Aspects of F-Theory-engineered Quantum Field Theories Memoria de Tesis Doctoral realizada por Federico Carta presentada ante el Departamento de F´ısicaTe´orica de la Universidad Aut´onomade Madrid para optar al T´ıtulode Doctor en F´ısicaTe´orica Tesis Doctoral dirigida por Dr. Fernando Marchesano, Cient´ıficoTitular del Instituto de F´ısicaTe´orica Departamento de F´ısicaTe´orica Universidad Aut´onomade Madrid Instituto de F´ısicaTe´oricaUAM/CSIC Mayo de 2018 Ai miei maestri. ABSTRACT In this thesis we discuss three examples of quantum field theories engineered from the IIB superstring theory and F-Theory. Firstly we consider a model of SU(5) GUT in F-Theory, with E7 enhancement. Yukawa couplings for the two heaviest families of MSSM are computed, as well as one CKM entry. Realistic masses for the fermions can be obtained by considering certain values of the parameters entering in the model. Secondly, we discuss the phenomenon of supersymmetry enhancement in QFTs, in which a theory with 4 supercharges flows in the IR to a theory with 8 supercharges. New systematic scans are performed in order to find more theories showing this peculiar feature. Furthermore, an explanation of this SUSY enhancement is given by geometrically engineering the QFT of a D3 probing a F-Theory singularity corresponding to a T-brane background. Thirdly, we consider mixed branches of 3d N = 4 QFTs. We devise a way to compute the Hilbert Series of a generic mixed branch of a particular set of theories of this kind, called T [SU(N)]. In order to understand how the usual formulae for a Coulomb Branch Hilbert series get modified in the mixed branch case, it was crucial to engineer the T [SU(N)] in IIB superstring theory, by the use of an Hanany-Witten setup. i RESUMEN En esta tesis damos de tres ejemplos de teorias cu´anticas de campos que pueden ser realizadas en la teor´ıade supercuerdas IIB, y teor´ıaF. Para empezar, consideramos un modelo modelo de Gran Unificaci´on(GUT) de grupo gauge SU(5) en teor´ıaF, con E7 enhancement. Se calculan los acoplos de Yukawa para las dos familias m´aspesadas del MSSM, como tambi´enuna entrada de la matriz CKM. Masas realistas para los fermiones pueden ser encontradas fijando algunos valores de los par´amentros que entran en el modelo. A continuaci´on,consideramos el fen´omenode incremento de supersimetr´ıaen Teor´ıasCu´anticas de Campos (QFTs), en que una teor´ıacon 4 supercargas fluye en el IR a una teor´ıacon 8 su- percargas. Hacemos nuevas b´usquedassistematicas para encontrar m´asteor´ıasque tienen esta propriedad particular. Adem´as,damos una explicaci´onde este fen´omeno a trav´esdel geometri- cal engineering de la QFT con una D3 que explora una singularidad en teor´ıaF que corresponde a una configuraci´ongauge no-Abeliana conocida como T-brana. Por ultimo, consideramos ramas mixtas de algunas teor´ıascu´anticas de campos 3d N = 4. Damos una manera para calcular la Serie de Hilbert de una cualquier rama mixta para un conjunto dado de teorias, llamadas T [SU(N)]. Para entender como se modifica la f´ormula de la Serie de Hilbert del Coulomb Branch en el caso de un mixed branch, es crucial pensar a la teor´ıa T [SU(N)] en el contexto de la teor´ıade supercuerdas IIB, a trav´esuna construcci´onde Hanany-Witten. ii ACKNOWLEDGEMENTS The first person I would like to thank is my supervisor, Fernando Marchesano, for having guided me through these years of the PhD. Thanks for having explained me numerous things in the field of String Phenomenology, having shared many interesting ideas with me, and having been always there when there was a need of any kind. It is a pleasure to have been your student! A second person I would like to thank is Hirotaka Hayashi. Basically everything I now know about the dynamics of supersymmetric QFTs, modulo a zero-mesure set of knowledge, I owe to him. I also thank him for having been an excellent collaborator over the years, always motivating me to proceed, in one way or another, trough the many struggles of scientific research. A third person I feel the urge to thank is Raffaele Savelli. Working with him I learned way too many interesting things in physics and mathematics, most importantly F-theory, which plays a major role in this thesis. Also thanks to the many other amazing collaborators I had over these years: Wieland, Gianluca, Simone, Cumrun and Daniel. It has been (and I am sure it will still be) a great opportunity and a great fun to work with all of you again. Thanks! I would like to thank the other seniors at the IFT, especially Luis and Angel, for all the physics related discussions, all the interesting conferences organized, and more importantly for having made the IFT a very enjoiable and stimulating environmenet to be in. Also thank you for writing The Book, which has been on my desk basically every day since I started this journey. Also, I would like to thank all the other students and postdocs at the Institute. The older generation, in particular Miguel, Irene, Ander, Gianluca, Sjoerd, Aitor, Wieland, Clemens, Francisco, Michael, Pramod and the newer generation as well. Thanks for having made, and still making the IFT a great and fun place! Especially important people in this PhD adventure have been my office mates: Paco, Sebastian, Eduardo and Nicol´o. With them I shared many everyday life moments, unforgettable discussions and trips around the world, during these years. Also special thanks to Eduardo, for helping me with the spanish chapters in this thesis. iii I would also like to thank Tom, Igor and Mirjam for having organized TASI 2017, as well as all the other speakers, students and people involved in that PhD school. TASI has been a unique experience for me: by far the most scientifically interesting and happiest month of my PhD life. I also feel the urge to acknowledge Isabel, Monica V, Monica E, Rebeca, Susana, Maria, Laura, Andr´es,Marcos, Emilio and all the other people working in the administration, outreach and IT at the IFT. The number of times they helped me during these years, and in verious different ways, is uncountable. Last but not least, I would like to thank my family: Rita, Marcello, Alice and Dario, zii e nonne. For all the love and support during these years, and always. iv CONTENTS 1 General Introduction 1 2 Yukawa couplings 5 2.1 The Standard Model and particles' masses. .5 2.2 Grand Unified Theories . .7 2.2.1 Gauge coupling unification . .8 2.2.2 Georgi-Glashow SU(5) GUT . .9 2.2.3 Anomaly cancellation. 11 2.2.4 GUTs' first prediction: Proton decay . 12 2.2.5 GUTs' second prediction: 't Hoft- Polyakov monopoles . 13 2.3 A brief introduction to F-Theory . 15 2.3.1 F-Theory from IIB superstring . 16 2.3.2 F-Theory from M-Theory . 18 2.3.3 F-Theory from Heterotic . 19 2.3.4 Elliptic curves, fibrations, and their singularities . 20 2.4 Yukawas and exceptional groups in F-theory GUTs . 23 2.4.1 SU(5) models with E7 enhancement . 27 2.4.2 Yukawa hierarchies in the E7 model . 29 2.4.3 Higgs background . 31 2.4.4 Primitive fluxes . 34 2.4.5 Residue formula for Yukawa couplings . 37 v 2.4.6 Holomorphic Yukawa couplings for the E7 model . 38 2.4.7 Normalization factors and physical Yukawas . 40 2.4.8 Perturbative wavefunctions . 41 2.4.9 Normalization factors . 43 2.4.10 Non-perturbative corrections to the wavefunctions . 44 2.4.11 Fitting fermion masses and mixing angles . 46 2.4.12 Fermion masses . 47 2.4.13 Quark mixing angles . 55 3 Supersymmetry enhancement 58 3.1 Generalities of 4d N = 2 QFTs . 58 3.2 The Seiberg-Witten curve. 63 3.3 Generalities of 4d N = 2 SCFTs. 65 3.4 Maruyoshi-Song flows. 71 3.5 New scans: looking for MS flows. 78 3.6 A geometric picture for the enhancement. 79 4 Mixed Branches of 3d N = 4 QFTs 85 4.1 3d field theories with 8 supercharges: generalities. 85 4.2 Quiver gauge theories . 92 4.3 A look at the moduli space . 94 4.4 Hilbert Series for Moduli Spaces of 3d N = 4 Theories . 96 4.4.1 Higgs branch moduli space . 97 4.4.2 Coulomb branch moduli space . 99 4.5 The Hanany-Witten cartoon . 100 4.6 3d mirror symmetry . 103 4.7 T [SU(N)] theory and its relation to class-S. 105 4.8 Mixed Branches of the T [SU(N)] Theory . 106 4.8.1 Hilbert series for the Coulomb branch factor . 112 4.8.2 Hilbert series for the Higgs branch factor . 114 4.8.3 The Restriction Rule for the Hilbert Series . 115 5 General conclusions 122 6 Conclusiones generales 128 vi 7 Appendices. 131 7.1 Dynkin Label notation . 131 7.2 E7 machinery . 132 7.3 Local chirality and doublet-triplet splitting . 135 7.3.1 Model A . 135 7.3.2 Model B . 136 7.4 Zero mode wavefunctions . 136 7.4.1 Wavefunctions in holomorphic gauge and Yukawa couplings . 136 7.4.2 Wavefunctions in real gauge . 138 7.4.3 Holomorphic Yukawa matrix . 145 7.5 Coulomb branch examples of restriction rule. 146 7.6 Higgs branch examples of the restriction rule . 158 7.7 Nilpotent Orbits . 165 7.7.1 A sketch of the classification .