On the Genus-One Gromov-Witten Invariants of a Quintic Threefold

TL;DR

This paper presents a direct, geometric derivation of relations between genus-one and genus-zero Gromov-Witten invariants of a quintic threefold, relying on a widely believed but unproven rigidity conjecture.

Contribution

It offers a more direct and geometric derivation of the genus-one GW-invariants relation, reducing dependence on external results compared to previous methods.

Findings

Derived a relation between genus-one and genus-zero GW-invariants for a quintic threefold.

Relied on a widely believed but unproven rigidity conjecture in Calabi-Yau threefolds.

Provided a more geometric and less assumption-dependent derivation.

Abstract

We rederive a relation between the genus-one GW-invariants of a quintic threefold in $\Pf$ and the genus-zero and genus-one GW-invariants of $\Pf$. In contrast to the more general derivation in a separate paper, the present derivation relies on a widely believed, but still unproven, statement concerning rigidity of holomorphic curves in Calabi-Yau threefolds. On the other hand, this paper's derivation is more direct and geometric. It requires a bit more effort, but relies on less outside work.