A Conjectural Formula for Genus One Gromov-Witten Invariants of a Class of Local Calabi-Yau n-folds

TL;DR

This paper proposes a conjectural formula for genus one Gromov-Witten invariants of certain local Calabi-Yau manifolds, with partial proofs and checks on BPS number integrality.

Contribution

It introduces a new conjectural formula for genus one invariants of local Calabi-Yau manifolds and verifies it in specific cases.

Findings

Conjectural formula for genus one Gromov-Witten invariants.

Partial proofs in special cases.

Checks on BPS number integrality.

Abstract

We conjecture a formula for the generating function of genus one Gromov-Witten invariants of the local Calabi-Yau manifolds which are the total spaces of splitting bundles over projective spaces. We prove this conjecture in several special cases, and assuming the validity of our conjecture we check the integrality of genus one BPS numbers of local Calabi-Yau 5-folds defined by A. Klemm and R. Pandharipande.