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Heterotic and M-Theory Compactifications for String Heterotic and M-theory Compactifications for String Phenomenology Lara Briana Anderson arXiv:0808.3621v1 [hep-th] 27 Aug 2008 Magdalen College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity 2008 “Although the universe is under no obligation to make sense, students in pursuit of the PhD are.” -Robert P. Kirshner Heterotic and M-theory Compactifications for String Phenomenology Lara Briana Anderson Magdalen College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity 2008 In this thesis, we explore two approaches to string phenomenology. In the first half of the work, we investigate compactifications of M-theory on spaces with co-dimension four, orbifold 2 singularities. We construct M-theory on C /ZN by coupling 11-dimensional supergravity to a seven-dimensional Yang-Mills theory located on the orbifold fixed-plane. The resulting action is supersymmetric to leading non-trivial order in the 11-dimensional Newton constant. We thereby 2 reduce M-theory on a G2 orbifold with C /ZN singularities, explicitly incorporating the additional gauge fields at the singularities. We derive the K¨ahler potential, gauge-kinetic function and super- potential for the resulting N = 1 four-dimensional theory. Blowing-up of the orbifold is described by a Higgs effect induced by continuation along D-flat directions. Using this interpretation, we show our results are consistent with the corresponding ones obtained for smooth G2 spaces. Fi- nally, we consider switching on flux and Wilson lines on singular loci of the G2 space, and discuss the relation to N = 4 SYM theory. In the second half, we develop an algorithmic framework for E E heterotic compactifications 8 × 8 with monad bundles, including new and efficient techniques for proving stability and calculating particle spectra. We begin by considering cyclic Calabi-Yau manifolds where we classify positive monad bundles, prove stability, and compute the complete particle spectrum for all bundles. Next, we generalize the construction to bundles on complete intersection Calabi-Yau manifolds. We show that the class of positive monad bundles, subject to the heterotic anomaly condition, is finite ( 7000 models). We compute the particle spectrum for these models and develop new techniques ∼ for computing the cohomology of line bundles on CICYs. There are no anti-generations of particles and the spectrum is manifestly moduli-dependent. We further investigate the slope-stability of positive monad bundles and develop a new method for proving stability of SU(n) vector bundles on CICYs. Acknowledgements I would like to acknowledge the excellent support and guidance of my supervisors Pro- fessor Philip Candelas and Professor Andr´eLukas. Your help, patience and insight over the past years has been invaluable. I am very lucky to have had the opportunity to work with both of you and I sincerely grateful for all that you have done for me. I would also like to thank my collaborators Adam Barrett and Yang-Hui He. Special appreciation goes to Yang, who has provided endless help and enthusiasm in the course of our work together. I would also like to acknowledge the significant contributions of Babiker Hassanain, Tanya Elliott, and Mario Serna of Office 6.2, who have taught me so much, preserved my sanity, and explored physics with me. For interesting and helpful conversations I am grateful to James Gray, Bal´azs Szendr¨oi, Lionel Mason, Dominic Joyce, Adrian Langer, Xenia de la Ossa, Graham Ross, Vincent Bouchard, and Eran Palti. I gratefully acknowledge the Rhodes Trust, the US National Science Foundation, and Magdalen College, Oxford for granting me scholarships and making my studies possible. I dedicate this thesis to my family in gratitude for their constant support and kindness (and for being their extraordinary selves) and to Dr. Wheeler, for showing me what was possible. Contents 1 Introduction 1 1.1 TheProblemofStringPhenomenology. ......... 1 1.2 AnOverview ...................................... 2 2 M-theory on manifolds of G2 holonomy 10 2.1 Introduction.................................... ..... 10 2.2 Manifolds of G2 holonomy................................. 13 2.2.1 An example G2 manifold ............................. 15 2.3 M-theory compactified on a smooth G2-manifold .................... 16 2.4 M-theory on singular G2-spaces.............................. 18 2.4.1 Arguments from String dualities . ....... 19 2.4.2 Co-dimension 4 Singularities . ....... 20 2.4.3 Co-dimension 7 singularities . ........ 21 2.5 Review of Hoˇrava-Witten Theory . ......... 23 2 3 M-theory on the orbifold C /ZN 26 3.1 Introduction.................................... ..... 26 3.2 Eleven-dimensional supergravity on the orbifold . ................ 28 3.3 Truncating the bulk theory to seven dimensions . ............. 31 3.4 General form of the supersymmetric bulk-brane action . ............... 35 3.5 Theexplicitbulk/branetheory . ......... 38 3.6 Discussionandoutlook. ....... 45 4 Four-dimensional Effective M-theory on a Singular G2 manifold 46 4.1 Introduction.................................... ..... 46 4.2 The four-dimensional effective action on a G2 orbifold ................. 49 4.2.1 Background solution and zero modes . ....... 50 i 4.2.2 Calculation of the four-dimensional effective theory ............... 53 4.2.3 Finding the superpotential and K¨ahler potential . ............. 55 4.2.4 Comparison with results for smooth G2 spaces ................. 58 4.3 SymmetryBreakingandDiscussions . ......... 60 4.3.1 Wilsonlines................................... 60 4.3.2 G- and F -Flux................................... 62 4.3.3 Relation to =4 Supersymmetric Yang-Mills Theory . 64 N 4.4 Conclusionandoutlook . .. .. .. .. .. .. .. .. ...... 67 5 Heterotic compactifications on Calabi-Yau three-folds 69 5.1 Model building in E E heteroticstringtheory. 69 8 × 8 5.2 The 10-dimensional effective heterotic lagrangian and supersymmetry . 70 5.3 Calabi-Yauthree-folds . ........ 71 5.4 AnomalyCancellation . .. .. .. .. .. .. .. .. ...... 73 5.5 Supersymmetry and vector bundle stability . ............ 74 5.6 TheStandardEmbedding . .. .. .. .. .. .. .. .. ..... 75 5.7 GeneralEmbeddings ............................... ..... 77 5.7.1 Wilson Lines and Symmetry Breaking . ...... 79 5.8 Particle Spectra in Heterotic Calabi-Yau Compactifications .............. 80 5.8.1 Net Families of Particles . ...... 81 5.9 Heterotic Compactification and Physical Constraints . ................ 82 6 An algorithmic approach to heterotic compactification: Monad bundles and Cyclic Calabi-Yau manifolds 84 6.1 Introduction.................................... ..... 84 6.2 Monad Construction of Vector Bundles . .......... 86 6.2.1 TheCalabi-YauSpaces . 86 6.2.2 ConstructingtheMonad. 88 6.2.3 Stability of Monad Bundles . ..... 91 6.3 ClassificationandExamples . ........ 94 6.3.1 Classification of Configurations . ........ 94 6.3.2 E6-GUTTheories ................................. 98 6.3.2.1 Particle Content . 98 6.3.3 SO(10)-GUTTheories. .. .. .. .. .. .. .. .. ..100 ii 6.3.3.1 Particle Content . 101 6.3.4 SU(5)-GUTTheories ...............................102 6.3.4.1 Stability for Rank 5 Bundles on the Quintic . 103 6.3.4.2 The Co-dimension 2 and 3 Manifolds . 104 6.3.4.3 Particle Content . 105 6.4 Conclusion ...................................... 106 7 Compactifying on Complete Intersection Calabi-Yau Manifolds 108 7.1 Introduction.................................... 108 7.2 Complete Intersection Calabi-Yau Threefolds . ..............109 7.2.1 Configuration Matrices and Classification . ..........110 7.2.2 FavorableConfigurations . 111 7.3 LinebundlesonCICYs .............................. 112 7.4 ThemonadconstructiononCICYs . .......114 7.5 Classification of positive monads on CICYs . ...........117 7.6 Stability....................................... 121 7.7 ComputingtheParticleSpectrum. .........124 7.7.1 BundleCohomology .............................. 124 7.7.1.1 The number of families and anti-families: H1(X, V ) and H1(X, V ∗) 124 7.7.1.2 Singlets and H1(X, V V ∗) ......................127 ⊗ 7.8 ConclusionsandProspects. ........129 8 Cohomology of Line Bundles on CICYs 131 8.1 TheKoszulResolution. .. .. .. .. .. .. .. .. 131 8.2 TheSpectralSequence. .. .. .. .. .. .. .. .. 132 8.3 CohomologyoflinebundlesonCICYs . ........133 8.3.1 Flag Spaces the Bott-Borel-Weil Theorem . .........134 8.3.2 Computing the Ranks and Kernels of Leray maps . ........136 8.3.2.1 IndividualMaps .............................137 8.3.3 Symmetric Tensors and Polynomials . .......139 8.3.3.1 TheMaps.................................139 8.4 Anexample....................................... 143 8.5 Anoutlineofthealgorithm . .......145 iii 9 Stability 146 9.1 Stable vector bundles in heterotic theories . ..............146 9.2 Sub-linebundlesandStability. ..........148 9.2.1 Hoppe’scriteria............................... 150 9.3 CohomologyofSub-lineBundles . ........151 9.3.1 Proof that H0(X, (k)V )=0 ...........................152 ∧ 9.4 Anewstabilitycriteria. ........155 9.4.1 TwoK¨ahlerModuli .............................. 155 9.4.2 The cohomology conditions . 157 9.5 AnExample....................................... 159 9.6 Conclusionandfuturework . .......162 10 Conclusion 164 10.1 Concluding remarks and future directions . .............164 A Appendix1 168 A.1 SpinorConventions............................... 168 A.2 Some group-theoretical properties . ............171 A.3 Einstein-Yang-Mills supergravity in seven dimensions ..................174 A.3.1
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