## Disk enumeration on the quintic 3-fold

HTML articles powered by AMS MathViewer

- by R. Pandharipande, J. Solomon and J. Walcher PDF
- J. Amer. Math. Soc.
**21**(2008), 1169-1209 Request permission

## 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

- Paul S. Aspinwall and David R. Morrison,
*Topological field theory and rational curves*, Comm. Math. Phys.**151**(1993), no. 2, 245–262. MR**1204770**, DOI 10.1007/BF02096768
cogp P. Candelas, X. de la Ossa, P. Green and L. Parkes, - Kenji Fukaya and Kaoru Ono,
*Arnold conjecture and Gromov-Witten invariant*, Topology**38**(1999), no. 5, 933–1048. MR**1688434**, DOI 10.1016/S0040-9383(98)00042-1 - Alexander B. Givental,
*Equivariant Gromov-Witten invariants*, Internat. Math. Res. Notices**13**(1996), 613–663. MR**1408320**, DOI 10.1155/S1073792896000414
g2 A. Givental, - 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**, DOI 10.4171/JEMS/99
HWZ2 H. Hofer, K. Wysocki and E. Zehnder, - 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**, DOI 10.4310/ATMP.2001.v5.n1.a1 - 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**, DOI 10.1007/978-1-4612-4264-2_{1}2 - 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** - Bong H. Lian, Kefeng Liu, and Shing-Tung Yau,
*Mirror principle. I*, Asian J. Math.**1**(1997), no. 4, 729–763. MR**1621573**, DOI 10.4310/AJM.1997.v1.n4.a5
melissa M. Liu, - Hirosi Ooguri and Cumrun Vafa,
*Knot invariants and topological strings*, Nuclear Phys. B**577**(2000), no. 3, 419–438. MR**1765411**, DOI 10.1016/S0550-3213(00)00118-8 - 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** - Yongbin Ruan and Gang Tian,
*A mathematical theory of quantum cohomology*, J. Differential Geom.**42**(1995), no. 2, 259–367. MR**1366548**
SP. Seidel, personal communication based on a remark of D. Joyce and a talk of K. Fukaya at Northwestern in spring 2004.
jake1 J. Solomon, - 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**, DOI 10.1007/s00222-005-0445-0 - 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**, DOI 10.1215/S0012-7094-04-12713-7
WWK. Wehrheim and C. Woodward, - 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**

*A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory*, Nuclear Physics

**B359**(1991), 21-74. FOOO K. Fukaya, Y.-G. Oh, H. Ohto and K. Ono,

*Lagrangian intersection Floer theory, anomaly and obstruction*, Kyoto University, preprint, 2006.

*Elliptic Gromov-Witten invariants and the generalized mirror conjecture*, math.AG/9803053. grzas T. Graber and E. Zaslow,

*Open string Gromov-Witten theory: calculation and a mirror theorem*, hep-th/0109075.

*A General Fredholm Theory II: Implicit Function Theorems,*arXiv:0705.1310.

*Moduli of J-holomorphic curves with Lagrangian boundary conditions and open Gromov-Witten invariants for a $S^1$-equivariant pair*, math.SG/0210257.

*Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions*, math.SG/0606429. jake2 J. Solomon,

*Virtual manifolds,*to appear. Walcher J. Walcher,

*Opening mirror symmetry on the quintic*, hep-th/0605162.

*Orientations for pseudo-holomorphic quilts,*preprint.

## Additional Information

**R. Pandharipande**- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 357813
- 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
- MR Author ID: 656979
- Email: walcher@ias.edu
- Received by editor(s): May 29, 2007
- Published electronically: February 12, 2008
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**21**(2008), 1169-1209 - MSC (2000): Primary 53D45, 14N35; Secondary 14J32
- DOI: https://doi.org/10.1090/S0894-0347-08-00597-3
- MathSciNet review: 2425184