The Reduced Genus-One Gromov-Witten Invariants of Calabi-Yau Hypersurfaces

TL;DR

This paper computes the reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces, confirming the BCOV prediction for a quintic threefold and providing a systematic approach applicable to other cases.

Contribution

It introduces a systematic method combining localization and previous results to compute genus 1 invariants, confirming long-standing theoretical predictions.

Findings

Confirmed BCOV prediction for quintic threefolds

Derived explicit hypergeometric series expressions

Applicable to other complete intersections and higher-genus cases

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.