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The reduced genus  Gromov-Witten invariants of Calabi-Yau hypersurfaces
Author(s):
Aleksey
Zinger
Journal:
J. Amer. Math. Soc.
22
(2009),
691-737.
MSC (2000):
Primary 14N35, 53D45
Posted:
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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Additional Information:
Aleksey
Zinger
Affiliation:
Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
Email:
azinger@math.sunysb.edu
DOI:
10.1090/S0894-0347-08-00625-5
PII:
S 0894-0347(08)00625-5
Keywords:
Gromov-Witten invariants,
genus $1$,
BCOV prediction
Received by editor(s):
July 23, 2007
Posted:
October 2, 2008
Additional Notes:
The author was partially supported by a Sloan Fellowship and DMS Grant 0604874
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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