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

HTML articles powered by AMS MathViewer

- by Aleksey Zinger PDF
- J. Amer. Math. Soc.
**22**(2009), 691-737 Request permission

## 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.## References

- M. F. Atiyah and R. Bott,
*The moment map and equivariant cohomology*, Topology**23**(1984), no. 1, 1–28. MR**721448**, DOI 10.1016/0040-9383(84)90021-1 - M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa,
*Holomorphic anomalies in topological field theories*, Nuclear Phys. B**405**(1993), no. 2-3, 279–304. MR**1240687**, DOI 10.1016/0550-3213(93)90548-4 - K. Behrend and B. Fantechi,
*The intrinsic normal cone*, Invent. Math.**128**(1997), no. 1, 45–88. MR**1437495**, DOI 10.1007/s002220050136 - Aaron Bertram,
*Another way to enumerate rational curves with torus actions*, Invent. Math.**142**(2000), no. 3, 487–512. MR**1804158**, DOI 10.1007/s002220000094 - Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes,
*A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory*, Nuclear Phys. B**359**(1991), no. 1, 21–74. MR**1115626**, DOI 10.1016/0550-3213(91)90292-6 - David A. Cox and Sheldon Katz,
*Mirror symmetry and algebraic geometry*, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. MR**1677117**, DOI 10.1090/surv/068 - 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 - Andreas Gathmann,
*Absolute and relative Gromov-Witten invariants of very ample hypersurfaces*, Duke Math. J.**115**(2002), no. 2, 171–203. MR**1944571**, DOI 10.1215/S0012-7094-02-11521-X - Alexander Givental,
*The mirror formula for quintic threefolds*, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc., Providence, RI, 1999, pp. 49–62. MR**1736213**, DOI 10.1090/trans2/196/04
[HKlQ]HKlQ M. Huang, A. Klemm, and S. Quackenbush, - M. Kontsevich and Yu. Manin,
*Gromov-Witten classes, quantum cohomology, and enumerative geometry*, Comm. Math. Phys.**164**(1994), no. 3, 525–562. MR**1291244** - Y.-P. Lee,
*Quantum Lefschetz hyperplane theorem*, Invent. Math.**145**(2001), no. 1, 121–149. MR**1839288**, DOI 10.1007/s002220100145 - Jun Li and Gang Tian,
*Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds*, Topics in symplectic $4$-manifolds (Irvine, CA, 1996) First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 1998, pp. 47–83. MR**1635695**
[LiZ]LiZ J. Li and A. Zinger, - 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 - Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow,
*Mirror symmetry*, Clay Mathematics Monographs, vol. 1, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2003. With a preface by Vafa. MR**2003030** - Yongbin Ruan and Gang Tian,
*A mathematical theory of quantum cohomology*, J. Differential Geom.**42**(1995), no. 2, 259–367. MR**1366548** - Ravi Vakil and Aleksey Zinger,
*A desingularization of the main component of the moduli space of genus-one stable maps into $\Bbb P^n$*, Geom. Topol.**12**(2008), no. 1, 1–95. MR**2377245**, DOI 10.2140/gt.2008.12.1
[ZaZ]ZaZ D. Zagier and A. Zinger, - Aleksey Zinger,
*On the structure of certain natural cones over moduli spaces of genus-one holomorphic maps*, Adv. Math.**214**(2007), no. 2, 878–933. MR**2349722**, DOI 10.1016/j.aim.2007.03.009 - Aleksey Zinger,
*Intersections of tautological classes on blowups of moduli spaces of genus-1 curves*, Michigan Math. J.**55**(2007), no. 3, 535–560. MR**2372615**, DOI 10.1307/mmj/1197056456
[Z4]bcov0 A. Zinger, - Aleksey Zinger,
*Standard versus reduced genus-one Gromov-Witten invariants*, Geom. Topol.**12**(2008), no. 2, 1203–1241. MR**2403808**, DOI 10.2140/gt.2008.12.1203

*Topological string theory on compact Calabi-Yau: modularity and boundary conditions*, hep-th/0612125. [KlPa]KlPa A. Klemm and R. Pandharipande,

*Enumerative geometry of Calabi-Yau 4-folds*, Comm. Math. Phys. 281 (2008), no. 3, 621–653.

*On the genus-one Gromov-Witten invariants of complete intersections*, math/0507104.

*Some properties of hypergeometric series associated with mirror symmetry*, to appear in Modular Forms and String Duality, The Fields Institute Communications, Volume 54. [Z1]g1comp2 A. Zinger,

*Reduced Genus-One Gromov-Witten Invariants*, math/0507103.

*Genus-zero two-point hyperplane integrals in the Gromov-Witten theory*, math/0705.2725.

## Additional Information

**Aleksey Zinger**- Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
- Email: azinger@math.sunysb.edu
- 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
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2505298