An Extended Lecture on Mirror Symmetry
Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Conférences de M. Audin, D. Bernard, A. Bilal, B. Enriquez, E. Frenkel, F. Golse, M. Katz, R. Lawrence, O. Mathieu, P. Von Moerbeke, V. Ovsienko, N. Reshetikhin, S. Theisen, Tome 47 (1995), Exposé no. 2, 44 p.
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Hosono, S.; Klemm, A.; Theisen, S. An Extended Lecture on Mirror Symmetry. Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Conférences de M. Audin, D. Bernard, A. Bilal, B. Enriquez, E. Frenkel, F. Golse, M. Katz, R. Lawrence, O. Mathieu, P. Von Moerbeke, V. Ovsienko, N. Reshetikhin, S. Theisen, Tome 47 (1995), Exposé no. 2, 44 p. http://archive.numdam.org/item/RCP25_1995__47__33_0/

[1] For introductions, see e.g. M. Green, J. Schwarz and E. Witten, Superstring Theory, Vols. I and II, Cambridge University Press, 1986; | Zbl

D. Lust and S. Theisen, Lectures on String Theory, Springer Lecture Notes in Physics, Vol. 346, 1990; | MR | Zbl

M. Kaku, Introduction to Superstrings, Springer 1988 | MR | Zbl

[2] E. Verlinde, H. Verlinde, Lectures on String Perturbation Theory, in Superstrings 88, Trieste Spring School, Ed. Greene et al. World Scientific (1989) 189 | MR

[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 | MR

[4] D. Nemeschanski and A. Sen, Phys. Lett. 178B (1986) 365

[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. | MR

[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 | MR

[7] W. Lerche, C. Vafa and N. Warner, Nucl. Phys. B324 (1989) 427 | MR

[8] B. Greene and M. Plesser, Nucl. Phys. B338 (1990) 15 | MR

[9] Essays on Mirror Manifolds (ed. S.-T. Yau), Int. Press, Hong Kong, 1992

[10] R. Dijkgraaf, E. Verlinde and H. Verlinde, Utrecht preprint THU-87/30; | MR

A. Shapere and F. Wilczek Nucl. Phys. B320 (1989) 669; | MR

A. Giveon, E. Rabinovici, G. Veneziano, Nucl. Phys. B322 (1989) 169

[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 1990

[12] P. Candelas, X. De La Ossa, P. Green and L. Parkes, Nucl. Phys. B359 (1991) 21

[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; | MR | Zbl

A. Klemm and S. Theisen, Nucl. Phys. B389 (1993) 153; | MR

A. Font, Nucl. Phys. B391 (1993) 358 | MR

[14] A. Libgober and J. Teitelbaum, Duke Math. Journ., Int. Res. Notices 1 (1993) 29; | MR | Zbl

A. Klemm and S. Theisen, Mirror Maps and Instanton Sums for Complete Intersections in Weighted Projective Space, preprint LMU-TPW 93-08 | MR | Zbl

[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 | Zbl

[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. | MR | Zbl

[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) | MR | Zbl

[18] P. Aspinwall. B. Greene and D. Morrison, Space-Time Topology Change: the Physicsof Calabi-Yau Moduli Space, preprint IASSNS-HEP-93-81; | Zbl

P. Aspinwall. B. Greene and D. Morrison Calabi-Yau Moduli Space, Mirror Manifolds and Space-Time Topology Change in String Theory, preprint IASSNS-HEP-93-38: | MR | Zbl

P. Aspinwall. B. Greene and D. Morrison The Monomial-Divisor Mirror Map, preprint IASSNS-HEP-93-43 | MR | Zbl

[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: | Zbl

P. Berglund, E. Derrich, T. Hübsch and D. Jancic, On Periods for String Compactification. preprint HUPAPP-93-6 | Zbl

[20] P. Berglund and S. Katz, Mirror Symmetry for Hypersurfaces in Weighted Projective Space and Topological Couplings, preprint IASSNS-HEP-93-65 | MR | Zbl

[21] P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B258 (1985) 46 | MR

[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

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 | MR

P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons 1978; | MR | Zbl

K. Kodaira, Complex Manifolds and Deformation of Complex Structures, Springer 1986 | MR | Zbl

[23] A good and detailed introduction to Calabi-Yau manifolds is the book by T. Hübsch, Calabi-Yau Manifolds. World Scientific 1991

[24] S. T. Yau, Proc. Natl. Acad. Sci. USA 74 (1977), 1798; | Zbl

S. T. Yau Comm. Pure Appl. Math. 31 (1978) 339 | MR | Zbl

[25] K. Kodaira, L. Nirenberg and D. C. Spencer, Ann. Math. 68 (1958) 450 | MR | Zbl

[25] K. Kodaira, L. Nirenberg and D. C. Spencer, Ann. Math. 68 (1958) 450 | MR | Zbl

[26] G. Tian, in Mathematical Aspects of String Theory, ed. S. T. Yau, World Scientific, Singapore (1987) | MR | Zbl

[27] A. Strominger, Phys. Rev. Lett. 55 (1985) 2547

A. Strominger in Unified String Theory, eds. M. Green and D. Gross, World Scientific 1986; | MR

A. Strominger and E. Witten, Comm. Math. Phys. 101 (1985) 341 | MR

[28] L. Dixon, D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B282 (1987) 13; | MR

S. Hamidi and C. Vafa, Nucl. Phys. B279 (1987) 465; | MR

J. Lauer, J. Mas and H. P. Nilles Nucl. Phys. B351 (1991) 353; | MR

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) 379 | MR

[29] D. Gepner, Phys. Lett. 199B (1987) 380 | MR

D. Gepner and Nucl. Phys. B311 (1988) 191 | MR

[30] J. Distler and B. Greene, Nucl. Phys. B309 (1988) 295 | MR

[31] M. Dine, N. Seiberg, X-G. Wen and E. Witten, Nucl. Phys. B278 (1987) 769, Nucl. Phys. B289 (1987) 319 | MR

[32] S. Ferrara, D. Lust, A. Shapere and S. Theisen, Phys. Lett. 225B (1989) 363 | MR

[33] P. Aspinwall and D. Morrison, Comm. Math. Phys. 151 (1993) 245 | MR | Zbl

[34] E. Witten, Comm. Math. Phys. 118 (1988) 411 | MR | Zbl

[35] T. Banks, L. Dixon, D. Friedan and E. Martinec, Nucl. Phys. B284 (1988) 613 | MR

[36] M. Ademollo et al., Nucl. Phys. B11 (1976) 77 and Nucl. Phys. B114 (1976) 297 and Phys. Lett. 62B (1976) 105

[37] A. Schwimmer and N. Seiberg, Phys. Lett. 184B (1987) 191 | MR

[38] M. Dine and N. Seiberg, Nucl. Phys. B301 (1988) 357 | MR

[39] E. Witten, Nucl. Phys. B202 (1982) 253;

R. Rohm and E. Witten, Ann. Phys. 17C (1986) 454 | MR

[40] P. Candelas, E. Derrick and L. Parkes, Nucl. Phys. B407 (1993) 115 | MR

[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 | MR

[42] N. Seiberg, Nucl. Phys. B303 (1988) 286 | MR

[43] S. Cecotti, S. Ferrara and L. Girardello, Int. J. Mod. Phys. A4 (1989) 2475; | MR

S. Ferrara, Nucl. Phys. (Proc. Suppl.) 11 (1989) 342 | MR | Zbl

[44] P. Candelas, P. Green and T. Hübsch, Connected Calabi-Yau compactifications, in Strings '88, J. Gates et al. (eds), World Scientific 1989 | Zbl

P. Candelas, P. Green and T. Hübsch Nucl. Phys. B330 (1990) 49;

T. Hübsch and R. Schimmrigk, Nucl. Phys. B329 (1990) 582

[45] L. Dixon, V. Kaplunovsky and J. Louis, Nucl. Phys. B329 (1990) 27 | MR

[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; | MR

J. P. Derendinger, S. Ferrara, A. Masiero and A. Van Proeyen, Phys. Lett. 140B (1984) 307; | MR

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) 385

[47] M. Kreuzer and H. Skarke, Nucl. Phys. B388 (1993) 113;

A. Klemm and R. Schimmrigk, Nucl. Phys. B411 (1994) 559 | MR

[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) 105

[49] V. Batyrev and D. Van Straten, Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties, preprint 1992 | MR | Zbl

[50] I. Dolgachev, Weighted Projective Varieties in Lecture Notes in Mathematics 956 Springer-Verlag (1992) 36 | MR | Zbl

[51] A. R. Fletcher, Working with Weighted Complete Intersections, Max-Planck-Institut Series, No. 35 (1989) Bonn

[52] D. G. Markushevich, M. A. Olshanetsky and A. M. Perelomov, Comm. Math. Phys. 111 (1987) 247 | MR | Zbl

[53] J. Erler and A. Klemm, Comm. Math. Phys. 153 (1993) 57 | MR

[54] L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B261 (1985) 678 | MR

L. Dixon, J. Harvey, C. Vafa and E. Witten and Nucl. Phys. B274 (1986) 285 | MR

[55] S. S. Roan and S. T. Yau, Acta Math. Sinica (NS) 3 (1987) 256; | MR | Zbl

S. S. Roan, J. Diff. Geom. 30 (1989) 523 | MR | Zbl

[56] M. Kreuzer and H. Skarke. Nucl. Phys. B405 (1993) 305;

M. Kreuzer, Phys. Lett. 314B (1993) 31; | MR

A. Niemeyer, Diplom Thesis, TU-München, 1993

[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; | MR | Zbl

V. I. Danilov, Russian Math. Surveys, 33 (1978) 97;

M. Audin, The Topology of Torus Actions on Symplectic Manifolds, Progress in Mathematics, Birkhäuser 1991; | MR | Zbl

W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, Princeton University Press 1993; | MR | Zbl

V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differntial Maps, Birkhäuser 1985, Vol. II, Chapter 8 | MR

[58] V. Batyrev, Duke Math. Journal 69 (1993) 349 | MR | Zbl

[59] V. Batyrev, Journal Alg. Geom., to be published

[60] V. Batyrev, Quantum Cohomology Rings of Toric Manifolds, preprint 1992 | MR | Zbl

[61] V. Batyrev and D. Cox, On the Hodge Structure of Projective Hypersurfaces in Toric Varieties, preprint 1993 | MR | Zbl

[62] F. Hirzebruch, Math. Annalen 124 (1951) 77 | MR | Zbl

[63] M. Beltrametti and L. Robbiano, Expo. Math. 4 (1986) 11. | MR

[64] D. Aspinwall, B. Greene and D. Morrison, Phys. Lett. 303B (1993) 249

D. Aspinwall, B. Greene and D. Morrison and The Monomial-Divisor Mirror Map, preprint IASSNS-HEP-93/43 | Zbl

[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 | MR

[66] E. L. Ince, Ordinary Differential Equations, Dover 1956 | JFM | MR | Zbl

[67] M. Yoshida, Fuchsian Differential Equations, Braunschweig (1987) Vieweg | MR | Zbl

[68] P. A. Griffiths, Bull. Amer. Math. Soc. 76 (1970) 228 | MR | Zbl

[69] P. Griffiths, Ann. of Math. 90 (1969) 460 | MR

[70] P. Candelas, Nucl. Phys. B298 (1988) 458 | MR

[71] I. M. Gel'Fand, A. V. Zelevinsky and M. M. Kapranov, Func. Anal. Appl. 28 (1989) 12

I. M. Gel'Fand, A. V. Zelevinsky and M. M. Kapranov and Adv. Math. 84 (1990) 255 | MR

[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

[73] M. Reid, Decomposition of Toric Morphisms, in Progress in Mathematics 36, M. Artin and J. Tate, eds, Birkhäuser 1983 | MR | Zbl

[74] S. Ferrara and A. Strominger, in the Proceedings of the Texas A & M Strings '89 Workshop; ed. R. Arnowitt et al., World Scientific 1990 | MR | Zbl

[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) 455

[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, 1983 | MR | Zbl

[77] A. Strominger, Comm. Math. Phys. 133 (1990) 163 | MR | Zbl

[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) 240 | MR

[79] A. Ceresole, R. D'Auria, S. Ferrara, W. Lerche and J. Louis, Int. J. Mod. Phys. A8 (1993) 79 | Zbl

[80] X.-G. Wen and E. Witten, Phys. Lett. 166B (1986) 397 | MR