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Prologus Terræ Sanctæ

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The Calabi–Yau Landscape

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2293))

Abstract

An invitation to the world of complex geometry and Calabi–Yau manifolds, extended to the student of physics and to the student of mathematics. It assumes no prior knowledge of string theory or of algebraic geometry and very quickly introduces some terminology.

Cæterum ad veridicam, sicut iam polliciti sumus, Terræ Sanctæ descriptionem, stylum vertamus. —Burchardus de Monte Sion1

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Notes

  1. 1.

    “Thus, as promised, let us now turn, to the description of the Holy Land”. Burchard of Mount Sion, from the Prologue to his Description of the Holy Land (Prologus & Descriptio Terrae Sanctae, 1285).

  2. 2.

    By fusing the bosonic string (whose critical dimension is 26) with the superstring (whose critical dimension is 10) in the process of “heterosis” by assigning them to be, respectively, left and right moving modes of the string, the fact that 26 − 10 = 16 gives 16 internal degrees of freedom to furnish a gauge symmetry. Beautifully, in 16 dimensions there are only two even self-dual integral lattices in which quantized momenta could take value, viz., the root lattices of E 8 × E 8 and of \(D_{16} = \mathfrak {so}(32)\).

  3. 3.

    According to Witten, in the sense of string phenomenology, “heterotic compactification is still the best hope for the real world”.

  4. 4.

    Recently, Li–Yau [281] showed that this is equivalent to ω being balanced, i.e., \(d\left ( ||\Omega || g^2 \right ) = 0\).

  5. 5.

    The reader might wonder how curious it is that two utterly different developments, one in mathematics, and another in physics, should both lead to Rome. Philip Candelas recounts some of the story. The paper [74] started life as two distinct manuscripts, one by Witten, and the other, by Candelas–Horowitz–Strominger, the latter began during a hike to Gibraltar reservoir, near Santa Barbara, in the fall of 1985. Both independently came to the conclusion that supersymmetry required a covariantly constant spinor and thus vanishing Ricci curvature for the internal manifold M.

    It was most timely that Horowitz had been a postdoctoral researcher of Yau, and Strominger also coincided with Yau when both were at the IAS, thus a phone call to Yau settled the issue about the relation to SU(3) holonomy and hence Calabi–Yau. Thus the various pieces fell, rather quickly, into place, the two manuscripts were combined, and the rest was history. In fact, it was the physicists who named these Ricci-flat Kähler manifolds as “Calabi–Yau”.

  6. 6.

    Serious effort, however, has been made by various groups, e.g., the UPenn-Oxford collaboration, to use stable holomorphic vector bundles V on CY3, whose index gives the number of particle generations. The standard embedding is the special case where V is the tangent bundle. A review of some of this direction in recent times is in [200].

  7. 7.

    Another way to see this is that for M simply connected, the first fundamental group π 1(M) vanishes. Then, H 1(M), being the Abelianization of π 1(M), also vanishes. Hence, h 1 = 0.

  8. 8.

    In almost all cases, they are both positive integers. The case of h 2, 1 = 0 is called rigid because here the manifold would afford no complex deformations. The Hodge number h 1, 1, on the other hand, is at least 1 because M is at least Kähler.

References

  1. R. Altman, J. Gray, Y.H. He, V. Jejjala, B.D. Nelson, A Calabi-Yau database: threefolds constructed from the Kreuzer-Skarke list. J. High Energy Phys. 1502, 158 (2015) [arXiv:1411.1418 [hep-th]]

    Google Scholar 

  2. L.B. Anderson, J. Gray, A. Lukas, E. Palti, Two hundred heterotic standard models on smooth Calabi-Yau threefolds. Phys. Rev. D 84, 106005 (2011) [arXiv:1106.4804 [hep-th]]; L.B. Anderson, J. Gray, A. Lukas, E. Palti, Heterotic line bundle standard models. J. High Energy Phys. 1206, 113 (2012) [arXiv:1202.1757 [hep-th]]; L.B. Anderson, J. Gray, A. Lukas, E. Palti, Heterotic standard models from smooth Calabi-Yau three-folds. PoS CORFU 2011, 096 (2011); A. Constantin, A. Lukas, Formulae for line bundle cohomology on Calabi-Yau threefolds (2019). arXiv:1808.09992 [hep-th]

    Google Scholar 

  3. A. Ashmore, Y.H. He, B.A. Ovrut, Machine learning Calabi-Yau metrics. Fortsch. Phys. 68(9), 2000068 (2020) [arXiv:1910.08605 [hep-th]]

    Google Scholar 

  4. P.S. Aspinwall, K3 surfaces and string duality, in Differential Geometry Inspired by String Theory, ed. by S.T. Yau (1996), pp. 1–95 [hep-th/9611137]

    Google Scholar 

  5. M. Atiyah, E. Witten, M theory dynamics on a manifold of G(2) holonomy. Adv. Theor. Math. Phys. 6, 1 (2003) [hep-th/0107177]

    Article  MathSciNet  Google Scholar 

  6. A.C. Avram, M. Kreuzer, M. Mandelberg, H. Skarke, The web of Calabi-Yau hypersurfaces in toric varieties. Nucl. Phys. B 505, 625 (1997) [hep-th/9703003]

    Article  MathSciNet  Google Scholar 

  7. A. Beauville , Complex Alg. Surfaces. LMS. Texts, vol. 34 (Cambridge University Press, Cambridge, 1996)

    Google Scholar 

  8. K. Becker, M. Becker, J.H. Schwarz, String Theory and M-Theory: A Modern Introduction (Cambridge University Press, Cambridge, 2007)

    MATH  Google Scholar 

  9. M. Berger, Sur les groupes d’holonomie homogènes des variétés a connexion affines et des variétés riemanniennes. Bull. Soc. Math. France 83, 279–330 (1953)

    MATH  Google Scholar 

  10. J.L. Bourjaily, Y.H. He, A.J. Mcleod, M. Von Hippel, M. Wilhelm, Traintracks through Calabi-Yaus: amplitudes beyond elliptic polylogarithms (2018). arXiv:1805.09326 [hep-th]

    Google Scholar 

  11. J.L. Bourjaily, A.J. McLeod, M. von Hippel, M. Wilhelm, A (bounded) bestiary of Feynman integral Calabi-Yau geometries (2018). arXiv:1810.07689 [hep-th]

    Google Scholar 

  12. R. Bousso, J. Polchinski, Quantization of four form fluxes and dynamical neutralization of the cosmological constant. J. High Energy Phys. 06, 006 (2000) [arXiv:hep-th/0004134 [hep-th]]; S. Kachru, R. Kallosh, A.D. Linde, S.P. Trivedi, De Sitter vacua in string theory. Phys. Rev. D 68, 046005 (2003) [arXiv:hep-th/0301240 [hep-th]]

    Google Scholar 

  13. I. Brunner, M. Lynker, R. Schimmrigk, Unification of M theory and F theory Calabi-Yau fourfold vacua. Nucl. Phys. B 498, 156 (1997) [hep-th/9610195]; A. Klemm, B. Lian, S.S. Roan, S.T. Yau, Calabi-Yau fourfolds for M theory and F theory compactifications. Nucl. Phys. B 518, 515 (1998) [hep-th/9701023]

    Google Scholar 

  14. E. Calabi, The space of Kähler metrics. Proc. Int. Congress Math. Amsterdam 2, 206–207 (1954); E. Calabi, On Kähler manifolds with vanishing canonical class, in Algebraic Geometry and Topology. A Symposium in Honor of S. Lefschetz, ed. by Fox, Spencer, Tucker. Princeton Mathematical Series, vol. 12 (PUP, Princeton, 1957), pp. 78–89

    Google Scholar 

  15. P. Candelas, G.T. Horowitz, A. Strominger, E. Witten, Vacuum configurations for superstrings. Nucl. Phys. B 258, 46 (1985)

    Article  MathSciNet  Google Scholar 

  16. P. Candelas, A.M. Dale, C.A. Lutken, R. Schimmrigk, Complete intersection Calabi-Yau manifolds. Nucl. Phys. B 298, 493 (1988)

    Article  MathSciNet  Google Scholar 

  17. P. Candelas, X. de la Ossa, Y.H. He, B. Szendroi, Triadophilia: a special corner in the landscape. Adv. Theor. Math. Phys. 12, 429 (2008) [arXiv:0706.3134 [hep-th]]

    Google Scholar 

  18. P. Candelas, X. de la Ossa, F. Rodriguez-Villegas, Calabi-Yau manifolds over finite fields. 1 & 2. hep-th/0012233; Fields Inst. Commun. 38, 121 (2013) [hep-th/0402133]

    Google Scholar 

  19. J. Conlon, Why String Theory (CRC, Boca Raton, FL, 2015). ISBN: 978-1-4822-4247-8

    MATH  Google Scholar 

  20. B. de Mont Sion, Prologus (sive Descriptiones) Terrae Sanctae (1283), in Rudimentum Noviciorum (Lübeck, Brandis, 1475)

    Google Scholar 

  21. R. Deen, Y.H. He, S.J. Lee, A. Lukas, Machine learning string standard models (2020) [arXiv:2003.13339 [hep-th]]

    Google Scholar 

  22. W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann, Singular 4-1-1 — a computer algebra system for polynomial computations (2018). http://www.singular.uni-kl.de

  23. F. Denef, M.R. Douglas, Computational complexity of the landscape. I. Ann. Phys. 322, 1096–1142 (2007) [arXiv:hep-th/0602072 [hep-th]]

    Google Scholar 

  24. M. Dine, Supersymmetry and String Theory: Beyond the Standard Model (Cambridge University Press, Cambridge, 2007)

    Book  Google Scholar 

  25. S. Donaldson, Scalar curvature and projective embeddings, I. J. Different. Geom. 59(3), 479–522 (2001)

    MathSciNet  MATH  Google Scholar 

  26. S. Donaldson, Some numerical results in complex differential geometry (2005). arXiv:math/0512625

    Google Scholar 

  27. F. Gmeiner, R. Blumenhagen, G. Honecker, D. Lust, T. Weigand, One in a billion: MSSM-like D-brane statistics. J. High Energy Phys. 0601, 004 (2006) [hep-th/0510170]

    Article  MathSciNet  Google Scholar 

  28. D. Grayson, M. Stillman, Macaulay2, a software system for research in algebraic geometry. Available at https://faculty.math.illinois.edu/Macaulay2/

  29. D.J. Gross, J.A. Harvey, E.J. Martinec, R. Rohm, The heterotic string. Phys. Rev. Lett. 54, 502 (1985)

    Article  MathSciNet  Google Scholar 

  30. M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory. Cambridge Monographs on Mathematical Physics, vol. 1 & 2 (Cambridge University Press, Cambridge, 1987)

    Google Scholar 

  31. J. Halverson, P. Langacker, TASI lectures on Remnants from the string landscape. PoS TASI 2017, 019 (2018) [arXiv:1801.03503]

    Google Scholar 

  32. J. Halverson, F. Ruehle, Computational complexity of vacua and near-vacua in field and string theory. Phys. Rev. D 99(4), 046015 (2019) [arXiv:1809.08279 [hep-th]]

    Google Scholar 

  33. J. Hauenstein, A. Sommese, C. Wampler, Bertini: software for numerical algebraic geometry. www.nd.edu/~sommese/bertini

  34. Y.H. He, An algorithmic approach to heterotic string phenomenology. Mod. Phys. Lett. A 25, 79 (2010) [arXiv:1001.2419 [hep-th]]

    Google Scholar 

  35. Y.-H. He, D. Mehta (Organizers), Applications of computational and numerical algebraic geometry to theoretical physics, in SIAM Conference on Applied Algebraic Geometry, Colorado, 2013; Y.-H. He, R.-K. Seong, M. Stillman, Applications of Computational Algebraic Geometry to Theoretical Physics (SIAM AG, Daejeon, 2015)

    Google Scholar 

  36. Y.-H. He, P. Candelas, A. Hanany, A. Lukas, B. Ovrut (eds.) Computational Algebraic Geometry in String and Gauge Theory, Hindawi, Special Volume (2012). ISBN: 1687-7357

    Google Scholar 

  37. Y.H. He, V. Jejjala, C. Matti, B.D. Nelson, M. Stillman, The geometry of generations. Commun. Math. Phys. 339(1), 149 (2015) [arXiv:1408.6841 [hep-th]]

    Google Scholar 

  38. H. Hopf, Differentialgeometry und topologische Gestalt. Jahresbericht der Deutschen Mathematiker-Vereinigung 41, 209–228 (1932); S.-S. Chern, On curvature and characteristic classes of a Riemann manifold. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 20, 117–126 (1966)

    Google Scholar 

  39. K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, E. Zaslow, Mirror Symmetry (AMS, Providence, 2000). ISBN: 0-8218-2955-6

    Google Scholar 

  40. C. Hull, Superstring compactifications with torsion and space-time supersymmetry, in Turin 1985 Proceedings. Superunification and Extra Dimensions, vol. 347(375) (World Scientific Publishers, Singapore, 1986)

    Google Scholar 

  41. L.E. Ibáñez, A.M. Uranga, String Theory and Particle Physics: An Introduction to String Phenomenology (Cambridge University Press, Cambridge, 2012)

    MATH  Google Scholar 

  42. S. Katz, Enumerative geometry and string theory, in SML, vol. 32 (AMS, Providence, 2006). ISBN: 978-0-8218-3687-3

    Google Scholar 

  43. R. Laza, M. Schütt, N. Yui, Calabi-Yau Varieties: Arithmetic, Geometry and Physics (Springer, New York, 2015), ISBN: 978-1-4939-2830-9

    Book  Google Scholar 

  44. W. Lerche, D. Lust, A.N. Schellekens, Ten-dimensional heterotic strings from Niemeier lattices. Phys. Lett. B 181, 71 (1986). A.N. Schellekens, The Landscape ‘avant la lettre’. physics/0604134

    Google Scholar 

  45. J. Li, S.-T. Yau, The existence of supersymmetric String theory with torsion. J. Differ. Geom. 70(1), 143–181 (2005)

    Article  MathSciNet  Google Scholar 

  46. Magma Comp. Algebra System (2021). http://magma.maths.usyd.edu.au/

  47. J.M. Maldacena, The large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38, 1113 (1999) [Adv. Theor. Math. Phys. 2, 231 (1998)] [hep-th/9711200]

    Google Scholar 

  48. D.R. Morrison, C. Vafa, Compactifications of F theory on Calabi-Yau threefolds. 1 & 2. Nucl. Phys. B 473, 74 & 476, 437 (1996) [hep-th/9602114], [hep-th/9603161]

    Google Scholar 

  49. J. Polchinski, Dirichlet Branes and Ramond-Ramond charges. Phys. Rev. Lett. 75, 4724 (1995) [hep-th/9510017]

    Article  MathSciNet  Google Scholar 

  50. J. Polchinski, String Theory, vols. 1& 2 (CUP, Cambridge, 1998)

    Google Scholar 

  51. D. Rickles, A Brief History of String Theory: From Dual Models to M-Theory (Springer, New York, 2014). ISBN-13: 978-3642451270

    MATH  Google Scholar 

  52. SageMath, The Sage Mathematics Software System (The Sage Developers, Thousand Oaks, CA). http://www.sagemath.org

  53. M. Schuett, Arithmetic of K3 surfaces (2008). arXiv:0808.1061

    Google Scholar 

  54. A. Strominger, Superstrings with Torsion. Nucl. Phys. B 274, 253 (1986)

    Article  MathSciNet  Google Scholar 

  55. The GAP Group, GAP – Groups, Algorithms, and Programming, version 4.9.2 (2018). https://www.gap-system.org

  56. The Particle Data Group (2010). cf. http://pdg.lbl.gov/

    Google Scholar 

  57. S.-T. Yau, Calabi’s conjecture and some new results in algebraic geometry. Proc. Natl. Acad. U. S. A. 74(5), 1798–1799 (1977). S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampére equation I. Commun. Pure Appl. Maths 31(3), 339–411 (1978)

    Google Scholar 

  58. S.-T. Yau, A survey of Calabi–Yau manifolds, in Geometry, Analysis, and Algebraic Geometry: Forty Years of the Journal of Differential Geometry. Surveys in Differential Geometry, vol. 13 (International Press, Somerville, MA, 2009), pp. 277–318

    Google Scholar 

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Authors and Affiliations

  1. London Institute, Royal Institution of Great Britain; City, University of London; School of Physics & Chern Institute, Nankai University; Merton College, University of Oxford, Oxford, UK

    Yang-Hui He

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He, YH. (2021). Prologus Terræ Sanctæ. In: The Calabi–Yau Landscape. Lecture Notes in Mathematics, vol 2293. Springer, Cham. https://doi.org/10.1007/978-3-030-77562-9_1

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