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

Zinger, Aleksey

doi:10.1090/S0894-0347-08-00625-5

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

J. Amer. Math. Soc. 22 (2009), 691-737

DOI: https://doi.org/10.1090/S0894-0347-08-00625-5

Published electronically: October 2, 2008

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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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