α (2.47)

When we quantize the scalar action in (2.45), obtaining thus a Quantum Field Theory, the functions defined in (2.46) become the wavefunction prefactor that accompanies the † creation and annihilation operators of the QFT, an,` and an,`. The resulting quantum field, which we will denote as Φ, is then simply obtained as in reference Kaplan (2018, p. 31), namely

X ∗ † Φ(t, ρ, Ω) = φ n,`,J (t, ρ, Ω) an,` + φ n,`,J (t, ρ, Ω) an,` (2.48) n,`,J

with φ n,`,J given in equation (2.46). Due to the Breitenlohner-Freedman inequality it is possible to have vacuum states for our QFT that are local maxima of the Hamiltonian in addition to local minima. In flat spaces, an expansion around these vacuum states would result in tachyonic states of negative mass, as we mentioned in section 2.4, but these states are non-tachyonic in a QFT with AdS vacua, and thus the theory is well defined.

Asymptotically Anti de-Sitter spacetimes

A spacetime is said to be asymptotically AdS if the boundary structure of its Penrose diagram matches that of AdS.

In this thesis we will be particularly interested in those asymptotically AdSd+1 spacetimes that break the full (d + 1)-dimensional Poincaré symmetry in the bulk while still respecting a d-dimensional Poincaré subgroup. These kind of spacetimes are more naturally described in Poincaré patch coordinates, as they automatically single-out a d-dimensional flat subspace and a radial coordinate r that let us approach the boundary. The most general metric for this kind of spaces can thus be written as

ds2 = L2 e2A(r) dr2 + e2B(r) ds2 Mkwd (2.49) subject to the additional condition that at the boundary the curvature of the spacetime must be negative. By including the unknown real functions A(r) and B(r) as arguments of an exponential we guarantee that the sign of the metric remains unchanged throughout the spacetime.

The boundary of this asymptotically AdS space will be situated at the point r∗ for which the size of the Minkowsky slice of the metric grows infinitely large, that is, 60 CHAPTER 2. THEORETICAL BACKGROUND

limr→r∗ B(r) → ∞. We can calculate the Ricci scalar for the metric in (2.49) in d + 1 dimensions, for which we obtain

! d dB(r)2 d2B(r) dA(r) dB(r) R(r) = −e−2A(r) (d + 1) + 2 − 2 (2.50) L2 dr dr2 dr dr

which must be negative when evaluated at the boundary, r = r∗. It turns out that the function A(r) is redundant, as it always possible to redefine the radial coordinate such that its prefactor is exactly equal to one without changing the ∂p(r) A(r) underlying physics by choosing a new coordinate p = p(r) such that | ∂r | = e , or equivalently, assuming that we want to preserve the orientation of the radial coordinate, p(r) = R eA(r)dr. From its definition it is clear that p(r) is a monotonically increasing function of r, as eA(r) is non-negative, and thus its inverse r(p) is defined and well-behaved. In terms of the new coordinate p the metric is simply

ds2 = L2 dp2 + e2B(r(p)) ds2 = L2 dp2 + e2F (p) ds2 Mkwd Mkwd (2.51)

And in this case the Ricci scalar simplifies to

! d dF (p)2 d2F (p) R(p) = − (d + 1) + 2 (2.52) L2 dp dp2

AdS vacuum in supergravity theories

Some supergravity theories produce AdS vacua when compactified with specific internal spaces, as for example type IIB SUGRA on S5, or also the 11-dimensional supergravity theory, which produces maximally supersymmetric AdS vacua when compactified in either S4 or S7. AdS vacua require the theory to have a scalar potential for the scalar fields (which means that theory is in fact a gauged supergravity) which musttake negative values for the vacuum, so all supergravities with AdS solutions are in fact gauged supergravities. To find these solutions we will in general look for local minima of the Hamiltonian for which the scalar potential is negative (and consequently the spacetime will have negative cosmological constant), but it is also possible to have stable AdS spaces that are local maxima instead of minima as long as the states obtained in the series expansion around that point do not violate the Breitenlohner-Freedman bound. 2.7. ANTI DE-SITTER/CONFORMAL FIELD THEORY CORRESPONDENCE 61 2.7 Anti de-Sitter/Conformal Field Theory correspondence

The Anti de-Sitter/Conformal Field Theory correspondence, also known as the Gauge/Grav- ity duality, is a conjecture that postulates the existence of a non-perturbative duality linking a theory of quantum gravity in an AdS spacetime with a lower-dimensional Con- formal Field Theory without gravity, which can be interpreted to live on the boundary of the AdS space. According to Kaplan (2018, p. 4), the formal statement of the conjecture claims that both theories must have the same Hilbert space, which implies that both descriptions must be dynamically equivalent. Additionally, the global symmetries in the CFT side have to match the gauge symmetries in the AdS description (Zaffaroni, 2000, p. 24) Every Conformal Field Theory is hypothesized to be a suitable description for the boundary of some quantum gravity theory in an AdS space, but only those CFTs for which the central charges characterizing the conformal anomalies (which are only present in even dimensions) are equal, namely a = c, can have a weakly coupled limit in the gravity theory side that can be well approximated by a supergravity theory. The dual of a CFT with non-matching central charges will necessarily need a non-perturbative description in terms of a complete and background-independent Quantum String theory (Zaffaroni, 2000, p. 40). In this thesis we will restrict ourselves to CFTs whosedual is well described by a supergravity, and thus we must take a = c and also restrict the maximum spin of the massless states present in the conformal theory to be equal or lower than 2 for the reasons described in sections 2.2 and 2.5. The correspondence was first suggested by Juan Maldacena in 1997, and ithas since received a great deal of attention for its potential not only to allow us discover new properties that quantum gravity theories need to have, but also for the powerful computational techniques that it provides, as it helps us translate many non-perturbative calculations that appear in supergravities and CFTs into equivalent problems that can be studied perturbatively. Most physicists believe that the conjecture is probably true even though a formal proof has not been found yet, and the fact that no counterexamples have been found in the last 20 years despite the great deal of research that has focused on the correspondence certainly gives us hope. Some of the particular cases of the correspondence that have received the most 5 attention are: type IIB string theory living on AdS5 × S and its dual, N = 4 super 4 Yang-Mills in 4 dimensions; M-theory on AdS7 × S , equivalent to the so-called (2,0)- 7 theory in 6D, and M-theory on AdS4 × S , which relates to ABJM Superconformal Field Theory in 3 dimensions. The gauge-gravity duality is a particular realization of the holographic principle, 62 CHAPTER 2. THEORETICAL BACKGROUND which states that in a theory of quantum gravity all the information contained in a d- dimensional volume is in fact stored on its (d − 1)-dimensional boundary. The rationale is the same as that for the proposed solution for the black hole information paradox that we explained in section 2.1.3. According to Ammon and Erdmenger (2015, p. 1), the reason that asymptotic Anti de-Sitter spacetimes have received far more attention than any other possible geometry of spacetime is because the boundary of AdS spaces is especially easy to define and study.

Figure 2.17: AdS/CFT correspondence. Credit: philosophy-of-cosmology.ox.ac.uk

As we mentioned in the Introduction we can distinguish three limits in the parameter space in which we could postulate that the AdS/CFT correspondence has to hold, obtaining this way three distinctive levels of restriction for the hypothesis. In the Quantum gravity side, usually given by String theory, the relevant parameters are the string coupling, gs, and the ratio of the characteristic length of the string, ls, to the curvature of the target spacetime in that region, L. The strongest form of the correspondence states that the gauge/gravity duality must hold for any values of gs and ls/L, in which case the parameter space spans the whole domain of a Quantum String theory; we obtain the strong formulation if we assume that the duality only has to hold when gs → 0, that is, for Classical String theory; finally, the weak form only needs the correspondence to hold in the supergravity limit, when both gs → 0 and ls/L 1 (Ammon and Erdmenger, 2015, p. 182). In order to define the duality we need a map relating the observables inthe bulk of the Quantum gravity theory with those in the boundary; this map is usually referred to as the holographic dictionary of the correspondence, and can be stated either as a equivalence between CFT operators and AdS fields or as a relation between the generating funtionals of both theories. The strongest formulation of the correspondence establishes that the 2.7. ANTI DE-SITTER/CONFORMAL FIELD THEORY CORRESPONDENCE 63 source of a conformal operator O in the CFT side, which we will call h(0)(x) (here we are obviating the possible index structure, which must be included in the case of non-scalar objects), must be the boundary value of a 5-dimensional field, h(x, z), that holds the equations of motion of the AdS theory defined on the bulk, and their relation is given by lim h(x, z) = h(0)(x) f(z), where f(z) provides the correct asymptotic behavior of z→0 the field with respect to z. We can write this statement as a relationship between the generating functional in the CFT side, WCFT(h(0)), which provides the expectation values of the operators in the boundary (including composite operators), and the on- shell partition function of the AdS Quantum gravity theory, ZAdS, which codifies the dynamics in the bulk. Explicitly this relation reads

iW (h (x)) D i R O(x) h (x) ddxE D iS (h(x,z))E CFTd (0) (0) AdSd+1 e = e = e = ZAdSd+1 (h(x, z)) CFTd AdSd+1 (2.53)

with the additional condition that lim h(x, z) = h(0)(x) f(z). z→0 In the case of a Quantum String theory the partition function ZAdS is a very complicated object that at this time is not known explicitly, but in the supergravity limit

(gs → 0 and ls/L 1) this functional is dominated by the saddle points, that is, the classical solutions, and it can be approximated by the Classical supergravity partition

iSSUGRA+O(gs) function, ZAdS 'ZSUGRA + O(ls/L) ' e + O(ls/L). When looking at the correspondence between the observables in both theories we find that the AdS local fields h(x, z) that appear in the Quantum gravity action are the dual of the boundary CFT operators O(x) they couple with at the boundary. Both of these objects have to transform in the same irreducible representation of the global symmetry group, so they share the same quantum numbers, and they have to be observables of the theory and thus gauge invariant; they must then be represented by composite operators instead of fundamental gauge-dependent degrees of freedom (Ammon and Erdmenger, 2015, p. 190). The holographic dictionary identifies the energy-momentum tensor in CFT,

Tµν, with AdS metric fluctuations described by hµν, while the R-symmetry current Aµ is dual to a gauge field fluctuation in the five-dimensional AdS supergravity obtained from the Kaluza–Klein reduction, Jµ, and, for supersymmetric theories, the superconformal µ µ currents Sα are related to the gravitinos ψα; when the AdS theory has a dilaton field, φ, µν it is related to the gauge-invariant field strength Tr(F Fµν). As we said in the beginning of the section, some CFT operators may not have a corresponding supergravity dual as they would require a String theory description with no low-energy analogue (Ammon and Erdmenger, 2015, p. 194). Under a dilatation transformation we see that the radial coordinate in the AdS space, z, transforms in the same way as the energy scale of the operators in the CFT. We thus find that there is also an equivalence between z and E, which is even more clear when we 64 CHAPTER 2. THEORETICAL BACKGROUND integrate over the energy scale of the CFT: the contribution of the AdS modes situated at a certain coordinate z0 dominate the integral in the neighborhood of its energy dual,

µ0. Intuitively we can think that, as the metric diverges at the boundary, it is more energetically favorable for fields to penetrate more into the bulk, and as a consequence we obtain a correspondence relating the IR regime of the CFT with the horizon of the AdS space, and the UV regime with the boundary (Zaffaroni, 2000, p. 24). We can find another relationship by studying how the objects of each theory behave under a rescaling of the boundary. In AdS the mass of a field determines how it behaves asymptotically as it approaches the boundary, and consequently it governs how it transforms under scaling. In CFT this property is associated with the scaling dimension of the operator, ∆, so we conclude that there must be a relationship between the mass of an AdS field and the scaling dimension of its dual. The exact relation between these parameters depends both on the spin of the objects and on which dimension we are working in. We have written the formulas for fields of spin equal to or less than 2in table 2.1.

Spin-0 scalar m2L2 = ∆(∆ − d) Spin-1/2 gaugino |m|L = ∆ − d/2 Spin-1 vector m2L2 = (∆ − 1)(∆ − d + 1) Spin-3/2 gravitino |m|L = ∆ − d/2 Massive spin-2 m2L2 = ∆(∆ − d) Massless spin-2 graviton m2L2 = 0, ∆ = d p-form m2L2 = (∆ − p)(∆ + p − d) Rank s symmetric traceless tensor m2L2 = (∆ + s − 2)(∆ − s + 2 − d)

Table 2.1: Relationship between the mass of an AdSd+1 local field and the scaling dimension of its CFTd dual operator. Adapted from table 5.2 in Ammon and Erdmenger (2015, p. 193)

From table 2.1 we see that for a scalar field with mass m living in AdSd+1 we obtain two different solutions for the scaling dimension associated with each possible mass.We q d d2 2 2 will label them as ∆+ and ∆−, and they depend on the mass as ∆± = 2 ± 4 + m L . Due to the Breitenlohner-Freedman bound m2L2 ≥ −d2/4, which holds for any stable scalar solution in AdS, the scaling dimension will always be a real number and ∆+ will d always satisfy the unitarity condition for scalars in a CFT, given by ∆ ≥ 2 − 1 (Ammon and Erdmenger, 2015, p. 107). We will now try to find the conformal dual of the AdS scalar. From the properties of fields in AdS reviewed in section 2.6, and in particular from equation (2.46), we know that near the boundary a scalar field can be expanded as

∆+ ∆− φ(x, z) ' φ−(x) z + φ+(x) z + g(x, z) (2.54) 2.7. ANTI DE-SITTER/CONFORMAL FIELD THEORY CORRESPONDENCE 65

where the contribution from g(x, z) is subleading as z → 0 (Ammon and Erdmenger, 2015, p. 193).

∆− As ∆+ ≥ ∆− = d − ∆+, the φ+ z factor will be the leading-order term when z → 0, which in the notation that we have established before gives us the identification f(z) = z∆− ). We can see that if we define a boundary condition for the AdS field

φ(x, z) we will be fixing the value of φ+(x) (this is equivalent to choosing Dirichlet boundary conditions for the field φ); when we choose this boundary condition we will rename φ+ as φ(0), where the asymptotic behavior of the scalar field near the boundary ∆− is lim φ(x, z) = φ(0)(x) f(z) = φ+ z . The scaling dimension of the corresponding CFT z→0 operator dual, O, will be ∆+, its vacuum expectation value will be equal to φ−, and

φ(0) = φ+ will act as its source (Ammon and Erdmenger, 2015, p. 194). On top of this identification between φ and O∆+ , we can also find a different dual CFT operator to φ by fixing Neumann boundary conditions; this choice of boundary condition fixesthe value of φ−(x) instead, which becomes the source for the new operator, whose the scaling dimension will be equal to ∆− and its vacuum expectation value will be φ+. If we want the Conformal Field Theory to remain unitary it is only possible to take Neumann this boundary conditions when 1 − d2/4 ≥ m2L2 ≥ −d2/4, otherwise we will violate the unitarity bound.

Figure 2.18: Relationship between the mass of a scalar field in AdS and the scaling dimension of its dual for unitary operators. Credit: Benini (2018, p. 34)

We can also compute correlation functions in the CFT directly from the on-shell action for the AdS theory (Freedman and Van Proeyen, 2012, p. 542). The n-point correlator is obtained by applying n functional derivatives to the action with respect to their corresponding dual fields 66 CHAPTER 2. THEORETICAL BACKGROUND

n δ SAdS hO1(x1) O2(x2) ...On(xn)i = (2.55) δφ (x ) δφ (x ) . . . δφ (x ) 1 1 2 2 n n φ1=···=φn=0

In the supergravity limit the correlator functions defined in (2.55) can be calculated from tree-level diagrams of the AdS theory, which only require a truncation of the AdS action involving at most n-point interactions. It is useful to develop a pictorial language for these computations analogous to the Feynman diagrams for QFT, which in this case are commonly called Witten diagrams. In figure 2.19 we can see the propagators with less than 4 external sources and less than 2 vertices, which are enough to calculate any 4-point correlator.

Figure 2.19: Examples of Witten diagrams. Credit: Freedman and Van Proeyen (2012, p. 542)

Propagators between an external point and an internal point are called boundary-to- bulk propagators, K(x0−x, z), while those connecting two internal points are referred to as bulk-to-bulk, G(x0 −x, z0 −z). Divergent contributions and local terms can be reabsorbed and regularized, a procedure commonly known as holographic renormalization. If the Quantum gravity solution defines a spacetime that has a maximally symmetric Anti de-Sitter geometry, the operator correlators in its boundary are covariant under conformal transformations and so the dual theory is an exact CFT. However, if the bulk solution behaves like AdS as we approach the boundary but it does not conserve all isometries in its interior, then the conformal symmetry of the dual theory is broken. The asymptotic behavior of the AdS fields as they approach the boundary can act as sources for non-conformal operators that perturb the conformal theory and trigger renormalization-group flows. It is also possible to break the conformal symmetry by inducing non-vanishing one-point functions hO(x)i, in which case the dual CFT defined in the boundary is not in its ground state (Freedman and Van Proeyen, 2012, p. 528). Chapter 3

Calculations

For the Quantum gravity side of the correspondence we are going to choose a half- maximal gauged supergravity theory in 7 dimensions for which we know from the literature that it presents two different AdS vacua (as is shown for example in the analysis by Dibitetto and Petri (2017)), and whose dual is known to be a 6-dimensional SCFT without a Lagrangian formulation. Starting from the Lagrangian and the supersymmetric transformations for the 7-dimensional ungauged supergravity originally derived by Townsend and van Nieuwenhuizen (1989), we will first derive a pseudo-Lagrangian for the gauged theory and then we will find a family of supersymmetric BPS states by imposing that the supersymmetry transformations vanish when evaluated at these solutions. The BPS states describe a domain wall involving only the scalar field and the metric, where the spacetime that this second field describes is only asymptotically Anti de-Sitter, breaking the full 7-dimensional Poincaré group of the theory but preserving a 6-dimensional subgroup in the bulk. Consequently, the parameters of the fields involved in the domain wall can only depend on the radial coordinate of the AdS space and they must asymptotically approach a minimal energy solution at their endpoints. Afterwards we will write an effective one-dimensional Lagrangian for the domain wall from which we will obtain its tension and the effective mass of the scalar BPS in a particular point.

In section 3.2 we will derive the two non-conformal operators dual to the scalar BPS field in Anti de-Sitter space, and we will calculate their scaling dimension intermsof the effective mass of the AdS field. We will show that their one-point functions vanishin the SCFT, and we indicate how to compute their two-point and three-point functions.

Finally, in section 3.3 we will calculate the c-function and the beta function for the Quantum Field Theory described by the renormalization group-flow, we will study their asymptotic behavior and plot them.

67 68 CHAPTER 3. CALCULATIONS

3.1 Supersymmetric domain wall in AdS7

In order to have a Quantum gravity theory with a supergravity description and an AdS vacuum solution we will work with the 7-dimensional half-maximal (N = 1) gauged supergravity restricted to its minimal field content; in this section we will follow part of the analysis carried out in Dibitetto and Petri (2017). From the existing literature on the topic we know that the scalar potential in this theory presents two AdS extrema, one of which conserves the full supersymmetry of the theory while supersymmetry is spontaneously broken for the other. In this thesis we will study an instantonic solution interpolating the supersymmetric vacuum and the asymptotic minimum found when the radial coordinate goes to infinity. Before looking for this domain wall solution we will analyze the origin of our gauged supergravity and we will determine its Lagrangian description. As described in section 2.5, we can obtain a gauged supergravity from its ungauged counterpart by promoting a subgroup of the bosonic symmetry group to be a local symmetry, and for the theory of interest for us we will additionally include mass terms not present in the ungauged theory. There is a precise correspondence between gaugings and generalized fluxes which can be obtained through the embedding tensor formalism, as reviewedin Dibitetto (2012, p. 40), so a gauged supergravity theory automatically determines the background flux and the topology of the compactified dimensions (Dibitetto andPetri, 2017, p. 1). Besides gauging the free theory we can also obtain gauged supergravities by compactifying a higher-dimensional supergravity theory on an appropriate background, as described in section 2.5. Domain walls originate from branes in higher-dimensional theories when we dimensionally-reduce the coordinates transverse to the brane. The Lagrangian for our 7-dimensional minimal content half-maximal (N = 1) gauged supergravity can then be obtained by deforming the Lagrangian of the ungauged theory, as it was first done in Townsend and van Nieuwenhuizen (1989). For this reason we will first review the properties of the ungaged theory and we will write its Lagrangian explicitly before proceeding to gauge it. We will first calculate the number of supercharges present in the theory, whichaswe reviewed in section 2.2 is a function of the number of supersymmetries present in the theory and the number of irreducible components of the spinors in the given dimension. In 7 spacetime dimensions the spin-1/2 irreducible representation of the Poincaré group can be described using a Dirac spinor with real dimension 2bD/2c+1 = 24 = 16; for a theory whose metric has Lorentzian signature this spinorial representation cannot be further decomposed into invariant subgroups, which means that spin-1/2 spinors span an irreducible representation of the Poincaré group with dimension 16. Nevertheless, these spinors can alternatively be written in terms of 8 complex degrees of freedom, in which case we refer to them as symplectic-Majorana spinors (SM) (Dibitetto and Petri, 3.1. SUPERSYMMETRIC DOMAIN WALL IN ADS7 69

2017, p. 28). As explained in section 2.2, if in a 7-dimensional theory with Lorentzian signature we want our field content to solely present states with spin equal or smaller than 2 we need to impose Q ≤ 32, in which case the only supersymmetries allowed are either N = 1 or N = 2. In this work we will focus on the former, the so-called half-maximal theory, which only contains 16 real supercharges. In order to find the matter content present in the theory we will analyze the symmetry group in order to find all the possible multiplets allowed. Following the derivation in Dibitetto (2012, p. 42), we know that the internal symmetry group for a half-maximal + theory in 7 dimensions is G0 = R × SO(3, n), where n is the number of extra vector multiplets added along with the gravity multiplet, and the maximal compact subgroup of G0 is H = SO(3) × SO(n). The fermions in the theory transform under a bosonic symmetry called R-symmetry, which is a subgroup of H; in half-maximal theories we can write H = HR × SO(n), and we can see that HR is a proper subgroup of H when n > 1. In this thesis we will truncate the theory to its minimal field content, corresponding to n = 0, in which case the symmetry group is simply R+× SO(3). In this case the bosonic symmetry group is HR = SO(3), but because we are interested in the projective representations of the group we will work with its universal cover instead, SU(2); in an abuse of language we will simply write HR = SU(2).

The field content of this truncation of the theory comprises the graviton eµν, one i gravitino ψµα, three gauge fields Aµ, a three-form Bµνρ, a gaugino χα and a scalar X. With these ingredients it is straight-forward to write now the bosonic part of the top-form Lagrangian for the ungauged supergravity theory, which following Townsend and van Nieuwenhuizen (1989, p. 42) can be written as:

1 L = R (∗ 1) − 5X−2 (∗ dX) ∧ dX − X4 (∗ F ) ∧ F − ungauged (7) (7) 2 (7) (4) (4) 1 1 (3.1) − X−2 (∗ F i ) ∧ F i + F i ∧ F i ∧ B 2 (7) (2) (2) 2 (2) (2) (3)

where B(3) is the 3-form gauge potential and F(4) is its modified field strength, while

F(2) is the modified field strength for the gauge vector bosons. Starting from higher-dimensional supergravities, this theory can be obtained as the effective description of the non-compact dimensions of type I supergravity when reduced on a T3 internal manifold, or also as an expansion of type mIIA supergravity over an AdS7 × M3 background. From the Lagrangian for the ungauged theory we can finally obtain the gauged supergravity we are interested in by simultaneously gauging the bosonic symmetry SU(2), whose gauge coupling we will denote by g, and also including a Stückelberg- like deformation that introduces a mass parameter h for the 3-form Bµνρ; this second 70 CHAPTER 3. CALCULATIONS deformation can also be interpreted as a topological term, and thus it depends on the characteristics of the spacetime background. When both deformations are turned on at the same time they induce a scalar potential with two AdS extrema, while if any of the parameters is set to zero the potential will not have any extrema (Dibitetto and Petri, 2017, p. 3). All the derivatives involving gauge fields must also be substituted by covariant derivatives. Strictly speaking this gauged supergravity does not have a variational principle, as the Stückelberg-like term can be interpreted as a topological contribution and it requires a self-duality condition for consistency (Dibitetto and Petri, 2017, p. 5). However, we are still able to write a pseudo-action that we must supplement with the additional constraint that the only solutions that are physical are those for which the self-duality condition holds. The bosonic content of said pseudo-Lagrangian can then be obtained by modifying its ungauged counterpart (3.1), resulting in:

1 L = R (∗ 1) − 5X−2 (∗ dX) ∧ dX − X4 (∗ F ) ∧ F − V(X)(∗ 1)− gauged (7) (7) 2 (7) (4) (4) (7) 1 1 − X−2 (∗ F i ) ∧ F i − h F ∧ B + F i ∧ F i ∧ B 2 (7) (2) (2) (4) (3) 2 (2) (2) (3) (3.2)

where the scalar potential induced by the deformations is V(X) = 2h2X−8 − √ 4 2ghX−3 − 2g2X2. Thanks to supersymmetry we can rewrite the potential in terms of 4 2 2 2 a superpotential function as V(X) = (−6 W + X (∂X W ) ), from which we find the 5 √ 1 −4 superpotential to be equal to W = 2 (hX + 2 gX). Equivalently, we could have found the same Lagrangian by either reducing the 11- dimensional supergravity on a squashed S4 or by compactifying type IIA supergravity 3 on a squashed S . In terms of brane configurations, we can obtain7 AdS from the backre- action of a system of NS5-, D6- and D8-branes intersecting along 5 spatial directions and a timelike one and arranged as shown in figure 3.1 in the limit case when the NS5-branes are coincident.

Figure 3.1: Brane. Credit: Bobev et al. (2017, p. 5)

From the Lagrangian in (3.2) it is now straight-forward to derive the equations of motion for each of the degrees of freedom of the theory. In particular, the vacuum 3.1. SUPERSYMMETRIC DOMAIN WALL IN ADS7 71 solutions (that is, those solutions with minimal energy that cannot decay into other states) can be easily found by noticing that all terms in the Hamiltonian will be non- negative except for the scalar potential and thus all fields must have vacuum expectation values equal to zero except for the scalar and the metric tensor, whose values must be those that extremalize V(X). We can find the extrema for the potential by looking for the critical points with respect to the scalar X, that is ∂X V(X) = 0; imposing this condition 10p 2 −2 p5 −7 8 2 we find a local maximum at X+ = 8h g , at which V(X+) = −15 2 g h , and 10p 2 −2 a local minimum at X− = 2h g where the potential has a value of V(X−) = p5 −5 2g8h2. A plot of V(X) for h = g = 1 can be seen in figure 3.2.

Figure 3.2: Scalar potential plotted for h = g = 1

The expansion of the theory around X+ will result in states with negative mass, which in a theory in flat space would mean that the vacuum is unstable and can decay; however, we will see from the equations of motion for the metric tensor eµν that when hXi = X+ the spacetime has constant negative curvature, that is, we obtain an Anti de-Sitter spacetime, and as explained in section 2.6, in an AdS spacetime with curvature L the negative mass states are not tachyonic as long as their mass is above the limit imposed by the Breitenlohner-Freedman bound. If this inequality holds for all the fields that appear when expanding our supergravity around X+ then the vacuum solution with hXi = X+ will be stable.

From the equations of motion for eµν we can obtain a relationship between the value of the scalar potential and the Ricci tensor of the spacetime. Assuming that the vacuum expectation values of all other fields is equal to 0 in order to minimize the energy, the equation of motion for the vacuum state solution will read 72 CHAPTER 3. CALCULATIONS

−2 1 Rµν − 5X ∂µX∂νX − V(X)eµν = 0 (3.3) 5 When the profile of the scalar field is constant throughout spacetime, this equation implies that the spacetime must be an Einstein manifold with constant curvature R = 7/5 V(X); in particular we will be interested in the solution describing a maximally symmetric manifold with constant negative curvature, that is, an Anti de-Sitter spacetime (other Einstein manifolds, for example those containing singularities, would have also been valid solutions to equation (3.3), but we will discard them by properly choosing our initial conditions for the supergravity fields). The value of the potential V(X) at the extrema will determine the characteristic length of the corresponding AdS space, q L(X ) = − 30 which will be equal to vacuum V(Xvacuum) . We can see that the characteristic length increases as the potential becomes smaller, and in particular at X+ and X− the characteristic length is equal to

r √ r 10 4 10 16 L(X+) = 2 and L(X−) = 3 (3.4) g8h2 g8h2

It can be checked that both vacuum solutions turn out to have the exact same energy, which implies that its possible to find topological solutions (solitons) that interpolate between these vacua solutions. We will be interested in finding functions X(r) and eµν(r) that are non-constant in the radial direction r but that preserve the full 6-dimensional Poincaré symmetry in the other directions, which result in an asymptotically Anti de- Sitter space; this guarantees that the dual Conformal Field Theory preserves the full Poincaré invariance, and effectively renders the problem one-dimensional. Additionally we will restrict ourselves to supersymmetric solutions, which are easier to find and solve analytically. The family of solutions with this characteristic describe a scalar BPS flow (for Bogomol’nyi, Prasad and Sommerfield); if we consider all the solutionsin the family together we find that they keep the supersymmetry intact, but if wesingle out a particular solution it will only conserve 8 of the 16 supercharges unbroken. It is easy to find a differential expression for this family of curves by imposing thatthe supersymmetric transformations induced by an arbitrary spinor ζ vanish, which due to the strong constraints imposed by supersymmetry is enough to completely determine the shape of our fields; as these transformations are linear, the differential equations that result from them have to be linear too (Zaffaroni, 2000, p. 48). We will also need to impose the self-duality condition, as the theory’s Lagrangian does not give a full description of its dynamics, but as our solutions only involve the scalar field and the metric this condition is automatically satisfied. The supersymmetric transformations for this theory are found in Townsend and van Nieuwenhuizen (1989, p. 43); as we are only interested in minimum energy solutions, for which all fields except the scalar and 3.1. SUPERSYMMETRIC DOMAIN WALL IN ADS7 73 the metric are equal to zero, most transformations automatically vanish and the others simplify. The non-trivial equations that we obtain are

a a a b 1 a δζ ψµ = ∂µζ + ωµ bζ − W (X)γµ ζ = 0 (3.5) √ 5 a 5 −1 µν a X a δζ χ = X γµe ∂νX ζ − ∂X W (X) ζ = 0 (3.6) 2 5

where we have written the expressions in terms of the superpotential, W (X), which √ 1 −4 is equal to W (X) = 2 (hX + 2 gX).

It is possible to further simplify these equations by writing down an ansatz for the metric and the scalar fields, as we know that the spacetime must be asymptotically AdS and the scalar field can only depend on the radial coordinate r. By substituting these expressions can immediately see that for the Killing spinor that induces the supersymmetric transformations we need functional form that it is proportional to a constant spinor times a function that depends on r. Following Dibitetto and Petri (2017, p. 6) we will write

2 µ ν 2A(r) 2 2B(r) 2 ds7 = eµν(r) dx dx = e dr + e dsMkw6 (3.7) X = X(r) (3.8) a a ζ (r) = Y (r)ζ0 (3.9)

a where we additionally require that the constant spinor ζ0 satisfies the projection a a condition γr ζ0 = ζ0 . The function A(r) in (3.7) is redundant, as it can be eliminated by a proper redefinition of the fields, but it is convenient to keep it as a gauge freedom in order to be able to integrate these differential equations analytically later.

After substituting these ansatzs into the formula for the supersymmetric transformation (3.5) we obtain a differential equation for the metric parameter B(r) from its µ = t component and a differential equation for Y (r) from the µ = r component. Finally, equation (3.6) produces a differential equation for X(r), and we obtain the following system of three coupled differential equations 74 CHAPTER 3. CALCULATIONS

dB(r) 2 = eA(r) W (X(r)) (3.10) dr 5

dX(r) 2 A(r) 2 = − e X (r) ∂X W (X(r)) (3.11) dr 5 dY (r) Y (r) = eA(r) W (X(r)) (3.12) dr 5

The solution to these equations is completely determined once we fix the gauge for A(r), but in most cases it will be complicated to solve this system of equations analytically. In order to give a closed expression for these functions we will choose a gauge in which the differential equation for the scalar field is simply X0 = 1, which according to equa- √ A(r) −2 −1 3 5 tion (3.11) is achieved by setting e = −5/2 X (∂X W ) = −5X / 2gX − 4h .

In this gauge the scalar profile for the domain wall solution is simply X(r) = r + c0, where c0 is a constant that determines which member of the family of BPS solutions we are going to select. For simplicity we will choose c0 = 0, in which case the metric tensor reads (up to reparametrizatons)

3 2 r 2 µ ν 5r 2 r 2 ds = eµν dx dx = √ dr + √ ds (3.13) 2gr5 − 4h 2gr5 − 4h Mkw6

10p which is defined only for r > 8 h2g−2. We can see the behavior of these coefficients for h = g = 1 in figure 3.3.

Figure 3.3: Plot of the√ coefficients of the metric defined in (3.13) for h = g = 1 showing the divergence at r = 10 8

The boundary of this spacetime must be situated at the point of the radial coordinate r for which the coefficient in front of dsMkw6 diverges (which due to physical considerations 3.1. SUPERSYMMETRIC DOMAIN WALL IN ADS7 75

10p 2 −2 we will require to be a positive real number), that is r∗ = 8 h g . Due to the fact that X = r, we find that the boundary is situated at the point for which the scalar potential reaches its maximum, which we called X+ (in different gauges we would find the same result, but with a more contrived expression). We can see then that our solution tends asymptotically to the Anti de-Sitter vacuum for which the scalar potential reaches its maximum. The Ricci scalar for the metric can be calculated from the metric in (3.13) and, as expected, it only depends on the radial coordinate r. √ 6 2g2r10 + 6 2ghr5 − 5h2 R(r) = − (3.14) 5 r8

10p where we have to remember that the metric is only defined for r > 8h2g−2. We find that the Ricci scalar is always negative in the range of validity of our solution,and that there is a curvature singularity at r → ∞.

When we evaluate R(r) at r = r∗ we can obtain the curvature and the characteristic length of the asymptotic AdS space

r 8 2 r 21 5 g h 10 4 R(r∗) = − L(r∗) = 2 (3.15) 2 4 g8h2 These values are the same as those found for the maximally symmetric Anti de-Sitter spacetime for which X = X+, as expected.

3.1.1 Effective action for the domain wall

In order to further study all possible solitonic solutions to the theory with Lagrangian (3.2) involving only the scalar field and the metric, we can write an effective top-form La- grangian for these two degrees of freedom by substituting the expectation values for the other fields into the Lagrangian in (3.2). Because for the examples we are interested in these expectation values are equal to zero, most terms in the Lagrangian can be dropped. Additionally, as the top-form in any given dimension is unique, we can rewrite √ 7 the expression in terms of the 7-dimensional volume form, dVol7 = −e d x, resulting in the following effective Lagrangian

−2 Leﬀ = R (∗(7)1) − 5X (∗(7)dX) ∧ dX − V(X)(∗(7)1) = √ (3.16) −2 µν 7 = R − 5X e ∂µX∂νX − V(X) −e d x

We can find all solutions involving only X and eµν by solving the Euler-Lagrange equations for this Lagrangian, but these are second order differential equations that are not always easy to solve. 76 CHAPTER 3. CALCULATIONS

The problem can be further simplified if we are only interested in solutions witha non-trivial profile in the radial direction, r, in which case we can reduce the 7-dimensional Lagrangian to a one-dimensional function with a single variable. We will also write an effective action, obtained as the integral of the top-form Lagrangian (or equivalently ˜ R as the integral of the scalar function Leff times the volume form), Seff = Leff = √ R ˜ 7 Leff −e d x, and in order to perform the integration in the transverse coordinates we want to decompose the integral measure into a simple product of the r component with the 6-dimensional integral measure, where this second term will preferably by independent of r.

p 7 p 6 p dVol7 = |e7(r)| d x = |e6| d x |e1(r)|dr (3.17)

Let us now again take the ansatz for eµν(r) and X(r) that we described in (3.7) p A(r)+6B(r) and (3.8). For these fields the volume is equal to |e7(r)| = e , so we would p A(r)+6B(r) p like to have |e1(r)| = e so that |e6| can be independent of r; by a suitable rescaling of the coordinates it is possible to obtain this decomposition of the volume form, and as a result it is possible to take the 6-dimensional integral out of the integral over R √ 6 r. If we now denote the (formally divergent) transversal volume by Vol6 = −e6 d x and we write the Ricci scalar as a function of A(r) and B(r), which we already know 00 from equation (2.50) that it is equal to R(r) = −6 7(B0)2 + 2B − 2A0B0 e−2A, the action obtained from the Lagrangian top-form in equation (3.16) reduces to Seﬀ = R p R √ 6 L1D |e1(r)| dr −e6 d x = S1D Vol6, where

Z −2 µν p S1D = R(r) − 5X (r)e (r) ∂µX(r)∂νX(r) − V(r) |e1(r)| dr =

Z 00 = − 6 7(B0)2 + 2B − 2A0B0 e−A+6B +5X−2(X0)2 e−A+6B + V(X) eA+6B dr

(3.18)

and the prime indicates a derivative with respect to the radial coordinate r. We see that this action has two dynamical fields, B(r) and X(r), as the action includes kinetic terms for them, while A0 is non-dynamical and can be eliminated by a rescaling of the other fields. 00 We can integrate the term containing B by parts by taking into account that B0 = 0 00 at the boundary of an asymptotically AdS space, which results in R 2B e−A+6Bdr = R (2A0B0 − 12B02) e−A+6B dr. The total contribution to the action from R(r) is then simply R −30B02 e−A+6B dr; this integral has a functional form similar to a kinetic term but with the wrong sign, which means that B must be a ghost field and thus unphysical. It is possible to simplify the action in (3.18) by writing it à la Bogomol’nyi, where we complete squares up to a total derivative with respect to r. The total derivative 3.1. SUPERSYMMETRIC DOMAIN WALL IN ADS7 77 can be directly integrated by means of the Fundamental theorem of calculus, where the function is evaluated at the limits of the integral, rUV and rIR. Due to the fact that V(X) 4 2 2 2 is derived from a superpotential through the formula V(X) = (−6 W + X (∂X W ) ), √ 5 1 −4 where W (X) = 2 (hX + 2 gX), we can complete the squares in (3.18) in terms of 0 0 W and ∂X W by grouping the term containing B together with W and X with ∂X W , so that the resulting action is

"Z √ 2 0 −A/2 2 A/2 6B Seﬀ [B,X] = −Vol6 −6 5B e + √ W e e + 5 √ 2 ! −1 0 −A/2 2 A/2 6B + 5X X e + √ X ∂X W e e dr− (3.19) 5 Z # 0 6B 0 6B − 4W (6B e ) + 4X ∂X W e dr

0 0 6B As X ∂X W = W , the last term is the derivative with respect to r of 4W e , and thus it can be integrated directly. This term only depends on the asymptotic value of scalar field X(r) and not on its profile, and thus it is usually referred to as the topological charge, which we will denote by T0 (Shifman, 2012, p. 49). For the effective action

6B(rUV) 6B(rIR) in (3.19) the topological charge is equal to T0 = −4 W (rUV) e − W (rIR) e . Once we have obtained the reduced action for the wall we can also calculate its tension, a quantity defined as the difference between the energy per unit area ofthe domain wall and the vacuum, which can be written down as TDW ·Vol6 = Ewall −Evacuum (Shifman, 2012, p. 42). If we do not want to calculate these energies explicitly, it is also possible to obtain the tension of the wall as TDW = −Seﬀ [Bwall,Xwall]/Vol6, where the action is evaluated at the domain wall solution. The result is

Z " √ 2 √ 0 2# 0 −A/2 2 A/2 X − A 2 A 6B TDW = T0 + −6 5B e + √ W e + 5 e 2 + √ ∂X W e 2 e dr 5 X 5 (3.20)

As the terms in the action are positive definite, the tension of any domain wall will be greater or equal than the topological charge, which is referred to as the Bogomol’nyi inequality (Shifman, 2012, p. 49). From equation (3.18) it is also possible to study the theory perturbatively at a given point by expanding V(X) in series around a vacuum solution for the scalar field with 2 a expectation value of hXi = X0. We will interpret the term proportional to X as an effective mass that the scalar acquires near X0 and higher order terms in the expansion 78 CHAPTER 3. CALCULATIONS will then be interpreted as interactions between several scalar fields. We can expand V(X) as a power series in X by substituting it with its Taylor function of the desired order, where its first three terms are

2 3 dV(X) 1 d V(X) 2 1 d V(X) 3 V3(X) = V(X0)+ (X−X0)+ (X−X0) + (X−X0) dX 2! dX2 3! dX3 X=X0 X=X0 X=X0 (3.21)

The two points in the scalar field that we are most interested in expanding our dV(X) potential around are its extrema, the points for which dX = 0, which are situated 10p 3 2 −2 10p 2 −2 at hXi = X+ = 2 h g and hXi = X− = 2h g . From equation (3.22) we can write down the first three terms in the expansions around these two points, obtaining

r g8h2 √ rg V (X) = −15 5 − 5g2(X − X )2 − 30 10 27 g2 (X − X )3 + ... + 27 + h + (3.22) √ r p5 10 g V (X) = −5 2g8h2 + 10g2(X − X )2 − 40 29 g2 (X − X )3 + ... − − h −

By inserting the expression for V+(X) into (3.19) we obtain the following perturbative expansion around hXi = X+

Z −2 −2A 0 2 p S+[B,X] = − Vol6 R(r) + 5X+ e (X ) + V(X) |e1(r)|dr = r ! Z g8h2 = − Vol R(r) − 15 5 + 5X−2 e−2A(X0)2 − 6 27 + (3.23) √ rg −5g2(X − X )2 − 30 10 27 g2 (X − X )3 + ... p|e (r)|dr + h + 1

From this formula it is straight-forward to read off the effective mass of the scalar field 1 −2A 0 2 in the neighborhood of X+ if we realize that the canonical action must read 2 e (X ) + 1 m2 (X − X )2 2 X+ + , so the mass must be equal to

2 2 mX+ = −(g X+) (3.24)

2 2 Using the AdS curvature value obtained in (3.15) we can see that mX+ L(X+) = −8, a value that respects the Breitenlohner-Freedman bound, which for a scalar in a 7- 2 2 dimensional AdS theory reads mX+ L(X+) ≥ −9. 3.2. NON-CONFORMAL DEFORMATION OF THE CFT6 DUAL 79

3.2 Non-conformal deformation of the CFT6 dual

Using the AdS/CFT dictionary developed in section 2.7 we are going to find an explicit expression for the SCFT dual to the 1/2-BPS object described in section 3.1, and we will study some of its properties. The dual theory for N = 1 supergravity in 7 dimensions must be a SCFT in 6 dimensions with the same number of supercharges, due to the fact that gauge symmetries in the AdS side are represented by global (ungauged) symmetries in the SCFT, which in particular applies to supersymmetry. As we had 16 supercharges in the AdS side, that corresponds with a single supersymmetry in a 6-dimensional SCFT, so we conclude that our conformal dual is N = (1, 0) SCFT, which is known from the literature not to have a Lagrangian formulation. Due to the strict constraints imposed by the Poincaré supergroup, the global symmetry of a 6-dimensional SCFT must be an orthosymplectic group that contains the Poincaré group and the fermionic symmetry group, which in our case is OSp(6, 2 | 1) ⊃ SO(6,2) × Sp(2), as can be seen for example in reference (Freedman and Van Proeyen, 2012, p. 17). The symplectic group Sp(2) has the same algebra than SU(2) or SO(3), which was expected, as the gauge symmetries in the AdS side of the correspondence must be matched with a global symmetry in the CFT side. This theory describes the low-energy dynamics of the brane picture in figure 3.1 when the NS5-branes are coincident. As we saw in section 2.3, a 1/2-BPS object in SCFT is a conformal primary that is also annihilated by one of the superconformal operators Q, so the conditions that |BPSi must hold are

µ K |BPSi = 0 for µ ∈ [0, 6] and Qα|BPSi = 0 (3.25)

where only one of the Qα operators need to annihilate the state. We would like to calculate the scaling dimension, the two-point function and the three-point function of the BPS states dual to the scalar field that we found in section 3.1. The scaling dimension of such operators can be obtained from the effective mass of the scalar calculated around the maximum of the potential, which as we see in the formula for scalars in table 2.1, in 6 dimensions reads

2 2 L mX+ = ∆(∆ − 6) (3.26)

2 2 As we calculated in section 3.1, L mX+ = −8, so the two possible solutions for ∆ are

∆+ = 4 and ∆− = 2 (3.27) 80 CHAPTER 3. CALCULATIONS

We see that we can define two different operators dual to the scalar X+, one with scaling dimension ∆ = 4 that we will denote O4 and another one with scaling dimension

∆ = 2 which we will write as O2, both of which satisfy the unitarity bound for a CFT in 6 dimensions, ∆ ≥ 2. As we are working in a 6-dimensional theory, an operator is relevant if it has scaling dimension ∆ < 6, irrelevant if ∆ > 6 and marginal if ∆ = 6, so we conclude that both operators are relevant and as a consequence they define deformations of the theory that will drive us away from the CFT at low energies and large scales. Their scaling dimensions are exact at all scales, as they are protected from acquiring an anomalous dimension, γ, thanks to cancellations induced by supersymmetry. Due to the fact that our operators are BPS their scaling dimension only depends on their R-charge and thus it remains unrenormalized.

With the values that we have calculated for ∆+ and ∆− it is now possible to write down the asymptotic behavior of the scalar field X+ near the boundary of the AdS space, ∆+ z = 0; the lowest powers of z that appear in the series expansion of X+ are z and z∆− , so for z 1 we can write

∆+ ∆− 2 4 X(x, z) ' X+ + X∆− (x) z + X∆+ (x) z = X+ + X4(x) z + X2(x) z (3.28)

The function X2(x) acts as the source of the operator O2, while X4(x) corresponds with the source of O4; in a Conformal Field Theory we set both sources equal to zero. We can now calculate the n-point functions of the CFT by means of equation (2.55), where for each O2 operator in the CFT correlator we need to take a functional derivative of the action with respect to X2, and for each O4 we variate the action with respect to

X4, so schematically

n δ SAdS(X(x, z)) hOi1 (xi1 ) Oi2 (xi2 ) ...Oin (xin )i = (3.29) δX (x ) δX (x ) . . . δX (x ) i1 i1 i2 i2 in in X2=X4=0

where for the action SAdS we can take the series expansion around X+ that we calculated in (3.23), but in the gauge with radial coordinate z given in equation (2.40).

Because there is no linear term in X around the critical point X+, the one-point functions of both O2 and O4 vanish at the boundary, as required in a CFT. As we mentioned in section 2.7, the presence of non-vanishing one-point functions hO(x)i would indicate that the Conformal Field Theory is not in its ground state. To calculate higher-order correlators we need the exact solution for X and not just its asymptotic behavior, so in the gauge where the radial coordinate is z we need to use the boundary-to-bulk propagator K(x0 − x, z) to obtain X(x, z) in terms of its boundary value. This boundary value corresponds with the source of O4(x) (that is, X4(x)) if we 3.2. NON-CONFORMAL DEFORMATION OF THE CFT6 DUAL 81 choose Dirichlet boundary conditions, or the source of O2(x), X2(x), for von Neumann boundary conditions.

The two point functions of these operators are constrained by conformal symmetry to have the functional form

cij hOi(x)Oj(y)i = (3.30) |x − y|∆i+∆j

which is a symmetric function of Oi and Oj and is zero whenever ∆i 6= ∆j, so we immediately find that hO2(x)O4(y)i = 0. Thus in order to characterize these operators we only need to find c22 and c44. The two-point functions will have different values depending on the normalization of the corresponding operators, so we can use its value to fix the normalization of O2 and O4 in order to obtain standard results for all higher- order correlators.

If we had the exact solution to X in the z gauge in terms of its boundary values we could calculate these two constants by means of formula (3.29), where we make the action be on-shell by substituting the expression for X(x, z) inside the integral. After taking the functional derivatives of the on-shell action we would then set the sources to be equal to zero, and the result will be the correlator functin that we were looking for. Due to the fact that we will set the sources equal to zero at the end we only need to consider the terms in the action proportional to X2, which in our case are equal to

Z L7 S = −5g2(X(z) − X )2 d6x dz (3.31) (2) + z7

After obtaining fixing the normalization constant the operators O2 and O4 we would turn our attention to three-point functions, which are also constrained by conformal symmetry in their dependency on the spacetime coordinates so that the only free parameters are the constants cijk, which are characteristic of the three operators involved.

cijk hOi(xi)Oj(xj)Ok(xk)i = ∆ +∆ −∆ −∆ +∆ +∆ ∆ −∆ +∆ |xi − xj| i j k + |xj − xk| i j k + |xk − xi| i j k (3.32)

Just as in the case of the two-point functins, the structure constants can be calcu- 3 lated from the terms in the action proportional to X , which we denote as S(3), after substituting the exact solution for X in the z gauge in terms of its boundary value,

X4(x) or X2(x) respectively. In our theory S(3) reads 82 CHAPTER 3. CALCULATIONS

Z √ rg L7 S = −30 10 27 g2 (X(z) − X )3 d6x dz (3.33) (3) h + z7

Once we have all the structure constants of the fundamental operators in the CFT the theory is completely determined, as any other correlator can be calculated in terms of these constants through the Operator Product Expansion of the theory.

3.3 Equivalent Quantum Field Theory

The sources of the operators O2 and O4 deform the Superconformal Field Theory, modifying the dynamics of the theory in terms of the energy scale we probe it at, µ. The dependency on a dimensionful parameter effectively breaks the conformal symmetry, and as a result the theory will be a non-conformal Quantum Field Theory with Poincaré invariance instead. The internal symmetry group of the SCFT before the deformation is Sp(2) ∼= SU(2), and it turns out that this will also be the internal symmetry group of the resulting QFT because despite the fact that 8 of the 16 supercharges are broken when we choose a particular operator in the BPS family of solutions, the broken supersymmetry generators can be identified with the superconformal generators of the SCFT, S. The global symmetry group of the QFT that this renormalization group-flow describes is then SO(5,1) × SU(2). This family of theories describe the low-energy dynamics of the NS5-, D6-, D8-brane configuration described in figure 3.1 when the vacuum expectation value of the scalar in the tensor branch (related to the distance between NS5-branes) is non-zero. If our 6-dimensional Conformal Field Theory had a Lagrangian formulation we could account for the deformation induced by some relevant operators by adding an extra term R 6 to the CFT Lagrangian, LQFT = LCFT + O(x) X(x) d x, where O are the relevant operators and X their sources. Even though our original Superconformal Field Theory does not have a Lagrangian description to start with, it is still possible to write down the additional term that accounts for the deformation induced by the O operators, that is, ∆L = R OX d6x. With this term it is now possible to take functional derivatives with respect to these fields and their sources, and thus we can calculate all the quantities of interest that takes us away from the conformal point. c-function

In order to characterize the QFT we will study how the available degrees of freedom change as we integrate out higher energy fields. In a Conformal Field Theory this property is related with the value of the conformal anomalies, also called central charges, 3.3. EQUIVALENT QUANTUM FIELD THEORY 83 that modify the expectation value of the trace of the energy-momentum tensor in a quantum theory. The number of distinct central charges depends on the dimension of the CFT, and in particular these anomalies only appear in spacetimes with even dimensions. In 2 dimensions there is only one central charge, the Weyl anomaly commonly denoted by c, and it is proportional to the two-point function of the stress-energy tensor, as shown in Ammon and Erdmenger (2015, p. 117). In the presence of supersymmetry this constant is completely determined by the matter content of the theory (Freedman and Van Proeyen, 2012, p. 539). In 4 dimensions and higher we have a second charge which appears in the three-point function of Tµν accompanying the Euler tensor, the a central charge. As we mentioned in section 2.7, for conformal theories with a gravity dual the c and a central charges must by identical, so it is enough to calculate one of the two (Freedman and Van Proeyen, 2012, p. 558). Finally, in 6 dimensions there are three different invariants that can be built from the Weyl tensor, and thus we havethree central charges ci on top of the one associated with the Euler tensor, a (Cremonesi and Tomasiello, 2016, p. 17). It can be proven that in the weak holographic limit all four terms are proportional to each other, as discussed in Cremonesi and Tomasiello (2016, p. 24), so it is only necessary to compute one of them. When we study a renormalization group-flow interpolating between two Conformal Field Theories we can define a function that interpolates between their central charges that in a certain sense describes the number of degrees of freedom that can be excited at each energy scale µ. For renormalization group-flows with a gravity dual there exists a theorem, the c-theorem, which states that c UV > cIR, which can be interpreted as the fact that the number of degrees of freedom has to decrease as we lower the probing energy

(Freedman and Van Proeyen, 2012, p. 558). When both CFTUV and CFTIR exist, we can then define a continuous monotonic function in terms of the renormalization group scale, c(µ), that interpolates between c UV and cIR. Because the energy scale of the Conformal Field Theory, µ, is dual to the radial coordinate of the Anti de-Sitter spacetime, z, we can equivalently find the function c(z) = c(µ(z)) (Freedman and Van Proeyen, 2012, p. 562). Even when one of the conformal theories at the endpoints of the flow does not exist, as in our case of interest, the c-function is still well defined and it can be proven that it is still continuous and monotonic. A value of zero for the central charge implies that either all anomalies cancel exactly due to the symmetry properties of the theory or the theory becomes trivial in this limit. Let us find the explicit formula for the c-function in our case; from Freedman and 2 2 2F (p) Van Proeyen (2012, p. 563) we know that in the gauge where ds7 = dp + e dsMkw6 it is possible to compute the c function in terms of the derivative of F (p) with respect to p by means of the formula 84 CHAPTER 3. CALCULATIONS

π 1 c(r) = 0 3 (3.34) 8 G7 F (p)

2 2A(r) 2 2B(r) We can now change gauges so that the metric is equal to ds7 = e dr +e dsMkw6 , with A(r) and B(r) given in equation (3.13). The relationship between F 0 and B0 is dF dB dr dr −A(r) dp = dr dp , where dp = e , so in this gauge

π e3A(r) π 125 r12 √ c(r) = 0 3 = (3.35) 8 G7 B (r) 64 G7 (h + 2gr5)3 As the central charges are related to the trace anomaly of a theory, it is also possible to obtain this function in terms of the flow of the Trr component of the energy-momentum tensor in the r direction. The shape of the function in 3.35 can be plotted for given values of h and g, obtaining the graph in figure 3.4.

Figure 3.4: Plot of the c function for h = g = 1

q π 5 8 The c-function is equal to 3 12 at the boundary of the AdS, which corresponds 64 G7 h g with the value in the SCFT, and it tends to zero as r → ∞, which indicates that it becomes trivial at low energies.

β-function

Another quantity of great interest in a Quantum Field Theory is the beta function, β(µ), which accounts for the rate of change of the coupling parameters with respect to the energy scale. This function is related to the energy-momentum tensor flow, and thus it can be written as a function of the central charge of the theory. In the reference by Anselmi et al. (2000, p. 3) we find that for a single scalar field, as it is our case,both functions are related by c0(µ) = −2c(µ)β2(µ), which immediately lets us write 3.3. EQUIVALENT QUANTUM FIELD THEORY 85

s s √ c0(r) 3 4h − 2gr5 β(r) = − = − √ (3.36) 2c(r) 2 r(h + 2gr5)

10p The beta function is equal to zero when r = 8 h2g−2 and when r → ∞, as we expect for the fixed points at the end of the renormalization group-flow, and itsshape its represented in 3.5.

Figure 3.5: Plot of the β function for h = g = 1 86 CHAPTER 3. CALCULATIONS Bibliography

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