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 Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(e) ISSN 0894-0347(p)

     

Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians

Author(s): David R. Morrison
Journal: J. Amer. Math. Soc. 6 (1993), 223-247.
MSC: Primary 14J30; Secondary 14D07, 14J15, 32G20, 32G81, 32J17
MathSciNet review: 1179538
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Abstract: We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new $ q$-expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the ``mirror symmetry'' phenomenon recently observed by string theorists.

References:

[1]
P. S. Aspinwall and C. A. Lütken, Geometry of mirror manifolds, Nuclear Phys. B 353 (1991), 427-461. MR 1098649 (92g:32055)

[2]
-, Quantum algebraic geometry of superstring compactifications, Nuclear Phys. B 355 (1991), 482-510. MR 1103075 (92c:32021)

[3]
P. S. Aspinwall, C. A. Lütken, and G. G. Ross, Construction and couplings of mirror manifolds, Phys. Lett. B 241 (1990), 373-380. MR 1055062 (91g:81107)

[4]
F. A. Bogomolov, Hamiltonian Kähler manifolds, Dokl. Akad. Nauk SSSR 243 (1978), no. 5, 1101-1104. MR 514769 (80c:32024)

[5]
P. Candelas, Yukawa couplings between $ (2\,,\,1)$-forms, Nuclear Phys. B 298 (1988), 458-492. MR 928307 (88m:53122)

[6]
P. Candelas, X. C. de la Ossa, P. S. Green, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Phys. Lett. B 258 (1991), 118-126; Nuclear Phys. B 359 (1991), 21-74. MR 1115626 (93b:32029)

[7]
P. Candelas, M. Lynker, and R. Schimmrigk, Calabi-Yau manifolds in weighted $ {\mathbb{P}_4}$, Nuclear Phys. B 341 (1990), 383-402. MR 1067295 (91m:14062)

[8]
J. Carlson, M. Green, P. Griffiths, and J. Harris, Infinitesimal variations of Hodge structure. I, Compositio Math. 50 (1983), 109-205. MR 720288 (86e:32026a)

[9]
S. Cecotti, $ N = 2$ supergravity, type $ IIB$ superstrings, and algebraic geometry, Comm. Math. Phys. 131 (1990), 517-536. MR 1065895 (91g:32023)

[10]
-, $ N = 2$ Landau-Ginzburg vs. Calabi-Yau $ \sigma $-models: Non-perturbative aspects, Internat. J. Modern Phys. A 6 (1991), 1749-1813. MR 1098365 (92h:32038)

[11]
J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), 308-339. MR 554399 (81j:20028)

[12]
R. Dijkgraaf, E. Verlinde, and H. Verlinde, On moduli spaces of conformal field theories with $ c \geq 1$, Perspectives in String Theory (P. Di Vecchia and J. L. Petersen, eds.), World Scientific, Singapore, New Jersey, and Hong Kong, 1988, pp. 117-137. MR 1029802 (91c:81131)

[13]
M. Dine, N. Seiberg, X.-G. Wen, and E. Witten, Nonperturbative effects on the string world sheet. II, Nuclear Phys. B 289 (1987), 319-363. MR 895317 (88k:81195)

[14]
J. Distler and B. Greene, Some exact results on the superpotential from Calabi-Yau compactifications, Nuclear Phys. B 309 (1988), 295-316. MR 967477 (90j:81185)

[15]
L. J. Dixon, Some world-sheet properties of superstring compactifications, on orbifolds and otherwise, Superstrings, Unified Theories, and Cosmology 1987 (G. Furlan et al., eds.), World Scientific, Singapore, New Jersey, and Hong Kong, 1988, pp. 67-126. MR 1104035 (91m:81199)

[16]
G. Ellingsrud and S. A. Strømme, The number of twisted cubic curves on the general quintic threefold, University of Bergen Report no. 63-7-2-1992.

[17]
R. Friedman and F. Scattone, Type $ III$ degenerations of $ K3$ surfaces, Invent. Math. 83 (1986), 1-39. MR 813580 (87k:14044)

[18]
D. Gepner, Exactly solvable string compactification on manifolds of $ SU(N)$ holonomy, Phys. Lett. B 199 (1987), 380-388. MR 929596 (89h:83035)

[19]
B. R. Greene and M. R. Plesser, Duality in Calabi-Yau moduli space, Nuclear Phys. B 338 (1990), 15-37. MR 1059831 (91h:32018)

[20]
B. R. Greene, C. Vafa, and N. P. Warner, Calabi-Yau manifolds and renormalization group flows, Nuclear Phys. B 324 (1989), 371-390. MR 1025421 (91c:14042)

[21]
P. Griffiths (ed.), Topics in transcendental algebraic geometry, Ann. of Math. Stud., vol. 106, Princeton University Press, Princeton, NJ, 1984. MR 756842 (86b:14004)

[22]
S. Katz, On the finiteness of rational curves on quintic threefolds, Compositio Math. 60 (1986), 151-162. MR 868135 (88a:14047)

[23]
G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Math., vol. 339, Springer-Verlag, Berlin, Heidelberg, and New York, 1973. MR 0335518 (49:299)

[24]
A. Landman, On the Picard-Lefschetz transformations, Trans. Amer. Math. Soc. 181 (1973), 89-126. MR 0344248 (49:8987)

[25]
W. Lerche, C. Vafa, and N. P. Warner, Chiral rings in $ N = 2$ superconformal theories, Nuclear Phys. B 324 (1989), 427-474. MR 1025424 (91d:81132)

[26]
D. G. Markushevich, Resolution of singularities (toric method), appendix to: D. G. Markushevich, M. A. Olshanetsky, and A. M. Perelomov, Description of a class of superstring compactifications related to semi-simple Lie algebras, Comm. Math. Phys. 111 (1987), 247-274. MR 899851 (89a:83061)

[27]
E. J. Martinec, Algebraic geometry and effective Lagrangians, Phys. Lett. B 217 (1989), 431-437. MR 981536 (90c:81129)

[28]
-, Criticality, catastrophes, and compactifications, Physics and Mathematics of Strings (L. Brink, D. Friedan, and A. M. Polyakov, eds.), World Scientific, Singapore, New Jersey, London, and Hong Kong, 1990, pp. 389-433. MR 1104265 (93d:32058)

[29]
T. Oda, Convex bodies and algebraic geometry, Ergeb. Math. Grenzgeb. (3), Bd. 15, Springer-Verlag, Berlin, Heidelberg, and New York, 1988. MR 922894 (88m:14038)

[30]
M. Reid, Minimal models of canonical $ 3$-folds, Algebraic Varieties and Analytic Varieties (S. Iitaka, ed.), Kinokuniya, Tokyo, and North-Holland, Amsterdam, 1983, pp. 131-180. MR 715649 (86a:14010)

[31]
-, The moduli space of threefolds with $ K = 0$ may nevertheless be irreducible, Math. Ann. 278 (1987), 329-334. MR 909231 (88h:32016)

[32]
S.-S. Roan, On the generalization of Kummer surfaces, J. Differential Geom. 30 (1989), 523-537. MR 1010170 (90j:32032)

[33]
-, On Calabi-Yau orbifolds in weighted projective spaces, Internat. J. Math. 1 (1990), 211-232. MR 1060636 (91m:14064)

[34]
-, The mirror of Calabi-Yau orbifold, Max-Planck-Institut preprint MPI/91-1, Internat. J. Math. 2 (1991), 439-455. MR 1113571 (92f:14037)

[35]
W. Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211-319. MR 0382272 (52:3157)

[36]
C. Schoen, On the geometry of a special determinental hypersurface associated to the Mumford-Horrocks vector bundle, J. Reine Angew. Math. 364 (1986), 85-111. MR 817640 (87e:14039)

[37]
A. Strominger and E. Witten, New manifolds for superstring compactification, Comm. Math. Phys. 101 (1985), 341-361. MR 815189 (87c:81153)

[38]
J. G. Thompson, Some numerology between the Fischer-Griess monster and the elliptic modular function, Bull. London Math. Soc. 11 (1979), 352-353. MR 554402 (81j:20030)

[39]
G. Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric, Mathematical Aspects of String Theory (S. T. Yau, ed.), World Scientific, Singapore, 1987, pp. 629-646. MR 915841

[40]
A. N. Todorov, The Weil-Petersson geometry of the moduli space of $ SU(n \geq 3)$ (Calabi-Yau) manifolds. I, Comm. Math. Phys. 126 (1989), 325-246. MR 1027500 (91f:32022)

[41]
S. T. Yau, On Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1798-1799.-60pt MR 0451180 (56:9467)

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Additional Information:

DOI: 10.1090/S0894-0347-1993-1179538-2
PII: S0894-0347-1993-1179538-2
Copyright of article: Copyright 1993, American Mathematical Society




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