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Master Thesis Domain Wall Solutions in the Ads/CFT Correspondence Author Master thesis Domain wall solutions in the AdS/CFT correspondence Author: Eduardo Mateos González Supervisor: Giuseppe Dibitetto 2 Acknowledgments Firstly, I would like to express my sincere gratitude to my supervisor, Professor Giuseppe Dibitetto, for his continuous support during the elaboration of my Master thesis, for his patience, motivation, and immense knowledge. I also want to thank Professor Luis Miguel Nieto Calzada, who helped me initiate in the world of research and gave me his advice during my Bachelor studies. I am especially grateful to my parents, who have given me the opportunity to study what I enjoy and pursue a career in research, and they have always fully supported me, to my girlfriend, Elisa, who I can always count on, through thick and thin, to my grandparents and my cousin Samuel, for always being there when I needed to call them, and also to the rest of my family. Additionally I want to thank my fellow classmates for all the stimulating discussions, for the sleepless nights we were working together before deadlines, and for all the fun we have had these years. To all the friends who have helped me by reading parts of the thesis and proposing corrections, especially Jose, Maor, Souvanik and Carlos. Finally, to all the professors in the Theoretical Physics department in Uppsala University that helped me whenever I had a question, thank you. i ii ACKNOWLEDGMENTS Contents Acknowledgments i Abstract vii 1 Introduction 1 1.1 Current goals in Theoretical Physics . 1 1.2 Tools to study strongly-coupled systems . 3 1.3 AdS7, SCFT6 ................................. 6 2 Theoretical background 7 2.1 Quantum Field Theory (QFT) . 7 2.1.1 Quantum mechanics . 7 2.1.2 Special Relativity . 14 2.1.3 General Relativity . 19 2.1.4 Quantum Field Theory . 25 2.2 Supersymmetry (SUSY) . 32 2.3 Conformal Field Theory (CFT) . 35 2.3.1 Superconformal Field Theory (SCFT) . 40 2.4 String theory . 41 2.4.1 Superstring theory . 48 2.5 Supergravity (SUGRA) . 50 2.6 Anti de-Sitter spacetime (AdS) . 52 2.7 Anti de-Sitter/Conformal Field Theory correspondence . 61 3 Calculations 67 3.1 Supersymmetric domain wall in AdS7 .................... 68 3.1.1 Effective action for the domain wall . 75 3.2 Non-conformal deformation of the CFT6 dual . 79 3.3 Equivalent Quantum Field Theory . 82 Bibliography 87 iii iv CONTENTS List of Figures 1.1 Standard Model of particle physics. Credit: wikimedia.org ....... 1 1.2 Artist’s conception of a realization of the AdS/CFT correspondence. Credit: researchgate.net ......................... 4 2.1 Schematic description of a quantum measurement. Credit: csee.umbc. edu/~lomonaco/graphics/Lomonaco-Quantum-Measurement.jpg .... 8 2.2 Visual representation of the double cover of SO(3), the special unitary group SU(2). Credit: wikimedia.org .................... 14 2.3 Causal cone of an event in spacetime. Credit: researchgate.net .... 18 2.4 Example of mass being transformed into energy. Credit: wikimedia.org 19 2.5 Artist’s conception of the spacetime curvature. Credit: sciencenews.org 23 2.6 Visual aid for lattice QFT represented as harmonic oscillators in a lattice. Credit: ribbonfarm.com . 27 2.7 . Credit: slimy.com ............................. 30 2.8 An example of a Feynman diagram. Credit: Modified from Peskin and Schröder . 31 2.9 Minimal Supersymmetric Standard Model (MSSM). Credit: arstechnica. org ....................................... 33 2.10 Scale invariant system at four different distance scales. Credit: Douglas Ashton . 36 2.11 Pictorial representation of a string attached to a D-brane. Credit: David Tong...................................... 46 2.12 Pictorial representation of two closed strings interacting. Credit: slimy.com 47 2.13 String theory dualities. Modified from originals found in quantamagazine. org and physics.stackexchange.com ................... 49 2.14 An example of a Calabi-Yau manifold. Credit: wikimedia.org ...... 52 2.15 Penrose diagram for AdS2. Credit: researchgate.com .......... 55 2.16 Diagramatic representation of the Poincaré Patch for AdS3, corresponding with the shaded area in the figure. Credit: Freedman and Van Proeyen (2012, p. 494) . 56 v vi LIST OF FIGURES 2.17 AdS/CFT correspondence. Credit: philosophy-of-cosmology.ox.ac.uk 62 2.18 Relationship between the mass of a scalar field in AdS and the scaling dimension of its dual for unitary operators. Credit: Benini (2018, p. 34) . 65 2.19 Examples of Witten diagrams. Credit: Freedman and Van Proeyen (2012, p. 542) . 66 3.1 Brane. Credit: Bobev et al. (2017, p. 5) . 70 3.2 Scalar potential plotted for h = g = 1 .................... 71 3.3 Plot of the coefficients of the metric defined in (3.13) for h = g = 1 p showing the divergence at r = 10 8 ..................... 74 3.4 Plot of the c function for h = g = 1 ..................... 84 3.5 Plot of the β function for h = g = 1 ..................... 85 Abstract In this thesis we study a particular realization of the Domain wall/Quantum Field Theory correspondence, a modification of the Anti de-Sitter/Conformal Field Theory correspondence that is used to study deformations of a Conformal Field Theory. In the Quantum gravity side of the duality we analyze a N = 1 gauged supergravity theory in 7 dimensions which presents two different Anti de-Sitter vacua, one of which preserves the full supersymmetry and one that breaks half of the supercharges. We will find a scalar 1=2-BPS solitonic solution describing a domain wall in an asymptotically Anti de-Sitter spacetime which interpolates between the supersymmetric AdS vacuum and a divergent AdS space situated at infinity, and we will calculate its tension and the effective mass of the scalar field when evaluated at the AdS vacuum. The dual theory of ourgauged supergravity is the 6-dimensional N = (1; 0) Superconformal Field Theory, and the scalar 1=2-BPS field is dual to two relevant operators that induce a relevant deformation of the SCFT which can be identified with a renormalization group-flow. Here we will first compute the scaling dimension and the one-point functions of these operators in the SCFT, as well as indicating how to compute the two-point and three-point functions, and then we will study the c-function along the renormalization group-flow they induce and the beta function that characterizes this flow in order to derive some properties of the resulting Quantum Field Theory. vii viii ABSTRACT Chapter 1 Introduction 1.1 Current goals in Theoretical Physics Theoretical physicists are trying to find a theory that accommodates all the observed phenomena in our world in a single framework, usually referred to as a Theory of Everything, in order to advance our understanding of the universe and discover new results. A theory like this must at least be able to describe the four fundamental forces that we know of so far: Gravity, Electromagnetism, Strong interaction and Weak interaction. The last three have a unified description in terms a Quantum Field Theory known as the Standard Model, while the first one receives a satisfactory treatment using the geometrical viewpoint of General Relativity; however, we find difficulties when we try to reconcile both frameworks in a theory of quantum gravity. Figure 1.1: Standard Model of particle physics. Credit: wikimedia.org 1 2 CHAPTER 1. INTRODUCTION String theory has received a lot of attention from the scientific community for the last five decades as a possible candidate for a quantum gravity theory and as a consequence there has been a great deal of theoretical results, especially in the perturbative regime of the different string theories. Nevertheless, String theory is not a unique description of a system, but a broad framework that encompasses a wide variety of models, so it is possible to generate a very broad range of outputs by adjusting the initial conditions of the theory; as a result, String theories have low predictivity at scales much larger than the strings themselves, and the energies needed to test whether the predicted effects beyond the Standard Model are correct still remain far away from our reach. This situation contrasts strongly with the case of Quantum Field Theories (QFTs): the QFTs that we have built so far cannot describe all known physics, as we do not know how to accommodate a quantum description of gravity within our current formalism, but these theories have managed to predict with astonishing precision many phenomena in the energy regimes accessible with our prevailing technology, consistently matching experimental results. A particularly impressive accomplishment is the prediction for the anomalous magnetic dipole moment of the electron that can be computed within Quantum Electrodynamics (QED), whose theoretically calculated value matches the experimental result up to the ninth significant digit, an achievement not matched in any other realm of science. The Standard Model of particle physics proves to be our best tool to understand the nature of fundamental particles, and with only a handful of input parameters (19 in the original Standard Model, 26 when including masses for the neutrinos (Thomson, 2013, p. 499)) it offers testable predictions that have been verified using our best particle ac- celerators, and the solutions we obtain are in better agreement with current experiments than those from any alternative theory or any proposed extension of new physics beyond the Standard Model. However, despite its impressive achievements, we are presented with two shortcomings in its role as a fundamental description of the universe: first, it is not a Theory of Everything. Not only we do not know how to accommodate a quantum description of the gravitational interaction with our current QFT formalism, but also it is likely that there are other phenomena in Nature that we are not aware of yet, for example there could be additional fundamental interactions or extra dimensions that are not apparent at our energy scale.
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