An Extended Lecture on Mirror Symmetry

S. Hosono; A. Klemm; S. Theisen

Recherche Coopérative sur Programme n°25 (1995)

  • Volume: 47, page 33-76

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Hosono, S., Klemm, A., and Theisen, S.. "An Extended Lecture on Mirror Symmetry." Recherche Coopérative sur Programme n°25 47 (1995): 33-76. <http://eudml.org/doc/274892>.

@article{Hosono1995,
author = {Hosono, S., Klemm, A., Theisen, S.},
journal = {Recherche Coopérative sur Programme n°25},
language = {eng},
pages = {33-76},
publisher = {Institut de Recherche Mathématique Avancée - Université Louis Pasteur},
title = {An Extended Lecture on Mirror Symmetry},
url = {http://eudml.org/doc/274892},
volume = {47},
year = {1995},
}

TY - JOUR
AU - Hosono, S.
AU - Klemm, A.
AU - Theisen, S.
TI - An Extended Lecture on Mirror Symmetry
JO - Recherche Coopérative sur Programme n°25
PY - 1995
PB - Institut de Recherche Mathématique Avancée - Université Louis Pasteur
VL - 47
SP - 33
EP - 76
LA - eng
UR - http://eudml.org/doc/274892
ER -

References

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  1. [1] For introductions, see e.g. M. Green, J. Schwarz and E. Witten, Superstring Theory, Vols. I and II, Cambridge University Press, 1986; Zbl0619.53002
  2. D. Lust and S. Theisen, Lectures on String Theory, SpringerLecture Notes in Physics, Vol. 346, 1990; Zbl0716.53063MR1028064
  3. M. Kaku, Introduction to Superstrings, Springer1988 Zbl0655.58001MR954610
  4. [2] E. Verlinde, H. Verlinde, Lectures on String Perturbation Theory, in Superstrings88, Trieste Spring School, Ed. Greene et al. World Scientific (1989) 189 Zbl0985.81681MR1091556
  5. [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) 1846 MR819433
  6. [4] D. Nemeschanski and A. Sen, Phys. Lett.178B (1986) 365 
  7. [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. MR1288936
  8. [6] L. Dixon in Proc. of the 1987 ICTP Summer Workshop in High Energy Physics and Cosmology, Trieste, ed. G. Furlan et al., World Scientific MR1026264
  9. [7] W. Lerche, C. Vafa and N. Warner, Nucl. Phys. B324 (1989) 427 MR1025424
  10. [8] B. Greene and M. Plesser, Nucl. Phys. B338 (1990) 15 MR1059831
  11. [9] Essays on Mirror Manifolds (ed. S.-T. Yau), Int. Press, Hong Kong, 1992 
  12. [10] R. Dijkgraaf, E. Verlinde and H. Verlinde, Utrecht preprint THU-87/30; MR1002959
  13. A. Shapere and F. WilczekNucl. Phys. B320 (1989) 669; MR1004372
  14. A. Giveon, E. Rabinovici, G. Veneziano, Nucl. Phys. B322 (1989) 169 
  15. [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 Scientific1990 
  16. [12] P. Candelas, X. De la Ossa, P. Green and L. Parkes, Nucl. Phys. B359 (1991) 21 
  17. [13] D. Morrison, Picard-Fuchs Equations and Mirror Maps for Hypersurfaces, in Essays on Mirror Manifolds (ed. S.-T. Yau), Int. Press, Hong Kong, 1992; Zbl0904.32020MR1191426
  18. A. Klemm and S. Theisen, Nucl. Phys. B389 (1993) 153; MR1202211
  19. A. Font, Nucl. Phys. B391 (1993) 358 MR1208020
  20. [14] A. Libgober and J. Teitelbaum, Duke Math. Journ., Int. Res. Notices1 (1993) 29; Zbl0789.14005MR1201748
  21. A. Klemm and S. Theisen, Mirror Maps and Instanton Sums for Complete Intersections in Weighted Projective Space, preprint LMU-TPW 93-08 Zbl1020.32507MR1287096
  22. [15] P. Candelas, X. de la Ossa, A. Font, S. Katz and D. Morrison, Mirror Symmetry for Two Parameter Models I, preprint CERN-TH.6884/93 Zbl0899.14017
  23. [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. Zbl0814.53056MR1316509
  24. [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) Zbl1020.32508MR1319280
  25. [18] P. Aspinwall. B. Greene and D. Morrison, Space-Time Topology Change: the Physicsof Calabi-Yau Moduli Space, preprint IASSNS-HEP-93-81; Zbl0844.32016
  26. P. Aspinwall. B. Greene and D. MorrisonCalabi-Yau Moduli Space, Mirror Manifolds and Space-Time Topology Change in String Theory, preprint IASSNS-HEP-93-38: Zbl0899.32006MR1274435
  27. P. Aspinwall. B. Greene and D. MorrisonThe Monomial-Divisor Mirror Map, preprint IASSNS-HEP-93-43 Zbl0798.14030MR1253648
  28. [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: Zbl1009.32500
  29. P. Berglund, E. Derrich, T. Hübsch and D. Jancic, On Periods for String Compactification. preprint HUPAPP-93-6 Zbl0990.32501
  30. [20] P. Berglund and S. Katz, Mirror Symmetry for Hypersurfaces in Weighted Projective Space and Topological Couplings, preprint IASSNS-HEP-93-65 Zbl0990.32500MR1282649
  31. [21] P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B258 (1985) 46 MR800347
  32. [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 Scientific1986 
  33. 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, Springer1987 MR1034772
  34. P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons1978; Zbl0836.14001MR507725
  35. K. Kodaira, Complex Manifolds and Deformation of Complex Structures, Springer1986 Zbl0581.32012MR815922
  36. [23] A good and detailed introduction to Calabi-Yau manifolds is the book by T. Hübsch, Calabi-Yau Manifolds. World Scientific1991 
  37. [24] S. T. Yau, Proc. Natl. Acad. Sci.USA74 (1977), 1798; Zbl0355.32028
  38. S. T. YauComm. Pure Appl. Math.31 (1978) 339 Zbl0369.53059MR480350
  39. [25] K. Kodaira, L. Nirenberg and D. C. Spencer, Ann. Math.68 (1958) 450 Zbl0088.38004MR112157
  40. [25] K. Kodaira, L. Nirenberg and D. C. Spencer, Ann. Math.68 (1958) 450 Zbl0088.38004MR112157
  41. [26] G. Tian, in Mathematical Aspects of String Theory, ed. S. T. Yau, World Scientific, Singapore (1987) Zbl0651.00012MR915812
  42. [27] A. Strominger, Phys. Rev. Lett.55 (1985) 2547 
  43. A. Strominger in Unified String Theory, eds. M. Green and D. Gross, World Scientific1986; MR849129
  44. A. Strominger and E. Witten, Comm. Math. Phys.101 (1985) 341 MR815189
  45. [28] L. Dixon, D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B282 (1987) 13; MR869937
  46. S. Hamidi and C. Vafa, Nucl. Phys. B279 (1987) 465; MR867241
  47. J. Lauer, J. Mas and H. P. NillesNucl. Phys. B351 (1991) 353; MR1093574
  48. S. Stieberger, D. Jungnickel, J. Lauer and M. Spalinski, Mod. Phys. Lett. A7 (1992) 3859; 
  49. J. Erler, D. Jungnickel, M. Spalinski and S. Stieberger, Nucl. Phys. B397 (1993) 379 MR1219527
  50. [29] D. Gepner, Phys. Lett.199B (1987) 380 MR929596
  51. D. Gepner and Nucl. Phys. B311 (1988) 191 MR974275
  52. [30] J. Distler and B. Greene, Nucl. Phys. B309 (1988) 295 MR967477
  53. [31] M. Dine, N. Seiberg, X-G. Wen and E. Witten, Nucl. Phys. B278 (1987) 769, Nucl. Phys. B289 (1987) 319 MR895317
  54. [32] S. Ferrara, D. Lust, A. Shapere and S. Theisen, Phys. Lett.225B (1989) 363 MR1009091
  55. [33] P. Aspinwall and D. Morrison, Comm. Math. Phys.151 (1993) 245 Zbl0776.53043MR1204770
  56. [34] E. Witten, Comm. Math. Phys.118 (1988) 411 Zbl0674.58047MR958805
  57. [35] T. Banks, L. Dixon, D. Friedan and E. Martinec, Nucl. Phys. B284 (1988) 613 MR930976
  58. [36] M. Ademollo et al., Nucl. Phys. B11 (1976) 77 and Nucl. Phys. B114 (1976) 297 and Phys. Lett.62B (1976) 105 
  59. [37] A. Schwimmer and N. Seiberg, Phys. Lett.184B (1987) 191 MR879234
  60. [38] M. Dine and N. Seiberg, Nucl. Phys. B301 (1988) 357 MR936751
  61. [39] E. Witten, Nucl. Phys. B202 (1982) 253; 
  62. R. Rohm and E. Witten, Ann. Phys.17C (1986) 454 MR851628
  63. [40] P. Candelas, E. Derrick and L. Parkes, Nucl. Phys. B407 (1993) 115 MR1242064
  64. [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 93-47 MR1220210
  65. [42] N. Seiberg, Nucl. Phys. B303 (1988) 286 MR953719
  66. [43] S. Cecotti, S. Ferrara and L. Girardello, Int. J. Mod. Phys. A4 (1989) 2475; MR1017548
  67. S. Ferrara, Nucl. Phys. (Proc. Suppl.)11 (1989) 342 Zbl0957.81642MR1124909
  68. [44] P. Candelas, P. Green and T. Hübsch, Connected Calabi-Yau compactifications, in Strings '88, J. Gates et al. (eds), World Scientific1989 Zbl0938.81527
  69. P. Candelas, P. Green and T. HübschNucl. Phys. B330 (1990) 49; 
  70. T. Hübsch and R. Schimmrigk, Nucl. Phys. B329 (1990) 582 
  71. [45] L. Dixon, V. Kaplunovsky and J. Louis, Nucl. Phys. B329 (1990) 27 MR1036216
  72. [46] B. de Wit, P. G. Lauwers, R. Philippe, S. Q. Su and A. van Proeyen, Phys. Lett.134B (1984) 37; 
  73. B. de Wit and A. van Proeyen, Nucl. Phys. B245 (1984) 89; MR762191
  74. J. P. Derendinger, S. Ferrara, A. Masiero and A. van Proeyen, Phys. Lett.140B (1984) 307; MR746903
  75. B. de Wit, P. G. Lauwers and A. van Proeyen, Nucl. Phys. B255 (1985) 569; 
  76. E. Cremmer, C. Kounnas, A. van Proeyen, J. P. Derendinger, S. Ferrara, B. de Wit and L. Girardello, Nucl. Phys. B250 (1985) 385 
  77. [47] M. Kreuzer and H. Skarke, Nucl. Phys. B388 (1993) 113; 
  78. A. Klemm and R. Schimmrigk, Nucl. Phys. B411 (1994) 559 MR1257839
  79. [48] P. Candelas, A. Dale, A. Lütken and R. Schimmrigk, Nucl. Phys. B298 (1988) 493; 
  80. P. Candelas, A. Lütken and R. Schimmrigk, Nucl. Phys. B306 (1989) 105 
  81. [49] V. Batyrev and D. van Straten, Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties, preprint 1992 Zbl0843.14016MR1328251
  82. [50] I. Dolgachev, Weighted Projective Varieties in Lecture Notes in Mathematics956Springer-Verlag (1992) 36 Zbl0516.14014MR704986
  83. [51] A. R. Fletcher, Working with Weighted Complete Intersections, Max-Planck-Institut Series, No. 35 (1989) Bonn 
  84. [52] D. G. Markushevich, M. A. Olshanetsky and A. M. Perelomov, Comm. Math. Phys.111 (1987) 247 Zbl0628.53065MR899851
  85. [53] J. Erler and A. Klemm, Comm. Math. Phys.153 (1993) 57 MR1218933
  86. [54] L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B261 (1985) 678 MR818423
  87. L. Dixon, J. Harvey, C. Vafa and E. Witten and Nucl. Phys. B274 (1986) 285 MR851703
  88. [55] S. S. Roan and S. T. Yau, Acta Math. Sinica (NS)3 (1987) 256; Zbl0649.14024MR916270
  89. S. S. Roan, J. Diff. Geom.30 (1989) 523 Zbl0661.14031MR1010170
  90. [56] M. Kreuzer and H. Skarke. Nucl. Phys. B405 (1993) 305; 
  91. M. Kreuzer, Phys. Lett.314B (1993) 31; MR1237377
  92. A. Niemeyer, Diplom Thesis, TU-München, 1993 
  93. [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 Verlag1988; Zbl0628.52002MR922894
  94. V. I. Danilov, Russian Math. Surveys, 33 (1978) 97; 
  95. M. Audin, The Topology of Torus Actions on Symplectic Manifolds, Progress in Mathematics, Birkhäuser1991; Zbl0726.57029MR1106194
  96. W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, Princeton University Press1993; Zbl0813.14039MR1234037
  97. V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differntial Maps, Birkhäuser1985, Vol. II, Chapter 8 Zbl1297.32001MR777682
  98. [58] V. Batyrev, Duke Math. Journal69 (1993) 349 Zbl0812.14035MR1203231
  99. [59] V. Batyrev, Journal Alg. Geom., to be published 
  100. [60] V. Batyrev, Quantum Cohomology Rings of Toric Manifolds, preprint 1992 Zbl0806.14041MR1265307
  101. [61] V. Batyrev and D. Cox, On the Hodge Structure of Projective Hypersurfaces in Toric Varieties, preprint 1993 Zbl0851.14021MR1290195
  102. [62] F. Hirzebruch, Math. Annalen124 (1951) 77 Zbl0043.30302MR45384
  103. [63] M. Beltrametti and L. Robbiano, Expo. Math.4 (1986) 11. MR879909
  104. [64] D. Aspinwall, B. Greene and D. Morrison, Phys. Lett.303B (1993) 249 
  105. D. Aspinwall, B. Greene and D. Morrison and The Monomial-Divisor Mirror Map, preprint IASSNS-HEP-93/43 Zbl0798.14030
  106. [65] S.-S. Roan: Topological Couplings of Calabi-Yau Orbifolds, Max-Planck-Institut Serie No. 22 (1992), to appear in J. of Group Theory in Physics MR1191431
  107. [66] E. L. Ince, Ordinary Differential Equations, Dover1956 Zbl0063.02971MR10757JFM53.0399.07
  108. [67] M. Yoshida, Fuchsian Differential Equations, Braunschweig (1987) Vieweg Zbl0618.35001MR986252
  109. [68] P. A. Griffiths, Bull. Amer. Math. Soc.76 (1970) 228 Zbl0214.19802MR258824
  110. [69] P. Griffiths, Ann. of Math.90 (1969) 460 MR260733
  111. [70] P. Candelas, Nucl. Phys. B298 (1988) 458 MR928307
  112. [71] I. M. Gel'fand, A. V. Zelevinsky and M. M. Kapranov, Func. Anal. Appl.28 (1989) 12 
  113. I. M. Gel'fand, A. V. Zelevinsky and M. M. Kapranov and Adv. Math.84 (1990) 255 MR1080980
  114. [72] D. Morrison, Where is the Large Radius Limit, preprint IASSNS-HEP-93/68 and Compactifications of Moduli Spaces Inspired by Mirror Symmetry, preprint DUK-M- 93-06 
  115. [73] M. Reid, Decomposition of Toric Morphisms, in Progress in Mathematics36, M. Artin and J. Tate, eds, Birkhäuser1983 Zbl0571.14020MR717617
  116. [74] S. Ferrara and A. Strominger, in the Proceedings of the Texas A & M Strings '89 Workshop; ed. R. Arnowitt et al., World Scientific1990 Zbl1058.81629MR1148627
  117. [75] P. Candelas, P. Green and T. HübschNucl. Phys. B330 (1990) 49; 
  118. P. Candelas and X. C. de la Ossa, Nucl. Phys. B355 (1991) 455 
  119. [76] R. Bryant and P. Griffiths, Some observations on the Infinitesimal Period Relations for Regular Threefolds with Trivial Canonical Bundle, in Progress in Mathematics36, M. Artin and J. Tate, eds; Birkhäuser, 1983 Zbl0543.14005MR717607
  120. [77] A. Strominger, Comm. Math. Phys.133 (1990) 163 Zbl0716.53068MR1071240
  121. [78] L. Castellani, R. D'Auria and S. Ferrara, Phys. Lett.241B (1990) 57 and Class. Quantum Grav.7 (1990) 1767; 
  122. S. Ferrara and J. Louis, Phys. Lett.278B (1992) 240 MR1159065
  123. [79] A. Ceresole, R. D'Auria, S. Ferrara, W. Lerche and J. Louis, Int. J. Mod. Phys. A8 (1993) 79 Zbl0802.53055
  124. [80] X.-G. Wen and E. Witten, Phys. Lett.166B (1986) 397 MR827697

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