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The reduced genus $ 1$ Gromov-Witten invariants of Calabi-Yau hypersurfaces


Author: Aleksey Zinger
Journal: J. Amer. Math. Soc. 22 (2009), 691-737
MSC (2000): Primary 14N35, 53D45
DOI: https://doi.org/10.1090/S0894-0347-08-00625-5
Published electronically: October 2, 2008
MathSciNet review: 2505298
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Abstract: We compute the reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces. As a consequence, we confirm the 1993 Bershadsky-Cecotti-Ooguri-Vafa (BCOV) prediction for the standard genus 1 GW-invariants of a quintic threefold. We combine constructions from a series of previous papers with the classical localization theorem to relate the reduced genus 1 invariants of a CY-hypersurface to previously computed integrals on moduli spaces of stable genus 0 maps into projective space. The resulting, rather unwieldy, expressions for a genus 1 equivariant generating function simplify drastically, using a regularity property of a genus 0 equivariant generating function in half of the cases. Finally, by disregarding terms that cannot effect the non-equivariant part of the former, we relate the answer to an explicit hypergeometric series in a simple way. The approach described in this paper is systematic. It is directly applicable to computing reduced genus 1 GW-invariants of other complete intersections and should apply to higher-genus localization computations.


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  • [ABo] M. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1-28. MR 721448 (85e:58041)
  • [BCOV] M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B405 (1993), 279-304. MR 1240687 (94j:81254)
  • [BeFa] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45-88. MR 1437495 (98e:14022)
  • [Ber] A. Bertram, Another way to enumerate rational curves with torus actions, Invent. Math. 142 (2000), no. 3, 487-512. MR 1804158 (2001m:14077)
  • [CaDGP] P. Candelas, X. de la Ossa, P. Green, L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B359 (1991), 21-74. MR 1115626 (93b:32029)
  • [CoKa] D. Cox and S. Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, 68, Amer. Math. Soc., 1999. MR 1677117 (2000d:14048)
  • [FuO] K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999), no. 5, 933-1048. MR 1688434 (2000j:53116)
  • [Ga] A. Gathmann, Absolute and relative Gromov-Witten invariants of very ample hypersurfaces, Duke Math. J. 115 (2002), no. 2, 171-203. MR 1944571 (2003k:14068)
  • [Gi] A. Givental, The mirror formula for quintic threefolds, Amer. Math. Soc. Transl. Ser. 2, 196 (1999). MR 1736213 (2000j:14083)
  • [HKlQ] M. Huang, A. Klemm, and S. Quackenbush, Topological string theory on compact Calabi-Yau: modularity and boundary conditions, hep-th/0612125.
  • [KlPa] A. Klemm and R. Pandharipande, Enumerative geometry of Calabi-Yau 4-folds, Comm. Math. Phys. 281 (2008), no. 3, 621-653.
  • [KoM] M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525-562. MR 1291244 (95i:14049)
  • [Le] Y. P. Lee, Quantum Lefschetz hyperplane theorem, Invent. Math. 145 (2001), no. 1, 121-149. MR 1839288 (2002i:14049)
  • [LiT] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, Topics in Symplectic $ 4$-Manifolds, 47-83, First Int. Press Lect. Ser., I, Internat. Press, 1998. MR 1635695 (2000d:53137)
  • [LiZ] J. Li and A. Zinger, On the genus-one Gromov-Witten invariants of complete intersections, math/0507104.
  • [LLY] B. Lian, K. Liu, and S.T. Yau, Mirror Principle I, Asian J. of Math. 1, no. 4 (1997), 729-763. MR 1621573 (99e:14062)
  • [MirSym] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror Symmetry, Clay Math. Inst., Amer. Math. Soc., 2003. MR 2003030 (2004g:14042)
  • [RT] Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Diff. Geom. 42 (1995), no. 2, 259-367. MR 1366548 (96m:58033)
  • [VaZ] R. Vakil and A. Zinger, A desingularization of the main component of the moduli space of genus-one stable maps into $ {\mathbb{P}}^n$, Geom. and Top. 12 (2008), no. 1, 1-95. MR 2377245
  • [ZaZ] D. Zagier and A. Zinger, Some properties of hypergeometric series associated with mirror symmetry, to appear in Modular Forms and String Duality, The Fields Institute Communications, Volume 54.
  • [Z1] A. Zinger, Reduced Genus-One Gromov-Witten Invariants, math/0507103.
  • [Z2] A. Zinger, On the structure of certain natural cones over moduli spaces of genus-one holomorphic maps, Adv. Math. 214 (2007) 878-933. MR 2349722
  • [Z3] A. Zinger, Intersections of tautological classes on blowups of moduli spaces of genus-one curves, Mich. Math. 55 (2007), no. 3, pp 535-560. MR 2372615
  • [Z4] A. Zinger, Genus-zero two-point hyperplane integrals in the Gromov-Witten theory, math/0705.2725.
  • [Z5] A. Zinger, Standard vs. reduced genus-one Gromov-Witten invariants, Geom. and Top. 12 (2008), no. 2, 1203-1241. MR 2403808

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Additional Information

Aleksey Zinger
Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
Email: azinger@math.sunysb.edu

DOI: https://doi.org/10.1090/S0894-0347-08-00625-5
Keywords: Gromov-Witten invariants, genus $1$, BCOV prediction
Received by editor(s): July 23, 2007
Published electronically: October 2, 2008
Additional Notes: The author was partially supported by a Sloan Fellowship and DMS Grant 0604874
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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