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Applications of the Gauge/Gravity Duality (DRAFT: July 30, 2013)

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Applications of the Gauge/Gravity Duality (DRAFT: July 30, 2013) 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 dual- ity 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 holo- graphic 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 infla- tion, 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 dual- ity. 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 col- laboration 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 Cos- mology 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 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 2 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 q fluctuations. 29 2.5 Baryons . 31 2.A Ramond-Ramond forms . 32 3 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 3.5 Finite temperature . 52 3.A Alternate coordinates for q fluctuations . 56 4 Twisted Inflation . 58 4.1 Introduction . 58 4.2 Basic setup . 65 4.2.1 Weak coupling: l 1................... 66 2 4.2.2 Strong coupling: 1 l N 3 . 66 2 4.2.3 Very strong coupling: l & N 3 . 68 4.3 Effective action for the candidate inflaton . 69 4.3.1 Weak coupling . 70 2 4.3.2 Strong coupling: 1 l N 3 , the D4-brane . 73 2 4.3.3 Very strong coupling: l & N 3 , the M5-brane . 75 4.4 Coupling to gravity and slow-roll potentials . 76 4.4.1 Weak coupling: l 1................... 78 v 4.4.2 Strong coupling: l 1 .................. 79 4.5 Analyzing the inflationary potentials . 81 4.5.1 General results for V(f) = V0(1 − f (f=f0)) potentials . 82 4.5.2 Weak and strong coupling results for the inflationary pa- rameters . 84 4.5.3 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: l 1................... 90 4.7.2 Strong coupling: l 1 .................. 91 4.8 The h-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.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 Multiple fields . 105 5.1.1 Weak coupling . 106 5.1.2 Strong coupling . 110 5.2 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 6 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 spec- tral index ns, the running of the spectral index as, and the tensor/scalar ratio r. For l 1 we require N 1 for the 4 − 3 analysis to be reliable, and N . 10 l 5 to obtain enough e- foldings. Numerical values are given assuming Cosmic Mi- crowave 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. 9 Figure 2.2 Examples of brane embeddings (in the y−q plane) for v0 = 0. The asymptotic angle between the two ends of the brane is 2q¥, which ranges from p for the stable embedding which extends to the smallest values of y, down to some value 2qMax ≈ 2:088 in the limit y0 ! ¥. 19 Figure 2.3 Asymptotic angle q¥ on the sphere vs minimum brane posi- tion y0 in radial direction, for various values of v0. Angle q is defined to be zero at y = y0. 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. 21 Figure 2.5 Geometrical interpretation of Goldstone bosons for the spe- cial case q¥ = p=2. 22 Figure 2.6 Behaviour of x4 vs q for various values of v0 at y0 = 2. As a function of y, the slope dx4=dy approaches a constant in each case. 22 Figure 2.7 Effective potential V1(y) = C(y)B(y) for v0 = 0 and various values of y0. Lower graphs have smaller y0. 28 Figure 2.8 Effective potential V2(y) = −A(y)B(y) for v0 = 0 and vari- ous values of y0. Lower graphs have smaller y0. 29 viii Figure 2.9 Effective potential V4(y) = −A(y)B(y) for v0 = −¥ and y0 = 1. ................................ 30 Figure 2.10 Effective potential Vq (y) = −A(y)B(y) for v0 = −¥ and y0 = 1. ................................ 30 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 be- comes parametrically light. 37 Figure 3.2 Schematic of a brane embedding. Left picture shows a plane in the space formed by the radial direction and the S4 direc- tions. Right picture shows same embedding in radial and x4 directions. 41 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 q¥ in (p=5;p=2] correspond to stable em- beddings. 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 em- bedding with the largest y0 is stable. The rest are perturba- tively unstable to slipping around the sphere, as shown. 44 Figure 3.6 Geometrical interpretation of zero modes (massless mesons) for the special case q¥ = p=2. 44 Figure 3.7 Spectrum of mesons arising from fluctuations of the brane embedding in the y − q plane.
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