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Disk enumeration on the quintic 3-fold


Authors: R. Pandharipande, J. Solomon and J. Walcher
Journal: J. Amer. Math. Soc. 21 (2008), 1169-1209
MSC (2000): Primary 53D45, 14N35; Secondary 14J32
Published electronically: February 12, 2008
MathSciNet review: 2425184
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Abstract: Holomorphic disk invariants with boundary in the real Lagrangian of a quintic 3-fold are calculated by localization and proven mirror transforms. A careful discussion of the underlying virtual intersection theory is included. The generating function for the disk invariants is shown to satisfy an extension of the Picard-Fuchs differential equations associated to the mirror quintic. The Ooguri-Vafa multiple cover formula is used to define virtually enumerative disk invariants. The results may also be viewed as providing a virtual enumeration of real rational curves on the quintic.


References [Enhancements On Off] (What's this?)

  • 1. Paul S. Aspinwall and David R. Morrison, Topological field theory and rational curves, Comm. Math. Phys. 151 (1993), no. 2, 245–262. MR 1204770
  • 2. P. Candelas, X. de la Ossa, P. Green and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory, Nuclear Physics B359 (1991), 21-74.
  • 3. K. Fukaya, Y.-G. Oh, H. Ohto and K. Ono, Lagrangian intersection Floer theory, anomaly and obstruction, Kyoto University, preprint, 2006.
  • 4. Kenji Fukaya and Kaoru Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999), no. 5, 933–1048. MR 1688434, 10.1016/S0040-9383(98)00042-1
  • 5. Alexander B. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 13 (1996), 613–663. MR 1408320, 10.1155/S1073792896000414
  • 6. A. Givental, Elliptic Gromov-Witten invariants and the generalized mirror conjecture, math.AG/9803053.
  • 7. T. Graber and E. Zaslow, Open string Gromov-Witten theory: calculation and a mirror theorem, hep-th/0109075.
  • 8. H. Hofer, K. Wysocki, and E. Zehnder, A general Fredholm theory. I. A splicing-based differential geometry, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 841–876. MR 2341834, 10.4171/JEMS/99
  • 9. H. Hofer, K. Wysocki and E. Zehnder, A General Fredholm Theory II: Implicit Function Theorems, arXiv:0705.1310.
  • 10. Sheldon Katz and Chiu-Chu Melissa Liu, Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc, Adv. Theor. Math. Phys. 5 (2001), no. 1, 1–49. MR 1894336, 10.4310/ATMP.2001.v5.n1.a1
  • 11. Maxim Kontsevich, Enumeration of rational curves via torus actions, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 335–368. MR 1363062
  • 12. Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120–139. MR 1403918
  • 13. Bong H. Lian, Kefeng Liu, and Shing-Tung Yau, Mirror principle. I, Asian J. Math. 1 (1997), no. 4, 729–763. MR 1621573, 10.4310/AJM.1997.v1.n4.a5
  • 14. M. Liu, Moduli of J-holomorphic curves with Lagrangian boundary conditions and open Gromov-Witten invariants for a $ S^1$-equivariant pair, math.SG/0210257.
  • 15. Hirosi Ooguri and Cumrun Vafa, Knot invariants and topological strings, Nuclear Phys. B 577 (2000), no. 3, 419–438. MR 1765411, 10.1016/S0550-3213(00)00118-8
  • 16. Rahul Pandharipande, Rational curves on hypersurfaces (after A. Givental), Astérisque 252 (1998), Exp. No. 848, 5, 307–340. Séminaire Bourbaki. Vol. 1997/98. MR 1685628
  • 17. Yongbin Ruan and Gang Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), no. 2, 259–367. MR 1366548
  • 18. P. Seidel, personal communication based on a remark of D. Joyce and a talk of K. Fukaya at Northwestern in spring 2004.
  • 19. J. Solomon, Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions, math.SG/0606429.
  • 20. J. Solomon, Virtual manifolds, to appear.
  • 21. J. Walcher, Opening mirror symmetry on the quintic, hep-th/0605162.
  • 22. Jean-Yves Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195–234. MR 2198329, 10.1007/s00222-005-0445-0
  • 23. Jean-Yves Welschinger, Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants, Duke Math. J. 127 (2005), no. 1, 89–121. MR 2126497, 10.1215/S0012-7094-04-12713-7
  • 24. K. Wehrheim and C. Woodward, Orientations for pseudo-holomorphic quilts, preprint.
  • 25. E. Witten, Chern-Simons gauge theory as a string theory, The Floer memorial volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 637–678. MR 1362846

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Additional Information

R. Pandharipande
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: rahulp@math.princeton.edu

J. Solomon
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: jake@ias.edu, jake@math.princeton.edu

J. Walcher
Affiliation: School of Natural Science, Institute for Advanced Study, Princeton, New Jersey 08540
Email: walcher@ias.edu

DOI: https://doi.org/10.1090/S0894-0347-08-00597-3
Received by editor(s): May 29, 2007
Published electronically: February 12, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.