×

zbMATH — the first resource for mathematics

GKZ-generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces. (English) Zbl 0870.14028
It is known that the Picard-Fuchs equations for the periods of Calabi-Yau hypersurfaces in a toric Fano variety \({\mathbb{P}}_{\Delta}\) coincide with a generalized hypergeometric system introduced by Gelfand-Kapranov-Zelevinski [see V. V. Batyrev and D. van Straaten, Commun. Math. Phys. 168, No. 3, 493-533 (1995; Zbl 0843.14016)]. Similarly, the periods of the mirror family in \({\mathbb{P}}_{\Delta^*}\) satisfy a GKZ-hypergeometric system. Although this system is usually reducible, its irreducible part can be extracted by introducing an extended system which incorporates additional differential operators corresponding to the automorphism group of \({\mathbb{P}}_{\Delta^*}\). The authors show that the local properties of this system are determined by an algebro-combinatorial object, known as a toric ideal. Its Gröbner basis determines a finite set of differential operators for the local solutions of the GKZ-system. In the case of 3-dimensional varieties with \(h^{11}\leq 3\) they give some more explicit analysis of local solutions as well as some examples.

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14J70 Hypersurfaces and algebraic geometry
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Candelas, P., de la Ossa, X., Green, P., Parks, L.: Nucl. Phys. B359, 21 (1991) · Zbl 1098.32506 · doi:10.1016/0550-3213(91)90292-6
[2] Morrison, D.: Picard-Fuchs Equations and Mirror Maps For Hypersurfaces. In: Essays on Mirror Manifolds. ed. S.-T.Yau, Hong Kong International Press, 1992 · Zbl 0841.32013
[3] Klemm, A., Theisen, S.: Nucl. Phys. B389, 153 (1993) · doi:10.1016/0550-3213(93)90289-2
[4] Libgober, A., Teitelboim, J.: Duke Math. J., Int. Res. Notices1, 29 (1993) · Zbl 0789.14005 · doi:10.1155/S1073792893000030
[5] Font, A.: Nucl. Phys. B391, 358 (1993) · Zbl 1360.32009 · doi:10.1016/0550-3213(93)90152-F
[6] Klemm, A., Theisen, S.: Theor. Math. Phys.95, 583 (1993) · Zbl 0863.53053 · doi:10.1007/BF01017144
[7] Candelas, P., de la Ossa, X., Font, A., Katz, S., Morrison, D.: Nucl. Phys. B416, 481 (1994) · Zbl 1007.32502 · doi:10.1016/0550-3213(94)90322-0
[8] Hosono, S., Klemm, A., Theisen, S., Yau, S.-T.: Commun. Math. Phys.167, 301 (1995) · Zbl 0814.53056 · doi:10.1007/BF02100589
[9] Candelas, P., de la Ossa, X., Font, A., Katz, S., Morrison, D.: Nucl. Phys. B429, 629 (1994) · Zbl 1020.32506 · doi:10.1016/0550-3213(94)90155-4
[10] Hosono, S., Klemm, A., Theisen, S., Yau, S.-T.: Nucl. Phys. B433, 501 (1995) · Zbl 1020.32508 · doi:10.1016/0550-3213(94)00440-P
[11] Berglund, P., Katz, S., Klemm, A.: Mirror Symmetry and the Moduli Space for Generic Hypersurfaces in Toric Varieties. hep-th/9506091 · Zbl 0899.32007
[12] Kontsevich, M., Manin, Yu.: Commun. Math. Phys.164, 525 (1994) · Zbl 0853.14020 · doi:10.1007/BF02101490
[13] Lian, B.H., Yau, S.-T.: Arithmetic properties of the mirror map and quantum coupling. hep-th/9411234
[14] Klemm, A., Lerche, W., Myer, P.: K3-Fibrations and Heterotic-Type II String Duality. hep-th/9506112
[15] Lian, B.H., Yau, S.-T.: Mirror Maps, Modular Relations and Hypergeometric Series I,II. hep-th/9507151, 9507153
[16] Kachru, S., Vafa, C.: Exact Results forN=2 Compactifications of Heterotic Strings. hep-th/9505105 · Zbl 0957.14509
[17] Kachru, S., Klemm, A., Mayr, P., Lerche, W., Vafa, C.: Nonperturbative Results on the Point Particle Limit ofN=2 Heterotic String Compactification. hep-th/9508155 · Zbl 1003.81524
[18] Lerche, W., Vafa, C., Warner, N.: Nucl. Phys. B329, 163 (1990)
[19] Batyrev, V.: J., Algebraic Geometry3, 493 (1994)
[20] Roan, S.-S.: Int. J. Math.2, 439 (1991) · Zbl 0817.14018 · doi:10.1142/S0129167X91000259
[21] Distler, J., Greene, B.: Nucl. Phys. B309, 295 (1988) · doi:10.1016/0550-3213(88)90084-3
[22] Batyrev, V.: Duke Math. J.69, 349 (1993) · Zbl 0812.14035 · doi:10.1215/S0012-7094-93-06917-7
[23] Gel’fand, I.M., Zelevinski, A.V., Kapranov, M.M.: Funct. Anal. Appl.28, 94 (1989) · Zbl 0721.33006 · doi:10.1007/BF01078777
[24] Hosono, S., Lian, B.H., Yau, S.-T.: Optional appendix to alg-geom/9511001
[25] Klemm, A., Schimmrigk, R.: Nucl. Phys. B411, 559 (1994) · Zbl 1049.81601 · doi:10.1016/0550-3213(94)90462-6
[26] Kreuzer, M., Skarke, H.: Nucl. Phys. B388, 113 (1993) · doi:10.1016/0550-3213(92)90547-O
[27] Candelas, P., Dale, A.M., Lütken, C.A., Schimmrigk, R.: Nucl. Phys. B298, 493 (1988) Candelas, P., Lütken, C.A., Schimmrigk, R.: Nucl. Phys.B306, 113 (1988) · doi:10.1016/0550-3213(88)90352-5
[28] Batyrev, V., van Straten, D.: Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties. Alg-geom/9307010 · Zbl 0843.14016
[29] Batyrev, V., Borisov, L.: On Calabi-Yau Complete Intersections in Toric Varieties. alggeom/9412017 · Zbl 0908.14015
[30] Dais, D.I.: Enumerative Combinatorics of Invariants of Certain Complex Threefolds with Trivial Canonical Bundle. MPI-preprint (1994)
[31] Dolgachev, I.: In: Group actions and vector fields. Lecture Notes in Math.Vol.956, Berlin-Heidelberg-New York: Springer-Verlag (1991) P. 34 · Zbl 0766.14031
[32] Greene, B., Plesser, M.: Nucl. Phys. B338, 15 (1990) · doi:10.1016/0550-3213(90)90622-K
[33] Candelas, P., de la Ossa, X., Katz, S.: Nucl. Phys. B450, 267 (1995) · Zbl 0896.14023 · doi:10.1016/0550-3213(95)00189-Y
[34] Berglund, P., Hübsch, T.: Nucl. Phys. B393, 377 (1993) · Zbl 1245.14039 · doi:10.1016/0550-3213(93)90250-S
[35] Oda, T.: Convex bodies and Algebraic Geometry, An Introduction to the Theory of Toric Varieties. A Series of Modern Surveys in Mathematics, Berlin-Heidelberg-New York: Springer-Verlag, 1985
[36] Billera, L., Filliman, P., Strumfels, B.: Adv. in Math.83, 155–179 (1990) · Zbl 0714.52004 · doi:10.1016/0001-8708(90)90077-Z
[37] Oda, T., Park, H.S.: Tôhoku Math. J.43, 375–399 (1991) · Zbl 0782.52006 · doi:10.2748/tmj/1178227461
[38] Fulton, W.: Introduction to Toric Varieties. Ann. of Math. Studies131, Princeton, NJ: Princeton University Press, 1993 · Zbl 0813.14039
[39] See for example, Cox, D., Little, J., O’shea, D.: Ideals, Varieties, and Algorithms. UTM Berlin-Heidelberg-New York: Springer-Verlag, 1991
[40] Strumfels, B.: Tôhoku Math. J.43, 249 (1991) · Zbl 0725.14041 · doi:10.2748/tmj/1178227496
[41] Griffiths, P.: Ann. of Math. (2)80, 227 (1964) · Zbl 0119.38501 · doi:10.2307/1970392
[42] Batyrev, V.: Quantum Cohomology Ring of Toric Manifolds. alg-geom/9310004 · Zbl 0806.14041
[43] Aspinwall, P., Greene, B., Morrison, D.: Phys. Lett. B303, 249 (1993)
[44] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York: Wiley-Interscience 1978 · Zbl 0408.14001
[45] Witten, E.: Nucl. Phys. B403, 159 (1993) · Zbl 0910.14020 · doi:10.1016/0550-3213(93)90033-L
[46] Morrison, D., Plesser.: Nucl. Phys. B404, 279 (1995) · Zbl 0908.14014 · doi:10.1016/0550-3213(95)00061-V
[47] Aspinwall, P., Morrison, D.: Commun. Math. Phys.151, 245 (1993) · Zbl 0776.53043 · doi:10.1007/BF02096768
[48] Witten, E.: Mirror Manifolds and Topological Field Theory. In: ”Essays on Mirror Symmetry”, ed. S.-T. Yau, Hong Kong: International Press, 1992 · Zbl 0834.58013
[49] Strominger, A.: Commun. Math. Phys.133, 163 (1990) · Zbl 0716.53068 · doi:10.1007/BF02096559
[50] Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: (with an appendix by S.Katz), Nucl. Phys. B405, 279 (1993) · Zbl 0908.58074 · doi:10.1016/0550-3213(93)90548-4
[51] Strominger, A.: Massless Black Holes and Conifolds in String Theory. hep-th/9504090 · Zbl 0925.83071
[52] Greene, B., Morrison, D., Strominger, A.: Black Hole Condensation and the Unification of String Vacua. hep-th/9504145 · Zbl 0908.53041
[53] Wall, C.T.: Invent. Math.1, 355 (1966) · Zbl 0149.20601 · doi:10.1007/BF01389738
[54] Lynker, M., Schimmrigk, R.: Phys. Lett. B249, 237 (1990)
[55] Yomemura, T.: Tôhoku Math. J.42, 351 (1990) · Zbl 0733.14017 · doi:10.2748/tmj/1178227616
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.