A Mirror Theorem for Toric Complete Intersections

  • Alexander Givental
Part of the Progress in Mathematics book series (PM, volume 160)


We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable hypergeometric functions.


Modulus Space Vector Bundle Line Bundle Complete Intersection Toric Variety 
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  1. [1]
    M. Audin, The topology of torus actions on symplectic manifolds. Birkhäuser, Basel, 1991.Google Scholar
  2. [2]
    V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Alg. Geom. 3 (1994), 493–535.MathSciNetzbMATHGoogle Scholar
  3. [3]
    V. Batyrev, Quantum cohomology rings of toric manifolds. Preprint, alggeom/9310004.Google Scholar
  4. [4]
    V. Batyrev, D. van Straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties. Comm. Math. Phys. 168 (1995), 493–533.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    K. Behrend, Yu. Manin, Stacks of stable maps and Gromov-Witten invariants. Duke Math. J. 85 (1996), 1–60.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    T. Eguchi, K. Hori, C.-S. Xiong, Gravitational quantum cohomology. Preprint, 1996.Google Scholar
  7. [7]
    A. Givental, A symplectic fixed point theorem for toric manifolds. Preprint UC Berkeley, 1992. Published in: The Floer Memorial Volume. H. Hofer, C. H. Taubs, A. Weinstein, E. Zehnder (eds.), Progress in Math. 133, Birkhäuser, 1995, 445–4Google Scholar
  8. [8]
    A. Givental, Homological geometry I: projective hypersurfaces. Selecta Math., New series, v. 1, No. 2, 325–345.Google Scholar
  9. [9]
    A. Givental, Homological geometry and mirror symmetry. In: Proceedings of the ICM, Zürich 1994, Birkhäuser, 1995, v. 1, 472–480.Google Scholar
  10. [10]
    A. Givental, Equivariant Gromov-Witten invariants. IMRN, 1996, No. 13, 613–663.Google Scholar
  11. [11]
    A. Givental, Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture. Preprint, alg-geom/9612001.Google Scholar
  12. [12]
    M. Kontsevich, Enumeration of rational curves via toric actions. In: The moduli spaces of curves, Ft. Dijkgraaf, C. Faber, G. van der Geer (eds.), Progress in Math. 129, Birkhäuser, 1995, 335–368.Google Scholar
  13. [13]
    M. Kontsevich, Yu. Manin, Gromov-Witten classes, quantum cohomology and enumerative geometry. Comm. Math. Phys. 164 (1994), 525–562.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    J. Li, G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. Preprint, alg-geom/9602007.Google Scholar
  15. [15]
    J. Li, G. Tian, Virtual moduli cycles and Gromov-Witten inv ariants in general symplectic manifolds. Preprint, alg-geom/9608032.Google Scholar
  16. [16]
    D. Morrison, R. Plesser, Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties. Nucl. Phys. B440 (1995), 279–354 (hep-th/9412236).MathSciNetCrossRefGoogle Scholar
  17. [17]
    A. Schwarz, Sigma-models having supermanifolds as target spaces. Preprint, IHES, 1995.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Alexander Givental
    • 1
  1. 1.University of California at BerkeleyBerkeleyUSA

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