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Toroidal Orbifolds: Resolutions, Orientifolds and Applications in String Phenomenology

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Toroidal Orbifolds: Resolutions, Orientifolds and Applications in String Phenomenology TOROIDAL ORBIFOLDS: RESOLUTIONS, ORIENTIFOLDS AND APPLICATIONS IN STRING PHENOMENOLOGY Dissertation an der Fakult¨atf¨urPhysik der Ludwig-Maximilians-Universit¨at M¨unchen vorgelegt von SUSANNE REFFERT aus Z¨urich M¨unchen, Mai 2006 ii 1. Gutachter: Prof. Dr. Dieter L¨ust,LMU M¨unchen 2. Gutachter: Prof. Dr. Wolfgang Lerche, CERN Tag der m¨undlichen Pr¨ufung:21.7.2006 iii To my parents iv v Summary As of now, string theory is the best candidate for a theory of quantum gravity. Since it is anomaly–free only in ten space-time dimensions, the six surplus spatial dimensions must be compactified. This thesis is concerned with the geometry of toroidal orbifolds and their applica- tions in string theory. An orbifold is the quotient of a smooth manifold by a discrete 6 group. In the present thesis, we restrict ourselves to orbifolds of the form T /ZN or 6 T /ZN × ZM . These so–called toroidal orbifolds are particularly popular as compacti- fication manifolds in string theory. They present a good compromise between a trivial compactification manifold, such as the T 6 and one which is so complicated that explicit calculations are nearly impossible, which unfortunately is the case for many if not most Calabi–Yau manifolds. At the fixed points of the discrete group which is divided out, the orbifold develops quotient singularities. By resolving these singularities via blow– ups, one arrives at a smooth Calabi–Yau manifold. The systematic method to do so is explained in detail. Also the transition to the Orientifold quotient is explained. In string theory, toroidal orbifolds are popular because they combine the advantages of calculability and of incorporating many features of the standard model, such as non- Abelian gauge groups, chiral fermions and family repetition. In the second part of this thesis, applications in string phenomenology are discussed. The applications belong to the framework of compactifications with fluxes in type IIB string theory. Flux compactifications on the one hand provide a mechanism for supersymmetry breaking. One the other hand, they generically stabilize at least part of the geometric moduli. The geometric moduli, i.e. the deformation parameters of the compactification manifold correspond to massless scalar fields in the low energy effective theory. Since such massless fields are in conflict with experiment, mechanisms which generate a potential for them and like this fix the moduli to specific values must be investigated. After some preliminaries, two main examples are discussed. The first belongs to the category of model building, where concrete models with realistic 6 properties are investigated. A brane model compactified on T /Z2 × Z2 is discussed. The flux-induced soft supersymmetry breaking parameters are worked out explicitly. The second example belongs to the subject of moduli stabilization along the lines of the proposal of Kachru, Kallosh, Linde and Trivedi (KKLT). Here, in addition to the background fluxes, non-perturbative effects serve to stabilize all moduli. In a second step, a meta-stable vacuum with a small positive cosmological constant is achieved. Orientifold models which result from resolutions of toroidal orbifolds are discussed as possible candidate models for an explicit realization of the KKLT proposal. The appendix collects the technical details for all commonly used toroidal orbifolds and constitutes a reference book for these models. vi vii Acknowledgments It is a pleasure to express my profound gratitude and appreciation to my thesis advisor Dieter L¨ust: For his continued support and encouragement, for letting me be a part of his wonderful work group, and especially for sharing his ideas and views of physics. I am deeply indebted to Stephan Stieberger, for countless hours of explanations, as well as his friendship and support. I am happy to extend my special thanks to Emanuel Scheidegger, who has taught me my first steps in algebraic geometry and much, much more. I would like to thank all those I had the pleasure to collaborate with: Dieter L¨ust,Peter Mayr, Emanuel Scheidegger, Waldemar Schulgin, Stephan Stieberger and Prasanta Tripathy. Furthermore, I would like to thank everyone who was or is part of the String Theory group at Humboldt University in Berlin and later in Munich, for contributing to the hospitable and lively atmosphere of the group I have enjoyed and have greatly benefitted from. In particular, I would like to thank two of my former office mates: George Kraniotis for sharing his great enthusiasm for physics, and Domenico Orlando, for pleasant company and lively discussions. Finally, I would like to thank everyone who has taken the trouble to teach me or to share their insights, if only once and just for ten minutes. Sometimes small things make a big difference. Outside of physics, I would like to thank my family and Bernhard for their support, and my landlords for taking care of the animals when I was traveling. viii Contents 1 Introduction and Overview 1 I The Geometry of Toroidal Orbifolds 7 2 At the orbifold point 9 2.1 What is an orbifold? . 9 2.2 Why should I care? . 9 2.3 Point groups and Coxeter elements . 10 2 2.3.1 Example A: Z6−I on G2 × SU(3) . 12 2.4 The usual suspects . 12 2.5 Shape and size . 13 2.6 The metric and its deformations . 15 6 2 2.6.1 Example A: T /Z6−I on G2 × SU(3) . 16 2.7 Parametrizing the geometrical moduli . 16 6 2 2.7.1 Example A: T /Z6−I on G2 × SU(3) . 17 2.8 Fixed tori with non-standard volumes . 18 6 2.8.1 Example B: T /Z6−II on SU(2) × SU(6) . 19 2.9 Fixed set configurations and conjugacy classes . 21 6 2 2.9.1 Example A: T /Z6−I on G2 × SU(3) . 22 2.10 There’s more than meets the eye . 23 3 The smooth Calabi-Yau 25 3.1 From singular to smooth . 25 3.2 A lattice and a fan . 25 3 3.2.1 Example A.1: C /Z6−I ....................... 27 3.3 Resolving the singularities . 27 3 3.3.1 Example A.1: C /Z6−I ....................... 29 3.4 Mori cone and intersection numbers . 30 3 3.4.1 Example A.1: C /Z6−I ....................... 32 3.5 Divisor topologies, Part I . 33 3 3.5.1 Example A.1: C /Z6−I ....................... 34 3 3.5.2 Example B.1: C /Z6−II ...................... 35 3.6 The big picture . 36 ix x Contents 2 3.6.1 Example A: Z6−I on G2 × SU(3) . 37 6 3.6.2 Example C: T /Z6 × Z6 ...................... 37 3.7 The inherited divisors . 40 6 2 3.7.1 Example A: T /Z6−I on G2 × SU(3) . 43 3.8 The intersection ring . 44 6 2 3.8.1 Example A: T /Z6−I on G2 × SU(3) . 47 3.9 Divisor topologies, Part II . 49 6 2 3.9.1 Example A: T /Z6−I on G2 × SU(3) . 52 6 3.9.2 Example B: T /Z6−II on SU(2) × SU(6) . 53 3.10 The twisted complex structure moduli . 55 4 From Calabi–Yau to Orientifold 57 4.1 Yet another quotient . 57 (1,1) 4.2 When the patches are not invariant: h− 6=0.............. 58 6 4.2.1 Example B: T /Z6−II on SU(2) × SU(6) . 59 4.3 The local orientifold involution . 59 6 4.3.1 Example B: T /Z6−II on SU(2) × SU(6) . 61 4.4 The intersection ring . 63 6 4.4.1 Example B: T /Z6−II on SU(2) × SU(6) . 63 4.5 Global O–plane configuration and tadpole cancellation . 65 6 4.5.1 Example B: T /Z6−II on SU(2) × SU(6) . 66 II Applications in String Phenomenology 69 5 Preliminaries 71 5.1 The type IIB low energy effective action . 71 5.2 The K¨ahlerpotential . 72 6 5.2.1 Example D: T /Z2 × Z2 ...................... 73 5.3 The orientifold action . 73 5.4 Turning on background flux . 74 5.5 Invariant 3-forms . 76 6 5.5.1 Example D: T /Z2 × Z2 ...................... 77 6 5.5.2 Example E: T /Z7 ......................... 78 5.6 The effective potential from fluxes . 79 5.7 String–theoretical K¨ahlermoduli T i vs. field–theoretical fields T i ... 80 6 5.7.1 Example D: T /Z2 × Z2 ...................... 81 5.8 Superpotential, F –terms and scalar potential . 81 6 5.8.1 Example D: T /Z2 × Z2 ...................... 82 5.9 Supersymmetry conditions for the background flux . 84 6 5.9.1 Example D: T /Z2 × Z2 ...................... 84 Contents xi 6 Braneworld Scenarios: Model building and soft SUSY breaking 87 6.1 Branes carrying internal gauge flux . 88 6.2 Open string low-energy effective action and soft terms . 91 6.3 Structure of the soft supersymmetry breaking terms . 93 7 Moduli stabilization in the framework of KKLT 99 7.1 Introducing the KKLT-scenario . 99 7.2 The origin of the non-perturbative superpotential . 100 7.3 Witten’s criterion and the index on D3–branes . 101 7.4 Vacuum structure and stability . 104 7.5 Resolved toroidal orbifolds as candidate models for KKLT . 106 6 7.5.1 Example B: T /Z6−II on SU(2) × SU(6) . 108 8 Conclusions and Summary 111 III Appendix 113 A Resolutions of local orientifold singularities 115 3 A.1 Resolution of C /Z3 ............................ 115 3 A.2 Resolution of C /Z4 ............................ 116 3 A.3 Resolution of C /Z6−I ........................... 118 3 A.4 Resolution of C /Z6−II ..........................
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