Applications of the Gauge/Gravity Duality by Kevin Robert Leslie Whyte B. Applied Science, University of Waterloo, 2004 B. Mathematics, University of Waterloo, 2006 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES (Physics) The University Of British Columbia (Vancouver) August 2013 c Kevin Robert Leslie Whyte, 2013 Abstract While varied applications of gauge/gravity duality have arisen in literature from studies of condensed matter systems including superconductivity to studies of quenched Quantum Chromodynamics (QCD), this thesis focuses on applications of the duality to holographic QCD-like field theories and to inflationary model that uses a QCD -like field theory. In particular the first half of the thesis examines a holographic QCD-like field theory with scalar quarks closely related to the Sakai-Sugimoto model of holographic QCD. The behaviour of baryons and mesons in the model is examined to find a continuous mass spectrum for the mesons, and a baryon that can identified with a topological charge. It then slightly modifies the theory to further study the behaviour of holographic field theories. The second half of the thesis presents a novel model for early Universe inflation, using an SU(N) gauge field theory as the inflaton. The inflation model is studied at both weak coupling and strong coupling using the gauge/gravity duality. The robustness of model’s predictions to exciting multiple inflationary fields beyond the single field of its original proposal, and its possible role in breaking the supersymmetry of the Minimal Supersymmetric Standard Model (MSSM) is also explored. ii Preface This thesis includes previously published work. Chapter 2 is an edited version of the work published in the Journal of High Energy Physics under the title, “Baryon charge from embedding topology and a continuous meson spectrum in a new holographic gauge theory”[1]. It was a collaboration between the candidate’s supervisor and the candidate. Chapter 3 is also an edited version of an earlier draft of the work that makes up chapter 2. Chapter 4 is an edited version of the work published in the Journal of Cosmology and Astroparticle Physics under the title, “Twisted Inflation” [2]. It was a collaboration between the candidate’s supervisor, two postdoctoral fellows (Joshua Davis and Thomas Levi), and the candidate. Chapter 5 is the sole work of the candidate. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii 1 2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction to D-branes . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Applications to studies of nuclear matter . . . . . . . . . . . . . . 4 1.3 Applications to studies of cosmology . . . . . . . . . . . . . . . . 5 Baryon Charge from Embedding Topology . . . . . . . . . . . . . . 8 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Adjoint sector . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.2 Fundamental matter . . . . . . . . . . . . . . . . . . . . 14 2.3 Vacuum solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Meson spectrum and stability . . . . . . . . . . . . . . . . . . . . 20 iv 2.4.1 Scalar modes . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.2 Gauge modes . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.3 Converting to a quantum mechanics problem . . . . . . . 26 2.4.4 Gauge field fluctuations: a continuous spectrum . . . . . . 27 2.4.5 Transverse scalar fluctuations . . . . . . . . . . . . . . . 28 2.4.6 X4 and θ fluctuations. . . . . . . . . . . . . . . . . . . . . 29 Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.A Ramond-Ramond forms . . . . . . . . . . . . . . . . . . . . . . . 32 Bottom-Up Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 Adjoint sector . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.2 Fundamental matter . . . . . . . . . . . . . . . . . . . . 38 3.3 Vacuum solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 Meson spectrum and stability . . . . . . . . . . . . . . . . . . . . 45 3.4.1 Scalar fluctuations . . . . . . . . . . . . . . . . . . . . . 46 3.4.2 Interpretation of the light scalar . . . . . . . . . . . . . . 50 Finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.A Alternate coordinates for θ fluctuations . . . . . . . . . . . . . . 56 Twisted Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 Basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.1 66 2.5 3 3.5 4 4.2.2 2 3 66 . . . . . . . . . . . . . . . 68 Effective action for the candidate inflaton . . . . . . . . . . . . . 69 4.3.1 70 4.3.2 4.3.3 4.4 Strong coupling: 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 4.3 Weak coupling: λ λ Very strong coupling: λ N N 2 3 Weak coupling . . . . . . . . . . . . . . . . . . . . . . . Strong coupling: 1 λ Very strong coupling: λ 2 3 N , the D4-brane . . . . . . . 2 3 73 N , the M5-brane . . . . . . . 75 Coupling to gravity and slow-roll potentials . . . . . . . . . . . . 76 4.4.1 78 Weak coupling: λ 1 . . . . . . . . . . . . . . . . . . . v 4.4.2 4.5 Strong coupling: λ 1 . . . . . . . . . . . . . . . . . . 79 Analyzing the inflationary potentials . . . . . . . . . . . . . . . . 81 4.5.1 General results for V (φ ) = V0 (1 − f (φ /φ0 )) potentials . . 82 4.5.2 Weak and strong coupling results for the inflationary parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Other light fields . . . . . . . . . . . . . . . . . . . . . . 86 4.6 Predictions and observational constraints . . . . . . . . . . . . . . 88 4.7 The end of inflation and reheating . . . . . . . . . . . . . . . . . 90 4.7.1 Weak coupling: λ 1 . . . . . . . . . . . . . . . . . . . 90 4.7.2 Strong coupling: λ 1 . . . . . . . . . . . . . . . . . . 91 4.8 The η-problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.9 Related models . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.9.1 Multiple scalars and N-flation . . . . . . . . . . . . . . . 95 4.9.2 Other field theories . . . . . . . . . . . . . . . . . . . . . 96 4.9.3 Other boundary conditions . . . . . . . . . . . . . . . . . 96 4.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.A Field theory calculations . . . . . . . . . . . . . . . . . . . . . . 97 4.5.3 4.B Energy in the deconfined phase at strong coupling . . . . . . . . . 100 4.C The inflaton potential for quantum field theory in de Sitter space . 102 5 Extensions to Twisted Inflation . . . . . . . . . . . . . . . . . . . . . 105 5.1 5.2 6 Multiple fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.1.1 Weak coupling . . . . . . . . . . . . . . . . . . . . . . . 106 5.1.2 Strong coupling . . . . . . . . . . . . . . . . . . . . . . . 110 Supersymmetry breaking . . . . . . . . . . . . . . . . . . . . . . 111 5.2.1 Gravity-mediated supersymmetry breaking . . . . . . . . 113 5.2.2 Gauge-mediated supersymmetry breaking . . . . . . . . . 114 5.2.3 Anomaly-mediated supersymmetry breaking . . . . . . . 115 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.1 Future lines of research . . . . . . . . . . . . . . . . . . . . . . . 118 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 vi List of Tables Table 4.1 Weak and strong coupling predictions for the scalar spectral index ns , the running of the spectral index αs , and the tensor/scalar ratio r. For λ analysis to be reliable, and N 1 we require N 104 λ − 53 1 for the to obtain enough e- foldings. Numerical values are given assuming Cosmic Microwave Background (CMB) perturbations at the pivot scale left the horizon at NCMB = 60 e-foldings before the end of inflation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Table 4.2 Summary of inflationary potentials. . . . . . . . . . . . . . . . 82 Table 4.3 Summary of results for the inflationary parameters . . . . . . . 86 Table 4.4 Predictions for scalar spectral index and running parameter . . 88 Table 4.5 Kaluza-Klein scale in terms of field theory parameters . . . . . 89 Table 4.6 Predictions for the tensor-to-scalar ratio . . . . . . . . . . . . . 89 Table 4.7 Inflation scale in terms of field theory . . . . . . . . . . . . . . 90 Table 5.1 Weak coupling multi-field inflationary perturbations . . . . . . 108 Table 5.2 Predictions for the tensor-to-scalar ratio at weak-coupling . . . 109 Table 5.3 Strong coupling multi-field inflationary perturbations . . . . . 111 vii List of Figures Figure 2.1 Brane construction for the holographic field theory. . . . . . Figure 2.2 Examples of brane embeddings (in the y − θ plane) for v0 = 9 0. The asymptotic angle between the two ends of the brane is 2θ∞ , which ranges from π for the stable embedding which extends to the smallest values of y, down to some value 2θMax ≈ 2.088 in the limit y0 → ∞. . . . . . . . . . . . . . . Figure 2.3 19 Asymptotic angle θ∞ on the sphere vs minimum brane position y0 in radial direction, for various values of v0 . Angle θ is defined to be zero at y = y0 . . . . . . . . . . . . . . . . . . Figure 2.4 20 Example of multiple embeddings for the same asymptotic sphere angles. Only embeddings which do not “wrap” the sphere are stable. The rest are perturbatively unstable to slipping around the sphere, as shown. . . . . . . . . . . . . . Figure 2.5 Geometrical interpretation of Goldstone bosons for the special case θ∞ = π/2. . . . . . . . . . . . . . . . . . . . . . . . Figure 2.6 21 22 Behaviour of x4 vs θ for various values of v0 at y0 = 2. As a function of y, the slope dx4 /dy approaches a constant in each case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 2.7 Effective potential V1 (y) = C(y)B(y) for v0 = 0 and various values of y0 . Lower graphs have smaller y0 . . . . . . . . . . . Figure 2.8 22 28 Effective potential V2 (y) = −A(y)B(y) for v0 = 0 and various values of y0 . Lower graphs have smaller y0 . . . . . . . . . viii 29 Figure 2.9 Effective potential V4 (y) = −A(y)B(y) for v0 = −∞ and y0 = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Figure 2.10 Effective potential Vθ (y) = −A(y)B(y) for v0 = −∞ and y0 = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.1 30 Spectrum of low-lying bosonic mesons as a function of the embedding parameter y0 . For y0 = y∗ , we have massless Goldstone bosons associated with SU(2) × SU(2) → SU(2) symmetry breaking. For large y0 , a single scalar meson becomes parametrically light. . . . . . . . . . . . . . . . . . . . Figure 3.2 37 Schematic of a brane embedding. Left picture shows a plane in the space formed by the radial direction and the S4 directions. Right picture shows same embedding in radial and x4 directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.3 Asymptotic angle on S4 41 vs minimum radial position for probe D4-brane embeddings. Function asymptotes to infinity at y0 = 1. Values of θ∞ in (π/5, π/2] correspond to stable embeddings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Figure 3.4 Examples of stable brane embeddings. 43 Figure 3.5 Example of multiple embeddings for the same brane asymp- . . . . . . . . . . . . totics. There are additional embeddings with smaller y0 and more “windings” around the sphere, however only the embedding with the largest y0 is stable. The rest are perturbatively unstable to slipping around the sphere, as shown. . . . . Figure 3.6 Geometrical interpretation of zero modes (massless mesons) for the special case θ∞ = π/2. . . . . . . . . . . . . . . . . . Figure 3.7 48 Spectrum of mesons arising from fluctuations of the brane embedding in the transverse sphere directions. . . . . . . . . . Figure 3.9 44 Spectrum of mesons arising from fluctuations of the brane embedding in the y − θ plane. . . . . . . . . . . . . . . . . . Figure 3.8 44 49 Spectrum of mesons arising from fluctuations of the brane embedding in the compact x4 direction. . . . . . . . . . . . . ix 49 Figure 3.10 Spectrum of mesons arising from fluctuations of the gauge field on the probe D4-brane. . . . . . . . . . . . . . . . . . . 50 Figure 3.11 Schematic of “black hole” embeddings in which the brane extends down to the horizon. . . . . . . . . . . . . . . . . . . 54 Figure 4.1 Geometrical picture of inflation at strong coupling. . . . . . . 60 Figure 4.2 Energy scales for weak and strong coupling. The energy density V0 during inflation will be of order the Kaluza-Klein Figure 4.3 scale 1/R in both regimes. . . . . . . . . . . . . . . . . . . . 67 Various interpretations for the inflaton field. . . . . . . . . . . 70 jz˙2 for which εG < δ . . . . . Figure 4.4 Diagram showing values of z and Figure 5.1 Geometric interpretation of the scalar fields. . . . . . . . . . . 106 Figure 6.1 Geometrical interpretation of Goldstone bosons. . . . . . . . 118 Figure 6.2 Various interpretations for the inflaton field. . . . . . . . . . . 118 x 80 Glossary ADS Anti-de Sitter ADS / CFT Anti-de Sitter/Conformal Field Theory CMB Cosmic Microwave Background MSSM Minimal Supersymmetric Standard Model QCD Quantum Chromodynamics xi Acknowledgments I wish to thank my parents, Ray and Linda Whyte, for all their support. Through their love and guidance I have made it to where I am today. I want to thank my brother, Jeff Whyte, for being the best roommate one could have when starting graduate school in a new city. With deep gratitude, I thank my collaborators, Mark van Raamsdonk, Joshua Davis and Thomas Levi, for their intellectual brilliance and guidance. Lastly, this thesis is dedicated to Sandy Ho. Her love is my greatest joy. xii Chapter 1 Introduction String theory was first studied as a theory of the strong nuclear force. In this form, it was hoped that string theory would explain the hadron spectrum and scattering properties of mesons. However there were problems with this approach: for mathematical consistency string theory needed more than the familiar four spacetime dimensions and predicted a massless particle with a spin of two. Combined with the convincing success of Quantum Chromodynamics (QCD), these problems could have meant the end to the study of string theory, but the massless particle with a spin of two proved too interesting to abandon. It was proposed that the particle be interpreted as the quantum particle of gravity, and the focus of string theory shifted from strong interactions to quantum gravity and the possibility of it containing a quantum theory of all four fundamental forces (electromagnetism, gravitation, strong and weak interactions)[3]. Since then, string theory has attracted its fair share of proponents and critics, and its study has ebbed and flowed with each setback and discovery. While even now, decades after its introduction, questions remain about string theory’s role in nature, unifying the fundamental forces, and whether or not it is the theory of quantum gravity, many ideas have originated from its inquiry and influenced other areas of physics: supersymmetry for example originated from early investigations into string theory[4, 5]. Another important idea coming from string theory, but with greater implications, is the gauge/gravity duality. This idea began [6] as the Anti-de Sitter/Con1 formal Field Theory (ADS / CFT) correspondence, a conjectured duality between string theory in an Anti-de Sitter (ADS) background geometry and a conformal field theory in one less dimension, living on the boundary of the ADS space. Here ADS is a well studied background geometry or spacetime for the string theory. It is a vacuum solution to the Einstein equations with negative cosmological constant in General Relativity. Meanwhile conformal field theory is a quantum field theory that respects a particular symmetry closely related to scale invariance, known as conformal symmetry. The gauge/gravity duality has since been expanded to include field theories that are not conformal and string backgrounds that are not ADS or even asymptotically ADS . While it is still frequently referred to as the ADS / CFT correspondence, be- cause of its application beyond ADS and conformal field theories it is also known as the gauge/gravity duality or the Maldacena conjecture. Furthermore, since the duality allows a quantum field theory to fully describe a string theory of one higher dimension, just as a 2-dimensional hologram presents a 3-dimensional image, research using the gauge/gravity duality is called the study of holography, and the field theories of the duality are called holographic field theories. As an example a field theory closly related to QCD and dual to a string theory might be called holographic QCD, as is the case for the Sakai-Sugimoto model of holographic QCD[7]. What makes this duality so useful is that a strongly coupled field theory corresponds to a weakly curved background, and vice versa that a weakly coupled field theory corresponds to a strongly curved background (typically with strong string coupling). This allows us to ask difficult questions in strongly coupled field theories using techniques of classical gravity with weak curvature, or – just as interesting – ask questions about quantum gravity using techniques of perturbation theory on a weakly coupled field theory. Varied applications of the duality have arisen in the literature, from studies of condensed matter systems including superconductivity [8, 9] to studies of quenched QCD and nuclear matter [7, 10, 11]. This thesis examines some of these applica- tions, specifically it uses the gauge/gravity duality to study field theories closely related to QCD and nuclear matter, and to study an inflationary field theory model at weak and strong coupling. 2 1.1 Introduction to D-branes A key component to studying string theory is an extended object known as a Dbrane. String theory is the study of small objects known as strings. Strings are 1-dimensional bundles of energy, just as a particle in regular quantum field theories are zero-dimensional points of energy. These strings can form loops making what is known as closed strings, or the string ends could remain separate to make open strings. In a theory with open strings, the end points of a string land on a D-brane. Thus when studying the gauge/gravity duality in string theory, D-branes are unavoidable, and for the benefit of the reader this section provides a brief introduction. To understand the origin of D-branes, consider an open string theory where the locations of string end points are unrestricted. In such a theory, there will be a vector gauge-field sourced by the string-ends that is not present in the otherwise similar closed string theory. Further assume that one of the spacial directions of this open string theory is compactified to a radius R. There exists a duality relating this open string theory to another one with a compactified dimension of radius R1 , the T-duality. Taking the T-dual of the open string theory currently under consideration, results in another dual open string theory. In this dual theory, the string ends are no longer free in every direction, but rather they are fixed for the direction in which the T-dual was taken. Not only are the string end points fixed in that direction, but the gauge theory also loses a degree of freedom, the component associated with the direction of the T-dual. What is happening is that there is a Dp-brane (a D-brane with p spatial dimensions) that fills all p spatial dimensions of the original open string theory. After taking the T-dual, the Dp-brane goes to a D(p − 1)-brane in the dual theory. The gauge field theory sourced by the string ends lives on the D-branes. Hence, when the D-brane goes down a dimension, the gauge theory on the brane loses a vector component in the T-dual direction. This degree of freedom of the gauge-field is not lost however. The D-brane itself is dynamical, and the D-brane’s position in the transverse direction is the degree of freedom that replaces the one lost from the gauge-field. 3 In this light, the close connection between the fields induced by the strings and the D-branes is apparent. D-branes are a key part of the non-perturbative definition of string theory, and they can be thought of as soliton-like configurations of string theory. 1.2 Applications to studies of nuclear matter As mentioned above, the gauge/gravity duality has been used to study the difficult subject of QCD. Now it is well known that coupling constants of many gauge theories run, in that the strength of the coupling depends on the energy scale at which the coupling is observed. In the case of QCD, the coupling actually decreases at high energies and short distances, giving rise to what is known as asymptotic freedom. The corollary of that statement is that the coupling increases in strength at lower energies. Should one wish to study the theory at lower energies, to study neutrons, atomic nuclei and other baryons and mesons for example, then one is forced to consider a strongly interacting theory. Thus a perturbation expansion in the coupling is no longer a valid approach, making one look for non-perturbative techniques. One of the longstanding non-perturbative approaches is the use of numerical lattice simulations [12]. This approach has even made contact with experiment [13], but it has its own limitations and approximations. Thus it makes sense to search for other approaches, such as gauge/gravity duality, and there has been steady progress towards using the duality to study nuclear matter [7, 11]. However a gravity background dual to QCD as we know it in the standard model of particle physics had not yet been discovered, and one may question whether one actually exists, or if it does whether its study would be any simpler than that of QCD itself. That said, duals to theories that approximate QCD have been found and studied [7, 11], and it makes sense to ask what features of these theories are related to their approximation of QCD and what features are related to their holographic nature. With that thought in mind, in chapter 2 we study a holographic field theory closely related to the ones introduced before [7, 11], but replace the branes responsible for fermionic fundamental matter in the holographic QCD model with 4 D4-branes transverse to the D4-colour branes. In this D4-D4 model, we directly study the gravitational brane picture without defining an explicit limit to move to the gauge picture from the gravitational one, and thus it is difficult to say exactly what kind of fundamental matter we are studying by introducing transverse D4-branes. However, we argue in the introduction to chapter 2 that the matter introduced by the transverse D4 branes are fundamental matter, most likely a scalar in behaviour. We go on to find the setup leads to interesting behaviour such as a continuous spectrum for the mesons and the baryons in the model being connected with a topological charge. Following the study of the D4-D4 model in chapter 2, in chapter 3 we modify the model’s gravitational Lagrangian. With this modification we obtain a different gravitational model, whose dual field theory should be similar to the dual field theory of the previous model in chapter 2, but even more removed from an explicit brane construction that might guide us to what the properties of the dual field theory might be. However, this bottom-up approach to ADS / CFT is not uncommon and is already performed to form gravitational duals of interesting real world phenomena, such as superconductivity and superfluidity [9, 14–16]. Here the benefit to taking the bottom-up approach is a gravitational model whose geometry provides an easy and intuitive way to understand complex field theoretic behaviour, including the appearance of a Goldstone boson. 1.3 Applications to studies of cosmology One of the challenges in string theory, especially for it to be considered as the theory of the four fundamental forces and matter, is the incorporation of inflation [17]. Inflation is the rapid expansion of the early Universe, typically driven by a field called the inflaton. During this time, microscopic quantum fluctuations of the fields that fill space were blown up to galactic proportions producing a power spectrum that is nearly scale invariant, with a slight red tilt, and predictions from the standard inflationary model of cosmology (the ΛCDM model) have wonderfully matched up with observations [18, 19]. However, despite the success of inflationary theory and the ΛCDM model in particular, it is still unknown what is the physical degree of freedom that makes up 5 the inflaton field. In many models the inflaton is assumed to be scalar, but it need not be, as example models have been proposed where multiple scalar fields drive inflation. Even with the microscopic nature of the inflaton still unknown, its success has made it a widely accepted and important ingredient of modern cosmology; for reviews on the subject of inflation see [20, 21]. It thus makes sense to look for ways to produce inflation from within string theory [22], not only to borrow from its success but also perhaps explain its physical origin. To that end, in chapter 4 we study a new model for inflation that we call Twisted Inflation, in the context of a supersymmetric field theory with anti-periodic boundary conditions for the fermions on an extra compactified direction. These boundary conditions result in the fermions gaining a mass, breaking supersymmetry. We then assume a large classical expectation value for one of the components of a nonabelian gauge field (resulting in a moduli scalar) and derive an effective potential for it by integrating out other components of the non-abelian gauge field and other fields. Interestingly enough, had supersymmetry been preserved then contributions to the effective potential by the bosonic degrees of freedom and their fermionic superpartners, which are related to the expectation value we gave the moduli scalar, would have exactly cancelled. In the case of broken supersymmetry however, the contributions of the bosonic degrees of freedom are offset relative to their superpartners, by about the mass scale defined by the supersymmetry-breaking compactified direction. This mass offset stops the fermionic and bosonic degrees of freedom from exactly cancelling, giving rise to an effective potential for the moduli scalar. Now if the expectation given to the scalar is taken to be sufficiently large, then the mass scale defined by compactification will be small in comparison. Thus for large values of the scalar, the effect of the offset in contributions will be minimal and the scalar’s effective potential will be nearly flat, almost as it was in the supersymmetric case. This nearly flat potential makes this scalar an excellent candidate to be an inflaton in slow-roll inflation [20]. Then, having studied the inflation that arises from this gauge theory at weak coupling it is natural to ask whether one would see the same behaviour at stronger coupling. Fortunately the considered gauge theory is very amenable to study under 6 the gauge/gravity duality, and its gravity dual in the vacuum state has been studied in [10] and forms the basis of various QCD-like theories [7, 11]. In fact the strongly coupled gravitational dual is the gravity background used in chapters 2 and 3. Through the gauge/gravity duality we are able to study the the proposed inflation model with the field theory at strong coupling. This provides two major benefits. First by looking at the theory from both the weak and strong coupling regimes, we are able to make robust predictions of its behaviour across a wide coupling range, and not be limited by perturbation theory. Second, in the strong coupling regime it describes a slow-roll inflation scenario that is fully embedded in string theory. Given that inflation so accurately predicts the Universe we observe [20], for string theory to be considered a possible realistic theory for quantum gravity and more, it needs models of inflation within it. While certainly our model is not the only model of inflation in string theory [23], it does offer another alternative and demonstrate that inflation is a very flexible paradigm. Finally, after examining the Twisted Inflation model at both weak and strong coupling in section 4.6 and finding that they offer similar physical predictions, in chapter 5 we examine extensions to the inflationary model. First we see how robust the model’s predictions are with respect to turning on multiple components of the gauge field theory. In chapter 4 only one component of the gauge field was taken to a large classical value and allowed to inflate spacetime. In chapter 5 multiple components are assumed to be initially large and their effects on physical predictions are examined. While the components are made large in a very symmetric way to ease analysis, the physical predictions remain robust to the changes with encouraging results. The next extension to Twisted Inflation considered is the possibility that the broken supersymmetry inherent in our model that drives the inflaton could also be used to break supersymmetry in the Minimal Supersymmetric Standard Model (MSSM). Different mechanisms are considered, but mass scale restrictions to avoid the hierarchy problem in the MSSM make it unlikely that Twisted Inflation could be used as a hidden supersymmetry-breaking sector. 7 Chapter 2 Baryon Charge from Embedding Topology 2.1 Introduction Gauge-theory / gravity duality [24] provides a powerful tool to construct and study strongly coupled field theory systems. In recent years, the set of field theories constructed and analyzed in this way has grown to include examples which are qualitatively similar to systems of great physical interest, including QCD (see e.g. [7, 10, 11, 25–27]), superconductors , superfluids, quantum Hall systems, and cold atom systems (see [8, 16] for recent reviews of applications to condensed matter systems). While it may be too optimistic to expect that we will be able to find gravitational systems that are exactly dual to real QCD or specific real-world condensed matter systems, these model systems can provide significant qualitative insight into generic phenomena that arise in strongly coupled systems similar to the real-world examples. There are already examples (e.g. the very low viscosity to entropy ratio for the quark-gluon plasma produced in heavy ion collisions) where the insight gained from holographic models offers the best theoretical understanding of an experimentally measured phenomenon (see e.g. [28–30]). With such potential for new theoretical insight into physically interesting systems, it seems fruitful to explore a wide variety of holographically constructed field theories. In doing so, we may uncover new qualitative phenomena in strongly cou8 Nc D4 01234 N D4 f 01235 X6 Figure 2.1: Brane construction for the holographic field theory. pled field theories that could help explain real-world physical phenomena or, more generally, lead to an improved understanding of quantum field theory at strong coupling. In addition, amassing a large number of detailed examples will help reveal which features of these systems are generic (and thus more likely to apply to other systems for which we may not have a precise gravity dual), and which are peculiar to specific constructions. Motivated by these considerations, we study in this chapter a holographic field theory closely related to the Sakai-Sugimoto model [7] of holographic QCD. Specifically, our theory has the same adjoint sector, but a different fundamental sector since we use probe D4-branes instead of probe D8-branes. Thus, our model is based on a brane construction where both the “colour branes” (which give rise to the adjoint sector) and the “flavour branes” are D4-branes. The relative orientation for the two sets of branes is: 0 1 2 3 4 5 6 7 8 9 Nc D4 × × × × × N f D4 × × × × × where we have listed the 10 spacetime dimensions of the underlying string theory and crossed-off the dimensions each brane fills. This configuration is shown in figure 2.1. At weak coupling, this system has a (complex scalar) tachyon in its 9 low-energy spectrum coming from the D4-D4’ strings. However, by separating the branes in a transverse direction (e.g. the 6 direction), we can arrange for this tachyon to become either massless or massive. The lightest D4-D4’ fermions have string scale masses in this case. For the original theory with no transverse separation, the geometrical SO(5) symmetry present in the adjoint sector of the theory is broken to SO(4), while the theory defined with a transverse separation between the two sets of branes has this symmetry broken to SO(3). In order to obtain a decoupled field theory, we want to take a low-energy decoupling limit in this brane setup. We would like to do this in such a way that the lightest modes of the D4-D4’ strings survive. It is plausible that we can tune the transverse separation of the two sets of branes as we take the limit to achieve this. In practice, we do not actually define an explicit decoupling limit starting from a brane configuration in asymptotically flat space. Instead, we do something much simpler (following the Sakai-Sugimoto example). We will always consider the limit where N f Nc , so that the fundamental matter does not affect the physics of the adjoint sector (i.e. the quenched approximation is accurate). Then the addition of fundamental matter can be achieved simply by adding probe D4-branes into the geometry dual to the field theory describing the low-energy degrees of freedom of the Nc D4-branes. Thus, instead of starting with colour and flavour branes in asymptotically flat space, we just look for a stable configuration of probe D4-branes in the geometry dual to the colour branes such that the configuration preserves the desired symmetries. Since we are working in a consistent background of string theory, we can say that the model we describe is some fully consistent quantum field theory sharing many qualitative features with QCD. Based on the weak coupling picture, it is tempting to suggest that the model we describe is one where the fundamental quarks are purely scalar, since the D4-D4’ fundamental fermions have string scale masses when the branes are tuned to make the lightest scalar massless or massive. Achieving a model with scalar quarks was one of the original motivations for studying this model, since we were interested to look at the qualitative similarities and differences between our model and the Sakai-Sugimoto model (where the fundamental matter is fermionic), and also to see whether any new qualitative phenomena appear in the physics of strongly coupled fundamental bosons. How10 ever, since our actual construction is less direct than an explicit decoupling limit, we cannot say definitively that the model includes only scalar quarks. Outline and Summary After a review of the basic setup in section 2.2, we carry out the analysis of probe brane embeddings in section 2.3, focusing on the case N f = 1. We find a twoparameter family of D4-brane embeddings.1 These are labeled by a parameter y0 that measures how far into the IR of the geometry the probe brane reaches and a parameter v0 that controls how much the brane is tilted into the compact direction of the field theory. The embeddings in this family correspond to the vacuum solutions for a two-parameter family of field theories. In each case, the solution preserves SO(3) ∼ SU(2) global symmetry, but for a one parameter family, this symmetry arises via a spontaneous breaking from SO(4) ∼ SU(2) × SU(2). For this special case, there is a family of solutions with the same SO(4)-preserving asymptotics. Each of the solutions preserves only SO(3), so we would expect an SO(3) vector of massless scalar Goldstone bosons associated with the broken symmetry. However, we do not find an ordinary discrete spectrum of mesons as in other models, but rather a continuous spectrum (as we would have in a conformal field theory)2 . This is related to the fact that the action governing small fluctuations of the gauge field on the probe brane is the ordinary Maxwell action in Minkowski space despite the nontrivial embedding of the probe brane in the curved background geometry. We do not have a good interpretation for this from the field theory point of view. However, the result crucially depends on the relative normalization of the ChernSimons and Born-Infeld terms in the brane action; if this normalization is changed at all, we get an ordinary discrete spectrum of mesons. Note that our results refer only to the small fluctuation analysis (keeping quadratic terms in the action for fluctuations of the probe brane about its equilibrium configurations). It is possible that including the effects of higher-order terms in the Born-Infeld action may 1 A holographic field theory corresponding to a different class of probe D4-brane embeddings was studied in [31] among other examples. Another model with D4-brane probes in a different background was studied in [32]. 2 A continuous meson spectrum was found also in [33] for a defect theory, but it is not clear whether the underlying mechanism is the same. 11 have interesting effects (in particular, they break the “accidental” five-dimensional Lorentz invariance present in the quadratic action for the gauge field), but this analysis is beyond the scope of the present work. One of the most interesting features of our model that the baryonic sector can be studied very reliably, as we discuss in section 2.5. As in the Sakai-Sugimoto model, baryon charge arises from D4-branes which wrap an S4 in the geometry (see section 2.5 for a review). But in our D4-D4 system, these wrapped D4-branes can smoothly reconnect with the probe D4-branes of the original embedding. Thus, states in the field theory with non-zero baryon number correspond to smooth D4-brane embeddings of different topology from the vacuum embedding. Baryon charge in the field theory corresponds in the bulk to a topological charge π4 (S4 ) for the embedding (relative to the original embedding). These smooth embeddings can be studied reliably using the abelian Born-Infeld action, so properties such as baryon mass and nuclear binding energies should be under complete control in the model. This is in contrast to the Sakai-Sugimoto model, where baryons are described by instantonlike configurations of the Yang-Mills fields on the probe D8-branes whose size is string-scale. In that case, higher α corrections to the Born-Infeld action should be important for a completely reliable treatment of baryon physics.3 On the other hand, given the results for the meson sector, our model is clearly much further from real QCD than the Sakai-Sugimoto model. 2.2 Setup In this section, we review the basic construction of our holographic field theory. We begin by describing the adjoint sector, and then describe the addition of flavour fields via the embedding of probe branes in the dual geometry. 2.2.1 Adjoint sector The adjoint sector of our model was originally proposed by Witten [10] as a construction of non-supersymmetric Yang-Mills theory. It is defined by the low-energy decoupling limit of N ≡ Nc D4-branes wrapped on a circle of length 2πR with 3 There have nevertheless been many studies of baryons and baryon physics in the Sakai-Sugimoto model making use of various approximations, see e.g. [34–38] for some of the early work. 12 anti-periodic boundary conditions for the fermions. This part of the theory has two dimensionless parameters, Nc and a coupling constant λ= λD4 , 2πR where λD4 = gY2 M N . The dimensionless parameter λ is the effective four-dimensional coupling at the Kaluza-Klein scale. For small λ , this coupling runs to strong coupling at a smaller scale ΛQCD ∼ 1 −c e λ R where the physics should be exactly that of pure 3+1 dimensional Yang-Mills theory (thanks to fermion masses generated by the anti-periodic boundary conditions and scalar masses generated at one loop). For large λ , the dual gravity theory becomes weakly curved, and physics is well described by type IIA supergravity on a background 2 ds = U R4 3 2 µ ν (ηµν dx dx + R4 U f (U)dx42 ) + 3 2 ( 1 dU 2 +U 2 dΩ24 ) f (U) 3 e φ F4 U 4 = gs R4 3 (2π)3 Nc (α ) 2 = ε4 . ω4 (2.1) Here ω4 = 38 π 2 is the volume of a unit 4-sphere, ε4 is the volume form on S4 , and f (U) = 1 − U0 U 3 . (2.2) The x4 direction, corresponding to the Kaluza-Klein direction in the field theory, is taken to be periodic, with coordinate periodicity 2πR. However, it is important to note that this x4 circle is contractible in the bulk since the x4 and U directions form a cigar-type geometry. The parameters R4 and U0 appearing in the supergravity solution are related to 13 the string theory parameters by R34 = πgs Nls3 U0 = 4 π gs Nls3 9 R2 while the four-dimensional gauge coupling λ is related to the string theory parameters as λ = 2π gs Nls . R In terms of the field theory parameters, the dilaton and string-frame curvature at √ 3 the tip of the cigar (the IR part of the geometry) are of order λ 2 /N and λ , so as usual, supergravity will be a reliable tool for studying the infrared physics when both λ and N are large (in this case, with N 2.2.2 3 λ 2 ). Fundamental matter The addition of fundamental matter manifests itself through the appearance of N f probe D4-branes in the geometry dual to the adjoint sector. These D4-branes are extended along the xµ directions, and are described by a one-dimensional path in the remaining radial, sphere, and x4 directions. It is convenient to redefine coordinates so that the metric in the radial and sphere directions takes the form α(ρ)(dρ 2 + ρ 2 dΩ24 ) . (2.3) These coordinates should satisfy dU dρ = . ρ U f (U) From this, we find the map U = U0 1 2 ρ ρ0 3 2 1 + 2 ρ0 ρ 3 2 2 3 , 2 where ρ0 = U0 2− 3 . Locally, the metric (2.3) is conformally equivalent to R5 , however we should note that the space has an infrared end at ρ = ρ0 where the X 4 circle contracts to a point. Thus, the ball ρ < ρ0 is not part of the geometry. 14 The equilibrium brane configurations come in from radial infinity, reach some minimum value of the radial coordinate, and go back out to radial infinity. These configurations asymptote to two specific directions on the S4 . The boundary conditions generically break the SO(5) symmetry to SO(3), and we expect that the minimum action embeddings do not break the symmetry further. In other words, we expect that the stable configurations will lie in a single plane in the R5 appearing in (2.3), so it will sometimes be convenient to use coordinates dρ 2 + ρ 2 dΩ24 = dr2 + r2 dθ 2 + dxT2 , in terms of which the equilibrium D4-brane configurations will be specified by xT = 0 and r(θ ) = ρ(θ ) (note that ρ = r for xT = 0). To write the action for the probe D4-branes, we focus on the case of a single brane, for which we can use the abelian Born-Infeld action S = −µ4 d 5 σ e−φ − det(gab + F˜ab ) , (2.4) together with the Chern-Simons part: S = µ4 ˜ ∑ C ∧ eF , (2.5) where F˜ = 2πα F . We choose static gauge X µ = σ µ for the field theory directions, and describe the nontrivial part of the embedding by functions X 4 (σ ), r(σ ), θ (σ ), XiT (σ ), where σ parameterizes the remaining coordinate along the brane. The pull-back metric appearing in the Born-Infeld action is then given explicitly by gµν = Gµν + G44 ∂µ X4 ∂ν X4 + Grr ∂µ r∂ν r + Gθ θ ∂µ θ ∂ν θ + Gi j ∂µ XTi ∂ν XTj gµσ = G44 ∂µ X4 ∂σ X4 + Grr ∂µ r∂σ r + Gθ θ ∂µ θ ∂σ θ + Gi j ∂µ XTi ∂σ XTj gσ σ = G44 ∂σ X4 ∂σ X4 + Grr ∂σ r∂σ r + Gθ θ ∂σ θ ∂σ θ + Gi j ∂σ XTi ∂σ XTj For now, we are interested in equilibrium brane configurations, which we assume 15 have XTi = 0, so we keep only terms in the action involving X4 (σ ), r(σ ) and θ (σ ). With this simplification, the Born-Infeld part of the action becomes S=− µ4 gs dσ d 4 x H(r(σ )) r2 where 3 H(r) = ρ3 1 + 30 r r2 3 R2 2 dθ dσ + dr dσ 2 , (2.6) 5 3 . We now turn to the Chern-Simons part of the action. Since the background we are considering involves a non-zero Ramond-Ramond four-form flux, the potentials C3 and the dual C5 are non-zero. For the configurations that we are considering (which are translation-invariant in the field theory directions), the pull-back of C3 is zero, but we have a non-zero pull-back for C5 . We find (see appendix 2.A for a derivation): 3 πN(α ) 2 3 C5 = (U −U03 )dt ∧ dx1 ∧ dx2 ∧ dx3 ∧ dx4 , 6 R4 so the Chern-Simons term in the action is SCS = µ4 C5 3 = µ4 2.3 πN(α ) 2 R64 dσ d 4 x (U 3 −U03 ) ∂ X4 . ∂σ Vacuum solutions For our calculations of the vacuum configurations, it is convenient to fix the remaining reparametrization invariance by choosing σ = θ . If we also define y= r , ρ0 x= 16 ρ0 X4 , R34 the resulting action is 5 S = µ4 ρ02 gs R 23 dθ h(y) y2 + (y )2 + g(y)(x )2 + − dθ q(y)x , 4 where 1 5 )3 , y3 (y3 − 1)2 g(y) = y 4 , (y3 + 1) 3 (y3 − 1)2 . q(y) = y3 3 h(y) = y 2 (1 + Since the resulting Lagrangian density does not depend explicitly on θ , we have a θ -independent quantity (analogous to energy for a time-independent Lagrangian density) given by y hy2 ∂S ∂S +x −S = ∂y ∂x y2 + (y )2 + g(y)(x )2 . Since the geometry caps off smoothly at some finite value of y, smooth brane configurations must have some minimal value of y for which y = 0. Calling this value y0 , and calling the derivative x at this point v0 , we have h(y)y2 y2 + (y )2 + g(y)(x )2 = h(y0 )y20 ≡B. y20 + g(y0 )v20 The action also does not depend explicitly on x (only on x ), so we have another constant −h(y)g(y)x y2 + (y )2 + g(y)(x )2 + q(y) = q(y0 ) − 17 h(y0 )g(y0 )v0 y20 + g(y0 )v20 ≡C . From the equations above, we can eliminate x to get: y2 B2 dy = ±y dθ We also find dx =± dy h2 (y) − (q(y) −C)2 g(y) −1 . (2.7) . (2.8) , (2.9) y(q(y) −C) y2 B2 Bg(y) 2 h2 (y) − (q(y)−C) g(y) −1 These two equations can be integrated to find θ (y) and x(y). Integrating, we find y θ (y) = 1 d y˜ y0 y˜2 B2 y˜ 2 ˜ h2 (y) ˜ − (q(y)−C) g(y) ˜ −1 where we define θ = 0 to be the angle at which the brane embedding reaches its minimum value of y. From this expression, it is straightforward to check that for any value y0 > 1 and v0 ,4 θ approaches a finite value as y goes to infinity. Thus, the brane configurations asymptote to lines of constant θ , as shown in figure 2.2. The relation between y0 and the maximal value of θ is given by 1 ∞ θ∞ (y0 ) = d y˜ y0 y˜ y˜2 B2 2 ˜ h2 (y) ˜ − (q(y)−C) g(y) ˜ . (2.10) −1 and plotted in figure 2.3 for various values of v0 . For any given v0 , we find that for large y0 , the asymptotic angle approaches a limiting value θMax (v0 ), given by v0 < 0 , 0 θMax (v0 ) = 1.044144565 v0 = 0 , π v0 > 0 . 2 For each v0 , there is a special value y0 = y∗ (v0 ) for which the two asymptotic ends of the brane go towards diametrically opposite points on the sphere. As y0 approaches 1, θ∞ increases without bound, corresponding to brane embeddings 4 Recall that y = 1 represents the IR end of the geometry. 18 Figure 2.2: Examples of brane embeddings (in the y − θ plane) for v0 = 0. The asymptotic angle between the two ends of the brane is 2θ∞ , which ranges from π for the stable embedding which extends to the smallest values of y, down to some value 2θMax ≈ 2.088 in the limit y0 → ∞. that wrap multiple times around the θ direction. However, for y0 < y∗ (v0 ) these embeddings are perturbatively unstable to “unwrapping,” i.e. slipping over the spherical hole in the geometry, as seen in figure 2.4. The perturbative instability will be demonstrated explicitly in the next section. For the special case θ∞ = π/2, the two ends of the probe brane go to diametrically opposite points on the S4 . For these asymptotics, we actually have a family of embeddings related by the SO(4) rotations that fix these diametrically opposite points on the sphere as shown in figure 2.5. This case corresponds to a spontaneous breaking of SO(4) → SO(3) (equivalently SU(2) × SU(2) → SU(2)), and we must therefore have an SO(3) vector of massless Goldstone bosons associated with the broken symmetry. It is these bosons that become tachyonic if we increase θ∞ be- 19 1.6 1.4 1.2 v0=2/3 v0=1/3 1 θ∞ v0=0 v =−1/ 0 0.8 3 v0=−2/3 0.6 0.4 0.2 0 2 4 6 8 10 y0 12 14 16 18 20 Figure 2.3: Asymptotic angle θ∞ on the sphere vs minimum brane position y0 in radial direction, for various values of v0 . Angle θ is defined to be zero at y = y0 . yond π/2 for fixed v0 . This is very similar to the naive brane picture in figure 2.1, where a tachyon develops if the transverse separation between the branes becomes too small, and we will revisit this massless Goldstone in section 3.3 when studying a closely related model in chapter 3. The behaviour of the embedding in the X4 direction can be obtained by integrating (2.8). We find that for any value of y0 , the x4 (y) asymptotes to a constant positive slope dx/dy, so that the brane continues to wrap the x4 direction as we go out to y = ∞. Because of the Chern-Simons coupling, the probe brane “prefers” a positive slope dx/dy; we see that the asymptotic slope is positive even if slope dx4 /dθ is negative at y = y0 . The behaviour of x4 is shown in figure 2.6. 2.4 Meson spectrum and stability In this section, we consider small fluctuations about the equilibrium brane configurations found in the previous section. We would like to determine which of the embeddings are perturbatively stable, and for these embeddings, to determine the 20 Figure 2.4: Example of multiple embeddings for the same asymptotic sphere angles. Only embeddings which do not “wrap” the sphere are stable. The rest are perturbatively unstable to slipping around the sphere, as shown. spectrum of small fluctuations that gives the meson spectrum for the theory. To determine the fluctuation spectrum, we start with the brane action (2.4) and expand to quadratic order about a chosen solution, parameterized by (y0 , v0 ). We consider all possible bosonic fluctuations, which include fluctuations in the x4 direction, fluctuations in the three transverse directions along the sphere (which we label by an SO(3) triplet of scalar fields XT ), fluctuations in the r − θ plane, and the gauge field fluctuations. In general, for the scalar field modes, the action for small fluctuations about the 21 Figure 2.5: Geometrical interpretation of Goldstone bosons for the special case θ∞ = π/2. y0 = 2 2 1.5 1 x4 0.5 0 −0.5 v0=2 v0=1 −1 v0=0 v0=−1 −1.5 v0=−2 −2 −1.5 −1 −0.5 0 θ 0.5 1 1.5 Figure 2.6: Behaviour of x4 vs θ for various values of v0 at y0 = 2. As a function of y, the slope dx4 /dy approaches a constant in each case. 22 vacuum solution takes the form S = −C 1 ∂φ A(y) 2 ∂ xµ ∞ 4 d x dy y0 2 1 ρ0 ∂φ + B(y) 2 R3 ∂y 2 + 1 ρ0 C(y)φ 2 , 2 R3 (2.11) while for the gauge fields, we have d4x S = −C ∞ dy y0 1 1 ρ0 A(y)Fµν F µν + B(y)Fµy F µy 4 2 R3 . It is convenient to define the functions A, B, and C using the function R(y) = (1 + y2 where dθ0 dy and dx0 dy dθ0 2 dx0 2 − 1 + g(y) ) 2, dy dy refer to the background embedding functions and are given in terms of y, y0 , and v0 by the equations (2.7) and (2.8). Explicitly, we have R(y) = 1− B2 (q(y) −C)2 − , y2 h2 (y) h2 (y)g(y) where B(y0 , v0 ) and C(y0 , v0 ) are defined in the previous section. In the special case where v0 = 0, we obtain: 10 R(y) = 1− y5 (y30 + 1) 3 10 y50 (y3 + 1) 3 (y3 − 1)2 − 3 (y + 1)2 y3 (y3 − 1)2 1 − 3 30 y0 (y − 1)2 2 . For the transverse scalar (XT ) fluctuations, we find A(y) = B(y) = C(y) = (y3 + 1) 9 2 R−1 (y) , y 5 (y3 + 1) 3 R(y) , 7 y2 2 2 1 (y3 + 1) 3 6y(y3 − 1)(y6 + 1) dx0 2 3 2 dθ0 R(y) (3y − 7)(1 + y ) + 4 2 y 112 dy dy (y3 + 1) 3 −3 dx0 (y3 + 1)(y3 − 1) , dy y5 23 where the last term in C comes from the Chern-Simons action. For the gauge field fluctuations, we find 1 A(y) = 1 2 1 2 1 (1 + y3 ) 3 R−1 (y) ; y 1 B(y) = y (1 + y3 ) 3 R(y) . Finally, the θ and X4 fluctuations mix with each other, and the fluctuation action for the combination (θ , x) is given as above where now A and B are matrices: (y3 +1) 5 2 0 (1 + g(y) dx dy ) y2 0 dθ0 − dx dy dy A(y) = R(y) 5 (y3 +1) 3 3 y2 B(y) = R3 (y) 0 dθ0 − dx dy dy (y3 −1)2 (y3 −1)2 1 3 1 7 y 2 (y3 +1) 3 y 2 (y3 +1) 3 1 3 y 2 (y3 +1) 3 2 0 (1 + y2 dθ dy ) 0 dθ0 (y − dx dy dy 1 (y3 −1)2 (y3 +1) 3 1 (y3 +1) 3 (y3 −1)2 1 5 y2 y2 In this special cases v0 = ±∞, we have dθ0 dy ; 1 3 −1)2 (y3 +1) 3 2 0 (1 + g(y) dx dy ) 0 dθ0 − dx dy dy (y3 −1)2 1 y2 2 0 (1 + y2 dθ dy ) . = 0, and so these matrices become diagonal. 2.4.1 Scalar modes For the scalar modes, the fluctuation actions above give rise to an equation of motion −A(y) ∂φ ∂ 2 φ ρ0 ∂ − 3 B(y) 2 ∂ xµ R ∂ y ∂y + ρ0 C(y)φ = 0 . R3 We look for solutions of the form φ (x, y) = eik·x f (y) where f (y) falls off fast enough so that the integral over y in the action converges (i.e. so that φ is a normalizible fluctuation). With this ansatz, the equation reduces to − ρ0 ∂ ∂f B(y) 3 R ∂y ∂y +( ρ0 C(y) − λ A(y)) f = 0 , R3 where λ = m2 represents the four-dimensional mass of the fluctuation. 24 (2.12) 2.4.2 Gauge modes In order to solve the gauge field fluctuation equations, it is convenient to choose a gauge ∂µ Aµ + A−1 ∂y (BAy ) = 0 . With this choice, the equations of motion for the various components of A decouple. For the components in the field theory directions, we have ∂ 2 Aν + A−1 ∂y (B∂y Aν ) = 0 , (2.13) while for the y component, we have ∂ 2 Ay + ∂y (A−1 ∂y (BAy )) = 0 . (2.14) This set of equations has a residual gauge invariance under transformations Aµ → Aµ + ∂µ λ , Ay → Ay + ∂y λ , (2.15) for any λ satsfying ∂ 2 λ + A−1 ∂y (B∂y λ ) . (2.16) This allows us to make a further gauge choice Ay = 0. The gauge field fluctuation modes are then captured by solutions to the equation (2.13). These can be found by separation of variables, considering solutions of the form Aµ = εµ (k)eik·x a(y) , where we require k · ε(k) = 0 by our original gauge condition, and where a(y) is a normalizible solution to −k2 A(y)a + ∂y (B∂y a) . This eigenvalue equation is the same type (2.12) as we obtain from the scalar equation. 25 2.4.3 Converting to a quantum mechanics problem For both gauge and scalar modes, we need to determine the values of λ for which normalizible solutions to the equation (2.12) exist. We can convert this into a simple quantum mechanics problem as follows. First, note that the equation arises from an action 2 1 ∂φ B(y) 2 ∂y ∞ S= dy y0 1 + (C(y) − λ˜ A(y))φ 2 2 , (2.17) where we have defined λ˜ = R3 λ /ρ0 . Now, we define a new variable z such that y z= y0 such that d y˜ , B(y) ˜ dy = B(y) , dz z(y0 ) = 0 . In the new variables, the action becomes z∞ S= dz 0 1 2 ∂φ ∂z 2 1 + B(y(z))(C(y(z)) − λ˜ A(y(z)))φ 2 2 where ∞ z∞ = y0 , (2.18) d y˜ . B(y) ˜ This gives rise to the time-independent Schrodinger equation for E = 0, − f (z) +V (z) f (z) = 0 , so our problem is reduced to determining for which values of λ˜ the Schrodinger equation with potential V (z) = B(y(z))(C(y(z)) − λ˜ A(y(z))) has a bound state with zero energy. 26 (2.19) 2.4.4 Gauge field fluctuations: a continuous spectrum We consider first the gauge field fluctuations. Here, we note that C = 0 and A(y)B(y) = 1, so our quantum mechanics potential is simply VA (z) = −λ . Also, in this case, we find z∞ = ∞. So we do not have any bound states for any λ , though there are zero-energy solutions √ λz fλ (z) = e±i to the Schrodinger equation for any λ ≥ 0. These do not fall off fast enough at large z to be normalizible, but we can superpose solutions with different λ to get normalizible solutions. Since the four-momentum in the field theory directions is related to λ by −k2 = λ , these superpositions will not be eigenstates of fourmomentum. Thus there are no true particle states in the field theory arising from the gauge mode fluctuations. To understand this better, we note that with the new radial variable, the action governing small fluctuations of the gauge field is exactly the Maxwell action in 4 + 1 dimensional Minkowski space (despite the nontrivial embedding of the brane in a non-trivial curved space!): S∝ 1 d 4 xdz{− FAB F AB } . 4 Configurations with finite energy in the field theory correspond to solutions of these 5D Maxwell equations that fall off sufficiently rapidly for large z and x. These are wavepackets obtained by appropriate superpositions of plane waves AA = εA (k, kz )eik·x+ikz ·z . So the meson sector of our field theory (at least, the part coming from the gauge field fluctuations) behaves more like a conformal field theory with a continuous spectrum than a massive field theory with particles. It is interesting to note that the behaviour we find depends crucially on the rel27 2 dy 0.2 0.4 0.6 0.8 1 0 –2 –4 V1 –6 –8 –10 Figure 2.7: Effective potential V1 (y) = C(y)B(y) for v0 = 0 and various values of y0 . Lower graphs have smaller y0 . ative normalization of the Chern-Simons and Born-Infeld terms in the probe D4brane action. If we change the relative coefficient even slightly, the function R(y) changes its asymptotic behaviour. In the effective quantum mechanics problem, the effective potential is still VA (z) = −λ , but now z∞ is finite. Since the fluctuation must vanish at z = ±z∞ , the quantum mechanics problem now has a discrete spectrum (that of an infinite square well), and we would have a discrete spectrum of mesons in the dual field theory. 2.4.5 Transverse scalar fluctuations For the transverse scalar fluctuations, the effective potential (2.19) has a λ independent part and a term proportional to λ , plotted in figures 2.7 and 2.8 respectively.5 For the λ -independent part of the potential V1 = BC, we see that for large enough y0 there will be no bound states, only a continuous spectrum with E ≥ 0. For these values of y0 , there will be a zero energy (albeit non-normalizible) solution to the Schrodinger equation for all λ ≥ 0 and no negative λ . Thus, the situation is similar to that for the gauge field modes. 5 Note that we are plotting the potentials in this section as a function of y rather than as a function of z, thus, the actual effective potential in the effective quantum mechanical problems will be related to the ones show by a reparametrization of the horizontal axis. 28 dy 0.2 0.4 0.6 0.8 1 0 –1 –2 –3 V2 –4 –5 –6 –7 –8 Figure 2.8: Effective potential V2 (y) = −A(y)B(y) for v0 = 0 and various values of y0 . Lower graphs have smaller y0 . For y0 small enough, the potential V1 will have one or more bound states with E < 0. For the full potential V1 + λV2 , these bound state energies increase as we decrease λ below zero, and so we will have a bound state with zero energy for one or more negative values of λ . Thus, the field theory will be unstable for y0 below some critical value y∗ (v0 ) at which the potential V1 develops a normalizible bound state. We anticipated this instability in the previous section as the tendency for certain brane configurations to “slip” over the sphere and lower their action. Based on that intuition, we expect that the critical value y∗ (v0 ) will be the same value for which the asymptotic behaviour of the brane configuration has θ∞ = π/2. 2.4.6 X4 and θ fluctuations. In general, the spectrum of fluctuations in the X4 and θ directions is more complicated to obtain, since the equations are coupled, but we can analyze the fluctuations in the simple case where v0 = ±∞. For v0 = −∞, the effective potentials for the X4 and θ fluctuations are shown in figures 2.9 and 2.10. Again, we multiply each of these by λ and ask for which values of λ the effective quantum mechanics problem has a zero energy eigenvalue. The result is again that any value λ ≥ 0, but no negative values of λ will work. Thus, at 29 y 2 4 6 8 10 0 –0.2 –0.4 –0.6 –0.8 –1 Figure 2.9: Effective potential V4 (y) = −A(y)B(y) for v0 = −∞ and y0 = 1. 0.5 1 y 1.5 2 2.5 3 0 –50 –100 V –150 –200 –250 –300 Figure 2.10: Effective potential Vθ (y) = −A(y)B(y) for v0 = −∞ and y0 = 1. 30 least for some values of parameters, we have shown that the model is tachyon-free. Since we do have modes for which the quadratic action is arbitrarily small, higher order terms (e.g. quartic in the fluctuations) may be important, but an analysis of these lies beyond the scope of the present work. 2.5 Baryons By the gauge-theory / gravity dictionary, gauge fields in the bulk can be associated with conserved currents in the boundary theory. As in the Sakai-Sugimoto model, we have a gauge field living on the probe branes associated with our fundamental matter. This corresponds to the conserved baryon number (more precisely, quark number) current in the dual field theory. Specifically, the boundary value of the electrostatic potential A0 (equivalently, the non-normalizible mode) corresponds to a chemical potential for baryon charge, while the electric flux at the boundary (equivalently, the normalizible mode of A0 ) corresponds to the expectation value of baryon charge. In order to have a state in the field theory with baryon charge, we need a source for electric flux on the probe brane. The simplest such source is a fundamental string endpoint (recall that the string action has a boundary term A where A is the gauge field on the brane). We can think of such an endpoint as corresponding to a single fundamental quark. If a string has both of its ends on the probe brane, we have two endpoints, but with opposite orientations, so this corresponds to a mesonic state with a quark and anti-quark. In order to have a baryon state, we must have N strings with the same orientation ending on the probe brane. For a finite energy state, these strings must begin at some other source in the bulk. In our background, such a source is provided by D4-branes wrapped on S4 [39]. These necessarily have N string endpoints, since the background D4-brane flux gives rise to N units of charge on the spherical D4-branes, so we need N units of the opposite charge (coming from the string endpoints) to satisfy the Gauss law constraint. Thus, a finite energy configuration with a single unit of baryon charge is given by a D4-brane wrapped on S4 , together with N fundamental strings stretched between this D4-brane and the probe D4-branes. A special feature of our model is that these wrapped D4-branes can smoothly combine with the probe D4-brane (af- 31 ter shrinking the strings to zero size) to give a configuration with lower energy. In the final configuration, there are no explicit fundamental strings; we simply have a configuration of the probe brane that now wraps the S4 . In the final configuration, the source for the electric field on the brane is the bulk flux of the Ramond-Ramond four-form, via the coupling a ∧ F4 . Mathematically, the baryon charge in the field theory corresponds to an element of π4 (S4 ) = Z associated with the embedding. To see this, note that the probe brane embeddings correspond to mappings to the bulk space from R4 , topologically equivalent to a ball if we add the sphere at infinity. Given any probe brane embedding E with the same asymptotic behaviour as the vacuum embedding E0 , we can define a map from a topological S4 to the bulk spacetime by splitting the S4 into two balls along an S3 and using the maps E and E0 to define the maps from the two balls. By considering only the sphere directions in the bulk, we can project this down to a mapping S4 → S4 , and such mappings may be associated with elements of the homotopy group π4 (S4 ) ∼ Z. This integer gives the baryon number of the configuration in the field theory. In order to find the actual bulk embedding corresponding to a single baryon, it is necessary to find the probe brane embedding with a single unit of winding on the S4 (relative to the vacuum embedding) which has the minimum energy. Similarly, to find the bulk embedding corresponding to a nucleus with n baryons, we want to find the minimal energy brane embedding with n units of winding on the S4 . In general, a numerical analysis will be required, but it should be possible to obtain precise results for the masses of small nuclei. 2.A Ramond-Ramond forms The results of our analysis depend crucially on the relative normalization of the Born-Infeld and Chern-Simons parts of the brane action. Since there is some variation in the literature for the normalization of the Ramond-Ramond fields in the Witten background, we include here a derivation of the correct result (consistent with a subset of previous papers). We use the fact that with the correct RamondRamond flux, a D4-brane wrapped on S4 should have induced N units of charge, so that a configuration with N string endpoints (of the correct orientation) on the 32 wrapped D4-brane should have zero integrated charge and thus satisfy the Gauss Law constraint for the compact surface. The brane action is µ4 (2πα )A ∧ F4 , while the action for each string endpoint is simply Ap , where A p is the pullback of the gauge field to the worldline of the string endpoint. We know that the four form is constant on the sphere, i.e. we have F4 = Cε4 where ε4 is the volume form on the sphere whose integral is 8π 2 /3. From this information, we find that the charge density on a wrapped D4-brane with string endpoints fixed at various locations Ωi is ρ = µ4 (2πα )Cε4 − ∑ δ (Ω − Ωi ) , i where the delta functions are defined as four-forms localized at the indicated point that integrate to 1. Integrating the density over the sphere and setting the result to zero gives: µ4 (2πα )Cω4 = N . Using the result that µ4 = (2π)−4 (α )−5/2 , we conclude that the Ramond-Ramond four-form is 3 F4 = 3πN(α ) 2 ε4 . From this, we can calculate that the dual six-form in the background is given by (the sign here is related to a choice of convention for the direction of the flux) (F6 )01234U 1 (F4 )θ1 θ2 θ3 θ4 = −G00 G11 G22 G33 G44 GUU √ −G 3 U2 3πN(α ) 2 , = 6 F4 33 and using F6 = dC5 , we obtain the five-form C5 = 3 U 3 −U03 πN(α ) 2 dt ∧ dx1 ∧ dx2 ∧ dx3 ∧ dx4 , 6 F4 where we have fixed the constant of integration so that the form is non-singular at U = U0 . 34 Chapter 3 Bottom-Up Mesons 3.1 Introduction In chapter 2 we introduced a holographic field theory closely related to the SakaiSugimoto model [7, 40]. The studied field theory included the same adjoint sector, but the probe D8 branes used to form the fundamental matter in the Sakai-Sugimoto model were replaced by probe D4 branes. We argued that this replacement meant the fundamental sector of the field theory in chapter 2 consisted of scalars instead of fermions. However, we never specified an exact low-energy decoupling limit. Even with no exact low-energy decoupling limit, we argued that the model in chapter 2 described some fully consistent quantum field theory sharing many qualitative features with QCD. In this chapter we move even further away from the comfort of the Sakai-Sugimoto model by removing the background Ramond-Ramond flux in addition to changing the probe branes from D8 branes to D4 branes. Thus out starting point is no longer a well known consistent background of string theory, but nonetheless it we believe in the gauge / gravity duality it is quite likely we are describing some field theory. This bottom-up approach is not without precedent and is typical when building phenomenological models of superconductors and superfluids[9, 14–16]. Here we have no particular guiding phenomenon we wish to describe, but are simply curious, as we were in chapter 2, to study a model closely related to the holographic QCD models already known. One of the interesting features we will see in this model is a geometric representation of a Goldstone 35 boson. Finally before describing the outline and summary of this chapter, we should mention that the expectation that this geometric model in this chapter corresponds to a field theory closely related to QCD influences the language used. References to adjoint and fundamental sectors, to mesons and baryons are all the direct analogues to those same aspects that we studied in the model of chapter 2. Outline and Summary After a review of the basic setup in section 3.2, we carry out the analysis of probe brane embeddings in section 3.3, focusing on the case where the number of probe branes is N f = 1. We find a one-parameter family of locally-stable D4-brane embeddings. These are labeled by a parameter y0 ∈ [y∗ , ∞) that measures how far into the IR of the geometry the probe brane reaches (y = 1 corresponds to the IR end of the geometry, while y∗ = 1.09038 . . . is the minimum value for a stable embedding). The embeddings in this family correspond to the vacuum solutions for a one-parameter family of field theories. In each case, the solution preserves SO(3) ∼ SU(2) global symmetry, but for y = y∗ , this symmetry arises via a spontaneous breaking from SO(4) ∼ SU(2) × SU(2). For this special case, there is a family of solutions with the same SO(4)-preserving asymptotics. Each of the solutions preserves only SO(3), and we have an SO(3) vector of massless scalar Goldstone bosons associated with the broken symmetry. These mesons gain mass as we increase y0 , and for all larger values of y0 , all the mesons (which correspond to small fluctuations about the original brane configuration) are massive. For all but one of the meson states, we find that the mass behaves like m2 ∝ y0 for large y0 , consistent with an interpretation of y0 as the mass-squared of the fundamental scalar quarks in the model. However, there is a single scalar particle (an SO(3) singlet) whose mass decreases for large y0 as m2 ∝ 1/y20 . Thus, for large y0 we have a single scalar particle that is parametrically lighter than all the other mesons. This lightness is not a consequence of any approximate symmetry, but is related to the “generalized conformal symmetry” of the near-horizon D4-brane background, which involves a scaling of the radial coordinate together with a transformation of the string coupling. In the limit y0 → ∞ where the mass goes to zero, the meson 36 30 xi x4 y A 25 unstable m2 20 15 10 5 0 0 y∗ 2 3 y0 4 5 6 Figure 3.1: Spectrum of low-lying bosonic mesons as a function of the embedding parameter y0 . For y0 = y∗ , we have massless Goldstone bosons associated with SU(2) × SU(2) → SU(2) symmetry breaking. For large y0 , a single scalar meson becomes parametrically light. sector completely decouples from the theory. A detailed discussion of this light scalar may be found in section 3.4, which begins with our presentation of the complete fluctuation analysis to determine the bosonic meson spectrum. Our results for the spectrum are summarized in figure 3.1. In section 3.5, we study the theory at finite temperature. We recall that above a critical temperature, the theory is in a deconfined phase [10, 41], manifested in the bulk picture by a black brane geometry. In similar holographic field theories, probe brane embeddings in a deconfined phase either sit completely outside the black hole (the “Minkowski embeddings” where we still have non-dissociated mesons) or extend all the way to the horizon (the “black hole embeddings” where all mesons are dissociated) [27]. In our case, the theories that admit stable embeddings at low 37 temperature have only black hole embeddings at high temperature, so all mesons “melt” at the deconfinement transition. 3.2 Setup In this section, we review the basic construction of our holographic field theory. We begin by describing the adjoint sector, and then describe the addition of flavour fields via the embedding of probe branes in the dual geometry. 3.2.1 Adjoint sector The adjoint sector of our model is based on the same one used in chapter 2 and that was originally proposed by Witten [10] as a construction of non-supersymmetric Yang-Mills theory. For the full description please refer back to section 2.2.1. The difference here is that we remove the background Ramond-Ramond flux. This removal leaves us with only the background geometry and the dilaton which are both given by: 2 ds e φ = = gs U R4 3 2 µ ν (ηµν dx dx + U R4 R4 U f (U)dx42 ) + 3 2 ( 1 dU 2 +U 2 dΩ24 ) , f (U) 3 4 . (3.1) Where we have used f (U) = 1 − U0 U 3 . (3.2) And also just as before, the x4 direction, corresponding to the Kaluza-Klein direction in the field theory, is taken to be periodic, with coordinate periodicity 2πR, however, it is important to note that this x4 circle is contractible in the bulk since the x4 and U directions form a cigar-type geometry. 3.2.2 Fundamental matter The addition of fundamental matter manifests itself exactly as it did in chapter 2, through the appearance of N f probe D4-branes in the geometry. Thus to describe their setup again, these D4-branes are extended along the xµ directions, sit at a point 38 in the x4 direction, and are described by a one-dimensional path in the remaining radial and sphere directions. It is convenient to redefine coordinates so that the metric in these directions takes the form α(ρ)(dρ 2 + ρ 2 dΩ24 ) . (3.3) These coordinates should satisfy dU dρ . = ρ U f (U) From this, we find the map U = U0 1 2 ρ ρ0 3 2 1 + 2 ρ0 ρ 3 2 2 3 . 2 where ρ0 = U0 2− 3 . Locally, the metric (3.3) is conformally equivalent to R5 , however we should note that the space has an infrared end at ρ = ρ0 where the X 4 circle contracts to a point. Thus, the ball ρ < ρ0 is not part of the geometry. We expect that the stable configurations will lie in a single plane in the R5 appearing in (3.3), so it will sometimes be convenient to use coordinates dρ 2 + ρ 2 dΩ24 = dr2 + r2 dθ 2 + dxT2 , in terms of which the equilibrium D4-brane configurations will be specified by xT = 0 and r(θ ) = ρ(θ ) (note that ρ = r for xT = 0). To write the action for the probe D4-branes, we focus on the case of a single brane, for which we can use the abelian Born-Infeld action S = −µ4 d 5 σ e−φ − det(gab + F˜ab ) , where for this setup we have F˜ = 2πα F = 0 . 39 (3.4) We choose static gauge X µ = σ µ for the field theory directions, and describe the nontrivial part of the embedding by functions X 4 (σ ), r(σ ), θ (σ ), XiT (σ ), where σ parameterizes the remaining coordinate along the brane. The pull-back metric appearing in the Born-Infeld action is then given explicitly by gµν = Gµν + G44 ∂µ X4 ∂ν X4 + Grr ∂µ r∂ν r + Gθ θ ∂µ θ ∂ν θ + Gi j ∂µ XTi ∂ν XTj gµσ = G44 ∂µ X4 ∂σ X4 + Grr ∂µ r∂σ r + Gθ θ ∂µ θ ∂σ θ + Gi j ∂µ XTi ∂σ XTj gσ σ = G44 ∂σ X4 ∂σ X4 + Grr ∂σ r∂σ r + Gθ θ ∂σ θ ∂σ θ + Gi j ∂σ XTi ∂σ XTj . For now, we are interested in equilibrium brane configurations, which we assume have X 4 = XTi = 0, so we keep only terms in the action involving r(σ ) and θ (σ ). With this simplification, the action becomes S = −µq dσ H(r(σ )) r2 2 dθ dσ + dr dσ 2 (3.5) , where 3 H(r) = 3.3 5 3 ρ3 1 + 30 r r2 3 R2 . Vacuum solutions For our calculations of the vacuum configurations, it is convenient to fix the remaining reparametrization invariance by choosing σ = θ . If we also define y= r , ρ0 the resulting action is S = −µ4 ρ0 3 2 ρ0 R3 dθ h(y) y2 + (y )2 , where h(y) = y 3 2 1 1+ 3 y 40 5 3 . S4 X4 θ y 0 Figure 3.2: Schematic of a brane embedding. Left picture shows a plane in the space formed by the radial direction and the S4 directions. Right picture shows same embedding in radial and x4 directions. Since the resulting action does not depend on θ , we have a θ -independent quantity y h(y)y2 ∂S −S = ∂y y2 + (y )2 . The brane configurations that we find all have some minimal value of y for which y = 0. Calling this value y0 , we have h(y)y2 y2 + (y )2 = h(y0 )y0 , which gives h2 (y)y2 −1 . h2 (y0 )y20 dy = ±y dθ Integrating, we find y θ (y) = 1 dx y0 x 10 3 10 3 5 x (1+y0 ) 3 y50 (1+x3 ) , (3.6) −1 where we define θ = 0 to be the angle at which the brane embedding reaches its minimum value of y. From this expression, it is straightforward to check that for 41 π 2 1. 4 1. 2 θ∞ 1 0. 8 π 0.56 0. 4 0. 2 0 0 y∗ 2 4 6 y0 8 10 12 Figure 3.3: Asymptotic angle on S4 vs minimum radial position for probe D4-brane embeddings. Function asymptotes to infinity at y0 = 1. Values of θ∞ in (π/5, π/2] correspond to stable embeddings. any value y0 > 1,1 θ approaches a finite value as y goes to infinity, as indicated in figure 3.2. Thus, the brane configurations asymptote to lines of constant θ .2 The relation between y0 and the maximal value of θ is given by 1 ∞ θ∞ (y0 ) = dy y0 y 10 3 10 5 3 y0 (1+y ) 3 y50 (1+y3 ) (3.7) −1 and plotted in figure 3.3. We see that for large y0 , the asymptotic angle approaches the limiting value θ∞ = π5 . For y0 = y∗ , the two asymptotic ends of the brane go towards diametrically opposite points on the sphere. As y0 approaches 1, θ∞ increases without bound, corresponding to brane embeddings that wrap multiple 1 Recall that y=1 represents the IR end of the geometry. is straightforward to check that configurations with θ = const are also solutions to the embedding equations. However, the brane cannot simply end at y = 1, so the only way to make such solutions (with fixed x4 physical is to patch them on to another such solution which sits at the diametrically opposite point on the x4 circle. This type of embedding and its generalizations were studied in the earlier work [31]. 2 It 42 Figure 3.4: Examples of stable brane embeddings. times around the θ direction. Some representative embeddings are shown in figure 3.4. As shown in figure 3.5, there exist discrete families of embeddings with the same asymptotics, which therefore correspond to different vacua in the same field theory. However, as we will see in the next section, all the embeddings with θ∞ > π/2 are unstable to “unwrapping,” in that they might slip over the spherical hole in the geometry, as seen in figure 3.5. Thus, each field theory has at most one stable vacuum, and the stable theories are those with brane asymptotics θ∞ ∈ ( π5 , π2 ]. For the special case θ∞ = π/2 (corresponding to y0 = y∗ = 1.0903795 . . .), the two ends of the probe brane go to diametrically opposite points on the S5 . For these asymptotics, we actually have a family of embeddings related by the SO(4) rotations that fix these diametrically opposite points on the sphere as shown in figure 3.6. This case corresponds to a spontaneous breaking of SO(4) → SO(3) (equivalently SU(2) × SU(2) → SU(2)), and we must therefore have an SO(3) vector of massless Goldstone bosons associated with the broken symmetry. It is these bosons that become tachyonic if increase θ∞ beyond π/2 (or equivalently, try to decrease the minimum radial position y0 below the value y∗ ), just as they did in 43 Figure 3.5: Example of multiple embeddings for the same brane asymptotics. There are additional embeddings with smaller y0 and more “windings” around the sphere, however only the embedding with the largest y0 is stable. The rest are perturbatively unstable to slipping around the sphere, as shown. Figure 3.6: Geometrical interpretation of zero modes (massless mesons) for the special case θ∞ = π/2. 44 section 2.3 of chapter 2. 3.4 Meson spectrum and stability In this section, we consider small fluctuations about the equilibrium brane configurations found in the previous section. We would like to determine which of the embeddings are perturbatively stable, and for these embeddings, to determine the spectrum of small fluctuations that gives the meson spectrum for the theory. To determine the fluctuation spectrum, we start with the brane action (3.4) and expand to quadratic order about a chosen solution, parameterized by y0 . We consider all possible bosonic fluctuations, which include fluctuations in the x4 direction, fluctuations in the three transverse directions along the sphere (which we label by an SO(3) triplet of scalar fields XT ), fluctuations in the r − θ plane, and the gauge field fluctuations. In general, for the scalar field modes, the action for small fluctuations about the vacuum solution takes the form S = −C 2 1 ∂φ A(y) 2 ∂ xµ ∞ d4x dy y0 + 2 1 ρ0 ∂φ B(y) 3 2R ∂y + 1 ρ0 C(y)φ 2 2 R3 , (3.8) while for the gauge fields, we have ∞ d4x S = −C dy y0 1 ρ0 1 A(y)Fµν F µν + B(y)Fµy F µy 4 2 R3 . There are no mixing terms between gauge field and scalar fields at quadratic order. For the various types of fluctuations, we find the following: Mode X4 XT θ Aµ A(y) B(y) (y3 −1)2 1 7 (y3 +1) 3 y 2 R−1 (y) C(y) 1 (y3 −1)2 (y3 +1) 3 5 5 (y3 +1) −1 R (y) (y3 +1) 3 (y3 +1) (y3 +1) 3 9 y2 5 y2 7 y2 5 R(y) (y3 +1) 3 11 y2 5 R(y) 1 1 R(y) 0 y2 1 y 2 (1+y3 ) 3 3 y2 R3 (y) 1 3 1 2 R−1 (y) y (1 + y3 ) R(y) 45 0 5 y3 −1 2 y3 +1 − 1 R−1 (y) Here 10 R(y) = 3.4.1 1− y5 (y30 + 1) 3 10 y50 (y3 + 1) 3 . Scalar fluctuations The scalar fluctuation action above gives rise to an equation of motion −A(y) ∂ 2 φ ρ0 ∂ ∂φ − 3 B(y) 2 ∂ xµ R ∂ y ∂y + ρ0 C(y)φ = 0 . R3 We look for solutions of the form φ (x, y) = eik·x f (y) , where f (y) falls off fast enough so that the integral over y in the action converges (i.e. so that φ is a normalizible fluctuation). With this ansatz, the equation reduces to − ρ0 ∂ ∂f B(y) 3 R ∂y ∂y +( ρ0 C(y) − λ A(y)) f = 0 , R3 where λ = m2 represents the four-dimensional mass of the fluctuation. The resulting Schrodinger-like equation can now be solved numerically to determine the values of λ for which normalizible solutions exist. These eigenfunctions are either even or odd functions in the θ coordinates. We have used three different methods to determine the eigenvalues. The first is the “shooting method” in which the differential equation is numerically integrated starting from either even or odd boundary conditions at y = y0 . The parameter λ is then varied until the result drops to zero at large y.3 In the second method, we choose some complete basis of normalizible functions hn (y) (chosen to correspond to either even or odd boundary conditions at θ = 0) and approximate N f (y) = ∑ fn hn (y) . n=1 3 In practice, one looks for values of λ at which the solution switches from having positive values at large y to having negative values at large y or vice versa. 46 The action reduces to S= 1 ∑ fm (λ Amn + Bmn +Cmn ) fn , 2 m,n where Amn = dyA(y)hm (y)hn (y) Bmn = dyB(y)hm (y)hn (y) Cmn = dyC(y)hm (y)hn (y) . The approximate eigenvalues are determined by extremizing the action, which gives the matrix eigenvalue equation A−1 (B +C) f = λ f . As we increase N, the eigenvalues determined in this way should converge to the exact ones (though the accuracy is much better for the eigenvalues λn with n N). In the third method, we determine the power series solutions of the differential equation about the points y = y0 and y = ∞. We choose boundary conditions at y = y0 corresponding to either an even or an odd function of θ , and boundary conditions at y = ∞ corresponding to a normalizible solution (the appropriate falloff can be determined from the large y behaviour of the differential equation). It can be shown that the large-y power series converges in the interval (y0 , ∞), while the √ small y power series converges in the interval [y0 , y0 + min(2y0 − y10 , |y0 − 21 − 3 2 i|]. Thus, by choosing enough terms in each power series, there is a range of y values for which both series accurately approximate the corresponding exact solutions. Choosing one of these y values,4 we now search for values of λ for which the value and first derivative of the two power series can be made to match by a choice of normalization. For these values of λ the two power series can be patched together to give a good approximate solution, which is normalizible by construction.5 This method can be used to quickly generate very accurate results, except for values of y0 close to 1, where the radius of convergence of the power series around y0 is 4 In practice we choose a value for which addition an additional term to either power series gives the same fractional change in the function. At this point, the combined error from the two power series is minimized. 5 Using Maple, we can calculate the power series coefficients analytically without approximation, so it is no problem to include 50 or more terms. 47 50 45 40 unstable 35 m2y 30 25 20 15 10 5 0 y∗ 0 2 4 6 y0 8 10 12 Figure 3.7: Spectrum of mesons arising from fluctuations of the brane embedding in the y − θ plane. small, and an impractical number of terms in the large y power series is required to achieve accurate results. Our results for the eigenvalues are summarized in figures 3.7, 3.8, 3.9 and 3.10, and the summary figure 3.1 in the Introduction. As we expected, we find that the lightest XT fluctuation becomes massless at the value y0 ≈ y∗ = 1.0903795 . . .. This corresponds to θ∞ = π 2 where the two ends of the brane are diametrically opposite from each other and there is a family of embeddings with the same asymptotics related by SO(4) transformations. The massless modes are Goldstone bosons for the spontaneous breaking of SO(4) (the symmetry preserved by the boundary conditions for the probe brane) to SO(3) (the symmetry preserved by the bulk embedding). For smaller values of y0 , this mode becomes tachyonic, so the corresponding vacua in the field theory are perturbatively unstable. For y0 > y∗ , we find that all fluctuations are massive. For almost all modes, we find that the large y0 behaviour of the mass goes like m2 ∝ y0 , consistent with an identification between y0 and the bare quark mass in the field theory Lagrangian. However, strikingly, for one single scalar mode, the lightest fluctuation mode in the r − θ plane, we find a mass that goes to zero for large y0 , m2 ∝ y−2 0 . One other notable feature of the spectrum for large y0 is that the odd r − θ modes become 48 100 90 80 unstable 70 i mx 2 60 50 40 30 20 10 0 y∗ 0 2 4 6 y0 8 10 12 Figure 3.8: Spectrum of mesons arising from fluctuations of the brane embedding in the transverse sphere directions. 100 90 80 unstable 70 m2x4 60 50 40 30 20 10 0 0 y∗ 2 4 6 y0 8 10 12 Figure 3.9: Spectrum of mesons arising from fluctuations of the brane embedding in the compact x4 direction. 49 30 25 unstable m2A 20 15 10 5 0 0 y∗ 2 4 6 y0 8 10 12 Figure 3.10: Spectrum of mesons arising from fluctuations of the gauge field on the probe D4-brane. degenerate with the even x4 modes in the limit, as we can see in figure 3.1. 3.4.2 Interpretation of the light scalar It is quite surprising that we should find a parametrically light scalar particle for large y0 . In this limit, all the other meson states (and also the baryon states) become heavy, so the behaviour is similar to a limit in QCD where the fundamental quark masses become large. If large y0 corresponds to large quark mass in our case (as the naive brane configuration would also suggest), the lightness of our scalar mode must be due to a large binding energy for the quarks in this meson due to strong interactions. Typically, we would expect that such a parametric lightness would be associated with an approximate symmetry (e.g. the case of a pseudo-Goldstone boson or broken SUSY). However, this is not quite true in our case. From the bulk perspective, the crucial feature that leads to the appearance of this light mode is the fact that the brane asymptotics (i.e. the values of θ∞ ) are very similar for different large values of y0 (recall that θ∞ approaches a limit π/5 for large y0 ). Thus, for large y0 , we can get to a new extremum of the action with a very slight change in the asymptotics. In a sense, these different embeddings with 50 nearby large values of y0 are related by almost-normalizible fluctuations. If we did have normalizible fluctuations relating a continuous family of stable embeddings, these fluctuations would correspond to massless particles. So a way to understand why we have such a light mode is that we can get to another extremum of the action by a fluctuation which is almost normalizible. The actual light mode corresponds to a normalizible fluctuation that is very similar to this one (we have verified this explicitly). Further insight into our light mode can be gained by noting that at large y0 , the geometry is very similar to the supersymmetric near-horizon D4-brane geometry obtained by setting f (U) = 1 in (3.2). In this geometry, dual to D4-branes compactified with periodic boundary conditions for fermions, the probe brane action simplifies to S=− const gs 3 dθ y 2 y2 + (y )2 . (3.9) It is straightforward to show that all solutions to the corresponding equations of motion have the form y = y0 f (θ ), where y diverges at the same asymptotic angle θ∞ regardless of y0 . The fluctuations that relate these solutions are normalizible, and correspond to an exactly massless scalar meson in the field theory. The situation is similar to that of reference [42], where a similar massless scalar could be understood as a Goldstone boson since the probe brane configuration spontaneously broke a conformal symmetry of the background geometry.6 However, in our case, the low-energy D4-brane theory is not conformally invariant, and the massless mode is not a Goldstone boson. The existence of a family of solutions with the same asymptotics in our case is related to the “generalized conformal symmetry” of the D4-brane background [43], in which a scaling of the radial coordinate y is combined with a change in the string coupling. Explicitly, we can see that that the transformation 5 gs → α 2 gs , y → αy , leaves the action (3.9) invariant. This is not a symmetry of the action, since we are changing the parameter gs , but it does correspond to a symmetry of the classical 6 We thank Ofer Aharony for pointing out this work, and providing the crucial insights that led to the interpretation in this paragraph. 51 equations of motion, which explains why there is a family of solutions related by the scaling y → αy. If we include gs corrections (important when we go away from Nc = ∞), different terms in the quantum effective action will have different dependence on gs , and so the scaling y → αy will no longer be a symmetry of the (quantum corrected) equations of motion. To summarize, a modification of our field theory in which the adjoint sector is supersymmetric would have an exactly massless mode in the large N limit, but this is not a Goldstone boson since the associated “symmetry” involves changing a parameter of the theory (alternately, because the mode should receive a mass by quantum corrections when N is finite.) The light mode that we find for large y0 arises because the geometry in our case is asymptotically the same as the geometry in the supersymmetric case. Interestingly, the limit y0 → ∞ where our meson would become massless is the limit where the probe branes go off to radial infinity in the geometry. So despite having a mass that goes to zero, our light meson completely decouples from the adjoint degrees of freedom in this limit. 3.5 Finite temperature In this section, we will investigate the behaviour of our model at finite temperature. In the N f Nc limit that we are considering, thermodynamics is dominated by the adjoint sector of the theory. At a temperature Tc = 1/(2πR), the model undergoes a deconfinement phase transition, in which the minimum action bulk solution changes from the original low-temperature solution to a black brane solutions with a horizon. This transition is easy to understand in the Euclidean picture, where both the x4 direction and the Euclidean time direction of the field theory are compactified with anti-periodic boundary conditions for the fermions. The lengths of the two compact directions are 2πR and 1/T respectively. For each value of T , we have two possible bulk solutions. The first is obtained from the Euclidean version of the zero temperature solution (3.1) simply by making an identification τ ∼ τ + 1/T . The second is obtained by the replacements 2πR ↔ 1/T and x4 ↔ τ. In the first solution, the x4 direction pinches off in the bulk geometry, while in the second solution, the τ direction pinches off in the bulk geometry (giving a horizon in the 52 corresponding Minkowski solution). It turns out that the minimum action solution for a given R and T is the one where the smaller radius circle pinches off. Thus, the solution with a contractible time direction becomes dominant for T > 1/(2πR). For this high-temperature phase, the presence of a contractible time circle in the Euclidean solution means that the corresponding Minkowski solution is a blackbrane solution with a horizon. The metric is given by 2 ds = U R4 3 2 2 i (− f (U)dt + δi j dx dx j + dx42 ) + R4 U 4π 3 R3 UT where now UT f (U) = 1 − ( )3 U 3 2 1 dU 2 +U 2 dΩ24 ) , f (U) (3.10) ( 1 2 = 1 . T As we did for the zero temperature case, we can now study the probe-brane action in this background and look for stable embeddings. Following the same steps, we find S = −µq dθ dσ r2 dσ H(r) 2 where HT (r) = r 3 2 R 3 2 1+ ρT3 r3 5 3 r ρT r ρT 2 dr + dσ 3 3 , (3.11) −1 . (3.12) +1 Now that we have a horizon, there are two qualitatively different types of embeddings for the D4-brane. Either the brane can remain entirely outside the black hole horizon (in what is known as a “Minkowski embedding”), or the brane can extend into the horizon (in a “black hole embedding”). In our model, the black hole embeddings are extremely simple, with each half of the brane sitting at some constant point on the S4 , and extending in the radial direction from the horizon to r = ∞, as shown in figure 3.11. These embeddings exist for any value of the asymptotic angle. To see that these configurations are possible, we simply note that the equations of motion for the action (3.12) are satisfied if To investigate solutions for which dθ dσ 53 dθ dσ = 0. = 0, we can fix the reparametrization Figure 3.11: Schematic of “black hole” embeddings in which the brane extends down to the horizon. invariance by identifying σ = θ . Defining also y = r/ρT , we get S = − 5 2 µ4 3 gs R 2 dθ hT (y) ρT y2 + dy dθ 2 where 2 7 hT (y) = y− 2 (1 + y3 ) 3 (y3 − 1) . Again, we can use the θ -independence of the action to deduce a conserved quantity y hT (y)y2 ∂S −S = ∂y y2 + (y )2 The black hole solutions above correspond to a value of 0 for this quantity (since y = ∞ for those). For any non-zero value, the brane cannot reach the horizon, since hT (y) = 0 at the horizon (where y = 1). Thus, all other embeddings are of the Minkowski type. For these, the brane reaches some minimal value y = y0 where y =0. We then have h(y)y2 y2 + (y )2 = h(y0 )y0 , which gives dy = ±y dθ h2 (y)y2 −1 . h2 (y0 )y20 54 Integrating, we find y θ (y) = y0 g(x) −1 g(y0 ) dx x , (3.13) where 4 g(x) = x−5 (x3 − 1)2 (x3 + 1) 3 , and as before, we define θ = 0 to be the angle at which the brane embedding reaches its minimum value of y. The maximum value of θ for the embedding with minimum brane position y0 is ∞ θ∞ = y0 dx x g(x) −1 g(y0 ) . This increases from θ∞ = 0 as y0 approaches 1 to θ∞ = (3.14) π 5 as y0 goes to infinity. Thus, we find that Minkowski embeddings exist for precisely the theories where there is no stable vacuum. Furthermore, it is straightforward to verify that the black hole solution with the same asymptotic behaviour always has a lower action. Hence, there is no regime of parameters for which the Minkowski embedding describes the equilibrium behaviour of the field theory. Summary For the range of parameters where our holographic field theory has a stable vacuum solution at zero temperature, we have found that the only solution to the equations of motion for the probe brane in the high-temperature geometry is a black hole embedding, where the two halves of the D4-brane extend directly into the horizon along the radial direction at constant position on the S4 . Physically, the interpretation is that mesons “melt” as soon as the deconfinement transition is reached. This is in contrast to the Sakai-Sugimoto model and other models, where (for certain parameter choices) there can be a window in temperature for which the theory is deconfined but there is still a well-defined meson spectrum. One other consequence of our analysis is that for the special case where the two halves of the D4-brane come in from diametrically opposite points on the sphere, the full SO(4) symmetry is restored in the high-temperature phase (recall that it was spontaneously broken to SO(3) for low temperatures). 55 3.A Alternate coordinates for θ fluctuations In this appendix, we present the action for fluctuation of our probe D4-brane in the y − θ plane using an alternate parametrization that can be more convenient for numerical study. Parameterizing fluctuations in the embedding function y(θ ) in terms of y as we have done above can be an inconvenient choice for a numerical analysis, since the “even” fluctuations result in a change of y0 , and the choice of coordinates above essentially assumes that y0 is fixed. A more straightforward choice is to take θ as the parameter. The original action governing the embedding and the fluctuations in the r, θ directions is S = −C dθ H(y) y2 + dy dθ where 3 H(y) = y 2 (1 + and f (y) = 2 − f (y) dy dt 2 , 1 5 )3 , y3 2 R3 y(1 + y3 )− 3 . ρ0 Here, we have not included terms involving spatial derivatives in the field theory directions since these can be restored at the end by Lorentz invariance. Expanding about one of the solutions to the equations of motion, we get an action for small fluctuations given by S =C dθ 1 1 1 A(y)(∂t y) ˜ 2 − B(y)(∂θ y) ˜ 2 − C(y)y˜2 − D(y)y∂ ˜ θ y˜ 2 2 2 We can integrate by parts and replace the last term with 1 2 y˜ ∂θ D . 2 56 . Thus, it is equivalent to use an action S =C dθ 2 ∂ y˜ 1 A(y) 2 ∂ xµ 2 ∂ y˜ 1 − B(y) 2 ∂θ 1 − C(y)y˜2 2 , where A and B are as above and C is replaced by C − ∂θ D. Explicitly, we find that A(y) = B(y) = C(y) = Hf 1 M2 Hy2 3 M2 ∂y2 H M 1/2 y2 + ∂y H H (2y3 − y2 y + 4y(y )2 ) + 5 (−2yy (y )2 + y3 y + 2(y )4 − y2 (y )2 ) , 3/2 M M2 where M = y2 + (y )2 . For numerical evaluation, it is convenient to use H 2 y4 − y2 H02 y20 y = and y = −y + HH y4 2H 2 y3 + 2 2 . H02 y20 H0 y0 The function y is determined implicitly in terms of θ by the equation y θ (y) = 1 dx y0 x 57 H 2 (x)x2 H 2 (y0 )y20 . −1 Chapter 4 Twisted Inflation 4.1 Introduction Cosmic inflation [44–46] provides an attractive explanation for the flatness and homogeneity of the observable Universe and gives rise in a natural way to primordial inhomogeneities that lead to structure formation (for a recent review, see [21]). Many effective field theory models of inflation have been suggested, however, one would like to have some guiding principles that narrow the field and therefore allow more specific predictions. From the field theory point of view, an attractive condition to demand is naturalness: the parameters in the effective field theory should not require a significant amount of fine-tuning. Another possible set of constraints may arise by demanding that the theory has a consistent UV completion including gravity. One possibility for such a UV completion is string theory; if we assume that string theory gives the correct microscopic theory of our Universe, then the correct model of inflation must arise in some way from string-theoretic degrees of freedom. In this chapter, we consider a new scenario for inflation that can be naturally embedded in string theory, but more generally can be understood as a mechanism to generate a naturally flat inflaton potential from supersymmetry-breaking compactification of higher-dimensional supersymmetric gauge theory. The model has a strongly-coupled version best described as inflation coming from wrapped branes on a non-supersymmetric throat geometry, and a weakly-coupled version that can 58 be analyzed directly in field theory. For any coupling, the model has a natural embedding in string theory based on the low energy dynamics of D-branes or M5branes. We begin by describing the basic idea in a general context. The brane inflation scenario Suppose that as part of some warped compactification of string theory, we have a supersymmetric throat geometry which includes a cycle upon which a brane is wrapped (it is also extended in the four non-compact directions). This is schematically depicted in figure 4.1. Suppose further that there is a moduli space for the brane, so that it can move freely up and down the throat. Now, consider a related geometry for which we choose some different supersymmetry-breaking boundary conditions on this cycle, such that in the modified non-supersymmetric geometry, the cycle contracts to zero size at the tip of the throat. Far away from this tip, where the cycle is large, the geometry is very similar to the supersymmetric case, so the radial potential for the brane is almost flat. But eventually, the brane will roll down to the tip and disappear, giving all its energy to the other degrees of freedom in the theory when it contracts. Thus, the scalar field describing the radial position of the brane is a natural candidate for an inflaton field. The field theory scenario Consider a higher-dimensional supersymmetric field theory with a moduli space (i.e. flat directions in the potential) parameterized by some scalar fields. Now, compactify to 3+1 dimensions on a manifold for which supersymmetry is broken by boundary conditions (e.g. a circle with anti-periodic boundary conditions for fermions). Suppose that the compactified theory flows in the IR to a massive (confining) theory. In the uncompactified theory, if we take one of the moduli scalar fields φ to have a large expectation value, then typically there will be some W-bosons (and their superpartners) that become very massive. With enough supersymmetry, integrating these out does not spoil the existence of a moduli space, so the contribution to the effective potential coming from the massive bosons must exactly cancel the contribution from the fermions. This remains true in a supersymmetric compactification, but is generically modified by boundary conditions that break supersym59 Figure 4.1: Geometrical picture of inflation at strong coupling. metry. To give a concrete example, consider a “twisted” circle compactification with periodic boundary conditions for the bosons and anti-periodic boundary conditions for the fermions on circle. The mode numbers for the Kaluza-Klein (KK) modes of the W-bosons are shifted relative to their fermionic superpartners, so the masses are Mb = Mf = n2 , R2 (n + 21 )2 (g4 φ )2 + , R2 (g4 φ )2 + (4.1) where 1/R = MKK is the KK compactification scale, and g4 is a Yang-Mills coupling in four dimensions. Thus, there is no longer an exact cancelation in calculating the effective potential. However, boson and fermion modes with n g4 φ R will be almost degenerate. As a result, cancelations return for large g4 φ R, and the potential becomes flat here. Thus, scalar fields which would be moduli in a supersymmetric compactification give rise to fields that can act as inflaton fields in the twisted compactification. If these fields start with large values g4 φ 1/R, they will roll slowly back towards the origin of moduli space. At the origin, we get to the theory which is massive in the IR, so there are no more light excitations. 60 A specific model In the bulk of this chapter, we will illustrate these two scenarios in a specific model. We will see that the brane inflation picture and the field theory picture represent strongly coupled and weakly coupled versions of the same mechanism. The model we consider is maximally supersymmetric 4+1 dimensional U(N) Yang-Mills theory compactified on a spatial circle of radius R = 1/MKK , with antiperiodic boundary conditions for fermions [10].1 The effective dimensionless ’t Hooft coupling of the compactified theory is λ . We consider initial conditions in which one of the adjoint scalar fields is X = diag(g4 φ , 0, · · · , 0). With supersymmetry-preserving boundary conditions, the scalar φ would parameterize an exact flat direction in the potential. In the case of broken supersymmetry, a potential is generated in the quantum effective action, but this turns out to be quite flat for φ Kaluza-Klein scale (λ MKK . When the theory is weakly coupled at the 1), we can compute the effective potential by integrating out the W-bosons and their superpartners at one loop. We find that the potential for large φ (canonically normalized) takes the form V (φ ) = V0 1 − A(φ /φ0 )2 e − φφ 0 , (4.2) which quickly approaches a constant for large φ . At strong coupling (λ 1), the field theory has a gravity dual description as type IIA string theory on a non-supersymmetric D4-brane throat geometry or M-theory on a non-supersymmetric M5-brane throat at very strong coupling.2 In either case, the circle direction corresponding to the anti-periodic compactified direction of the field theory contracts smoothly to a point at the IR end of the geometry. Assuming that N is large, taking one scalar to have an expectation value corresponds to introducing a probe brane in this throat geometry which is extended along all the field theory directions, including the compact ones, as shown in figure figure 4.1. This is the wrapped brane that we described in general above. The 1 In the UV, this can be completed as the 5+1 dimensional (0, 2) CFT (the low-energy theory of M5-branes) compactified on a torus with periodic boundary conditions on one cycle and anti-periodic boundary conditions on the other cycle. 2 We will make these statements more precise in the body of the chapter. 61 λ 1 λ 2 ns = 1 − NCMB ∼ 0.967 αs = − N 22 1 ns = 1 − 5N8CMB ∼ 0.973 αs = − 5N82 ∼ −0.00055 CMB ∼ −0.00044 CMB 3 r ∼ 7.2 × 10−14 (N/λ 2 ) r ∼ 2.5 × 10−9 (λ N) 7 Table 4.1: Weak and strong coupling predictions for the scalar spectral index ns , the running of the spectral index αs , and the tensor/scalar ratio r. For 3 λ 1 we require N 1 for the analysis to be reliable, and N 104 λ − 5 to obtain enough e-foldings. Numerical values are given assuming CMB perturbations at the pivot scale left the horizon at NCMB = 60 e-foldings before the end of inflation. quantum effective potential for our scalar is then calculated using the effective action for this probe brane (Born-Infeld plus Chern-Simons type couplings to the form fields) in the throat geometry. In either the type IIA string theory regime or the M-theory regime, we find a potential that behaves for large φ as V (φ ) = V0 (1 − φ03 /φ 3 ) , (4.3) again asymptotically approaching a constant. Starting with the effective actions we derive and minimally coupling to gravity,3 it is straightforward to show that we can achieve slow-roll inflation with an acceptable spectrum of fluctuations for natural values of the parameters. For example, the predictions for the scalar spectral index ns range continuously from ns ≈ 0.967 in the weakly coupled model to ns ≈ 0.973 at strong coupling. The predictions and constraints for the minimally coupled model are summarized in table 4.1. On the other hand, it is well known that once the field theory is coupled to gravity and treated in an inflating spacetime, there will generically be H 2 φ 2 terms in the effective potential (here H is the Hubble parameter during inflation) that can spoil slow-roll inflation. The coefficient of such a term will in general depend on 3 A matter field is minimally coupled to gravity if its action does not involve explicit curvature tensors. 62 the UV completion of our model (the full string compactification in the brane picture). Thus, before taking the predictions seriously, one should investigate whether there exists a more complete theory, including gravity, for which the coefficient of H 2 φ 2 is either naturally small or can be fine-tuned to an acceptable level. Even in cases where such corrections do not spoil inflation, they can change the predictions significantly. The end of inflation A particularly nice feature of our mechanism is that inflation ends naturally – with no remaining light degrees of freedom in the inflationary sector – without having to invoke any additional fields, antibranes, etc. In the string theory scenario, this occurs since the brane self-annihilates when it reaches the IR, where the cycle on which the brane is wrapped contracts to a point. In the field theory scenario, the theory at the origin of moduli space is a confining theory, in which the confinement scale can be far below the scale of inflation.4 It is conceivable that the nonsupersymmetric confining gauge theory left over at the end of inflation could be part the Standard Model (e.g. the Yang-Mills part of QCD), or perhaps a separate sector that gives rise to the observed dark matter. Relation to previous work There exist many previous discussions of inflation models based on supersymmetric gauge theory (see [20] for an early review), and of string theory models based on branes in warped throat geometries (for reviews of inflation in the context of string theory, see [22, 47–49]), but none employ the basic mechanism we have described. From a field theory point of view, the work in reference [50] is somewhat related in that it uses a higher-dimensional gauge theory with a supersymmetrybreaking compactification, but their candidate inflaton comes from the Wilson line of the higher-dimensional gauge field around the circle. In string theory, models based on wrapped branes have been discussed before [51], but here the wrapped branes already have a (relatively steep) potential in a supersymmetric throat ge4 In this case, the theory will be heated well into its deconfined phase when inflation ends. In the string theory scenario, the scale of inflation is generally not high enough to deconfine the gauge theory dual to the throat geometry. For details see appendix 4.B . 63 ometry. The well-known brane-antibrane inflation model [23] is similar in that the inflaton field is the radial position of a brane in a warped throat, and the potential is generated only after supersymmetry breaking. In that case, the supersymmetry breaking occurs by the addition of an anti-brane at the tip of the throat, and inflation ends when the brane and antibrane annihilate. Our model is slightly simpler in that the only brane required is the one that gives rise to the inflaton. Another rather novel feature of our model relative to models studied previously is that we are able to carry out the analysis both in a weakly coupled limit, where we can use field theory methods, and at strong coupling, where we exploit the gravitational dual descriptions. Outline The outline of the chapter is as follows. We begin in section 4.2 by describing the field theory on which our model is based, and giving the gravity dual descriptions that may be used to analyze the theory at strong coupling. In section 4.3, we derive the effective action for our candidate inflaton field. We obtain the effective action by directly integrating out W-bosons at one loop and by a dual gravity calculation at strong coupling. The effective actions are in general not of canonical form, but we show in section 4.4 that during any possible period of inflation, the theory is well-approximated by an effective action with canonical kinetic terms and a simple potential, which we derive in each limit. In section 4.5, we write down the conditions that arise from demanding that slow-roll parameters are small, and derive the predictions for inflationary perturbations in terms of the parameters of our model, assuming a minimal coupling to gravity. In section 4.6, we compare these predictions with observations, thereby constraining the model parameters. We find that our theory gives a viable model of inflation consistent with observations for a broad range of parameters, and summarize the final predictions as a function of the parameters once all observational constraints have been taken into account. In section 4.7, we make some comments about the end of inflation for the two regimes of our model, and discuss a few scenarios for how this model may be integrated with the degrees of freedom of the Standard Model in a more complete theory. In section 4.8, we recall that a consistent UV completion of our field theory may include terms beyond the minimal coupling to gravity that we have assumed, and comment 64 on the potential “η-problem” in which some of these terms may spoil slow-roll inflation. In section 4.9, we discuss various ways in which our model may be generalized. In section 4.10, we make some concluding remarks and discuss future directions. 4.2 Basic setup The field theory we use for our main example is maximally supersymmetric 4+1 dimensional Yang-Mills theory compactified to 3+1 dimensions on a circle with anti-periodic boundary conditions for the fermions. This is the theory describing the low-energy physics of a stack of D4-branes (compactified on a circle with twisted boundary conditions) in type IIA string theory. This model was originally introduced by Witten as a construction of pure 3+1 dimensional Yang Mills theory via compactification of maximally supersymmetric 4+1 dimensional gauge theory [10]. We will summarize the relevant aspects of the construction here. To define a UV complete theory, we start with a 5+1 dimensional theory, the (0, 2) conformal field theory describing the low-energy physics of M5-branes in M-theory. We begin by compactifying this on a circle of size 2πRM with supersymmetry-preserving boundary conditions. This gives a theory in 4+1 dimensions whose IR physics is described by the 4+1 dimensional maximally supersymmetric Yang-Mills theory. We further compactify the theory on a circle of size 2πR with anti-periodic boundary conditions for fermions. This theory flows in the IR to a non-supersymmetric confining gauge theory in 3+1 dimensions. The final theory has two dimensionless parameters: a dimensionless coupling λ= 2πNRM , R (4.4) which is the ’t Hooft coupling in the 3+1 dimensional gauge theory at the KaluzaKlein scale 1/R, and the parameter N, the rank of the gauge group (which comes from the number of branes). Once the theory is coupled to gravity, the ratio MKK /Mp = 1/(RMp ) will provide a third dimensionless parameter. 65 4.2.1 Weak coupling: λ 1 When λ is small, the theory is weakly coupled at the Kaluza-Klein scale MKK = 1/R (the much larger scale 1/RM where the 6D physics becomes important will not be relevant in this case). The action is obtained by dimensional reduction of the d = 10 N = 1 supersymmetric U(N) Yang-Mills theory and gives S = 1 g25 1 1 1 d 5 x − Tr(Fµν F µν ) − Dµ X I Dµ X I + Tr([X I , X J ]2 ) 4 2 4 (4.5) i 1 I+4 I ¯ µ Dµ ψ − ψΓ ¯ − ψΓ [X , ψ] 2 2 where the scalars X I (I = 1, .., 5) and fermions ψ transform in the adjoint representation of the gauge group, and we use ten-dimensional notation for the spinors and gamma matrices. The dimensionful coupling g5 is related to RM by g25 = (2π)2 RM , the dimensionless four-dimensional ’t Hooft coupling λ is related to g5 by λ = g25 N/(2πR) and the four-dimensional Yang-Mills coupling g4 by λ = g24 N. As argued by Witten [10], the theory flows to pure Yang-Mills theory in the IR with a confinement scale Λcon f ∼ 1 −c e λ R (4.6) for some constant c of order 1. At low energies, we have the physics of glueballs with mass of order Λcon f . The relevant energy scales in this regime are summarized on the left hand side of figure 4.2. 4.2.2 For λ Strong coupling: 1 2 λ N3 1, the theory is strongly coupled at all scales, but we can analyze the theory using a gravitational dual description.5 In the regime 1 2 λ N 3 , the physics is well described by type IIA supergravity on a background [52] ds2 = U R4 3 2 (ηµν dxµ dxν + f (U)dx42 ) + R4 U 3 2 ( 1 dU 2 +U 2 dΩ24 ) f (U) 5 Geometries dual to branes compactified on a circle with anti-periodic boundary conditions for fermions may be obtained by double analytic continuation of the corresponding black p-brane solutions [52]. 66 , Weak Coupling ( λ<<1 ) M Strong Coupling ( λ>>1 ) M P P 1/4 1/R~Λ c~ V 0 1/4 V ~1/R 0 Λc Figure 4.2: Energy scales for weak and strong coupling. The energy density V0 during inflation will be of order the Kaluza-Klein scale 1/R in both regimes. 3 e ϕ F4 U 4 = R4 = 3R34 ε4 . (4.7) where ε4 is the volume form on the unit S4 , and f (U) = 1 − U0 U 3 . (4.8) The x4 direction, corresponding to the KK-direction in the field theory, is taken to be periodic, with coordinate periodicity 2πR. It is important to note that this x4 circle is contractible in the bulk, so the x4 and U directions form a cigar-type geometry. For this circle to contract smoothly requires R2 = 4R34 . 9U0 (4.9) The property that the space ends smoothly at U = U0 is the gravitational signature that the dual gauge theory is confining [10]. The parameters appearing in the metric are related to the field theory parame- 67 ters by N= where ls = R34 , πls3 λ 2πls = , N R (4.10) √ α is the string length. For our purposes it will be useful to encode the Ramond-Ramond flux in terms of the Poincar´e dual five-form potential, F4 = dC5 , with C5 = 3 U R4 f (U)dx0 ∧ dx1 ∧ dx2 ∧ dx3 ∧ dx4 . (4.11) The gauge in (4.11) has been chosen so that C5 vanishes at the tip of the geometry. In terms of the field theory parameters, the dilaton and the string-frame radius 3 of curvature at the tip of the cigar (the IR part of the geometry) are of order λ 2 /N √ and λ s respectively. As claimed, supergravity is valid (i.e. the string coupling and the curvature in string units will be small) in the IR region of the geometry 2 when 1 λ N 3 . Note that the string coupling still becomes large in the UV, 4 reaching O(1) at U/U0 ∼ N 3 /λ 2 . However, for an inflation scenario, we imagine that we have only some finite region of the throat U < Umax as part of a consistent compact manifold.6 In order to trust calculations based on type IIA supergravity, 4 we require that Umax /U0 < N 3 /λ 2 . 4.2.3 Very strong coupling: λ In the case where λ 2 N3 2 N 3 , the string coupling in the type IIA throat becomes large even in the IR part of the geometry, so type IIA supergravity no longer gives a valid description. However, in this regime, the geometry has a weakly curved eleven-dimensional geometrical description, as long as N is assumed to be large. In this description, the eleven-dimensional metric (obtained from a double analytic continuation of the thermal near-horizon M5-brane solution) is [53] ds2 = ρ L2 2 (ηµν dxµ dxν + dxM + f (ρ)dx42 ) + 2 f −1 (ρ)dρ 2 + L2 dΩ24 , L ρ 6 The (4.12) limit where the throat becomes infinite corresponds to taking the compactification volume to infinity, which sends the four-dimensional Planck mass to infinity and decouples gravity. 68 where f (ρ) = 1 − ρ03 . ρ3 (4.13) We take the x4 and xM coordinates to be periodic with periodicities 2πR and 2πRM respectively. The parameters in the metric are defined as L9 = N 3 2 κ11 , 27 π 5 ρ0 = 4L3 , 9R2 (4.14) where κ11 is the eleven-dimensional gravitational coupling. As above, the coupling λ in the field theory is related to the metric parameters as λ= 2πNRM . R (4.15) The metric is sourced by a four-form field strength F4 = 3L3 ε4 , where ε4 is the volume form on the sphere. Again, it is useful for our purposes to describe the flux in terms of the Poincar´e dual potential, F4 = dA6 , which gives A6 = 4.3 ρ L 3 f (ρ)dx0 ∧ dx1 ∧ dx2 ∧ dx3 ∧ dx4 ∧ dxM . (4.16) Effective action for the candidate inflaton Had we defined our theory with periodic boundary conditions for fermions, the resulting 3+1 dimensional theory would have been supersymmetric, and the theory would have had a moduli space (a family of vacua with zero energy) parameterized by mutually commuting scalar matrices. In the picture where our field theory is understood as the low-energy physics of a stack of N D-branes, these scalar field configurations correspond to configurations of D-branes that are parallel but not coincident: the eigenvalues of the commuting scalar matrices correspond to the coordinates of the branes in the transverse space. With twisted (anti-periodic) boundary conditions for fermions, the classical potential in (4.5) still vanishes whenever the scalar fields are described by a set of commuting matrices, but since supersymmetry is broken we will find that a nontrivial potential is generated in the quantum effective action. However, when the 69 Figure 4.3: Various interpretations for the inflaton field. scalar field expectation values are significantly greater than the scale of supersymmetry breaking – the Kaluza-Klein scale – this potential becomes quite flat. In this chapter, we will mostly consider the simplest nontrivial configuration XI ∼ diag(φ I , 0, . . . , 0), which in the brane picture corresponds to separating off a single brane from the rest. The absence of supersymmetry results in a net attractive force which tends to bring the isolated brane back to the stack. In this section we will calculate the quantum effective action for the field φ I . For small λ , we can do this by a direct field theory calculation (assuming that |φ | > MKK ). For large λ , the physics is described by considering a single brane as a probe rolling down towards the tip of the geometry (4.7) or (4.12), which describes the physics of the remaining branes in the stack (see figure 4.3). In all of these regimes, we will show that the potential governing the motion on this lifted branch is naturally rather flat for |φ | 4.3.1 MKK , making |φ | a good candidate for an inflaton field. Weak coupling In this section, we consider the gauge theory (4.5) in the case of small λ . We begin with the action (4.5), and expand the theory about a background X I = diag(g4 φ I , 0, . . . , 0) . (4.17) We then derive the effective action for the inflaton field, φ ∼ |φ I | by a direct field theory calculation. To make the formulae simpler, we consider the U(N +1) theory. We find that all the fields have off-diagonal modes (the (1, n) and (n, 1) matrix 70 elements with 2 ≤ n ≤ N + 1) with masses of order |g4 φ |. These are the W-bosons (together with their scalar and fermionic partners) associated with the breaking of gauge symmetry U(N + 1) → U(N) ×U(1). During inflation, we will assume |φ | to be initially much larger than the Kaluza-Klein scale 1/R (which is roughly the scale at which inflation occurs), so it makes sense to integrate these out to find an effective action for φ I . As we show in appendix 4.A , for |g4 φ | 1/R, the kinetic term for φ I is dom- inated by the canonical one coming from the tree-level action, so we can focus on calculating the effective potential (i.e. non-derivative terms), which has no treelevel contribution. A properly gauge-fixed treatment is discussed in appendix 4.A but the result for the effective potential can be reasoned out without lengthy calculation. The one-loop correction to the potential is given as usual by a trace over the logarithm of the inverse propagator V (1) (φ ) = 1 2 d4k (2π)4 ∑ log k2 + Mb2 − ∑ log k2 + M 2f , (4.18) f ermions bosons where the sum is over all real field modes being integrated out. In the 5D picture, we have N complex W-bosons with three polarizations, 5N complex scalars, and 8N complex fermions, all with mass (g4 φ )2 = g24 φ I φ I . In the 4D picture, each of these leads to a KK-tower of particles, with masses Mb = Mf = n2 , R2 (n + 21 )2 (g4 φ )2 + . R2 (g4 φ )2 + (4.19) The masses are different because the anti-periodic boundary conditions imply that the allowed momenta in the compact direction are different for bosons and fermions. Thus, while we start with a supersymmetric theory in five dimensions, the potential (4.18) will be non-zero when we compactify to four dimensions with twisted boundary conditions. 71 Evaluating (4.18) in our case, we find that the one-loop effective potential is7 V (1) (φ ) = 8N ∑ n∈Z d4k (2π)4 log k2 + (g4 φ )2 + n2 R2 − log k2 + (g4 φ )2 + (n + 21 )2 R2 (4.20) This gives a good approximation to the full effective potential V , since the treelevel potential for φ vanishes and higher-loop contributions are suppressed by powers of λ . The integral in (4.20) is divergent but the finite piece can be isolated by differentiation with respect to (g4 φ )2 , d4k (2π)4 ∂V = 8N ∂ (g4 φ )2 2πR (g4 φ )2 + k2 sinh(2πR k2 + (g4 φ )2 ) , (4.21) where we have used 1 = ∑ n2 + q2 n∈Z 1 = ∑ (n + 1 )2 + q2 n∈Z 2 π coth(πq) , q π tanh(πq) . q (4.22) Integrating, we find that8 V (φ ) = 2NCW W (2πg4 φ R) , π 2 (2πR)4 (4.23) where we define the function W (y) = 1 CW y 0 q3 dq ∞ dxx 0 q2 + x2 sinh( q2 + x2 ) 7 In , (4.24) string theory language, the coefficient here simply counts the number of fundamental string excitations which stretch from the stack of N D4-branes to the single brane which has been separated from the stack. Each such string is characterized by the member of the stack on which it ends, as well as its ten-dimensional polarization, of which there are eight (complex) physical modes for both bosons and fermions. There are thus 8N such strings. 8 Here, we are assuming that the vacuum energy V (φ = 0) vanishes. In reality, this should be equal to the present day vacuum energy, but this would have a negligible effect during inflation. 72 . and the constant ∞ CW = 0 q3 dq ∞ dxx 0 q2 + x2 sinh( q2 + x 2 ) = 93 ζ (5) ≈ 12.054285 , 8 (4.25) so that W (∞) = 1. For large y, the function W behaves as W (y) = 1 − 4 2 (y + 3y + 3)e−y + O(e−2y ) . CW (4.26) Thus, the leading behaviour of the effective potential for large φ is V (φ ) = V0 1 − A φ φ0 2 e − φφ 0 , (4.27) where 2NCW , π 2 (2πR)4 1 φ0 = , 2πg4 R 4 . A = CW V0 = (4.28) Since we show in appendix 4.A that φ is canonically normalized, the effective action at weak coupling is S=− 4.3.2 Strong coupling: 1 d4x 1 (∂µ φ )2 +V (φ ) . 2 (4.29) 2 λ N 3 , the D4-brane For large λ , a more geometrical language is appropriate in describing the dynamics of the inflaton field. The degrees of freedom in the unbroken part of the gauge theory are well described by gravitational physics in the geometry (4.7). The remaining relevant light degree of freedom, corresponding to the scalar field φ , is described by adding a probe D4-brane [25] in the geometry of (4.7), extended along the xµ directions and wrapped on the direction x4 . In a static gauge, the 73 probe brane’s worldvolume is parameterized by the spacetime coordinates xµ and x4 . We consider an ansatz where the brane is localized on the S4 and has a radial profile U(xµ ). Note that we assume that U is independent of the twist circle x4 . The induced metric on the probe is given by9 2 ds = 3 2 U R4 3 R4 ηµν + U ∂µ U∂ν U f (U) dxµ dxν + f (U)dx42 . (4.30) The Ramond-Ramond form has a non-trivial pull-back which is (4.11) with U now a function of xµ . The action for a single probe D4-brane is given by the abelian Born-Infeld action S = −µ4 d 5 σ e−ϕ − det(gab + F˜ab ) , (4.31) together with the Chern-Simons term ˜ ∑ Cp ∧ eF , S = µ4 (4.32) p where F˜ = 2πls2 F , and µ4 = (4.33) 1 . (2π)4 ls5 (4.34) For the scenario at hand, in which we do not excite the worldvolume gauge field, the only relevant terms in the brane action are S = −µ4 d 4 x dx4 e−ϕ − det(gab ) + µ4 C5 . (4.35) Evaluating this action for the induced metric (4.30) and form field (4.11) yields S = (2V0 ) d 4 x −z3 1 − z−3 1+ 9 In 9R2 1 4 z3 − 1 (∂µ z)2 + z3 − 1 , (4.36) this section, we are deriving the effective potential in Minkowski space. During inflation, the background spacetime is approximately de Sitter, so the ηµν factor in this metric will be replaced by a de Sitter factor. This will be discussed in detail in section 4.4 and appendix 4.C . 74 where we have introduced the dimensionless coordinate z = U/U0 and the constant V0 = 2λ N 36 π 2 R4 . (4.37) We have expressed the final action entirely in terms of the field theory parameters N, λ , and R, having made use of (4.9) and (4.10). 4.3.3 2 Very strong coupling: λ N 3 , the M5-brane In this section we show the same effective action is valid in the M-theory regime. Here, the rolling scalar corresponds to the radial location of a single M5-brane wrapped on the x4 and the xM directions in the geometry (4.12). Assuming that the worldvolume two-form field is set to zero, the action for a single M5-brane is S = −T5 −det(gab ) + T5 A6 , (4.38) where A6 is the six-form field given in (4.16). Here, the five-brane tension is T5 = 1 M6 , (2π)5 11 (4.39) where the eleven-dimensional Planck mass10 is related to κ11 by 2 −9 2κ11 = (2π)8 M11 . (4.40) Evaluating the action in the background above, and integrating over the compact x4 and xM directions, we get S= T5 (2πR)(2πRM )ρ03 L3 d 4 x −z3 1+ L3 −1 f (z)(∂µ z)2 ρ0 z3 f (z) + (z3 − 1) , (4.41) where we have defined z = ρ/ρ0 . Expressing everything in terms of the field theory parameters, we find the action (4.36). Thus, the effective action (4.36) can be considered to be valid for all values λ 10 We 1. use the conventions of [54]. 75 4.4 Coupling to gravity and slow-roll potentials In the previous section, we computed the effective action for our system assuming a Minkowski space background for the geometry in the four non-compact dimensions and without dynamical gravity (i.e. with Mp = ∞ ). To study inflation, we need to include dynamical gravity. In this case, the ratio between the KK-scale and the Planck scale MKK /Mp = 1/(RMp ) becomes a third dimensionless parameter in the model, together with λ and N. In the strong coupling picture, studying the field theory without gravity corresponds to considering an infinitely long throat in the dual description (as we did above). In a theory with dynamical gravity, we would have only a finite portion of the throat, appearing as part of some consistent compactification of string theory / M-theory to four dimensions (see figure 4.1). In this case, the full spacetime (in the eleven-dimensional picture) is described by a warped metric ds2 = w(yi )gµν (x)dxµ dxν + g˜i j (y)dyi dy j , (4.42) where the yi are the internal directions, and the four-dimensional Planck mass is given as an integral over the compact directions (including the throat region), Mp2 = 1 κ11 2 d 7 y w(yi ) g˜ . (4.43) A similar formula holds in the IIA picture. In this section (and in section 4.5 and section 4.6), we will consider the inflationary dynamics of our field theory minimally coupled to gravity with a standard Einstein action. However, starting with a consistent string compactification (or more generally, a consistent UV complete theory that includes dynamical gravity) the complete effective action will generally also include operators suppressed by powers of the Planck mass or other UV scales, as well as couplings between the fields and spacetime curvature tensors. We postpone a discussion of these corrections to section section 4.8, however the reader should bear in mind that these corrections can significantly affect the predictions of an inflationary model, or or even spoil inflation in what is known as the “η-problem.”11 11 For the impatient reader, the upshot will be that for a given compactification, there are several 76 To analyze the behaviour of the scalar and find the gravitational backreaction we will make use of a generalized formalism for inflation models with general Lagrangians that are functions of a single scalar field and its first derivative [55]. At weak coupling, the scalar field is already canonical and so can be treated by the standard slow-roll analysis. At strong coupling, a more detailed analysis is necessary, and we will show that under a suitable field redefinition we can reduce this model to the canonical slow-roll form, albeit with a different potential than at weak coupling. We will briefly sketch the generalized formalism; for a more detailed treatment the reader is directed to [55] and references therein. Consider an action 1 S = Mp2 2 √ d 4 x −gR + √ d 4 x −gP(X, z) , (4.44) where z is the inflaton field and X ≡ − 21 gµν ∂µ z∂ν z. Since we are interested in inflationary solutions, we make the ansatz that the metric in the four non-compact directions can be written in Friedmann-Robertson-Walker (FRW) form ds2 = −dt 2 + a2 (t) dx32 , (4.45) where t is comoving time and we have assumed spatial flatness. The Einstein and inflaton equations of motion reduce to (a dot indicates a derivative with respect to the time t) 3Mp2 H 2 = E , E˙ = −3H(E + P) , (4.46) (4.47) where E ≡ 2XP,X −P is the “energy” of the inflaton and H = a/a ˙ is the Hubble scale factor. The action (4.44) can give rise to inflation if the “generalized slowdifferent corrections that can give order one contributions to η, with either sign. Given that we only require |η| ∼ 10−2 or smaller to obtain slow-roll inflation, we would expect that some reasonable fraction of compactifications have this property, though one might generally still expect significant corrections to the predictions from our analysis with minimal coupling to gravity. More optimistically, there may be a class of compactifications in which this cancellation happens naturally. 77 roll parameters” (not to be confused with the canonical ones we will define below), εG ≡ − are all H˙ 3(E + P) = , H2 2E ηG ≡ ε˙ , εH sG ≡ c˙s , cs H (4.48) 1, where the “speed of sound” cs is defined by c2s = P,X dP = . dE P,X +2XP,XX (4.49) We will shortly show how to relate these to the usual slow-roll parameters when we are in that regime. We will analyze our system at both weak and strong coupling and show that there are inflationary solutions for all values of λ . 4.4.1 Weak coupling: λ 1 At weak coupling we showed in the previous section that the effective action for g4 φ R 1 is S = − 1 (∂µ φ )2 +V (φ ) , 2 d4x V (φ ) = V0 1 − A φ φ0 2 − φφ e 0 , (4.50) (4.51) where V0 , A and φ0 are given in (4.28). We minimally couple this to gravity: 1 S = Mp2 2 √ d 4 x −gR − √ 1 (∂µ φ )2 +V (φ ) . d 4 x −g 2 (4.52) This is of the form (4.44) with P = X − V . The inflaton is already canonically normalized, so V is the canonical potential. We find cs = 1 and εG = 3X . X +V (4.53) Requiring εG and |ηG | to be small reduces in this case to the usual slow-roll conditions since φ is the canonically normalized scalar. The generalized slow-roll 78 parameters are related to the usual slow-roll parameters by ε = εG = Mp2 2 V V 2 , η =− ηG V + 2εG = Mp2 , 2 V (4.54) where the quantities without subscripts are the usual slow-roll parameters. In this case, the conditions on the generalized parameters guarantee a solution of the canonical ones. In section 4.5 and 4.6 we will analyze the potential (4.51) in the context of slow-roll inflation. 4.4.2 Strong coupling: λ 1 We have shown in section 4.3.2 and 4.3.3 that both the D4 and M5 probe branes in flat space can be described by the same effective action (4.36). The minimal coupling to gravity for these probes is 1 S = Mp2 2 √ d 4 x −gR +V0 √ d 4 x −g −h(z) 1 + j(z)(∂µ z)2 + q(z) , (4.55) where h(z) = 2z3 1 − z−3 , q(z) = 2(z3 − 1) , 9R2 1 j(z) = . 4 z3 − 1 (4.56) The above action is of the form (4.44) with P = V0 −h(z) 1 − 2 j(z)X + q(z) , E = V0 E + P = V0 h(z) − q(z) 1 − 2 j(z)X 2h(z) j(z)X 1 − 2 j(z)X . , (4.57) We would like to search for inflationary solutions in this action. The scalar field z is non-canonical and so we must start with the generalized slow-roll parameters. 79 j (z) z˙ 2 2δ 3 ǫ G> δ ǫ G< δ z Figure 4.4: Diagram showing values of z and jz˙2 for which εG < δ . First, we require εG = 3h(z) j(z)˙z2 2(h(z) − q(z) 1 − j(z)˙z2 ) 1. (4.58) We will now show that whenever this is satisfied, it is a good approximation to keep only the leading terms in the Taylor expansion of the square root. Considering εG as a function of z and j(z)˙z2 , it is straightforward to show that in the region where εG < δ (for any δ constraint εG 1) we have jz˙2 < 2δ /3, as shown in figure 4.4. Thus, the 1 can only be satisfied if jz˙2 1.12 This simplifies our analysis considerably, because it is then a good approximation to expand the square root in the action 1 − j(z)˙z2 ≈ 1 − 1 j(z)˙z2 , 2 (4.59) and we find the action for z minimally coupled to gravity is S ≈ G(z) = √ 1 2 Mp d 4 x −gR + 2 9V R2 √ 0 , 2 1 − z−3 V (z) = 2V0 (1 − z3 ) + z3 √ 1 d 4 x −g − G(z)gµν ∂µ z∂ν z −V (z) (4.60) , 2 1 − z−3 . This action is quadratic in derivatives, but the kinetic term is still not canonical. 12 In particular, DBI inflation is not a viable option here. 80 We can put it in canonical form with a field redefinition φ (z) = ξM 9V0 R2 2 z d z˜ 1 4 9V0 R2 2 ≈ (1 − z˜−3 ) Γ(2/3)Γ(3/4) = − ≈ −0.779936 , Γ(5/12) 1 z + ξM − 1 + O(z−5 ) 8z2 , (4.61) where the expansion has been carried out at large z since that is the region where we seek a slow-roll solution. This field redefinition gives a canonically normalized scalar field with potential at large φ , V (φ ) = V0 1 − φ03 φ3 + O(φ −4 ) , (4.62) where V0 is given in (4.37) and φ0 = 1 2 23 √ 9R2V0 λN = 2 . 2 2 3 9πR (4.63) Note that φ0 and V0 here are different from those quantities at weak coupling. We thus have a canonical scalar field with a slow-roll potential, allowing us to carry out the standard analysis. We will do this in section 4.5 and 4.6. 4.5 Analyzing the inflationary potentials In the previous section, we have seen that in both the weakly coupled and strongly coupled regimes, the effective action for our candidate inflaton field φ may be treated as a slow-roll model with canonical kinetic term and potential of the form V0 (1 − f (φ /φ0 )) where f (x) → 0 for x (4.64) 1. The constants V0 and φ0 , and the function f (x) depend on λ . The results we have found for small and large λ are summarized in table 4.2 where CW = 93 ζ (5)/8 ≈ 12.05. For intermediate values of λ , these parameters should interpolate from the small λ values to the large λ values, but we do not have any analytic methods to compute them for general λ . 81 λ 1 V0 NCW 8π 6 R4 φ0 1 2πg4 R 2λ N 36 π 2 R4 √ λN 2 2 3 9πR 1 x3 4 2 −x CW x e f (x) 1 λ Table 4.2: Summary of inflationary potentials. In this section we apply a standard slow-roll inflation analysis to determine which values of our model parameters give a viable model of inflation and to investigate the predictions for the inflationary observables. We will also favour writing results in terms of the ’t Hooft coupling λ and the rank of the gauge group N instead of the Yang-Mills coupling g4 = 4.5.1 λ N. General results for V (φ ) = V0 (1 − f (φ /φ0 )) potentials We begin with a general analysis for potentials V (φ ) which approach a constant V0 for φ much larger than some value φ0 . We define x = φ /φ0 and assume that during inflation f (x) 1 and ( f (x))2 f (x), which will be true for the specific cases we consider. The standard slow-roll parameters, which must be small to give an acceptable model of inflation, are Mp2 V 2 Mp2 ≈ 2 ( f (x))2 , 2 V 2φ0 2 M V p ≈ 2 f (x) . η = Mp2 V φ0 ε = Since ( f (x))2 (4.65) f (x) in our case, ε will be small as long as η is small. Using a standard slow-roll result, the number of e-foldings during inflation is given by Ne ≡ tf H(t)dt ≈ ti 82 1 Mp2 φi φf V dφ , V (4.66) where we integrate between the final and initial values of φ . In our case, this gives the final result Ne ≈ − φ02 Mp2 xi dx . f (x) xf (4.67) We must demand that this is at least 60 in order to solve the standard problems of big bang cosmology. For the functions f (x) appearing in our potential for weak and strong coupling, the condition that Ne is large enough is essentially the same as the condition that η is small. It will also be useful to relate the value of x at an arbitrary time during inflation to N , the number of e-foldings before the end of inflation. By the same arguments, we find N ≈− φ02 Mp2 x d x˜ . f (x) ˜ xf (4.68) We can now use standard slow-roll results to calculate predictions for the inflationary perturbations. The primordial scalar and gravitational wave power spectra are given as V3 1 V0 φ02 1 1 ≈ , 12π 2 Mp6 (V )2 12π 2 Mp4 Mp2 ( f (x∗ ))2 2V 2 V0 = Pζh = 2 4 ≈ 2 4 , 3π Mp 3π Mp ∆2s = Pk = ζ ∆t2 (4.69) from which we can compute the tensor-to-scalar ratio r≡ Mp2 ∆t2 ≈ 8 ( f (x∗ ))2 . ∆2s φ02 (4.70) These quantities are to be evaluated at x∗ , the value of x when the perturbations of the desired scale left the horizon. For the perturbations that lead to the observed CMB anisotropies, this occurred at a number of e-foldings NCMB ∼ 60 before the end of inflation. From (4.68), we obtain the relation NCMB ≈ − φ02 Mp2 x∗ xf dx . f (x) (4.71) The scale dependence of the power spectra is conveniently parameterized by scalar 83 and tensor spectral indices ns and nt , and the running parameter αs . Their definitions and their expressions for our class of potentials are ζ d ln Pk d ln k = Mp2 −3 d ln Pkh nt ≡ d ln k dns αs ≡ d ln k V2 −Mp2 2 V ns − 1 ≡ = = Mp4 −6 V V2 +2 2 V V ≈− Mp2 φ02 ≈ −2 Mp2 φ02 f (x∗ ) , (4.72) ( f (x∗ ))2 , V 2V VV V4 + 8 −2 2 4 3 V V V where for αs , we have assumed that | f (x)| ≈ −2 | f (x) f (x)| Mp4 φ04 f (x∗ ) f (x∗ ) , |( f (x))3 | as is the case for the specific potentials appearing in our model. 4.5.2 Weak and strong coupling results for the inflationary parameters Using the results of the previous section, we can specialize to the specific potentials that we obtained for small and large λ in our model, as summarized in table 4.2. First, the formula (4.68) relating the variable x to the number of e-foldings before the end of inflation gives for large x N λ 1 = CW N ex , 16π 2 λ (RMp )2 x2 N λ 1 = 1 x5 . (4.73) 2 5 2 (RM ) p 2 3 5π λN 4 3 Number of e-foldings Taking x to be the initial value of φ /φ0 in (4.73) gives Ne , the total number of efoldings during inflation. At first sight, it appears that we could obtain an arbitrarily large number of e-foldings by taking the initial value of x to be sufficiently large. However, at least in the strong coupling framework, we have a bound on x since φ is limited by the length of the throat in our compactification, and the size of the throat sets a lower bound on the 4D Planck mass by equation (4.43). Specifically, the integral in (4.43) must be at least as big as the contribution from the throat 84 region, which yields13 3 Mp , N φmax < (4.74) or equivalently 2 5 RMp xmax ≤ 2 3 3 2 π √ . λN (4.75) Combining this with the strong coupling result in (4.73), we obtain the constraint 15 Neλ 1 ≤ 22 3 2 π 3 (RMp )3 . 3 5 N 4λ 2 (4.76) For the weak coupling model, it is less clear that there should be an upper bound on x, though we might demand φ Mp in order that Planck-suppressed operators may be ignored. Applying this gives xmax 2πg4 RMp , (4.77) so (4.73) gives a constraint Neλ 1 ≤ N2 CW 2πRM p e 4 2 4 64π λ (RMp ) λ N . (4.78) Inflationary perturbations Starting again from (4.73), but taking N to be the number of e-foldings NCMB before the end of inflation when the CMB perturbations left the horizon, gives the value x∗ needed to compute the spectral parameters. Using this, we can express the results for all spectral parameters in terms of the field theory parameters λ , N, and 1/(RMp ) = MKK /Mp and the parameter NCMB which should be about 60. The results are given in table 4.3 below. We note in particular that ns and αs are independent of the parameters of the 13 Here, we are using the eleven-dimensional metric (4.12) in the integral (4.43) and assuming that the throat extends to a maximum value zmax . Note that this bound implies that the field is restricted to sub-Planckian values, as found quite generally for inflation models based on D3-branes in type IIB throats in reference [56]. As in that case, this bound implies that the production of gravitational waves (tensor modes) during inflation will be small [57, 58]. 85 1 λ 2 CW λ NCMB 6 24π (RM p )2 ∆t2 CW N 12π 8 (RM p )4 r 2 N 2 π 2 λ (RM p )2 NCMB 5 (λ N) 5 NCMB 55 1 8 2 8 ∆2s ns − 1 1 λ 2 5 35 π 14 5 (RM p ) 14 5 λN 4 37 π 4 (RM p )4 11 5 8 6 32 5 5 π 5 3 (λ N) 5 2 2 − NCMB 8 6 5 (RM p ) 5 NCMB − 5N8CMB 3 nt αs − 4π1 2 λ (RM N)2 N 2 p − CMB − N 22 (λ N) 5 1 4 8 6 2 5 32 5 5 π 5 6 8 5 (RM p ) 5 NCMB − 5N82 CMB CMB Table 4.3: Summary of results for the inflationary parameters model for large and small λ . Since the small and large λ values are different, the result for intermediate λ must be some nontrivial function of λ that approaches these two constants in the limits. 4.5.3 Other light fields Up to this point we have used the single field formalism to study the inflationary potentials. However, there are other light fields closely related to the identified inflaton in both the weak and strong coupling scenarios that need to be considered, to ensure they do not spoil our analysis. In the field theory language, we expanded about a background given by X I ∼ diag(φ I , 0, . . . , 0), corresponding to the separation of a single brane from the rest in the brane picture. Our inflaton is the magnitude of this separation, |φ | = φ Iφ I, but the angular components of φ I also correspond to light fields. In the strong coupling regime, these fields describe the position of the probe brane in the S4 86 directions. It is clear that the potential is independent of these angular components, since configurations with different values for these are related by a symmetry. Another light field arises from the Wilson line of the gauge field on the probe brane around the compact direction. In the field theory language, this is the Wilson line of the U(1) gauge field around the compact direction when the gauge symmetry is broken from U(N) to U(N − 1) ×U(1). From the four-dimensional point of view, this is a scalar field, which does not appear in the potential since a constant shift in the U(1) gauge field is a symmetry of the theory. The initial values of these extra fields, labeling one of them θ for illustration, have no effect on the potential, so we have a family of equivalent inflationary trajectories labeled by different θ . We could also consider more general trajectories by allowing the other fields to have an initial velocity14 but here, we restrict to the case where only the radial field varies. In this case we should still check whether the extra light fields contribute to the power spectrum. We have already made slow-roll assumptions on these fields, which allows us to make use of the machinery for multi-component inflation expounded in Sec. 4 of [20]. In this case, the spectrum is given by ∆2s ∝ V ∑ (Ne ,b )2 , (4.79) b∈fields where Ne ,b stands for the derivative of N with respect to the field indicated by b evaluated at the value of the field when the perturbations of the desired scale leave the horizon. For these massless fields we have Ne ,θ = 0, since the values of these fields may be changed by a symmetry transformation.15 . Thus, the only field that participates in the sum is the inflaton and (4.79) reduces to the single field result, justifying our analysis above. It is also important to ask whether bulk moduli can be ignored during inflation. While this question depends on the details of the compactification, a generic expectation is that the masses of scalar fields associated with the bulk moduli should not be lighter than the scale of supersymmetry breaking. Thus, we expect bulk moduli 14 We have checked that assuming zero initial velocity is not a requirement for inflation. is possible that in the context of a consistent compactification, the angular symmetry would be broken. In this case, a more detailed analysis would be required. 15 It 87 λ 1 λ 1 ns = 0.967 ns = 0.973 αs = −0.00055 αs = −0.00044 Table 4.4: Predictions for scalar spectral index and running parameter fields to be heavier than MKK ∼ 1/R. We will see that this is much larger than the 2 /M that controls thermal/quantum fluctuations during inflation, so scale H ∼ MKK p the bulk moduli fields can be safely ignored. 4.6 Predictions and observational constraints We are now ready to see which parameter values in our model give predictions consistent with current cosmological observations. To begin, we will assume that NCMB = 6016 . This immediately leads to the predictions shown in table 4.4 without any further assumptions about the parameters λ , N, or RMp . The results for both regimes are consistent with the best fits from the seven-year WMAP data [60], ns = 0.968 ± 0.01217 and −0.061 ≤ αs ≤ 0.017. It also close to the Planck data with ns = 0.9608 ± 0.0054 [61]. In fact the best fit of ns = 0.9611 from the Planck data would correspond to a few less e-foldings of NCMB = 51.4. The observed magnitude of scalar perturbations can be used to determine the Kaluza-Klein scale 1/(RMp ) = MKK /Mp in terms of the field theory parameters λ and N. To obtain the WMAP observed CMB normalization result ∆2s = (2.4 ± 0.2) × 10−9 [62] from the results above, we must have the Kaluza-Klein scales shown in table 4.5. A further constraint comes from demanding that the number of e-foldings is at least 60. In the weakly coupled regime, we see from (4.78) that we can obtain an essentially arbitrary number of e-foldings for any value of N and λ 1, since the exponent in the constraint on Ne is O(105 ). For λ 16 The 1, using the precise value depends on details of the history of the Universe between the end of inflation and the well-understood era after reheating. The value NCMB = 60 is typical for a reheat temperature not far below the scale of inflation, but this can be somewhat higher or lower. For a detailed discussion, see reference [59]. 17 This value assumes r = 0 in the fit. 88 1 λ 1 RM p = 1 λ −5 3.6×10 √ λ 1 RM p = 7.5×10−4 1 (λ N) 7 Table 4.5: Kaluza-Klein scale in terms of field theory parameters λ 1 λ 1 3 r = 7.2 × 10−14 (N/λ 2 ) r = 2.5 × 10−9 (λ N) 7 Table 4.6: Predictions for the tensor-to-scalar ratio values from table 4.5 for the Kaluza-Klein scale in (4.76) gives Neλ 1 ≤ 2.2 × 1014 15 λ 14 N 25 7 . (4.80) Demanding that the right side is at least 60 gives a constraint 3 104 . λ 5N (4.81) Even if we take very large λ of order N (corresponding to the theory derived from M5-branes on torus with cycles of similar size), we obtain a sufficient number of e-foldings as long as N is less than about 103 . Thus, we can obtain a sufficient number of e-foldings for a very broad range of parameters in the model. Finally, we note that using the results from table 4.5 for the Kaluza-Klein scale, the predicted tensor-to-scalar ratio in the two regimes is Taking into account the constraint (4.81), we find that the tensor-to-scalar ratio is never more than 10−6 . This is certainly consistent with the current experimental bound r < 0.24 [60], but also at a level that is beyond the sensitivity of planned experiments. The smallness of the tensor modes arises due to the relatively low scale for inflation. Explicitly, we list the inflation scales in table 4.7, where we recall that V0 gives the energy density during inflation. 89 λ 1 λ 1 1 1 V04 /Mp = 7.1 × 10−6 (N/λ 2 ) 4 1 3 V04 /Mp = 9.7 × 10−5 (λ N) 28 Table 4.7: Inflation scale in terms of field theory 4.7 The end of inflation and reheating An interesting feature of our model, both for strong and weak coupling, is that the field theory is confining. That is, expanding about φ = 0, all physical particles must be gauge-singlets, just as in QCD. In particular, since the inflaton field transforms non-trivially under the action of the gauge group, inflaton particles do not exist as finite energy excitations about the vacuum state (just as there are no free quarks in QCD). This is clear geometrically in the strong coupling picture, since the brane associated with the inflaton field self-annihilates at the end of inflation. In the weakly coupled regime, the IR physics is that of pure Yang-Mills theory, so the relevant excitations are glueballs with mass of order Λc . In either case, there is no leftover light scalar field to worry about at the end of inflation, and the inflaton should efficiently give up its energy to other degrees of freedom. The details of this process, and the implications of our scenario, will be somewhat different in the two regimes, as we now explain. 4.7.1 Weak coupling: λ 1 In the small λ regime, the energy scale during inflation is of order the KK-scale, which is larger than the confinement scale by a factor ec/λ . Thus, the energy of the inflaton field is certainly enough to drive a deconfinement transition in the field theory at the end of inflation.18 Eventually, we would have a transition back to a confined phase, with glueballs replacing the deconfined plasma. In a more complete model, our field theory must at least be supplemented with the degrees of freedom of the Standard Model. Here, there are a few interesting 18 If the confinement scale is sufficiently far below the KK-scale, the Hubble temperature during inflation can be larger than the confinement scale, in which case, the field theory will already be deconfined during inflation. 90 scenarios. First, it is possible that the non-abelian gauge fields in the inflation model are actually the same ones that appear in QCD. It is certainly possible that the Standard Model could arise from a higher-dimensional supersymmetric field theory compactified on a circle with anti-periodic boundary conditions (e.g. see reference [63] and references therein). If this higher dimensional field theory (before the compactification) has a moduli space parameterized by some scalar fields, then one of these scalar fields could become the inflaton in the compactified theory. In this scenario, reheating is particularly simple, since the energy of the inflaton would be dumped directly into Standard Model degrees of freedom. Another interesting scenario would be that the field theory we have discussed is part of a dark matter sector. In this case, the glueballs we obtain in this sector at the end of inflation would be a dark-matter candidate. There must be some coupling between the degrees of freedom in this sector with the Standard Model degrees of freedom, so that Standard Model particles are also created during reheating. In this scenario, supersymmetry breaking in the Standard Model sector may arise via these couplings from the explicit supersymmetry breaking in this inflaton/dark matter sector. Finally, we could have a scenario in which the Standard Model is a separate sector from the inflationary sector, but is coupled to this sector in such a way that the glueballs can decay into Standard Model (and dark matter) particles. In this case, no particles from the inflationary sector would be relevant to the current epoch. It would be interesting to investigate in greater detail the dynamics of reheating in these models, especially in the weak coupling scenario where the inflaton releases its energy directly into Standard Model degrees of freedom. We leave this for future work. 4.7.2 Strong coupling: λ 1 At large values of λ where the gravity picture is valid, the Kaluza-Klein scale is roughly the same as the confinement scale, so it is less clear whether deconfinement in the inflationary sector will occur at the end of inflation. To decide this, we can compare the energy density of the brane during inflation with the energy density of the deconfined plasma just above the deconfinement transition. As we show in 91 appendix 4.B , the latter energy density is edec = 2 · 5 N 2λ . 37 π 2 R4 (4.82) On the other hand, the energy density carried by the inflaton field at the start of inflation is approximately V0 , given in table 4.2, so we have ei = 2λ N 36 π 2 R4 . (4.83) Comparing these, we find that ei 3 = . edec 5N (4.84) Thus, in the range of parameters where a gravity description is valid, the brane will not have enough energy to drive a deconfinement transition at the end of inflation. In this case, the reheating phase of inflation will more directly involve the production of glueballs. As in the field theory scenario, these could either decay into Standard Model particles, or be left over as a potential dark-matter candidate.19 In this regime, the confinement scale is too high to associate the confining gauge theory used for inflation with QCD. 4.8 The η-problem In the previous sections, we introduced a particular field theory, computed the effective action in Minkowski space, minimally coupled this to gravity, and then studied the resulting theory as a potential model of inflation. Ideally, we would have started with a UV complete theory including gravity (perhaps arising from some consistent compactification of string theory), derived the effective action at the scale of inflation and proceeded to analyze that theory. In our minimal treatment, there are two types of terms in the effective action that we may have missed. Curvature Couplings The first type of corrections can be terms coupling the inflaton directly to the space19 For a current review of reheating see reference [64]. 92 time curvature, for example − α d 4 x Rφ 2 . 2 (4.85) These terms make no contribution when studying the field theory in Minkowski space, but clearly modify the potential during inflation, where the background spacetime is approximately de Sitter space with positive spacetime curvature. Since R is of order H 2 during inflation, the term (4.85) gives an effective mass term with m2 ∝ αH 2 for the inflaton field. When α is of order 1, these terms give order one contributions to the slow-roll parameter η, which therefore must be canceled by other contributions in order to maintain a viable model. For a field theory described in terms of a gravity dual, such as our model in the strong coupling regime, we can calculate these contributions directly by perturbing the infinite throat metric so that the boundary metric is de Sitter, and then calculating the inflaton effective action by studying the brane effective action in this background. We carry out this calculation for our model in appendix 4.C , and find a leading correction term 3 VH = H 2 φ 2 . 4 (4.86) This gives a contribution to η of ηH = In order to obtain η 3 H 2 Mp2 1 ∼ . 2 V0 2 (4.87) 1 in the complete theory, we must therefore have other contributions to η that cancel this one. Extra terms from a consistent UV completion with dynamical gravity The terms we have just described appear even if we study the quantum field theory in curved spacetime without adding dynamical gravity or any other additional UV physics (i.e. keeping Mp = ∞). The second type of correction arises from terms that must be added to obtain a consistent UV completion with dynamical gravity. These appear in the effective action as operators suppressed by powers of the Planck mass 93 or another UV scale. For field theories dual to gravity on a throat geometry, there is a simple geometrical picture for the origin of these correction terms. Going from a decoupled field theory to a field theory coupled to gravity corresponds to passing from an infinite throat geometry to that of a finite throat, which appears as part of some consistent compactification. Near the tip of the throat, the potential for a probe brane should be essentially the same as in the infinite throat case. However, as we go to the top of the throat, the geometry becomes modified relative to the infinite throat, and this results in modifications to the inflaton potential. At present, finding fully consistent compactifications which include an appropriate throat region is difficult, so we cannot systematically investigate all possible UV completions to determine the possibilities for the corrected inflaton potential. However, as pointed out in references [65, 66], one can at least classify the possible corrections that can occur by starting with the infinite throat geometry and studying all (non-normalizable) perturbations in the UV consistent with the supergravity equations of motion. One can then check to see which of these affect the inflaton potential and classify the possible corrections that might appear in the context of a true compactification. In the eleven-dimensional picture, our throat geometry in the UV approaches AdS7 /T 2 × S4 , where by AdS7 /T 2 we mean AdS7 with two directions compactified. Thus it may be straightforward to carry out this program using previous results (see reference [67] and references therein) for the perturbation spectrum of eleven-dimensional supergravity on AdS7 × S4 . One could then check, as was done in references [65, 66] for the case of the Klebanov-Strassler throat in IIB string theory, that there exist allowed perturbations which could (for the right compactification) cancel the effect of the curvature couplings discussed above. Assuming such perturbations are possible at all, one might expect that one in a hundred consistent UV completions would give rise to sufficient cancelation, since we only require η ∼ 10−2 . Here, we are assuming a uniform distribution of order one coefficients for the total H 2 φ 2 term. More optimistically, there could be a class of compactifications in which such cancelation occurs naturally. 94 4.9 Related models In this chapter, we have focused on a particular field theory (or particular brane configuration) and a particular initial condition for which a single scalar field eigenvalue begins with a non-zero value. However, there are many possible generalizations making use of the same basic idea. 4.9.1 Multiple scalars and N-flation First, we could choose a more generic initial condition in which many scalar field eigenvalues are non-zero (or equivalently, in which the U(N) gauge symmetry is broken not just to U(N − 1) × U(1) but to a more generic subgroup). In the supersymmetric case, any collection of mutually commuting scalar matrices X i gives zero potential, so we expect that any such configuration (with large enough eigenvalues) could be a viable initial condition for inflation. For the case where the number of non-zero scalar eigenvalues is still small compared with N, the potential would take the form V = ∑ V (φi ) (4.88) i with terms depending on differences of the scalar field eigenvalues suppressed by 1 N. In the strong coupling geometrical picture, this form arises because we have a number of branes independently falling down towards the tip of the same throat geometry. Models with multiple scalars and a potential of the form (4.88) have been dubbed “N-flation” [68] (for a specific choice of potential see reference [69]), and are generally even more advantageous for inflation, since each scalar feels the Hubble friction associated with the whole collection of inflatoin fields.20 In the most generic case where the gauge symmetry is broken to U(1)N , the scalar field potential will be a more complicated function that (4.88) depending on the differences of the eigenvalues. In this case, there will be no geometrical picture in the strongly coupled regime until the end of inflation when gauge symmetry is restored. 20 Further analysis would be required to see whether this “assisted” inflation effect works in our model beyond leading order. 95 4.9.2 Other field theories A much broader set of generalizations comes by starting with more general higherdimensional supersymmetric field theories. Taking some higher-dimensional theory with a moduli space and compactifying to four-dimensions so that supersymmetry is broken by boundary conditions, we again expect the potential far from the origin of moduli space to be quite flat. Another example starting with 16 supersymmetries would be the D5-brane field theory (maximally SUSY 5+1 dimensional Yang-Mills theory) compactified on a torus with anti-periodic boundary conditions for fermions along one direction. In the strong coupling picture, this leads to a potential of the form V (φ ) = V0 (1 − φ02 /φ 2 ). Starting with less supersymmetry, there are many more possibilities. For example, many N = 2 theories (i.e. eight supercharges) in four dimensions can be defined by compactifying the 5+1 dimensional (0, 2) CFT on a Riemann surface (possibly with defects) [70, 71]. These typically have a moduli space of vacua. If we change the boundary conditions around some cycle of the Riemann surface so that supersymmetry is broken, there should be a potential for the scalar fields that previously parameterized the moduli space. For examples with lower supersymmetry, there are fewer examples where the gravity dual to the strongly coupled theory is known, so a detailed analysis may be more difficult. 4.9.3 Other boundary conditions Finally, we could consider supersymmetry-breaking boundary conditions that are more general than the anti-periodic boundary conditions for fermions. In general, if the higher-dimensional theory has a global symmetry, we can consider boundary conditions in which the fields around some cycle come back to themselves only up to an element of the global symmetry group. A model of inflation based on this more general type of twisted boundary condition was discussed in [50]. 4.10 Discussion In this chapter, we have described a new mechanism for slow-roll inflation. To conclude, it is worth emphasizing a few of its special features: 96 • The model arises quite naturally with only two ingredients: supersymmetry and extra dimensions. In compactifying to four dimensions, supersymmetrybreaking boundary conditions are no less natural (and perhaps even more generic) than supersymmetry preserving boundary conditions. • These essential features are both crucial ingredients of string theory; thus the model arises quite naturally from string-theoretic degrees of freedom • As a brane-inflation model, it is very simple since it requires only a single brane on a natural throat geometry; no additional ingredients are required to ensure a graceful exit at the end of inflation. • The model can be treated analytically – and provides a viable model of inflation – over a very broad range of parameters. • There are no problematic light degrees of freedom in the inflationary sector left over at the end of inflation; the energy in the inflaton field is quickly transferred to other degrees of freedom at the end of inflation Of course, as for most models of inflation involving supersymmetry, our scenario may require some modest fine-tuning (of order one part in one hundred) to avoid the η-problem, as we have discussed in section 4.8. An interesting future direction would be to carry out an analysis of allowed perturbations in the UV part of our throat geometry (following references [65, 66]) to check that operators able to counteract the effects of the curvature couplings discussed in section 4.8 can arise in the context of consistent compactifications of string theory. Other interesting future directions include fleshing out some of the reheating scenarios discussed in section 4.7, and exploring some of the other models of section section 4.9. 4.A Field theory calculations In this appendix, we give details of the calculation of the one-loop effective potential for our inflaton field in the regime where the field theory is weakly coupled at the Kaluza-Klein scale. We begin with the action (4.5), and expand the theory about a background X I = BI = diag(g4 φ I , 0, . . . , 0) . 97 (4.89) We find that all the fields have off-diagonal modes with masses of order |g4 φ |, which we assume to be initially much larger than the Kaluza-Klein scale 1/R. Thus, it makes sense to integrate these out to find an effective action for φ I . We begin by calculating the effective potential (i.e. the non-derivative terms). For this, we may assume that the background φ is independent of the spacetime coordinates, which simplifies the calculation considerably. For the calculation, it is convenient to use a gauge-fixing term 1 L f ix = − (∂µ Aµ − i[BI , X I ])2 . 2 (4.90) in order to cancel cross terms between A and X. The corresponding ghost action is ¯ µ C + [BI , C][X ¯ I ,C] . Lghost = −∂µ CD (4.91) We can now write down the action for the heavy modes Y , η, W , and c (each N-component vectors) defined by X I = BI + Aµ = 0 YI Y I† 0 0 Wµ Wµ† , ψ= , 0 C= 0 η η† 0 0 c c† 0 , (4.92) . It is also convenient to combine the gauge field modes and the scalar modes into a single field YA where Yµ ≡ Aµ and YI ≡ Y I . Then expanding the action to quadratic order in the heavy fields (and any order in the background fields), we find Lbos = −|∂µ YA |2 − (g4 φ )2 |YA |2 , ¯ µ Dµ η + g4 ηΓ ¯ I+4 φI η , L f erm = −iηΓ Lghost = −|∂µ c|2 − (g4 φ )2 |c|2 . (4.93) The one-loop effective potential may be obtained from functional determinants of 98 the operators defining the quadratic action, and we obtain21 V = 8N (n + 12 )2 d4k n2 2 2 2 2 ln(k + (g φ ) + ) − ln(k + (g φ ) + ) 4 4 (2π)4 ∑ R2 R2 n , (4.94) as we found in section 3. To calculate the modifications to the kinetic terms we can use the same setup as above, but we can no longer work with a spacetime-independent background field. In this case, it is convenient to write the background field as X I = BI = diag(g4 φ I + K I , 0, . . . , 0) , where we take φ I to be a constant and K I to include all possible spacetime dependence of the background field. The action to quadratic order in the heavy fields (and any order in the background fields) is int Lbos = −|∂µ YA |2 − (g4 φ )2 |YA |2 + Lbos , µ I+4 ¯ Dµ η + g4 ηΓ ¯ L f erm = −iηΓ φI η + L fint erm , Lghost int = −|∂µ c|2 − (g4 φ )2 |c|2 + Lghost , (4.95) where the “interaction terms” involving the background field K are int = −YA† ((2g4 φ · K + K 2 )δAB + 2FAB )YB , Lbos ¯ I+4 KI η , L fint erm = ηΓ int Lghost = −c† (2g4 φ · K + K 2 )c . (4.96) Here, we define an antisymmetric quantity F as FµI = i∂µ KI , with Fµν = FIJ = 0. In our calculation, we will find it convenient to define the propagators for the heavy fields using the K-independent terms and treat the terms depending on K as vertices (with two legs). 21 A similar calculation was recently performed in reference [72]. 99 Then the propagators are YA (x)YB† (0) η(x)η † (0) c(x)c† (0) = = = 1 2πR n d4k ei(k·x+ R x p ) δAB , ∑ (2π)4 n k2 + n22 + (g4 φ )2 R d4k ei(k·x+ (n+ 12 ) R xp) (kµ Γµ + g4 φ I ΓI+4 ) 1 2πR (2π)4 ∑ n 1 2πR d4k . n2 2 2 (2π)4 ∑ n k + 2 + (g4 φ ) (n+ 21 )2 R2 n ei(k·x+ R x p ) k2 + , + (g4 φ )2 (4.97) R To find the kinetic terms, we do a perturbative expansion using the vertices defined above, keeping only terms that are quadratic in the derivatives of K. After some work, we find 1 loop Lkin = 4N −32N d 4 x(∂µ φI ∂ µ φI ) d 4 x(φ · ∂µ φ )(φ · ∂ν φ ) d4k 1 1 − n2 2 (n+ 1 )2 2 2 (2π)4 ∑ (k + (g4 φ ) + R2 ) n (k2 + (g4 φ )2 + R22 )2 k µ kν d4k k µ kν − n2 4 (n+ 12 )2 4 2 2 (2π)4 ∑ n (k + (g4 φ ) + 2 ) (k2 + (g φ )2 + ) R 4 where we have redefined g4 φ + K → g4 φ , so that now φ is the full spacetimedependent inflaton field. The sums over n can again be evaluated analytically and we find that the resulting integrands for k are exponentially suppressed for large k and for large φ R. Thus, for large φ , the kinetic term for the scalar field is just the canonical one coming from the tree-level action. 4.B Energy in the deconfined phase at strong coupling In this appendix, we calculate the energy density in the deconfined phase of the inflationary field theory at large λ . This determines the minimum initial energy in the inflaton field necessary to drive a deconfinement transition at the end of inflation. The deconfined phase of the gauge theory corresponds to a gravity solution (in 100 R2 the M-theory picture) [53] ds2 = ρ L2 (− f (ρ)dt 2 + dx2 ) + 2 f −1 (ρ)dρ 2 + L2 dΩ24 , L ρ where f (ρ) = 1 − ρ03 . ρ3 (4.98) (4.99) The temperature is related to ρ0 and L by 3 1 4π L 2 = . T 3 ρ 12 0 (4.100) The area of the horizon is 8π 2 4 ρ0 A= L 3 L 5 2 V, (4.101) where V is the five-dimensional coordinate volume in the xi directions. Using 2 , we find that the entropy density in the field the Bekenstein formula S = 2πA/κ11 theory is given by S 2 =2 V 3 6 π 3N 3T 5 . (4.102) Using dE = T dS, and V = (2πR)(2πRM )V3 , we obtain 5 E = V3 3 2 3 6 π 3 N 3 T 6 (2πR)(2πRM ) (4.103) for the three-dimensional energy density of the M5-brane theory on a torus with radii R and RM in its high-temperature phase. This phase is preferred for [41] T > Tc = 1 , 2πR (4.104) and the energy just above this critical temperature is edec = 2 · 5 N 2λ . 37 π 2 R4 101 (4.105) This is the minimum energy density for the deconfined plasma in the case where it is thermodynamically preferred. 4.C The inflaton potential for quantum field theory in de Sitter space In this appendix, we calculate the modification to the effective potential for the inflaton field in the strongly coupled regime that arises when we study the field theory in a de Sitter background (as we have to a good approximation during inflation), but with non-dynamical gravity (i.e. Mp = ∞). Similar computations in a different background were performed in references [73, 74]. From an effective field theory point of view, these corrections arise due to couplings involving the spacetime curvature tensors. We recall that the field theory in the regime of very strong coupling is dual to M-theory on the background with metric ds2 = ρ L2 2 (ηµν dxµ dxν + dxM + f (ρ)dx42 ) + 2 f −1 (ρ)dρ 2 + L2 dΩ24 L ρ (4.106) and N units of four-form flux through the sphere. Here f (ρ) = 1 − ρ03 . ρ3 (4.107) During inflation, the four-dimensional metric is not Minkowski space but rather an FRW-type metric that should be well approximated by de-Sitter space. Thus, the actual geometry in the throat region should be that of a warped de-Sitter compactification rather than a warped Minkowski compactification. We will use the fact that there is a consistent truncation of eleven-dimensional supergravity on a sphere (with four-form flux through the sphere) to seven-dimensional supergravity with a cosmological constant. Thus, we will search for a seven-dimensional solution of the form ds2 = ρ L2 2 (−dt 2 + eHt dx2 + ea(ρ) dxM + eq(ρ) dx42 ) + 2 f −1 (ρ)dρ 2 , L ρ 102 (4.108) satisfying 15 1 Rµν − gµν R = 2 gµν . 2 4L (4.109) Here, f should go to 1 at large ρ while q and a should go to zero, so that the boundary metric is dS4 × T 2 . With this ansatz, it turns out that the Einstein equations can be reduced to a set of decoupled ordinary differential equations for the functions f (ρ), a(ρ), and q(ρ). Setting L = 1 for now, the function f must satisfy −2ρ 4 f f +2ρ 4 ( f )2 −2ρ 3 f f −18ρ 3 f −36ρ 2 f +36ρ 2 = 9H 2 (ρ 2 f +2ρ f −4ρ)−9H 4 . (4.110) Given a solution for f , we must have (a + q) = 6 6 f 3H 2 − − + 2 , ρf ρ f fρ (4.111) and defining W = a − q , we must have 1 3 3H 2 d ln(W ) = − − − . dρ ρ ρ f 2 f ρ2 (4.112) We can solve these equations perturbatively for large ρ, starting with the exact solution where a = 0, q = ln(1 − ρ03 /ρ 3 ) and f is given by (4.107). We find (restoring dependence on L) 9 H 6 L9 9 H 4 L6 ρ03 9 H 2 L3 − − (3 ln(ρ) − 1) + O(ρ −4 ) , + 10 ρ 80 ρ 2 ρ 3 800 ρ 3 3H 2 L3 9H 4 L6 +... − 2ρ 20ρ 2 ρ03 +... . (4.113) ρ3 f (ρ) = 1 + a(ρ) + q(ρ) = a(ρ) − q(ρ) = We can use the perturbed metric in the M5-brane action to determine the corrections to the effective potential for the inflaton. We end up with a leading correction V /V0 → V /V0 + 103 3H 2 L3 2 ρ . 2ρ03 (4.114) In terms of the canonical field φ , 3 V → V + H 2φ 2 . 4 104 (4.115) Chapter 5 Extensions to Twisted Inflation In this chapter we look at some minor extensions to the Twisted Inflation model introduced in chapter 4. In Twisted Inflation the inflaton is a single eigenvalue of a SU(N) gauge field. In section 5.1 we will put all the eigenvalues in the weak coupling model in a circular distribution. Doing this, we move one step beyond the simple case of a single eigenvalue, and examine the effects it has on the predictions of the model. In the next section we note that Twisted Inflation explicitly breaks supersymmetry. We then study if this broken supersymmetry can be communicated to a MSSM sector. While not completely ruling it out, we find using Twisted Inflation as a hidden sector to break supersymmetry in the MSSM is impractical. 5.1 Multiple fields In chapter 4, a model for inflation was introduced where the inflaton field is a component of a five dimensional supersymmetric SU(N) Yang-Mills field, with one dimension compactified in a way that breaks supersymmetry. The component that makes up the inflaton is an eigenvalue of the adjoint scalar field in the SU(N) theory. Here we look to generalize the results from chapter 4 by turning on more than one eigenvalue. First we will attempt to turn on all the eigenvalues, but this will only be done in the context of weak coupling. For strong coupling, we make use of the gauge/- 105 Figure 5.1: Geometric interpretation of the scalar fields. gravity duality to study the theory, and the scalar fields in the field theory correspond geometrically to the position in directions transverse to their volume of the N branes that create the geometry at strong coupling. Turning on one eigenvalue corresponds to moving one brane away from the stack as first shown in chapter 4 and repeated in figure 5.1. Should all the eigenvalues be turned on, all the branes in the stack would spread out, destroying the geometry used to study the theory. So after studying the effects of turning on all the eigenvalues at weak coupling, the effects of turning multiple eigenvalues in strong are examined, but here we will assume that just a few eigenvalues are turned on. In other words rather than a single brane moved away from the stack, we shall move NP branes away from the stack with NP small enough that the probe approximation remains valid. 5.1.1 Weak coupling Our starting point for studying the effects of turning on all the eigenvalues at weak coupling will be the potential (4.23) derived in 4.3.1: V (φ ) = 2NCW W (2πg4 φ R) . π 2 (2πR)4 (5.1) This potential was derived assuming only one eigenvalue of the scalar fields was turned on with X I = diag(φ I , 0, . . . , 0) and φ = φ I φ I . However, turning on all the eigenvalues with X I = diag(φ1I , φ2I , . . . , φNI ) and following through the same derivation results in all the places where φ 2 = φ I φ I came in as a mass-like term to be replaced with differences between eigenvalues, φiI − φ jI 106 2 . This replacement leads to the potential, V= 2CW 2 π (2πR)4 ∑W 2πg4 R φiI − φ jI 2 , (5.2) i> j which we can then use to study multiple eigenvalues. Let us consider a scenario where half of the eigenvalues have a value of + √φN √ and the other half a value of − √φN where the factor N is added so that the kinetic term for φ remains canonical. Putting these scalar values into (5.2) returns the potential 2CW N 2 2φ W 2πg4 R √ π 2 (2πR)4 2 N 2 φ N , = V0 W √ 4 N φ0 V (φ ) = which is close enough to the original potential (4.23) that the analysis done in section 4.5.1 carries through as long as the simple substitutions of V0 → √ φ0 → φ0 N 2 V0 N 4 and are made. The results are summarized in table 5.1. Another distribution that we can readily study consists of positioning the N eigenvalues on an regular (N − 1)-simplex1 in the 5D space spanned by the five scalars in the field theory. Using a simplex simplifies the study as each eigenvalue would be equidistant from the others, however because the five scalars can only span a 5D space, at most we can have is a 5-simplex and so we restrict to N = 2, 3, 4, 5, 6 . Then by inscribing the (N −1)-simplex on a SN−2 of radius of the ratio of edge length to radius of an (N − 1)-simplex, 1A √φ and making use N 2N N−1 , and substituting N-simplex is a N-dimensional polytope with N + 1 vertices, thus a 2-simplex would be a triangle, and a 3-simplex would be a tetrahedron. A regular simplex is one where the vertices are equidistant from one another. 107 Single Eigenvalue Split Eigenvalues Simplex Distribution (N = 2, . . . , 6) ∆2s 2 CW λ NCMB 6 24π (RM p )2 2 CW λ NCMB 6 24π (RM p )2 2 CW λ NCMB 6 12π (RM p )2 ∆t2 CW N 12π 8 (RM p )4 CW N2 48π 8 (RM p )4 CW N(N−1) 12π 8 (RM p )4 r 2 N 2 π 2 λ (RM p )2 NCMB 1 N2 2 2 2π λ (RM p )2 NCMB N(N−1) 1 2 π 2 λ (RM p )2 NCMB ns − 1 2 − NCMB 2 − NCMB 2 − NCMB nt αs 2 − 4π1 2 λ (RM N)2 N 2 1 N − 16π 2 λ (RM )2 N 2 N(N−1) − 8π1 2 λ (RM )2 N 2 − N 22 − N 22 − N 22 p CMB p CMB CMB CMB p CMB CMB Table 5.1: Weak coupling multi-field inflationary perturbations into (5.2) leading to the potential, V (φ ) = 2CW N(N − 1)W 2 π (2πR)4 = V0 (N − 1)W φ 2πg4 R √ N 2 φ N − 1 φ0 2N N −1 . Comparing this potential to the original (4.23), we see just as before that it is quite similar and the only differences can be encapsulated by making the appropriate substitutions, V0 → V0 (N − 1) and φ0 → N−1 2 φ0 . The changes these substitutions introduce into the predictions are summarized in table 5.1. Effects of multiple eigenvalues on predictions In examining the differences between the columns in table 5.1 we see that the predictions for scalar perturbations remain very similar. The primordial scalar power spectrum is unchanged for the the split eigenvalues and only picks up a factor of 108 Single Eigenvalue r = 7.2 × 10−14 λN2 Split Eigenvalues r = 1.8 × 10−14 Simplex Distribution N 2 λ r = 1.8 × 10−14 N(N−1) λ2 Table 5.2: Predictions for the tensor-to-scalar ratio at weak-coupling two for the simplex distribution compared to the single eigenvalue case. Meanwhile the scalar spectral index remains unchanged between each eigenvalue distribution. These results demonstrate the robustness of the predictions for Twisted Inflation for different eigenvalue distributions. Unlike for the scalar perturbations, we do find a possibly significant difference arising in the predictions for tensor perturbations. The tensor power spectrum, and by extension the tensor-scalar ratio, and the tensor spectral index all gain a factor close to N. The split eigenvalue distribution gains a factor of simplex distribution gains of factor of N − 1 or N−1 2 . N 4, and the For small values of N, this difference will be largely insignificant, however for large enough values of N, this could drastically affect the possibility of observing primordial gravity waves in the CMB 2 . In section 4.6 we found by fixing the scalar power spectrum to match observations, the tensor-to-scalar ratio, r, could be written in terms of the field theory parameters N and λ . The results for a single eigenvalue at weak coupling are reproduced in table 5.2, and we found that r is likely too small to measure. However, looking at the results for multiple eigenvalue distributions in table 5.2 we see they pick up an extra factor of N. If it were to be that N ∼ λ −1 ∼ 103 then we would have r ∼ 10−2 . At r ∼ 10−2 , the signature of primordial gravity waves would be at the edge of detectability of currently running experiments [75]. 2 For a simplex distribution, we are restricted to N = 2, 3, 4, 5, 6 and taking N ∼ 103 is not possible. However, we can imagine a distribution of eigenvalues with slightly less symmetry, perhaps a regular spread of the eigenvalues on Sd with d ≤ 4. It seems quite reasonable to assume such a distribution would have a potential with similar form with a factor of N, but with a denominator larger than two. Future study could include an analysis of such a distribution, or perhaps a numerical analysis of a random distribution. 109 5.1.2 Strong coupling For strong coupling we take NP probe branes away from the stack under the approximation that they do not attract each other. With this approximation, the NP branes give us an action that is nothing more than NP times the the single brane Born-Infeld action. Thus our starting point for examining multiple branes on the throat is simply the single brane action (4.60) after canonicalization of the kinetic term, summing over each probe brane that we pull away: 1 S ≈ Mp2 2 NP √ d 4 x −gR + ∑ i=1 d4x 1 (∂µ φi )2 +V (φi ) , 2 (5.3) φ3 where V (φ ) ≈ V0 1 − φ03 . Looking at (5.3), we see that taking each of the branes out to φi = √φ NP will preserve a canonical kinetic term for φ and we’ll have the same form for the po√ tential V as long as we take V0 → NPV0 and φ0 → NP φ0 . Thus, just as it did in the weak coupling case, our analysis caries through just as it did before as long as the appropriate substitutions are made. The effect of these substitutions on physical parameters are summarized in table 5.3, and can be even more succinctly summarized by the substitution of λ → λ NP in formulae for physical parameters. A significant difference between the effects of multiple probe branes in the strong coupling scenario compared to multiple eigenvalues in the weak coupling scenario is the effect on the scalar power spectra, ∆2s . In table 5.1, the scalar power spectra is either unaffected or only picks up a factor of 1 2 for multiple eigenvalues, while for multiple probe branes in table 5.3 the scalar power spectra gains a factor 2 of NP5 . The effect of NP on the scalar power spectra, as well its addition to φ0 , carries through the same analysis demanding enough e-foldings that lead to the constraint (4.81), and leads to a tighter constraint 3 3 λ 5 NP5 N 104 . With N large, which it needs to be for to form the throat geometry used in this thesis, and λ large for strong coupling, little room is left to increase NP significantly to affect physical predictions. Thus multiple probe branes at strong coupling offers 110 Single Probe Brane 1 8 2 8 2 5 35 π 14 5 (RM p ) 1 2 5 35 π 11 5 8 6 32 5 5 π 5 3 11 5 8 6 32 5 5 π 5 (λ N) 5 2 r 14 5 3 αs − − 5N8CMB 4 3 (λ N) 5 1 8 6 8 6 5 (RM p ) 5 NCMB 3 nt 14 5 (λ NNP ) 5 2 8 6 5 (RM p ) 5 NCMB − 5N8CMB ns − 1 (RM p ) 4 λ NNP 37 π 4 (RM p )4 λN 4 37 π 4 (RM p )4 ∆t2 5 (λ NNP ) 5 NCMB 55 14 5 8 2 8 5 (λ N) 5 NCMB 55 ∆2s NP Probe Branes 6 8 2 5 32 5 5 π 5 (RM p ) 5 N 5 CMB − 5N82 − (λ NNP ) 5 1 4 8 6 6 8 2 5 32 5 5 π 5 (RM p ) 5 N 5 CMB − 5N82 CMB CMB Table 5.3: Strong coupling multi-field inflationary perturbations less advantages and flexibility compared to multiple eigenvalues at weak coupling. Furthermore, with the effect on physical parameters being nothing more than the substitution of λ → λ NP , any effect of multiple probe branes can be emulated by one probe brane and the scaling of λ . 5.2 Supersymmetry breaking We now turn our attention to using Twisted Inflation to break supersymmetry in the MSSM. In order to get an appropriate mass spectrum in the MSSM, there is a need for a separate supersymmetry-breaking sector that communicates the supersymmetry breaking to the MSSM sector [76]. In our Twisted Inflation model, we explicitly break supersymmetry by giving different periodic boundary conditions to the fermions and the bosons. We can then use Twisted Inflation as a hidden sector that communicates this broken supersymmetry to the MSSM through one of 111 three broadly classified mediation mechanisms, gravity-mediated supersymmetry breaking, gauge-mediated supersymmetry breaking and anomaly-mediated supersymmetry breaking. In dealing with each form of supersymmetry breaking mediation we will not detail an explicit mechanism to incorporate Twisted Inflation as the separate supersymmetry breaking sector, but simply take general considerations for each mechanism. In these considerations we’ll need the scale of supersymmetry breaking in Twisted Inflation which we take to be ΛSSB = R1 . To justify this, consider energy levels high above R1 . At these levels, the mass difference added to the fermion tower of states by the anti-periodic boundary compared to the boson tower of states, R1 , becomes negligible and supersymmetry is effectively restored. Below this level, the mass difference is quite significant and supersymmetry is fully broken. Then to evaluate the practicality of each mechanism we need a means to estimate the scale of ΛSSB . To this end we consider the hierarchy problem in the Standard Model. One of the troubling aspects of the Standard Model is how small the Higgs boson’s mass is compared to the Planck mass. At first glance this may not seem like much of a problem, but the Higgs mass receives quantum corrections from every particle that couples to the Higgs field. Furthermore, any new particles discovered beyond the Standard Model that couple to the Higgs will contribute to these corrections. With such corrections, one would expect the Higgs to have a enormous mass, but the corrections cancel just so to give the Higgs mass, implying a large degree of fine tuning. Supersymmetry offers a way to explain this fine tuning. The contributions to the Higgs boson’s mass from each particle is exactly cancelled by the particle’s superpartner. Thus the quantum corrections to the Higgs mass is kept at zero until supersymmetry is broken, at which point the corrections are related to the scale of supersymmetry breaking in the MSSM, mso f t . Taking these corrections into consideration, as well as hoping to use supersymmetry to avoid the hierarchy problem in the Standard Model while getting the correct observed known particle masses, one finds that mso f t should be at most around 1 TeV [76]. Then for each of the mediation mechanisms, we can connect this estimated bound for supersymmetry breaking in the MSSM to the supersymmetry breaking scale in Twisted Inflation, ΛSSB . 112 5.2.1 Gravity-mediated supersymmetry breaking In gravity-mediated (or Planck-scale-mediated) supersymmetry breaking the broken supersymmetry is communicated to the MSSM through gravitational interactions with no direct connection between the hidden supersymmetry breaking sector and the MSSM. Through dimensional analysis we can estimate the relation between mso f t and ΛSSB as mso f t ∼ Λ2SSB MP (5.4) where MP is the Planck mass [76]. This estimation can be justified by the fact that as we turn off supersymmetry breaking in the hidden sector, ΛSSB → 0, we should restore supersymmetry in the MSSM, mso f t → 0. Similarly as we turn off gravity, MP → ∞, the supersymmetry breaking in the hidden sector will not be communicated to the MSSM and we again find mso f t → 0. Now making use of (5.4) and our desire to have mso f t ∼ 1 TeV we find an estimate for the Kaluza-Klein scale in Twisted inflation: 1 ΛSSB ∼ = RMP MP Λ2SSB ∼ MP2 mso f t ∼ 10−8 . MP We can then relate the Kaluza-Klein scale to Twisted Inflation’s field theory parameters through the results in table 4.5, a relation found by requiring we get the observed CMB normalization for scalar perturbations. For weak coupling in Twisted Inflation, λ λ 1⇒ 1, we find: √ 3.6 × 10−5 1 √ = ∼ 10−8 ⇒ λ ∼ 3.6 × 103 ⇒ λ ∼ 107 , RMP λ giving us a contradiction. Thus using Twisted Inflation as a hidden supersymmetry breaking sector for the MSSM can not be done with gravitational mediation and weak coupling. Turning to strong coupling, λ λ 1⇒ 7.5 × 10−4 (λ N) 1 7 1, we find: 1 ∼ 10−8 ⇒ (λ N) 7 ∼ 105 ⇒ λ N ∼ 1035 . (5.5) This avoids the contradiction in the ’t Hooft coupling size, λ , we had in the weak 113 coupling case, but another problem arises. From (4.76) and demanding at least 60 3 e-foldings gives us a constraint on the field theory parameters (4.81), λ 5 N 104 . However our estimate for λ and N based on supersymmetry breaking concerns in the MSSM, (5.5), go well past this bound. Thus we find for both strong and weak coupling, using Twisted Inflation as a hidden supersymmetry breaking sector with the MSSM is impractical. 5.2.2 Gauge-mediated supersymmetry breaking We now turn our attention to gauge-mediated supersymmetry breaking. In this mechanism, the dominant supersymmetry breaking terms in the MSSM come from messenger particles with interactions with both the gauge fields in the MSSM and with the hidden sector, the Twisted Inflation fields in this case. Then assuming a coupling of αa and a mass of Mmsg for the messenger particles, a similar dimensional analysis to the one done for gravity-mediated supersymmetry breaking gives mso f t ∼ αa Λ2SSB . 4π Mmsg (5.6) Just like in the gravity-mediated supersymmetry breaking, here we see mso f t → 0 as we turn off the messenger particles, either αa → 0 or Mmsg → 0. If we assume Mmsg ∼ ΛSSB , setting mso f t to about 1 TeV in (5.6) gives ΛSSB ∼ 10 − 1000TeV, a much lower scale than the one for gravity mediation. From this we find 1 ΛSSB ∼ ∼ 10−15 − 10−13 . RMP MP Substituting this result into the results in table 4.5 for weak coupling gives us the same problem we see in gravity mediation: λ 1⇒ 3.6 × 10−5 1 √ = ∼ 10−15 − 10−13 ⇒ λ ∼ 1017 − 1021 . RM P λ In fact the contradiction is even greater here compared to the case in gravity medi- 114 ation. Similarly for strong coupling we find λ 1⇒ 7.5 × 10−4 (λ N) 1 7 ∼ 10−15 − 10−13 ⇒ λ N ∼ 1070 − 1084 , which is even more incompatible with the bound (4.81) imposed to give us enough e-foldings. Thus we find that using Twisted Inflation as the hidden supersymmetry breaking sector with gauge mediation suffers the same problems as gravity mediation. The difference is that gauge mediation allows for a lower supersymmetry breaking scale than gravity mediation. However this lower scale makes it even more difficult to match CMB predictions using results from table 4.5 with weak coupling, or to meet the bound (4.81) to get enough e-foldings with strong coupling. With the estimates used, gauge-mediated supersymmetry breaking is even more impractical than gravity-mediated supersymmetry breaking. To get closer to a more realistic supersymmetry breaking scenario, the desire here is to increase ΛSSB not lower it. 5.2.3 Anomaly-mediated supersymmetry breaking Finally, after examining gravity mediation and gauge mediation, we turn our attention to anomaly-mediated supersymmetry breaking [77]. In this model of mediation, one allows for extra dimensions with the MSSM confined to one brane or set of branes and the hidden supersymmetry breaking sector is confined to another brane or set of branes that are separated from the MSSM brane along the extra dimensions. The supersymmetry breaking on the hidden brane are then communicated to the MSSM brane through supergravity effects. Now as done for the other two mediation mechanisms, we wish to relate mso f t to ΛSSB . To this end we consider the gravitino mass, m 3 , which relates to ΛSSB in 2 anomaly-mediated supersymmetry breaking through [77], m3 ∼ 2 Λ2SSB . MP Next if we take mso f t to be about the same mass scale as the lightest supersymmetric partner, which in anomaly-mediated supersymmetry breaking is typically the 115 wino, we find αW m 3 ∼ 10−3 m 3 , 2 4π 2 mso f t ∼ mwino ∼ where αW is the weak fine structure constant. Setting mso f t to 1 TeV, as we have done we find for ΛSSB that 1 ΛSSB ∼ ∼ RMP MP m3 2 MP ∼ 10−6 . Substituting this scale for the compactified direction in Twisted Inflation into the results in table 4.5 for weak coupling we find the same contradiction as before: 1⇒ λ 1 3.6 × 10−5 √ = ∼ 10−6 ⇒ λ ∼ 103 . RMP λ Thus giving up on weak coupling and turning to strong coupling we find λ 1⇒ 7.5 × 10−4 (λ N) 1 7 ∼ 10−6 ⇒ λ N ∼ 1021 . (5.7) This result is again is too large for the bound, (4.81), imposed on the field theory parameters to get enough e-foldings in the strong coupling scenerario of Twisted Inflation. Thus, even combining Twisted Inflation with anomaly-mediated supersymmetry breaking is impractical. The general problem with using Twisted Inflation as the hidden sector in mediated supersymmetry breaking is that getting the right scale for supersymmetry restoration in the MSSM pushes the scale of supersymmetry restoration too low for Twisted Inflation to meet the requirements for the right CMB predictions or the required number of e-foldings. However, with anomaly-mediated supersymmetry breaking, we are closer to meeting the (4.81) bound than we were with gravity-mediated supersymmetry breaking. Noting the factor of 1 7 in (5.7), ΛSSB is only off by two or three or- ders of magnitude. While we still found using Twisted Inflation as a hidden sector with the MSSM to be impractical regardless of the chosen mediation mechanism, should future research revisit using Twisted Inflation for breaking supersymmetry, anomaly-mediated supersymmetry breaking should be the main focus. 116 Chapter 6 Conclusions In this thesis we have made use of the gauge/gravity duality to examine two different areas of physics. The first area of study was QCD and nuclear matter. While QCD itself was not directly studied, closely related holographic theories were stud- ied. In the study of these related holographic theories we saw a few main results of interest. First the holographic D4-D4 construction of chapter 2, which we surmise to be related to a field theory with scalar quarks, had a continuous meson spectrum. Further examining this spectrum, we found the appearance of three massless mesons that we identified with the Goldstone bosons of the spontaneous breaking of SO(4) symmetry to SO(3) by the embedding of the probe flavour brane that asymptotes to opposites sides of the S4 as seen in figure 6.1. This geometric interpretation of the Goldstone was found in both the holographic construction examined in chapter 2 and the simplified construction in chapter 3. The other main result from chapter 2 studying holographic QCD was the discovery that baryons in the system could be identified with a topological charge. This is different from the baryon believed to be appear as a point-like instanton in the Sakai-Sugimoto model. After examining holographic QCD-like theories in chapters 2 and 3, in chapters 4 and 5 we used the gauge/gravity duality to study an inflationary model with an SU(N) gauge field inflaton. The various interpretations of this model through the gauge/gravity duality can be seen in figure 6.2. The model produced physical predictions within the constraints of observation in section 4.6. 117 Figure 6.1: Geometrical interpretation of Goldstone bosons. Figure 6.2: Various interpretations for the inflaton field. In chapter 5, we then extended the analysis of the Twisted Inflation model of chapter 4 to consider its possible role in breaking supersymmetry in the MSSM and its robustness to using multiple eigenvalues of the gauge field to drive inflation. We found the examined symmetric distributions of multiple eigenvalues did not significantly change the physical predictions. However, we did unfortunately find that due to mass-scale constraints imposed by the hierarchy problem, using Twisted Inflation as a hidden sector to break supersymmetry of the MSSM is impractical. 6.1 Future lines of research Like most works of science, the work presented in this thesis is not complete. There are many lines of inquiry that can be followed for future research. 118 From the first half of the thesis, the most intriguing result was likely the discovery of the baryon as a topological charge. Given concerns related to the canonical interpretations of the baryon as an instanton in the Sakai-Sugimoto model [78, 79], any further insight into holographic baryons would be a valuable contribution to the study of holographic QCD. Thus solving the partial differential equations that define the embedding of a probe D4-brane wrapping the geometry to give a baryon would be an esteemed avenue of future research. While due to the model’s construction, this baryon is not the proton or neutron or any other baryon coming from real QCD, but the studied holographic field theory is closely related to QCD. Furthermore, just as we saw a nice geometrical interpretation of Goldstone bosons in chapters 2 and 3 that allowed for an easy visual explanation of why those same Goldstone bosons could go tachyonic (to slip over the center hole in the geometry), perhaps understanding the properties of these geometric baryons might allow for similar, easy to grasp visual insights. It is in this way that string theory is providing true contributions to physics. While it may or may not be the true theory of everything, from quantum gravity to the standard model of particle physics, it is already providing valuable mathematical tools useful in gaining theoretical insight in very difficult problems. In the second half of the thesis, the focus shifted from looking for theoretical insights into holographic QCD theories to the study of an inflationary model being compared to directly to physical observations from WMAP and Planck satellites [60, 61]. While attempting to use this inflation model as a hidden sector for supersymmetry breaking proved impractical, many more questions remain. For instance, the one most intriguing to the author would be the looking at other mechanisms of supersymmetry breaking in the model itself. The breaking of supersymmetry was how the model had a long flat potential perfect for slow-roll inflation. Rather than explicitly breaking supersymmetry in the model’s construction, as we did with the introduction of anti-periodic boundary conditions for the fermions, perhaps a dynamic breaking of supersymmetry could provide a similar mechanism for slow-roll inflation. Another inquiry might involve what happens to the inflaton after inflation ends. Here our inflaton was a component of a non-abelian gauge field. Once inflation has finished there remains this gauge field, and just as we can for any inflaton candidate, 119 we might ask where is it today. 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Applications of the gauge/gravity duality Whyte, Kevin Robert Leslie 2013
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Title | Applications of the gauge/gravity duality |
Creator |
Whyte, Kevin Robert Leslie |
Publisher | University of British Columbia |
Date Issued | 2013 |
Description | While varied applications of gauge/gravity duality have arisen in literature from studies of condensed matter systems including superconductivity to studies of quenched Quantum Chromodynamics (QCD), this thesis focuses on applications of the duality to holographic QCD-like ﬁeld theories and to inflationary model that uses a QCD-like ﬁeld theory. In particular the first half of the thesis examines a holographic QCD-like ﬁeld theory with scalar quarks closely related to the Sakai-Sugimoto model of holographic QCD. The behaviour of baryons and mesons in the model is examined to find a continuous mass spectrum for the mesons, and a baryon that can identified with a topological charge. It then slightly modifies the theory to further study the behaviour of holographic ﬁeld theories. The second half of the thesis presents a novel model for early Universe inflation, using an SU(N) gauge ﬁeld theory as the inflaton. The inflation model is studied at both weak coupling and strong coupling using the gauge/gravity duality. The robustness of model’s predictions to exciting multiple inflationary fields beyond the single ﬁeld of its original proposal, and its possible role in breaking the supersymmetry of the Minimal Supersymmetric Standard Model is also explored. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2013-08-07 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0074042 |
URI | http://hdl.handle.net/2429/44764 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2013-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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