Skip to main content
SpringerLink
Log in

Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We formulate general conjectures about the relationship between the A-model connection on the cohomology of ad-dimensional Calabi-Yau complete intersectionV ofr hypersurfacesV 1 ,...,V r in a toric varietyP Σ and the system of differential operators annihilating the special generalized hypergeometric series Φ0 constructed from the fan Σ. Using this generalized hypergeometric series, we propose conjectural mirrorsV′ ofV and the canonicalq-coordinates on the moduli spaces of Calabi-Yau manifolds.

In the second part of the paper we consider some examples of Calabi-Yau 3-folds having Picard number >1 in products of projective spaces. For conjectural mirrors, using the recurrent relation among coefficients of the restriction of the hypergeometric function Φ0 on a special line in the moduli space, we determine the Picard-Fuchs equation satisfied by periods of this special one-parameter subfamily. This allows to obtain some sequences of integers which can be conjecturally interpreted in terms of Gromov-Witten invariants. Using standard techniques from enumerative geometry, first terms of these sequence of integers are checked to coincide with numbers of rational curves on Calabi-Yau 3-folds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Batyrev, V.V.: Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties. Nov. 18 (1992) Preprint, Uni-Essen

  2. Batyrev, V.V.: Variations of the Mixed Hodge Structure of Affine Hypersurfaces in Algebraic Tori. Duke Math. J.69, 349–409 (1993)

    Google Scholar 

  3. Batyrev, V.V.: Quantum Cohomology Rings of Toric Manifolds. Preprint MSRI, (1993)

  4. Borisov, L.A.: Towards the Mirror Symmetry for Calabi-Yau Complete Intersection in Gorenstein Toric Fano Varieties. Preprint 1993, alg-geom 9310001

  5. Candelas, P., Dale, A.M., Lütken, C.A., Schimmrigk, R.: Complete Intersection Calabi-Yau Manifolds I, II. Nucl. Phys.B 298, 493–525 (1988), Nucl. Phys.B 306, 113–136 (1988)

    Google Scholar 

  6. Candelas, P., Lynker, M., Schimmrigk, R.: Calabi-Yau Manifolds in WeightedP 4. Nucl. Phys.B 341, 383–402 (1990)

    Google Scholar 

  7. Candelas, P., He, A.: On the Number of Complete Intersection of Calabi-Yau Manifolds. Commun. Math. Phys.135, 193–199 (1990)

    Google Scholar 

  8. Candelas, P., de la Ossa, X.: Nucl. Phys.342, 246 (1990)

    Google Scholar 

  9. Candelas, P., de la Ossa, X.C., Green, P.S., Parkes, L.: A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys.B 359, 21–74 (1991)

    Google Scholar 

  10. Ceresole, A., D'Auria, R., Ferrara, S., Lerche, W., Louis, J.: Picard-Fuchs Equations and Special Geometry. CERN-TH.6441/92, UCLA/92/TEP/8, CALT-8-1776, POLFIS-TH.8/92

  11. Deligne, P.: Letter to D. Morrison, 6 November 1991

  12. Dixon, L.: In Superstrings. Unified Theories and Cosmology 1987, G. Furlan et al., eds., Singapore: World Scientific, 1988

    Google Scholar 

  13. Ferrara, S., Louis, J.: Phys. Lett.B 273, 246 (1990)

    Google Scholar 

  14. Font, A.: Periods and duality symmetries in Calabi-Yau compactifications. Nucl. Phys.B 389, 153 (1993)

    Google Scholar 

  15. Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Hypergeometric functions and toric manifolds. Funct. Anal. Appl.28:2, 94–106 (1989)

    Google Scholar 

  16. Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants of polynomials in several variables and triangulations of Newton polytopes. Algebra i analiz (Leningrad Math. J.)2, 1–62 (1990)

    Google Scholar 

  17. Greene, B.R., Plesser, M.R.: Duality in Calabi-Yau moduli space. Nucl. Phys.B 338, 15–37 (1990)

    Google Scholar 

  18. Greene, B.R., Morrison, D.R., Plesser, M.R.: Mirror manifolds in higher dimension. In preparation

  19. Katz, S.: Rational curves on Calabi-Yau 3-folds. In: Essays on Mirror Manifolds (Ed. S.-T. Yau), Hong Kong: Int. Press Co., 1992 pp. 168–180

    Google Scholar 

  20. Katz, S.: Rational curves on Calabi-Yau manifolds: Verifying of predictions of Mirror Symmetry. Preprint, alg-geom/9301006 (1993)

  21. Klemm, A., Theisen, S.: Consideration of One Modulus Calabi-Yau Compactification: Picard-Fuchs Equation, Kähler Potentials and Mirror Maps. Nucl. Phys.B 389, 753 (1993)

    Google Scholar 

  22. Klemm, A., Theisen, S.: Mirror Maps and Instanton Sums for Intersections in Weighed Projective Space. LMU-TPW 93-08, Preprint April 1993

  23. Klemm, A., Schimmrigk, R.: Landau-Ginzburg String Vacua. CERN-TH-6459

  24. Kim, J.K., Park, C.J., Yoon, Y.: Calabi-Yau manifolds from complete intersections in products of weighted projective spaces. Phys. Lett.B 224, 108–114 (1989)

    Google Scholar 

  25. Kreuzer, M., Schimmrigk, R., Skarke, H.: Abelian Landau-Ginzburg Orbifolds and Mirror Symmetry. Nucl. Hys.B 372, 61–86 (1992)

    Google Scholar 

  26. Kreuzer, M., Skarke, H.: No Mirror Symmetry in Landau-Ginzburg Spectra!. CERN-TH 6461/92, to appear in Nucl. Phys. B

  27. Lerche, W., Vafa, C., Warner, N.: Nucl. Phys.B 324, 427 (1989)

    Google Scholar 

  28. Libgober, A., Teitelbaum, J.: Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard-Fuchs equations. Duke Math. J., Int. Math. Res Notices129, (1993)

  29. Lynker, M., Schimmrigk, R.: Phys. Lett.249, 237 (1990)

    Google Scholar 

  30. Manin, Yu.I.: Private communication

  31. Mathai, A.M., Saxens, R.K.: Generalized Hypergeometric Functions with Applications in Statistic and Physical Sciences. Lect. Notes Math348, (1973)

  32. Morrison, D.: Mirror symmetry and rational curves on quintic threefolds: A guide for mathematicians. J. Arn. Math. Soc.6, 223–247 (1993)

    Google Scholar 

  33. Morrison, D.: Picard-Fuchs equations and mirror maps for hypersurfaces. In: Essay on Mirror Manifolds. Ed. Yau, S.-T., Hong Kong: Int. Press. Co., 1992, pp. 241–264

    Google Scholar 

  34. Morrison, D.: Compactifications of moduli spaces inspired by mirror symmetry. Preprint (1993)

  35. Morrison, D.: Hodge-theoretic aspects of mirror symmetry. In preparation

  36. Roan, S.-S.: The mirror of Calabi-Yau Orbifold. Internat. J. Math.2, 439–455 (1991)

    Google Scholar 

  37. Roan, S.-S.: Topological Properties of Calabi-Yau Mirror Manifolds. Preprint (1992)

  38. Schimmrigk, R.: A new construction of a three-generation Calabi-Yau manifold. Phys. Lett.B 193, 175–180 (1987)

    Google Scholar 

  39. Slater, L.J.: Generalized Hypergeometric Functions. Cambridge: Cambridge University Press, XIII, 1966

    Google Scholar 

  40. Stienstra, J., Beukers, F.: On the Picard-Fuchs Equation and the Formal Brauer Group of Cer tain Elliptic K3-surfaces. Math. Ann.271, 269–304 (1985)

    Google Scholar 

  41. Witten, E.: Topological sigma models. Commun. Math. Phys.118, 411–449 (1988)

    Google Scholar 

  42. Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surveys in Diff. Geom.1, 243–310 (1991)

    Google Scholar 

  43. Witten, E.: Mirror Manifolds and Topological Field Theory. In: Essays on Mirror Manifolds, Ed. S.-T. Yau, Hong Kong: Int. Press Co., 1992, pp. 120–180

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. FB Mathematik, Universität-GHS-Essen, Universitätsstraße 3, D-45141, Essen, Germany

    Victor V. Batyrev

  2. FB Mathematik, Universität Kaiserslautern, Erwin-Schrödingerstraße 48, D-67663, Kaiserslautern, Germany

    Duco van Straten

Authors
  1. Victor V. Batyrev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Duco van Straten
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by S.-T. Yau

Rights and permissions

Reprints and permissions

About this article

Cite this article

Batyrev, V.V., van Straten, D. Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties. Commun.Math. Phys. 168, 493–533 (1995). https://doi.org/10.1007/BF02101841

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02101841

Keywords

Search

Navigation