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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
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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Aleksey Zinger
Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651

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.