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A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. (English) Zbl 1098.32506
Summary: We compute the prepotentials and the geometry of the moduli spaces for a Calabi-Yau manifold and its mirror. In this way we obtain all the sigma model corrections to the Yukawa couplings and moduli space metric for the original manifold. The moduli space is found to be subject to the action of a modular group which, among other operations, exchanges large and small values of the radius, though the action on the radius is not as simple as $R \to 1/R$. It is also shown that the quantum corrections to the coupling decompose into a sum over instanton contributions and moreover that this sum converges. In particular there are no `sub-instanton’ corrections. This sum over instantons points to a deep connection between the modular group and the rational curves of the Calabi-Yau manifold. The burden of the present work is that a mirror pair of Calabi-Yau manifolds is an exactly soluble superconformal theory, at least as far as the massless sector is concerned. Mirror pairs are also more general than exactly soluble models that have hitherto been discussed since we solve the theory for all points of the moduli space.

32G20Period matrices, variation of Hodge structure; degenerations
14J30Algebraic threefolds
32G05Deformations of complex structures
32G81Applications of deformations of analytic structures to physics
81T30String and superstring theories
81T40Two-dimensional field theories, conformal field theories, etc.
14J15Analytic moduli, classification (surfaces)
Full Text: DOI
[1] Candelas, P.; Lynker, M.; Schimmrigk, R.: Nucl. phys.. 341, 383 (1990)
[2] B.R. Greene and M.R. Plesser, (2,2) and (2,0) superconformal orbifolds, Harvard University report HUTP-89/B241; Duality in Calabi-Yau moduli space, Harvard University report HUTP-89/A043
[3] Aspinwall, P.; Lütken, A.; Ross, G. G.: Phys. lett.. 241, 373 (1990)
[4] Strominger, A.; Witten, E.: Commun. math. Phys.. 101, 341 (1985)
[5] Dine, M.; Seiberg, N.: Phys. rev. Lett.. 57, 2625 (1986)
[6] Dine, M.; Seiberg, N.; Wen, X. G.; Witten, E.: Nucl. phys.. 289, 319 (1987)
[7] Distler, J.; Greene, B. R.: Nucl. phys.. 309, 295 (1988)
[8] L. Dixon and D. Gepner, unpublished
[9] Lerche, W.; Vafa, C.; Warner, N. P.: Nucl. phys.. 324, 427 (1989)
[10] Candelas, P.: Nucl. phys.. 298, 458 (1988)
[11] Shapere, A.; Wilczek, F.: Nucl. phys.. 320, 669 (1989)
[12] Ferrara, S.; Lüst, D.; Theisen, S.: Phys. lett.. 242, 39 (1990)
[13] Kollar, J.: Bull. am. Math. soc.. 17, 211 (1987)
[14] Grisaru, M. T.; Van De Ven, A.; Zanon, D.: Nucl. phys.. 277, 409 (1986)
[15] M. Lynker and R. Schimmrigk, Landau-Ginzburg theories as orbifolds, University of Texas report UTTG-22-90; Santa Barbara Institute for Theoretical Physics report NSF-ITP-90-88
[16] P. Candelas, P.S. Green, M. Lynker and R. Schimmrigk, Calabi-Yau manifolds in weighted P4, University of Texas report, in preparation · Zbl 0962.14029
[17] Candelas, P.; Green, P.; Hübsch, T.: Nucl. phys.. 330, 49 (1990)
[18] Candelas, P.; De La Ossa, X. C.: Nucl. phys.. 342, 246 (1990)
[19] Bryant, R.; Griffiths, P.: Progress in mathematics 3. 36, 77 (1983)
[20] P. Candelas and X.C. de la Ossa, Moduli space of Calabi-Yau manifolds, University of Texas report UTTG-07-90 · Zbl 0732.53056
[21] A. Strominger, Special geometry, University of California at Santa Barbara report, UCSBTH-89-61
[22] Erdélyi, A.; Oberhettinger, F.; Magnus, W.; Tricomi, F. G.: Higher transcendental functions. (1953) · Zbl 0051.30303
[23] Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N.: Singularities of differentiable maps, vol. II. Monographs in mathematics 83 (1988)
[24] Slater, L. J.: Generalized hypergeometric functions. (1966) · Zbl 0135.28101
[25] Gepner, D.: Phys. lett.. 199, 380 (1987)
[26] Freeman, M. D.; Pope, C. N.; Sohnius, M. F.; Stelle, K. S.: Phys. lett.. 178, 199 (1986)
[27] Aspinwall, P.; Lütken, A.: Quantum algebraic geometry of superstring compactifications. Oxford university report (September 1990)
[28] Katz, S.: Compositio math.. 60, 151 (1986)
[29] Harris, J.: Duke math. J.. 46, 685 (1979)
[30] Clemens, H.: Some results on Abel-Jacobi mappings in topics in transcendental algebraic geometry. (1984) · Zbl 0575.14007
[31] Bender, C. M.; Orzag, S. A.: Advanced mathematics for scientists and engineers. (1978)
[32] Candelas, P.; Horowitz, G.; Strominger, A.; Witten, E.: Nucl. phys.. 258, 46 (1985)