×

Mirror symmetry for two-parameter models. II. (English) Zbl 0899.14018

Summary: [For part I see P. Candelas, X. de la Ossa, A. Font, S. Katz and D. R. Morrison, ibid. 416, No. 2, 481-538 (1993; see the preceding review).]
We describe in detail the space of the two Kähler parameters of the Calabi-Yau manifold \(\mathbb{P}{}_{4}^{(1,1,1,6,9)}\) [D. R. Morrison, in: Journeés de Géométrie algébrique, Orsay 1992, Astérisque 218, 243-271 (1993; Zbl 0824.14007)] by exploiting mirror symmetry. The large complex structure limit of the mirror, which corresponds to the classical large radius limit, is found by studying the monodromy of the periods about the discriminant locus, the boundary of the moduli space corresponding to singular Calabi-Yau manifolds. A symplectic basis of periods is found and the action of the \(\text{Sp}(6,\mathbb{Z})\) generators of the modular group is determined. From the mirror map we compute the instanton expansion of the Yukawa couplings and the generalized N=2 index, arriving at the numbers of instantons of genus zero and genus one of each bidegree. We find that these numbers can be negative, even in genus zero. We also investigate an \(\text{SL}(2,\mathbb{Z})\) symmetry that acts on a boundary of the moduli space.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
14D20 Algebraic moduli problems, moduli of vector bundles
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Candelas, P.; De La Ossa, X.; Font, A.; Katz, S.; Morrison, D. R.: Nucl. phys. B. 416, 481 (1994)
[2] Candelas, P.; De La Ossa, X.; Green, P.; Parkes, L.: Nucl. phys. B. 359, 21 (1991)
[3] Morrison, D. R.: Picard-Fuchs equations and mirror maps for hypersurfaces. Essays on mirror symmetry (1992) · Zbl 0841.32013
[4] Font, A.: Nucl. phys. B. 391, 358 (1993)
[5] Klemm, A.; Theisen, S.: Nucl. phys. B. 389, 153 (1993)
[6] Mirror maps and instanton sums for complete intersections in weighted projective space, preprint LMU-TPW 93-08 [hep-th/9304034].
[7] Libgober, A.; Teitelbaum, J.: Intern. math. Res. notices. 15 (1993)
[8] V. Batyrev and D. van Straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, Essen preprint [alg-geom/9307010]. · Zbl 0843.14016
[9] A. Ceresole, R. D’Auria and T. Regge, Duality group for Calabi-Yau 2-moduli space, preprint DFTT-34-93 [hep-th/9307151].
[10] Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C.: With an appendix by S. Katz nucl. Phys. B. Nucl. phys. B 405, 279 (1993)
[11] S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, preprint HUTMP-93/0801 [hep-th/9308122]. · Zbl 0814.53056
[12] Greene, B. R.; Plesser, M. R.: Nucl. phys. B. 338, 15 (1990)
[13] Aspinwall, P. S.; Greene, B. R.; Morrison, D. R.: Intern. math. Res. notices. 319 (1993)
[14] Oda, T.; Park, H. S.: To\check{}hoku math. J.. 43, 375 (1991)
[15] Billera, L. J.; Filliman, P.; Sturmfels, B.: Adv. math.. 83, 155 (1990)
[16] Aspinwall, P. S.; Greene, B. R.; Morrison, D. R.: Nucl. phys. B. 416, 414 (1994)
[17] P. Berglund, P. Candelas, X. de la Ossa, A. Font, T. Hübsch, D. Jančić and F. Quevedo, Periods for Calabi-Yau and Landau-Ginzburg vacua, preprint CERN TH. 6865/93, HUPAPP-93/3, NEIP 93-004, NSF-ITP-93-96, UTTG-13-93 [hepth-9308005].
[18] Dimca, A.: Singularities and topology of hypersurfaces. (1992) · Zbl 0753.57001
[19] D.R. Morrison, Compactifications of moduli spaces inspired by mirror symmetry, preprint DUK-M-93-06, [alg-geom/9304007].
[20] Cadavid, A. C.; Ferrara, S.: Phys. lett. B. 267, 193 (1991)
[21] Blok, B.; Varchenko, A.: Intern. J. Mod. phys. A. 7, 1467 (1992)
[22] Lerche, W.; Smith, D. J.; Warner, N. P.: Nucl. phys. B. 372, 87 (1992)
[23] Ceresole, A.; D’auria, R.; Ferrara, S.; Lerche, W.; Louis, J.: Intern. J. Mod. phys. A. 8, 79 (1993)
[24] Candelas, P.: Nucl. phys. B. 298, 458 (1988)
[25] Aspinwall, P. S.; Morrison, D. R.: Commun. math. Phys.. 151, 245 (1993)
[26] Erdélyi, A.; Oberhettinger, F.; Magnus, W.; Tricomi, F. G.: Higher transcendental functions. (1953) · Zbl 0051.30303
[27] Mcduff, D.: Invent. math.. 89, 13 (1987)
[28] Katz, S.: Rational curves on Calabi-Yau threefolds. Essays on mirror manifolds (1992) · Zbl 0835.14015
[29] Kuranishi, M.: Ann. math.. 75 (1962)
[30] Y. Ruan, Topological sigma model and Donaldson type invariants in Gromov theory, preprint. · Zbl 0864.53032
[31] Sommervoll, D. E.: Rational curves of low degree on a complete intersection Calabi-Yau threefold in P3 x P3. (June 1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.