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Heterotic String Models on Smooth Calabi-Yau Threefolds

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Heterotic String Models on Smooth Calabi-Yau Threefolds Heterotic String Models on Smooth Calabi-Yau Threefolds Andrei Constantin University College University of Oxford arXiv:1808.09993v2 [hep-th] 26 Sep 2018 A thesis submitted for the degree of Doctor of Philosophy Trinity Term 2013 2 Heterotic String Models on Smooth Calabi-Yau Threefolds Andrei Constantin University College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity Term 2013 This thesis contributes with a number of topics to the subject of string compactifications, especially in the instance of the E8 × E8 heterotic string theory compactified on smooth Calabi-Yau threefolds. In the first half of the work, I discuss the Hodge plot associated with Calabi-Yau threefolds that are hypersurfaces in toric varieties. The intricate structure of this plot is explained by the existence of certain webs of elliptic-K3 fibrations, whose mirror images are also elliptic-K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fiber. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fact, together with a remarkable relation on the additivity of Hodge numbers, give to the Hodge plot the appearance of a fractal. Moving on, I discuss a different type of web of manifolds, by looking at smooth Z3-quotients of Calabi-Yau three-folds realised as complete intersections in products of projective spaces. Non-simply connected Calabi-Yau three-folds provide an essential ingredient in heterotic string compactifications. Such manifolds are rare in the classical constructions, but they can be obtained as quotients of homotopically trivial Calabi-Yau three-folds by free actions of finite groups. Many of these quotients are connected by conifold transitions. In the second half of the work, I explore an algorithmic approach to constructing E8 × E8 het- erotic compactifications using holomorphic and poly-stable sums of line bundles over complete intersection Calabi-Yau three-folds that admit freely acting discrete symmetries. Such Abelian bundles lead to N = 1 supersymmetric GUT theories with gauge group SU(5) × U(4) and matter fields in the 10, 10, 5, 5 and 1 representations of SU(5). The extra U(1) symmetries are generically Green-Schwarz anomalous and, as such, they survive in the low energy theory only as global symmetries. These, in turn, constrain the low energy theory and in many cases forbid the existence of undesired operators, such as dimension four or five proton decay operators. The line bundle construction allows for a systematic computer search resulting in a plethora of models with the exact matter spectrum of the Minimally Supersymmetric Standard Model, one or more pairs of Higgs doublets and no exotic fields charged under the Standard Model group. In the last part of the thesis I focus on the case study of a Calabi-Yau hypersurface embedded in a product of four CP1 spaces, referred to as the tetraquadric manifold. I address the question of the finiteness of the class of consistent and physically viable line bundle models constructed on this manifold. Line bundle sums are part of a moduli space of non-Abelian bundles and they provide an accessible window into this moduli space. I explore the moduli space of heterotic compactifications on the tetraquadric hypersurface around a locus where the vector bundle splits as a direct sum of line bundles, using the monad construction. The monad construction provides a description of poly-stable S (U(4) × U(1)){bundles leading to GUT models with the correct field content in order to induce standard-like models. These deformations represent a class of consistent non-Abelian models that has co-dimension one in K¨ahlermoduli space. Acknowledgements This thesis has been shaped and grew along with its author inspired by the joint guidance of Prof. Philip Candelas and Prof. Andr´eLukas. I am very much indebted to them, considering myself blessed to be able to learn and work in their company. The ideas presented here wouldn't have acquired their fullness without the creative work of my co-laborators, Dr. Lara Anderson, Dr. Evgeny Buchbinder, Dr. James Gray, Dr. Eran Palti and Dr. Harald Skarke. I am grateful for their collaboration and friendship. Special thanks go to Dr. Volker Braun, Dr. Rhys Davis, Prof. Xenia de la Ossa, Prof. Yang Hui-He and Prof. Graham Ross for their advice, inspiration and accurate knowledge shared with me in our conversations. I am grateful for the friendship and support of many of my colleagues (some of which have already gradu- ated), especially of Maxime Gabella, Georgios Giasemidis, Michael Klaput, Cyril Matti, Challenger Mishra, Chuang Sun, Eirik Svanes and Lukas Witkowski. My DPhil studies and the present thesis wouldn't have been possible without the generous support received through the Bobby Berman scholarship offered by Univer- sity College, Oxford and complemented by the STFC. The final year of my studies was partly supported through the College Lectureship offered by Brasenose College and I would like to thank Prof. Laura Herz and Prof. Jonathan Jones for giving me the chance to partake in this facet of Oxford's academic life. Last but not least, I would like to express my sincere gratitude towards my wife Carmen Maria for her constant love, friendship and support and for being my constitutive Other; to our children Elisabeta, Clara Theodora and Cristian for brightening up my life in countless ways; to my parents Carmen and Marin, my sister Alina and our extended family for their unconditional love and support. There are of course many more people who contributed to my becoming during these years and close friends whose advice, help and presence meant a lot for me. To all these people I am deeply grateful and indebted. To my family. Contents 1 Introduction 1 1.1 The Heterotic String . .2 1.1.1 The Basics . .3 1.1.2 From 10 to 4 Dimensions . .4 1.2 Calabi-Yau Manifolds . .5 1.3 Overview and Summary . .5 I The Manifold 12 2 Elliptic K3 Fibrations 13 2.1 Half-mirror Symmetry and Translation Vectors . 15 2.2 The Language of Toric Geometry . 18 2.2.1 Reflexive polytopes and toric Calabi-Yau hypersurfaces . 18 2.2.2 A two dimensional example: the elliptic curve . 18 2.2.3 Fibration structures . 20 2.2.4 K3-fibrations in F-theroy/type IIA duality . 21 2.3 Nested Elliptic-K3 Calabi-Yau Fibrations . 22 2.3.1 Elliptic-K3 fibrations . 22 2.3.2 Composition of projecting tops and bottoms . 22 2.3.3 An additivity lemma for the Hodge numbers . 25 2.4 The E8×{1g K3 Polytope and Its Web of Fibrations . 29 i 2.4.1 Translation vectors and the Hodge number formula . 30 2.4.2 The web of E8×{1g elliptic K3 fibrations . 32 2.4.3 Searching for E8×{1g elliptic K3 fibrations . 33 2.4.4 Generating all E8×{1g elliptic K3 fibrations . 34 2.5 Other K3 Polytopes . 34 2.5.1 The web of E7×SU(2) elliptic K3-fibrations . 36 2.5.2 The web of E7×{1g and E8×SU(2) elliptic K3-fibrations . 38 2.5.3 Lists of tops . 39 2.6 Outlook . 40 3 Discrete Calabi-Yau Quotients 42 3.1 Introduction and Generalities . 42 3.1.1 Important aspects of CICY quotients . 45 3.1.2 Conifold transitions . 47 3.2 The New Z3 Quotients . 48 12;18 6;8 12;18 3.2.1 The manifold X with quotient X = X =Z3 ............ 48 9;21 5;9 9;21 3.2.2 The manifold X with quotient X = X =Z3 ............. 51 11;26 5;10 11;26 3.2.3 The manifold X with quotient X = X =Z3 ........... 53 8;29 4;11 8;29 3.2.4 The manifold X with quotient X = X =Z3 ............ 56 7;37 3;13 7;37 3.2.5 The manifold X with quotient X = X =Z3 ............ 58 11;29 5;11 11;29 3.2.6 The manifold X with quotient X = X =Z3 ........... 60 3.3 The Web of Z3 Quotients . 62 3.3.1 Conifold transitions between smooth quotients of Calabi-Yau threefolds . 62 3.3.2 The web of Z3 quotients . 63 3.4 The List of Smooth Z3 Quotients of CICYs . 65 ii II The Bundle 70 4 Line Bundle Models on CICY Quotients 71 4.1 Introduction . 71 4.2 Overview of the Construction . 75 4.2.1 Heterotic line bundle compactifications . 75 4.2.2 The GUT gauge group . 77 4.2.3 The GUT spectrum . 78 4.2.4 The Green-Schwarz mechanism and the extra U(1) symmetries . 80 4.3 The Manifolds . 81 4.3.1 CICY Manifolds . 82 4.3.2 Favourable configurations . 83 4.4 The Bundles . 84 4.4.1 Topological constraints . 84 4.4.2 Constraints from stability . 85 4.4.3 Constraints from the GUT spectrum . 85 4.4.4 Equivariance and the doublet-triplet splitting problem . 86 4.5 The Scanning Algorithm . 87 4.6 An Example . 91 4.7 Results and Finiteness . 92 4.8 Distribution of Models . 96 5 Line Bundle Models on the Tetraquadric Hypersurface 99 5.1 Introduction . 99 5.2 The Line Bundle Construction . 101 5.2.1 A brief review of SU(5) line bundle models . 101 5.2.2 Highlights in the moduli space: line bundle models on the tetraquadric . 102 5.3 The Tetraquadric Hypersurface . 105 iii 5.4 Topological Identities for Line Bundle on the Tetraquadric . 107 5.5 Finiteness of the Class of Line Bundle Models .
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