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Modified Gravitational Backgrounds: Horndeski's Theory and Non Modified gravitational backgrounds: Horndeski's theory and non-geometrical backgrounds in string theory Fabrizio Basti´anCordonier Tello M¨unchen2019 Modified gravitational backgrounds: Horndeski's theory and non-geometrical backgrounds in string theory Fabrizio Basti´anCordonier Tello Dissertation an der Fakult¨atf¨urPhysik der Ludwig{Maximilians{Universit¨at M¨unchen vorgelegt von Fabrizio Basti´anCordonier Tello aus Vi~nadel Mar, Chile M¨unchen, den 30. Juli 2019 Erstgutachter: Prof. Dr. Dieter L¨ust Zweitgutachter: PD Dr. Erik Plauschinn Tag der m¨undlichen Pr¨ufung:24. September 2019 To my family and friends, who bear with me in every step I take. Credits: NASA JP We succeeded in taking that picture [from deep space], and, if you look at it, you see a dot. That's here. That's home. That's us. On it, everyone you ever heard of, every human being who ever lived, lived out their lives. The aggregate of all our joys and sufferings, thousands of confident religions, ideologies and economic doctrines, every hunter and forager, every hero and coward, every creator and destroyer of civilizations, every king and peasant, every young couple in love, every hopeful child, every mother and father, every inventor and explorer, every teacher of morals, every corrupt politician, every superstar, every supreme leader, every saint and sinner in the history of our species, lived there on a mote of dust, suspended in a sunbeam. { Carl Sagan extract from speech at Cornell University October 13, 1994 Contents Zusammenfassungv Abstract ix List of Figures xiii List of Tables xv Acknowledgements xvii General Introduction1 Motivation.....................................1 Thesis' structure.................................3 I Horndeski's theory with non-vanishing torsion5 1 Introduction of Part I7 1.1 Motivation.................................. 10 1.2 Outline of Part I.............................. 11 2 A primer on Horndeski's theory 13 2.1 Horndeski's theory............................. 15 2.1.1 Horndeski's Lagrangian....................... 15 2.1.2 Theories contained inside Horndeski's Lagrangian........ 18 2.2 Horndeski after LIGO........................... 21 3 Horndeski's Lagrangian with torsion 25 3.1 First order formalism for Horndeski's theory............... 25 3.1.1 Preliminaries: contraction operators and vierbein........ 25 3.1.2 Horndeski's Lagrangian in differential form language...... 27 3.1.3 Equations of motion........................ 29 3.2 Wave operators, torsion and Weitzenb¨ock identities........... 33 3.3 Gravitational waves and torsional modes................. 37 3.3.1 Linear perturbations in first order formalism........... 37 3.3.2 Gravitational waves in the first-order formalism......... 40 3.3.3 Gravitational Waves and generic terms of Horndeski's Lagrangian 43 4 Summary and conclusions of Part I 47 II T-duality and the Open String 51 5 Introduction of Part II 53 5.1 Motivation.................................. 56 5.2 Outline of Part II.............................. 56 6 T-duality and non-geometrical Spaces 59 6.1 T-Duality.................................. 59 6.1.1 The case of S1 ............................ 59 6.1.2 On general background configurations............... 65 6.1.3 The case of TD ........................... 67 6.1.4 Buscher rules............................ 72 6.2 Non-Geometric Spaces........................... 80 6.2.1 T3 with H−flux........................... 81 6.2.2 Twisted torus............................ 82 6.2.3 T-fold................................ 83 6.2.4 R space............................... 84 6.3 Toroidal fibrations............................. 85 7 T-duality transformations for the Open String 89 7.1 The gauged non-linear sigma model.................... 89 7.1.1 Worldsheet action.......................... 89 7.1.2 The gauged worldsheet action................... 92 7.1.3 Recovering the original action................... 96 7.2 T-duality.................................. 98 7.2.1 The closed-string sector...................... 98 7.2.2 The open-string sector: Neumann directions........... 103 7.2.3 The open-string sector: Dirichlet directions............ 107 7.3 Examples: T3 with H−flux........................ 109 7.3.1 One T-duality transformation................... 109 7.3.2 Two T-duality transformations.................. 115 7.3.3 Three T-dualities.......................... 119 7.4 The Freed-Witten Anomaly and boundary conditions.......... 122 7.4.1 The Freed-Witten anomaly..................... 123 7.4.2 Well-definedness of boundary conditions............. 124 8 Developments on non-abelian T-Duality Transfomations 129 8.1 On non-abelian T-duality transformations................ 129 8.2 The WZW model for Lie group manifolds................. 130 8.3 T-duality.................................. 133 8.3.1 Dual action............................. 133 8.3.2 Vielbein algebra........................... 135 8.3.3 The case of SU(2)......................... 135 9 Summary and conclusions of Part II 139 III Close 143 General conclusion 145 A The bosonic string 147 Bibliography 163 Zusammenfassung Diese These befasst sich mit einem zweiteiligen Studium modifizierter Hintergr¨undein Theorien, die die Gravitation verallgemeinern. Im ersten Teil erkunden wir Horndeskis Lagrange-Dichte im Rahmen des Cartan Formalismus erster Ordnung, und wir studieren Gravitationswellen in Anbetracht einer nicht-verschwindenden Torsion. Im zweiten Teil gehen wir zur Stringtheorie ¨uber, und studieren T-Dualit¨ats-Transformationen zu einem konkreten nichtlinearen Sigma-Modell, das offene Strings enth¨alt. Dies motiviert das Studium der Hintergr¨unde,die sich aus solchen Transformationen ergeben. Im ersten Teil analysieren wir Horndeskis Lagrange-Dichte in Cartan Formalismus erster Ordnung. Dieser Formalismus erlaubt der Torsion, ungleich Null zu sein, w¨ahrendsie in der allgemeinen Relativit¨atstheorieeine verschwindende Quantit¨atist. Horndeskis Lagrange-Dichte ist die allgemeinste Lagrange-Dichte in vier Dimensionen, die jede m¨ogliche Wechselwirkung zwischen einem Skalarfeld φ und der Gravitation zul¨asst. Ihre Feldgleichungen sind bis zu zweiter Ordung partielle Differentialgleichungen. Diese Besonderheit jener Feldgleichungen verhindert die Existenz von Geistfeldern. Da diese Lagrange-Dichte bekannte modifizierte Theorien der Gravitation als Sonderf¨alle enth¨alt, konzentrieren wir uns auf die Rolle der Torsion und ihrer Wirkung auf die linearen St¨orungen der Felder. Um unsere Untersuchung handhabbar zu machen, formulieren wir Horndeskis Lagrange-Diche in die Sprache der Differentialformen um. Dem Cartan Formalismus folgend nehmen wir an, dass der Spin-Zusammenhang !ab und das Vierbein ea voneinander unabh¨angig sind. Wir nehmen die gesamte Horndeskis Lagrange-Dichte und leiten daraus die Feldgleichungen des Skalarfeldes, des Spin-Zusammenhanges und des Vierbeines ab. Um den torsionslosen Fall und die gew¨ohnliche allgemeine Relativit¨atstheorie zu erreichen, m¨ussenwir eine Nebenbedingung durch Lagrange-Mutiplikatoren einf¨uhren. Als Vorbereitung zur Analyse linearer St¨orungenunserer Felder, definieren wir mehrere Differentialoperatoren, um die Zeitraumstorsion unterscheiden zu k¨onnen. Diese Operatoren k¨onnen auf Lorentz-Indizes tragende p−Formen kovariant angewendet werden. Insbesondere beweisen wir eine Verallgemeinerung der Weitzenb¨ock-Identit¨at,die Torsion enth¨alt. Sp¨atererforschen wir Horndeskis Lagrange-Dichte unter linearen St¨orungendes Skalarfeldes, des Spin-Zusammenhanges und des Vierbeines. Wir entdecken, dass nicht-minimale Kopplungen und zweite Ableitungen des Skalarfeldes generische Quellen der Torsion sind. Das steht in Kontrast zu dem, was bekannt in Einstein-Cartan-Sciama-Kibble-Schema ist, bei dem sich die Torsion ausschließlich auf Fermionen bezieht. Tats¨achlich entdecken wir, dass die Hintergrundtorsion mit den sich ausbreitenden metrischen Freiheitsgraden koppelt. Das stellt eine M¨oglichkeit dar, Torsion durch Gravitationswellen zu falsifizieren. Im zweiten Teil behandeln wir T-Dualit¨ats-Transformationen durch Buschers Verfahren zum offenen String im Rahmen der Stringtheorie. Solche Transformationen f¨uhren zu interessanten Geometrien, die eine wichtige Rolle in der Stringtheorie spielen, wie beispielsweise in der Modulistabilisierung oder beim Aufbau von Inflationspotentialen. Diese Geometrien werden als nicht-geometrische Hintergr¨unde bezeichnet. In diesem zweiten Teil arbeiten wir technische Details aus, um L¨ucken in der Literatur zu schließen. Diese Details betreffen die Anwesenheit von D-Branen und die Wirkung von T-Dualit¨ats-Transformationen auf diese. Zu Beginn studieren wir hierf¨urein nichtlineares Sigma-Modell zum offenen String mit Feldern, die auf der Weltfl¨ache Σ des offenen Strings und ihrer Grenze @Σ definiert sind. Wir ziehen nicht-triviale Topologien f¨urdiese Weltfl¨ache in Betracht und pr¨asentieren die entsprechenden Grenzbedingungen f¨ur den offenen String. Buschers Verfahren zufolge nehmen wir gewisse Bedingungen f¨ur die Hintergrundskonfiguration an, und wir definieren zus¨atzliche Felder auf Σ und @Σ, um die Anwesenheit von D-Branen ebenfalls zu betrachten. Wir folgen Buschers Verfahren und f¨uhrenT-Dualit¨ats-Transformationen durch, indem wir eine Symmetrie der Weltfl¨ache eichen und Weltfl¨ache-Eichfelder entlang der Isometrierichtungen der Zielraummetrik integrieren. Wir erreichen somit die duale Konfiguration f¨urden offenen und geschlossenen Stringsektor. Insbesondere finden wir heraus, dass das duale Kalb-Ramond-Feld B einen verbleibenden Teil enth¨alt,der eine wichtige Rolle spielt,
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