×

Disk enumeration on the quintic 3-fold. (English) Zbl 1203.53086

Summary: 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.

MSC:

53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Paul S. Aspinwall and David R. Morrison, Topological field theory and rational curves, Comm. Math. Phys. 151 (1993), no. 2, 245 – 262. · Zbl 0776.53043
[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. · Zbl 1098.32506
[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. · Zbl 0946.53047 · doi:10.1016/S0040-9383(98)00042-1
[5] Alexander B. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 13 (1996), 613 – 663. · Zbl 0881.55006 · doi:10.1155/S1073792896000414
[6] A. Givental, Elliptic Gromov-Witten invariants and the generalized mirror conjecture, math.AG/9803053. · Zbl 0961.14036
[7] T. Graber and E. Zaslow, Open string Gromov-Witten theory: calculation and a mirror theorem, hep-th/0109075. · Zbl 1085.14518
[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. · Zbl 1149.53053 · doi:10.4171/JEMS/99
[9] H. Hofer, K. Wysocki and E. Zehnder, A General Fredholm Theory II: Implicit Function Theorems, arXiv:0705.1310. · Zbl 1217.58005
[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. · Zbl 1026.32028 · doi: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. · Zbl 0885.14028 · doi:10.1007/978-1-4612-4264-2_12
[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. · Zbl 0846.53021
[13] Bong H. Lian, Kefeng Liu, and Shing-Tung Yau, Mirror principle. I, Asian J. Math. 1 (1997), no. 4, 729 – 763. · Zbl 0953.14026 · doi: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. · Zbl 1036.81515 · doi: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.
[17] Yongbin Ruan and Gang Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), no. 2, 259 – 367. · Zbl 0860.58005
[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. · Zbl 1135.14030
[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. · Zbl 1082.14052 · doi: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. · Zbl 1084.14056 · doi: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. · Zbl 0844.58018
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.