Genus zero Gopakumar–Vafa type invariants for Calabi–Yau 4-folds II: Fano 3-folds

Abstract

In analogy with the Gopakumar–Vafa (GV) conjecture on Calabi–Yau (CY) 3-folds, Klemm and Pandharipande defined GV type invariants on Calabi–Yau 4-folds using Gromov–Witten theory and conjectured their integrality. In a joint work with Maulik and Toda, the author conjectured their genus zero invariants are [Formula: see text] invariants of one-dimensional stable sheaves. In this paper, we study this conjecture on the total space of canonical bundle of a Fano 3-fold [Formula: see text], which reduces to a relation between twisted GW and [Formula: see text] invariants on [Formula: see text]. Examples are computed for both compact and non-compact Fano 3-folds to support our conjecture.