Dissertation submitted to the Combined Faculties of Natural Sciences and Mathematics of the Ruperto-Carola-University of Heidelberg, Germany for the degree of Doctor of Natural Sciences Put forward by Florent Baume Born in Porrentruy (JU), Switzerland 21st June, 2017 Constraints on Four Dimensional Effective Field Theories From String and F-Theory Referees: Dr. Eran Palti Prof. Dr. J¨orgJ¨ackel Abstract: This thesis is a study of string theory compactifications to four dimensions and the constraints the Effective Field Theories must exhibit, exploring both the closed and open sectors. In the former case, we focus on axion monodromy scenarios and the impact the backreaction of the energy density induced by the vev of an axion has on its field excursions. For all the cases studied, we find that the backreaction is small up to a critical value, and the proper field distance is flux independent and at most logarithmic in the axion vev. We then move to the open sector, where we use the framework of F-theory. We first explore the relation between the spectra arising from F-theory GUTs and those coming from n a decomposition of the adjoint of E8 to SU(5) × U(1) . We find that extending the latter spectrum with new SU(5)-singlet fields, and classifying all possible ways of breaking the Abelian factors, all the spectra coming from smooth elliptic fibration constructed in the literature fit in our classification. We then explore generic properties of the spectra arising when breaking SU(5) to the Standard Model gauge group while retaining some anomaly properties. We finish by a study of F-theory compactifications on a singular elliptic fibration via Matrix Factorisation, and find the charged spectrum of two non-Abelian examples. Zusammenfassung: Diese Arbeit befasst sich mit Stringtheorie Kompaktifizierungen und ihren Bedingungen an effektive Feldtheorien in vier Dimensionen, wobei sowohl der Sektor geschlossener als auch offener Strings ber¨ucksichtigt wird. Im Fall geschlossener Strings liegt unser Fokus auf Axion Monodromie Szenarien und dem Einfluss der R¨uckkopplung der durch den Axion Vakuumerwartungswert induzierten Energiedichte auf die entsprechenden Feldwerte. In allen untersuchten Szenarien finden wir, dass die R¨uckkopplung bis zu einem kritischen Wert klein ist, wobei die Feldreichweite unabh¨angigvom Fluss ist und h¨ochstens logarithmisch vom Axion Vakuumerwartungswert abh¨angt. Den Sektor offener Strings betrachten wir im Rahmen von F-Theorie. Zun¨achst un- tersuchen wir den Zusammenhang zwischen den Spektren von F-Theorie GUTs und den entsprechenden Spektren aus Zerlegungen der adjungierten Darstellung von E8 nach SU(5)× U(1)n. Wir finden heraus, dass durch Erweiterung letzteren Spektrums durch zus¨atzliche SU(5) Singlet Felder und durch eine Klassifizierung aller M¨oglichkeiten, die zus¨atzlichen Abelschen Faktoren zu brechen, alle bereits in der Literatur konstruierten Spektren von glat- ten elliptischen Faserungen in unsere Klassifizierung fallen. Weiterhin untersuchen wir gener- ische Eigenschaften dieser Spektren nach Brechung von SU(5) auf die Eichgruppe des Stan- dardmodells, w¨ahrendwir einige Anomalie Eigenschaften beibehalten. Abschliessend be- fassen wir uns mit F-Theorie Kompaktifizierungen auf einer singul¨arenelliptischen Faserung mittels Matrix Faktorisierung und berechnen das geladene Spektrum f¨urzwei nicht-Abelsche Beispiele. A la m´emoire de l'Abner et du Raymond. ii Contents 1 Introduction 1 2 Four Dimensional Effective Actions and Geometry 9 2.1 Sigma-models and their Symmetries . 10 2.2 Compactification of Field Theories . 14 2.3 String Theory: Space-time as a Target Space . 17 2.4 Compactification of Type II Supergravity . 21 2.4.1 Flux Compactification . 23 2.4.2 Unification of String Supergravities . 24 3 Constraints From Field Ranges 27 3.1 Type IIA Flux Compactification . 30 3.2 Axion Monodromy and Backreaction . 34 3.2.1 Ramond-Ramond Axions . 35 3.2.2 Neveu-Schwarz Axions . 48 3.3 Summary . 51 4 From Type IIB Supergravity to F-theory and Back Again 53 4.1 Elliptic Fibrations . 55 4.2 Defining F-theory from M-theory . 59 4.3 Extracting Target Manifold Data From Singularities . 60 4.4 Going Back to Type IIB Supergravity: Sen's Limit . 66 5 The Role of E8 in F-theory GUTs 69 5.1 Global F-theory Models and E8 ......................... 71 5.1.1 An Example of Higgsing Beyond E8 ................... 73 5.1.2 Classifying Higgsing Beyond E8 ..................... 75 5.1.3 Embedding Known Models . 77 5.2 Summary . 85 6 Hypercharge Flux Breaking and Anomaly Unifications 87 6.1 General Properties of Hypercharge Flux GUT Breaking . 88 6.1.1 The General Spectrum . 89 iii 6.2 A General Mechanism . 92 6.2.1 The Pseudo-Nambu{Goldstone Boson . 94 6.3 Summary . 97 7 F-theory and Matrix Factorisation 99 7.1 A First Encounter with Matrix Factorisation . 101 7.2 Matrix Factorisation of Non-Abelian F-theory Models . 104 7.2.1 An SU(2) × U(1) Model . 105 7.2.2 Towards an Example Without Abelian Factors . 111 7.3 Summary . 116 8 Conclusions 119 A Mathematical Glossary 123 B Embedding in the Presence of Non-Flat Points 127 C Ext Groups for an SU(2) Model 131 iv Acknowledgments First and foremost, my thanks go to my supervisor, Eran Palti, for introducing me to string theory and its beautiful connection to mathematics. His supervision helped me navigate smoothly through the meanders of those marvelous but at first intimidating worlds. More- over, his enthusiasm and passion for research has been contagious, and I cannot thank him enough for that. For his support and help since I arrived in Heidelberg, he has my deepest gratitude. I am also grateful to my collaborators, Boaz Keren-Zur, Craig Lawrie, Ling Lin, Riccardo Rattazzi, Raffaele Savelli, Sebastian Schwieger, Roberto Valandro, and Lorenzo Vitale for many helpful discussions, and their patience towards my sometimes probably irritating lack of understanding. A special mention goes to Riccardo Rattazzi, my Masters supervisor, who guided and helped me find this position in Heidelberg, and whose advice to leave Lausanne and go abroad for my doctoral studies was one of the best I ever received. I would like to thank my friends and colleagues Craig, Fabrizio, Feng-Jun, Ling, Oskar, Pablo, Philipp, Sebastian (I, II, and III), and Viraf for creating a cordial atmosphere on the Philosophenweg through our coffee breaks, unfortunately short-lived ap´eros, and chats about physics or unrelated non-sense. I am also grateful to Profs. J¨orgJ¨ackel, Arthur Hebecker, Stefan Theisen, and Timo Weigand for helpful discussions. I am particularly grateful to Craig Lawrie, Viraf Mehta, and Sebastian Schenck to accept going through the job of proofreading this thesis, and our administrative staff, Cornelia, Melanie, and Sonja whose help understanding the German bureaucracy cannot be given enough recognition. Outside of the institute, I am also eternally grateful to my friends: From Heidelberg, Alessandro, Christos, Ma¨elle,Milan, Sabrina, and Raffaela, those from the Poly era, Alexis, Anne-Laur`ene,Antoine, (the multiple) Camille, Eleonie, Luc, and Olivier. Their unwavering support accompanied me through good, but more importantly bad, times during my stay in Germany and before. I will always look back to my time in Heidelberg with great fondness, and they are an integral part of the reasons why. Finalement, je remercie infiniment mes parents, Christiane et Vincent, ma soeur M´elanie, mes grands-parents Ginette, Germaine, Abner et Raymond, ainsi qu'au reste de ma famille pour leur soutien. Mes grands-p`eresne sont malheureusement plus l`apour me voir achever mon doctorat, mais sont l'une des principales sources de mon insatiable soif de connaissances. Nos multiples bricolages, exp´eriences, balades, et autres explications sur la Nature durant mon enfance ont petit `apetit developp´emon go^utpour la Science. Cette th`eseleur est d´edi´ee. v Chapter 1 Introduction In the late 19th century, most of the observed physical phenomena were described to good accuracy by Newtonian Physics. The motion of objects, from solids to stars, were governed by Newton's laws of motion, charged particles by Maxwell's electrodynamics, and the collective behaviour of complex systems could be described by the laws of thermodynamics. This led many members of the scientific community to think that they were close to the Holy Grail of Physics: a complete description of Nature in terms of first principles. In fact, an apocryphal quote often misattributed to William Thomson, 1st Baron Kelvin epitomises this school of thought: There is nothing new to be discovered in physics now. All that remains is more and more precise measurement. This way of thinking was in fact quite wrong, as within a few years, new experimental meth- ods led to observations that could not be explained by Newtonian Physics, demanding a new framework to explain them. Two paradigms arose simultaneously to try to understand the discrepancy between the theoretical and experimental worlds. On the one hand, Quantum Mechanics began explaining phenomena occurring at short distances, while Special Relativity arose as a description of those having a high velocity. These two new theories were not completely ex nihilo, in the sense that they did not replace Newtonian Physics, but rather built upon it to generalise it. Indeed, when quantum fluctuations are small compared to the size of the observed object, it can be described by the rules of Newtonian Physics. Similarly, if the velocity of an object is slow with respect to the speed of light, the Newtonian framework is sufficient to characterise it to good accuracy. These two new theories were soon combined into Quantum Field Theory, describing objects of high velocity exhibiting a quantum behaviour. This framework has been one of the most celebrated theoretical achievements of the past century, and is one of the two cornerstones on which our current understanding of the Universe is built.