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Lectures on mirror symmetry

  • S. Hosono
  • A. Klemm
  • S. Theisen
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 436)

Abstract

We give an introduction to mirror symmetry of strings on Calabi-Yau manifolds with an emphasis on its applications e.g. for the computation of Yukawa couplings. We introduce all necessary concepts and tools such as the basics of toric geometry, resolution of singularities, construction of mirror pairs, Picard-Fuchs equations, etc. and illustrate all of this on a non-trivial example.

Keywords

Yukawa Coupling Toric Variety Exceptional Divisor Hodge Number Complex Structure Modulo 
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References

  1. [1]
    For introductions, see e.g. M. Green, J. Schwarz and E. Witten, Superstring Theory, Vols. I and II, Cambridge University Press, 1986; D. Liist and S. Theisen, Lectures on String Theory, Springer Lecture Notes in Physics, Vol. 346, 1990; M. Kaku, Introduction to Superstrings, Springer 1988Google Scholar
  2. [2]
    E. Verlinde, H.Verlinde, Lectures on String Perturbation Theory, in Superstrings 88, Trieste Spring School, Ed. Greene et al. World Scientific (1989) 189Google Scholar
  3. [3]
    C. Callan, D. Friedan, E. Martinec and M. Perry, Nucl. Phys. B262 (1985) 593, A. Sen, Phys. Rev. D32 (1985) 2102 and Phys. Rev. Lett. 55 (1985) 1846Google Scholar
  4. [4]
    D. Nemeschanski and A. Sen, Phys. Lett. 178B (1986) 365Google Scholar
  5. [5]
    For review, see A. Giveon, M. Porrati and E. Rabinovici, Target Space Duality in String Theory, hep-th 9401139, to be published in Phys. Rep.Google Scholar
  6. [6]
    L. Dixon in Proc. of the 1987 ICTP Summer Workshop in High Energy Physics and Cosmology, Trieste, ed. G. Furlan et al., World ScientificGoogle Scholar
  7. [7]
    W. Lerche, C. Vafa and N. Warner, Nucl. Phys. B324 (1989) 427Google Scholar
  8. [8]
    B. Greene and M. Plesser, Nucl. Phys. B338 (1990) 15Google Scholar
  9. [9]
    Essays on Mirror Manifolds (ed. S.-T. Yau), Int. Press, Hong Kong, 1992Google Scholar
  10. [10]
    R. Dijkgraaf, E. Verlinde and H. Verlinde, Utrecht preprint THU-87/30; A. Shapere and F. Wilczek Nucl. Phys. B320 (1989) 669; A. Giveon, E. Rabinovici, G. Veneziano, Nucl. Phys. B322 (1989) 169Google Scholar
  11. [11]
    S. Ferrara and S. Theisen, Moduli Spaces, Effective Actions and Duality Symmetry in String Compactifications, in Proc. of the Third Hellenic School on Elementary Particle Physics, Corfu 1989, Argyres et al.(eds), World Scientific 1990Google Scholar
  12. [12]
    P. Candelas, X. De la Ossa, P. Green and L. Parkes, Nucl. Phys. B359 (1991) 21Google Scholar
  13. [13]
    D.Morrison, Picard-F~zchs Equations and Mirror Maps for Hypersurfaces, in Essays on Mirror Manifolds (ed. S.-T. Yau), Int. Press, Hong Kong, 1992; A. Klemm and S. Theisen, Nucl. Phys. B389 (1993) 153; A. Font, Nucl. Phys. B391 (1993) 358Google Scholar
  14. [14]
    A. Libgober and J. Teitelbaum, Duke Math. Journ., Int. Res. Notices 1 (1993) 29; A. Klemm and S. Theisen, Mirror Maps and Instanton Sums for Complete Intersections in Weighted Projective Space, preprint LMU-TPW 93–08Google Scholar
  15. [15]
    P. Candelas, X. de la Ossa, A. Font, S. Katz and D. Morrison, Mirror Symmetry for Two Parameter Models I, preprint CERN-TH.6884/93Google Scholar
  16. [16]
    S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces, to appear in Comm. Math. Phys.Google Scholar
  17. [17]
    S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror Symmetry, Mirror Map and Applications to Complete Intersection Calabi-Yau Spaces, preprint LMU-TPW-94-03 (to appear)Google Scholar
  18. [18]
    P. Aspinwall, B. Greene and D. Morrison, Space-Time Topology Change: the Physics of Calabi-Yau Moduli Space, preprint IASSNS-HEP-93-81; CalabiYau Moduli Space, Mirror Manifolds and Space-Time Topology Change in String Theory, preprint IASSNS-HEP-93-38; The Monomial-Divisor Mirror Map, preprint IASSNS-HEP-93-43Google Scholar
  19. [19]
    P. Berglund, P. Candelas, X. de la Ossa, A. Font, T. Hübsch, D. Jancic and F. Quevedo, Periods for Calabi-Yau and Landau-Ginsburg Vacua, preprint CERN-TH.6865/93; P. Berglund, E. Derrich, T. Hiibsch and D. Jancic, On Periods for String Compactification, preprint HUPAPP-93-6Google Scholar
  20. [20]
    P. Berglund and S. Katz, Mirror Symmetry for Hypersurfaces in Weighted Projective Space and Topological Couplings, preprint IASSNS-HEP-93-65Google Scholar
  21. [21]
    P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B258 (1985) 46Google Scholar
  22. [22]
    Easily accessible introductions to Calabi-Yau manifolds are G. Horowitz, What is a Calabi-Yau Space? in Unified String Theories, M. Green and D. Gross, editors, World Scientific 1986 and P. Candelas, Introduction to Complex Manifolds, Lectures at the 1987 Trieste Spring School, published in the proceedings. For a rigorous mathematical treatment we refer to A.L. Bessis, Einstein Manifolds, Springer 1987; P. Griffiths and J. Harris, Principles of Algebraic Geometry. John Wiley & Sons 1978; K. Kodaira, Complex Manifolds and Deformation of Complex Structures, Springer 1986Google Scholar
  23. [23]
    A good and detailed introduction to Calabi-Yau manifolds is the book by T. Hübsch, Calabi-Yau Manifolds, World Scientific 1991Google Scholar
  24. [24]
    S.T. Yau, Proc. Natl. Acad. Sci. USA 74 (1977), 1798; Comm. Pure Appl. Math. 31 (1978) 339Google Scholar
  25. [25]
    K. Kodaira, L. Nirenberg and D. C. Spencer, Ann. Math. 68 (1958) 450Google Scholar
  26. [26]
    G. Tian, in Mathematical Aspects of String Theory, ed. S. T. Yau, World Scientific, Singapore (1987)Google Scholar
  27. [27]
    A. Strominger, Phys. Rev. Lett. 55 (1985) 2547, and in Unified String Theory, eds. M. Green and D. Gross, World Scientific 1986; A. Strominger and E. Witten, Comm. Math. Phys. 101 (1985) 341Google Scholar
  28. [28]
    L. Dixon, D. Friedan, E. Martinec and S, Shenker, Nucl. Phys. B282 (1987) 13; S. Hamidi and C. Vafa, Nucl. Phys. B279 (1987) 465; J. Lauer, J. Mas and H. P. Nilles Nucl. Phys. B351 (1991) 353; S. Stieberger, D. Jungnickel, J. Lauer and M. Spalinski, Mod. Phys. Lett. A7 (1992) 3859; J. Erler, D. Jungnickel, M. Spalinski and S. Stieberger, Nucl. Phys. B397 (1993) 379Google Scholar
  29. [29]
    D. Gepner, Phys. Lett. 199B (1987) 380 and Nucl. Phys. B311 (1988) 191Google Scholar
  30. [30]
    J. Distler and B. Greene, Nucl. Phys. B309 (1988) 295Google Scholar
  31. [31]
    M. Dine, N. Seiberg, X-G. Wen and E. Witten, Nucl. Phys. B278 (1987) 769, Nucl. Phys. B289 (1987) 319Google Scholar
  32. [32]
    S. Ferrara, D. Lüst, A. Shapere and S. Theisen, Phys. Lett. 225B (1989) 363Google Scholar
  33. [33]
    P. Aspinwall and D. Morrison, Comm. Math. Phys. 151 (1993) 245Google Scholar
  34. [34]
    E. Witten, Comm. Math. Phys. 118 (1988) 411Google Scholar
  35. [35]
    T. Banks, L. Dixon, D. Friedan and E. Martinec, Nucl. Phys. B284 (1988) 613Google Scholar
  36. [36]
    M. Ademollo et al., Nucl. Phys. B11 (1976) 77 and Nucl. Phys. B114 (1976) 297 and Phys. Lett. 62B (1976) 105Google Scholar
  37. [37]
    A. Schwimmer and N. Seiberg, Phys. Lett. 184B (1987 191Google Scholar
  38. [38]
    M. Dine and N. Seiberg, Nucl. Phys. B301 (1988) 357Google Scholar
  39. [39]
    E. Witten, Nucl. Phys. B202 (1982) 253; R. Rohm and E Witten, Ann. Phys. 170 (1986) 454Google Scholar
  40. [40]
    P. Candelas, E. Derrick and L. Parkes, Nucl. Phys. B407 (1993) 115Google Scholar
  41. [41]
    R. Schimmrigk, Phys. Rev. Lett. 70 (1993) 3688 and Kähler manifolds with positive first Chern class and mirrors of rigid Calabi-Yau manifolds, preprint BONN-HE 93-47Google Scholar
  42. [42]
    N. Seiberg, Nucl. Phys. B303 (1988) 286Google Scholar
  43. [43]
    S. Cecotti, S. Ferrara and L. Girardello, Int. J. Mod. Phys. A4 (1989) 2475; S. Ferrara, Nucl. Phys. (Proc. Suppl.) 11 (1989) 342Google Scholar
  44. [44]
    P. Candelas, P. Green and T. Hübsch, Connected Calabi-Yau compactifications, in Strings `88, J. Gates et al. (eds), World Scientific 1989 and Nucl. Phys. B330 (1990) 49; P. Candelas, T. Hübsch and R. Schimmrigk, Nucl. Phys. B329 (1990) 582Google Scholar
  45. [45]
    L. Dixon, V. Kaplunovsky and J. Louis, Nucl. Phys. B329 (1990) 27Google Scholar
  46. [46]
    B. de Wit, P.G. Lauwers, R. Philippe, S.Q. Su and A. van Proeyen, Phys. Lett. 134B (1984) 37; B. de Wit and A. van Proeyen, Nucl. Phys. B245 (1984) 89; J.P. Derendinger, S. Ferrara, A. Masiero and A. van Proeyen, Phys. Lett. 140B (1984) 307; B. de Wit, P.G. Lauwers and A. van Proeyen, Nucl. Phys. B255 (1985) 569; E. Cremmer, C. Kounnas, A. van Proeyen, J.P. Derendinger, S. Ferrara, B. de Wit and L. Girardello, Nucl. Phys. B250 (1985) 385Google Scholar
  47. [47]
    M. Kreuzer and H. Skarke, Nucl. Phys. B388 (1993) 113; A. Klemm and R. Schimmrigk, Nucl. Phys. B411 (1994) 559Google Scholar
  48. [48]
    P. Candelas, A. Dale, A. Lütken and R. Schimmrigk, Nucl. Phys. B298 (1988) 493; P. Candelas, A. Lütken and R. Schimmrigk, Nucl. Phys. B306 (1989) 105Google Scholar
  49. [49]
    V. Batyrev and D. van Straten, Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties, preprint 1992Google Scholar
  50. [50]
    I. Dolgachev, Weighted Projective Varieties in Lecture Notes in Mathematics 956 Springer-Verlag (1992) 36Google Scholar
  51. [51]
    A. R. Fletcher, Working with Weighted Complete Intersections, MaxPlanck-Institut Series, No. 35 (1989) BonnGoogle Scholar
  52. [52]
    D. G. Markushevich, M. A. Olshanetsky and A. M. Perelomov, Comm. Math. Phys. 111 (1987) 247Google Scholar
  53. [53]
    J. Erler and A. Klemm, Comm. Math. Phys. 153 (1993) 57Google Scholar
  54. [54]
    L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B261 (1985) 678 and Nucl. Phys. B274 (1986) 285Google Scholar
  55. [55]
    S.S. Roan and S. T. Yau, Acta Math. Sinica (NS) 3 (1987) 256; S.S. Roan, J. Diff. Geom. 30 (1989) 523Google Scholar
  56. [56]
    M. Kreuzer and H. Skarke, Nucl. Phys. B405 (1993) 305; M. Kreuzer, Phys. Lett. 314B (1993) 31; A. Niemeyer, Diplom Thesis, TU-München, 1993Google Scholar
  57. [57]
    T. Oda, Convex Bodies and Algebraic Geometry: an Introduction to the Theory of Toric Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 15, Springer Verlag 1988; V.I. Danilov, Russian Math. Surveys, 33 (1978) 97; M. Audin, The Topology of Torus Actions on Symplectic Manifolds, Progress in Mathematics, Birkhäuser 1991; W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, Princeton University Press 1993; V.I. Arnold, S. M. Gusein-Zade and A.N. Varchenko, Singularities of Differntial Maps, Birkhäuser 1985, Vol. II, Chapter 8Google Scholar
  58. [58]
    V. Batyrev, Duke Math. Journal 69 (1993) 349Google Scholar
  59. [59]
    V. Batyrev, Journal Alg. Geom., to be publishedGoogle Scholar
  60. [60]
    V. Batyrev, Quantum Cohomology Rings of Toric Manifolds, preprint 1992Google Scholar
  61. [61]
    V. Batyrev and D. Cox, On the Hodge Structure of Projective Hypersurfaces in Toric Varieties, preprint 1993Google Scholar
  62. [62]
    F. Hirzebruch, Math. Annalen 124 (1951) 77Google Scholar
  63. [63]
    M. Beltrametti and L. Robbiano, Expo. Math. 4 (1986) 11.Google Scholar
  64. [64]
    D.Aspinwall, B.Greene and D.Morrison, Phys.Lett. 303B (1993) 249 and The Monomial-Divisor Mirror Map, preprint IASSNS-HEP-93/43Google Scholar
  65. [65]
    S.-S. Roan: Topological Couplings of Calabi-Yau Orbifolds, Max-PlanckInstitut Serie No. 22 (1992), to appear in J. of Group Theory in Physics Google Scholar
  66. [66]
    E.L. Ince, Ordinary Differential Equations, Dover 1956Google Scholar
  67. [67]
    M. Yoshida, Fuchsian Differential Equations, Braunschweig (1987) ViewegGoogle Scholar
  68. [68]
    P. A. Griffiths, Bull. Amer. Math. Soc. 76 (1970) 228Google Scholar
  69. [69]
    P. Griffiths, Ann. of Math. 90 (1969) 460Google Scholar
  70. [70]
    P. Candelas, Nucl. Phys. B298 (1988) 458Google Scholar
  71. [71]
    I.M.Gel'fand, A.V.Zelevinsky and M.M.Kapranov, Func. Anal. Appl. 28 (1989) 12 and Adv. Math. 84 (1990) 255Google Scholar
  72. [72]
    D. Morrison, Where is the Large Radius Limit, preprint IASSNS-HEP93/68 and Compactifications of Moduli Spaces Inspired by Mirror Symmetry, preprint DUK-M-93-O6Google Scholar
  73. [73]
    M. Reid, Decomposition of Toric Morphisms, in Progress in Mathematics 36, M. Artin and J. Tate, eds, Birkhäuser 1983Google Scholar
  74. [74]
    S. Ferrara and A. Strominger, in the Proceedings of the Texas A & M Strings '89 Workshop; ed. R. Arnowitt et al., World Scientific 1990Google Scholar
  75. [75]
    P. Candelas, P. Green and T. Hübsch Nucl. Phys. B330 (1990) 49; P. Candelas and X.C. de la Ossa, Nucl. Phys. B355 (1991) 455Google Scholar
  76. [76]
    R. Bryant and P. Griffiths, Some observations on the Infinitesimal Period Relations for Regular Threefolds with Trivial Canonical Bundle, in Progress in Mathematics 36, M. Artin and J. Tate, eds; Birkhäuser, 1983Google Scholar
  77. [77]
    A. Strominger, Comm. Math. Phys. 133 (1990) 163Google Scholar
  78. [78]
    L. Castellani, R. D'Auria and S. Ferrara, Phys. Lett. 241B (1990) 57 and Class. Quantum Grav. 7 (1990) 1767; S. Ferrara and J. Louis, Phys. Lett. 278B (1992) 240Google Scholar
  79. [79]
    A. Ceresole, R. D'Auria, S. Ferrara, W. Lerche and J. Louis, Int. J. Mod. Phys. A8 (1993) 79Google Scholar
  80. [80]
    X.-G. Wen and E. Witten, Phys. Lett. 166B (1986) 397Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • S. Hosono
    • 1
  • A. Klemm
    • 2
  • S. Theisen
    • 3
  1. 1.Department of MathemathicsToyama UniversityToyamaJapan
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA
  3. 3.Sektion Physik der Universität MünchenMünchenGermany

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