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Dynamics of D-Branes on Local Calabi-Yau Geometries

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Dynamics of D-Branes on Local Calabi-Yau Geometries Universidad Consejo Superior Aut´onoma de Madrid de Investigaciones Cient´ıficas Facultad de Ciencias Instituto de F´ısica Te´orica Departamento de F´ısica Te´orica IFT{UAM/CSIC Dynamics of D-branes on local Calabi-Yau geometries Memoria de Tesis Doctoral presentada ante la Facultad de Ciencias, Secci´onde Ciencias F´ısicas, de la Universidad Aut´onoma de Madrid, por I~naki Garc´ıa-Etxebarria, Trabajo dirigido por el Doctor D. Angel´ M. Uranga, Investigador Cient´ıfico del Instituto de F´ısica Te´orica IFT y Consejo Superior de Investigaciones Cient´ıficas. Madrid, 4 de febrero de 2008. Acknowledgements On the professional level I owe a lot to my advisor Angel Uranga. He was the one who patiently guided my first steps in string theory, always having time for explaining whatever basic concept had me confused at the time. When I started becoming more independent, he always supported and encouraged me to explore my own ideas. Working with someone with such a sharp mind and clear vision of so many aspects of string theory was always enormously pleasurable and didactic, and thanks to Angel's extremely nice personality, it was also a lot of fun. I also want to thank Fouad Saad, with whom I have spent so many hours working. Collaborating with him was always a very enriching experience, and I learnt a lot from our long discussions about any aspect of string theory that came in our way. On a more personal level, I also learnt very much from him as a person, and always enjoyed immensely interacting with him. Thanks go also to the institutions that kindly hosted me during my different stays abroad: the theory division of L'Ecole´ Polytechnique, the theory department of Princeton University, and the CERN theory division. I keep excellent memories of all these places. I also want to thank Sebastian Franco for a nice collaboration, and most importantly for generously devoting some of his time to being a nice host for my stay in Princeton in such a crucial moment of his life. I sincerely appreciate it. Special thanks go to the secretaries of the UAM, IFT and CERN theory departments: Juan Carlos, Isabel, Diana de Toth, and Nanie Perrin, who thanks to their competence (and what in my mind can only be ascribed to big amounts of white magic) many times made dealing with the administration something bearable, instead of the usual dadaist nightmare. I must also thank the Basque Government for funding my Ph.D. studies through a BFI Fellowship, and the European Union for support for part of my stay in CERN in the form of a Marie Curie grant. On a more personal level, I want to thank my parents, brother and grand- mother for support during the thesis, and always being ready to lend a hand whenever I needed some help with something. Also thanks for supporting my decisions, weird as they might have seen sometimes. i ii During these last three years I have met a lot of other people that deserve a lot of thanks for helping make me a better person. The list is long, but some of the names that come to mind most easily are Afri, Alfonso, Ant´on, Daniel, Davide, Elodie, Felipe, Jose O~norbe, Leila, Marc, Sergio Monta~nez and Tom´as. I am surely omitting quite a few people, so sorry if you are not on the list, and feel free to jump in. Finally, my most heartfelt thanks go to Nao, who with her kindness and intelligence made my days full of light and joy. Contents Contents iii 1 Introduction 1 2 Introducci´on 9 3 Metastable vacua in string theory 17 3.1 Type IIA configuration . 18 3.1.1 The susy breaking vacuum . 19 3.1.2 The metastable vacuum in the electric theory . 22 3.1.3 Longevity of the metastable vacuum . 22 3.2 M-theory curve . 23 3.2.1 An aside on hyper-K¨ahler geometry . 24 3.2.2 Back to M-theory . 27 3.2.3 Pseudomoduli stabilization . 29 3.3 Generalizations . 29 3.3.1 Symplectic and orthogonal gauge groups . 29 3.3.2 SU(Nc) with non-chiral matter in or ....... 31 3.3.3 SU(Nc) with chiral matter in the + + 8 .... 31 4 Susy breaking in runaway quiver theories 33 4.1 The ISS model revisited . 34 4.1.1 Basic amplitudes . 35 4.1.2 The basic superpotentials . 36 4.1.3 The ISS metastable minimum . 38 4.1.4 Goldstone bosons . 40 4.2 A case study: dP1 ........................ 42 4.3 The general case . 46 4.3.1 A general argument . 47 4.3.2 The detailed proof . 50 4.4 Some additional examples . 54 4.4.1 The complex cone over dP2 ............... 55 4.4.2 The complex cone over dP3 ............... 58 4.4.3 The Y p;q family . 59 iii iv CONTENTS 5 Resolved and deformed dimer models 63 5.1 Resolving the singularity . 63 5.1.1 Perfect matchings . 70 5.1.2 Gauge theory interpretation . 71 5.1.3 Details of the massive sector . 74 5.2 Deforming the singularity . 80 5.2.1 Perfect matchings . 85 5.2.2 Gauge theory interpretation . 86 6 Continuity of nonperturbative effects 91 6.1 Ooguri-Vafa geometries . 92 6.2 Non-gauge D-brane instanton effects . 94 6.2.1 O(1) instanton splitting as O(1) × U(1) instantons . 94 6.2.2 O(1) splitting as U(1) instanton . 100 6.2.3 A remark about the O(1) ! O(1) × O(1) (non)splitting103 6.3 Gauge D-brane instantons . 106 6.3.1 A Nf < Nc SQCD example: spacetime view . 106 6.3.2 Microscopic interpretation . 109 6.3.3 Adding semi-infinite D-branes . 113 6.3.4 Gauge theory instantons and Seiberg duality . 114 6.4 Exotic instantons becoming gauge instantons . 117 6.4.1 Dualizing the O(1) instanton . 118 6.4.2 A duality cascade example . 120 6.5 Topology changing transitions in F-theory . 124 7 A gauge mediated model 127 7.1 Overview of the construction . 128 7.2 Description of the mediator sector . 128 7.3 Our toy model . 130 7.4 Generalizations . 135 8 Conclusions and future directions 139 9 Conclusiones y direcciones futuras 143 A The ISS model 147 A.1 Overview ............................. 147 A.2 One-loop masses for the pseudomoduli . 149 B Toric Geometry 151 B.1 Toric varieties . 151 B.1.1 The conifold . 152 B.1.2 GLSM . 153 B.2 Toric diagrams . 154 B.2.1 Web diagrams . 155 CONTENTS v B.3 Acting on the variety . 157 B.3.1 Resolving the singularity . 158 B.3.2 Deforming the singularity . 159 B.3.3 Taking orbifolds . 160 C Introduction to dimer models 163 C.1 Branes at singularities . 163 C.2 Defining dimer models . 164 C.2.1 Perfect matchings . 166 C.3 String theory interpretation . 168 C.4 Adding flavor . 172 D A short introduction to D-instantons 175 D.1 Superpotentials from gauge D-brane instantons . 175 D.2 Non-gauge, \exotic" or \stringy" instantons . 177 D.3 Higher F-terms from D-brane instantons . 178 D.4 An example: dynamical mass terms . 178 Bibliography 181 Chapter 1 Introduction All experiments so far in particle physics support one theoretical model: the standard model of particle physics [1, 2, 3], extended with masses for the neutrinos to account for oscillations. Its predictions match experiment quite beautifully. Until evidence against it is presented, perhaps coming from the LHC, there is no need to modify it. Still, there are reasons for not feeling completely satisfied by it. Let us describe a few: • The modern accepted theory of gravity, general relativity, does not fit well within the theoretical framework of quantum field theory. Namely it is non-renormalizable. This means that the behavior at short scales is not well defined: the theory as formulated requires specifying an infinite number of unknown constants (the coefficients of the irrelevant operators) if it is going to describe arbitrarily large energies, so it loses predictivity in this regime. Building a theory that reproduces gravity at long scales and is still well defined at arbitrarily short scales proves to be very hard, with very few candidates to date. • For some time the renormalizability of the standard model was inter- preted as a necessary consistency requirement for the theory. Nowa- days the viewpoint has changed, and the renormalizability is under- stood as coming from the fact that the standard model is the low en- ergy effective theory valid only below some energy threshold. Near that threshold we should start seeing deviations that can be accounted for by including non-renormalizable terms suppressed by the new physics scale. A usual suspect for this scale is something like 1 TeV, the new physics being given by the Minimal Supersymmetric Standard Model (MSSM). Even if this is not the case, at the Planck scale quantum gravity effects are expected to become an essential part of the descrip- tion, so the standard model should break down there. • There are a few (around 30) free numerical parameters in the standard model once one fixes the gauge group and the representations. While 1 2 Chapter 1. Introduction this might not seem much taking into account the amount of results it accurately predicts, it would be nice to have a better understanding of the nature of these numbers: whether there are hidden relations between them due to some more fundamental theory or they are just environmental properties of the vacuum we happen to live in.
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