Calabi-Yau Threefolds and Heterotic String Compactification Rhys Davies University College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity Term 2010 Calabi-Yau Threefolds and Heterotic String Compactification Rhys Davies University College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity Term 2010 This thesis is concerned with Calabi-Yau threefolds and vector bundles upon them, which are the basic mathematical objects at the centre of smooth supersymmetric compactifications of heterotic string theory. We begin by explaining how these objects arise in physics, and give a brief review of the techniques of algebraic geometry which are used to construct and study them. We then turn to studying multiply-connected Calabi-Yau threefolds, which are of particular importance for realistic string compactifications. We construct a large number of new examples via free group actions on complete intersection Calabi-Yau manifolds (CICY's). For special values of the parameters, these group actions develop fixed points, and we show that, on the quotient spaces, this leads to a particular class of singularities, which are quotients of the conifold. We demonstrate that, in many cases at least, such a singularity can be resolved to yield another smooth Calabi-Yau threefold, with different Hodge numbers and fundamental group. This is a new example of the interconnectedness of the moduli spaces of distinct Calabi-Yau threefolds. In the second part of the thesis we turn to a study of two new `three-generation' manifolds, constructed as quotients of a particular CICY, which can also be represented as a hypersurface in dP6×dP6, where dP6 is the del Pezzo surface of degree six. After describing the geometry of this manifold, and especially its non-Abelian quotient, in detail, we show how to construct on the quotient manifolds vector bundles which lead to four-dimensional heterotic models with the standard model gauge group and three generations of particles. The example described in detail has the spectrum of the minimal supersymmetric standard model plus a single vector-like pair of colour triplets. Acknowledgements I have been lucky to benefit from the help and support of many people before and during my time at Oxford, and I am grateful to all of them, but I would particularly like to acknowledge the following people. • Most of all I thank my wife Emma, whose endless love and encouragement motivates and strengthens me in all pursuits. • A special thank-you goes to my parents Pam and Owen, who have always supported and encouraged me, even though my choices have taken me half a world away from them. • I thank my supervisors and collaborators Andre Lukas, Xenia de la Ossa, Philip Candelas, Volker Braun and Ron Donagi. They have taught me a great deal, and this thesis would not have come to be without their help. I also thank Mark Gross, Bal´azsSzendr}oiand Duco van Straten for sharing with me some of their expertise in algebraic geometry. • I would like to thank everybody with whom I have talked informally about physics and mathematics during my D.Phil., but especially Matt McCullough. Such discussions, espe- cially over cups of coffee or pints of ale, have been an invaluable and immensely enjoyable part of the whole experience. • Finally, I would like to acknowledge the financial support of the Sir Arthur Sims Travelling Scholarship, from the University of Melbourne, and the Oxford-Australia University College Old Members' Scholarship. Contents 1 Introduction 1 2 Aspects of Algebraic Geometry 7 2.1 Divisors and line bundles . .8 2.1.1 Divisors . .8 2.1.2 Holomorphic line bundles . .9 2.2 Holomorphic vector bundles and their cohomology . 10 2.2.1 Bundles on hypersurfaces . 11 2.2.2 Projective space . 12 2.3 Singularities and resolutions . 14 2.3.1 Blowing up . 14 2.3.2 Resolution of singularities . 15 2.3.3 The conifold, and conifold transitions . 16 2.4 Toric geometry . 17 2.4.1 Affine toric varieties . 18 2.4.2 Fans and toric varieties . 19 2.4.3 Homogeneous coordinates . 20 2.4.4 Toric geometry and Calabi-Yau manifolds . 22 3 New Calabi-Yau Manifolds with Small Hodge Numbers 23 3.1 Overview: CICY's and quotient manifolds . 23 3.1.1 Splitting and Contraction . 24 3.1.2 Configurations and diagrams . 27 3.1.3 Calculation of the Euler characteristic and Hodge numbers . 28 3.1.4 Checking transversality of the defining equations . 29 3.1.5 Webs of CICY's with freely acting symmetries . 30 3.2 Some examples of free actions by Z5 ......................... 45 3.2.1 X2;52; a split of the quintic threefold . 45 3.2.2 X7;27; a further split of the quintic . 46 3.3 Some manifolds admitting free actions by Z3 .................... 48 3;48 2 3.3.1 X ; a P split of the bicubic . 48 3.3.2 X4;40; a second split of the bicubic . 51 5 1;73 3.3.3 P [3; 3] .................................... 52 3.3.4 X3;39; splitting the two cubics . 53 3.3.5 X9;27; a second split of both cubics . 56 3.3.6 X5;50; a one-sided split of X1;73 ........................ 59 3.3.7 X8;44 ....................................... 60 3.3.8 The manifold X19;19 .............................. 66 3.4 Some manifolds with quaternionic symmetry . 66 7 3.4.1 The manifold P [2222]............................ 66 5;37 7 3.4.2 X , a symmetrical split of P [2 2 2 2] . 67 3.4.3 X4;68|the tetraquadric . 68 1;1 2;1 3.4.4 An H quotient of the manifold with (h ; h ) = (19; 19) . 69 4 Hyperconifold Singularities 71 4.1 A Z5 example . 73 4.1.1 The conifold and Z5-hyperconifold as toric varieties . 74 4.2 Local hyperconifold singularities in general . 75 4.2.1 Analysis of fixed points . 75 4.2.2 The hyperconifolds torically . 76 4.2.3 Topology of the singularities . 80 4.3 Global resolutions . 80 4.3.1 Blowing up a node . 80 4.3.2 The toric picture, and the Z2-hyperconifold . 81 4.3.3 The Z2M -hyperconifolds . 83 4.4 Hyperconifold transitions in string theory? . 85 5 New Three-Generation Manifolds 88 5.1 A manifold with a χ = −6 quotient . 88 5.1.1 The quotient manifold . 89 5.1.2 The group representations of Dic3 ...................... 93 5.1.3 Group action on homology . 93 5.2 Symmetries of X and XG ............................... 95 5.2.1 Symmetries of dP6×dP6 ............................ 96 5.2.2 Symmetries of the quotient . 96 5.3 Conifold transition to a manifold with (h1;1; h2;1) = (2; 2) . 99 5.3.1 The parameter space of 3-nodal quotients XG ................ 99 5.3.2 The nodes and their resolution . 100 5.3.3 Identifying the (2; 2) manifold with a quotient of X19;19 .......... 103 5.4 Toric considerations and the mirror manifold . 104 5.4.1 The Newton polyhedron and its dual . 104 5.4.2 Triangulations . 107 5.4.3 The mirror manifold . 110 5.5 The Abelian quotient . 113 5.5.1 Group action on homology . 113 5.6 Alternative representations . 113 5.6.1 The non-Abelian quotient . 114 5.6.2 The Z12 quotient . 114 6 Semi-Realistic Heterotic Models 115 6.1 Bundle deformations and the low-energy theory . 116 6.2 The deformed bundle . 117 6.2.1 Bundle stability and deformations . 118 6.2.2 Equivariant structures . 120 6.2.3 Constructing bundle deformations . 121 6.3 Calculating the cohomology/spectrum . 125 6.3.1 Calculating n10 and n10 ............................ 125 6.3.2 Calculating n5 and n5 ............................. 126 6.3.3 Group action on cohomology, and the low-energy spectrum . 128 6.4 Models on the Z12 quotient . 130 6.4.1 Wilson line breaking of SU(5)GUT ...................... 131 6.4.2 Bundle deformations and the spectrum . 131 6.5 Moduli stabilisation . 133 6.6 Summary . 134 A Cohomology Calculations 135 A.1 Cohomology on dP6 .................................. 135 A.1.1 The tangent bundle and related bundles . 135 A.2 Cohomology on X8;44 ................................. 137 A.2.1 Some cohomologies . 138 A.2.2 Deformations of T X8;44 ............................ 139 1. Introduction Superstring theory has many remarkable features which set it apart from any other attempt to describe fundamental physics. The consistent quantisation of the theory requires that the dimension of spacetime is ten | no other theory makes such a prediction. The spectrum of excitations of a string contains gravitons, Yang-Mills gauge fields, and charged fermions, all of which we know are required for a sensible description of our universe. Furthermore, string theory provides field equations for the vacuum configurations of these fields, which reduce at large distances to general relativity coupled to a Yang-Mills gauge theory. It also appears that the theory is finite, and therefore makes sense at all energy scales. This striking structure makes string theory a very appealing candidate for the correct description of physics at the most fundamental level. Nevertheless, there is still a lot of work to be done to properly understand the theory itself. Various non-perturbative features have been understood, but a complete non- perturbative definition of the theory is yet to be found. At least as important as understanding the foundations of string theory is attempting to connect it directly to experiment. The big problem for this pursuit, referred to broadly as `string phenomenology', is that at low energies, string theory is typically indistinguishable from quantum field theory; quintessentially `stringy' effects only show up around the string scale.