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

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 $\mathrm{D}{\mathrm{T}}_4$ 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 $Y$, which reduces to a relation between twisted GW and $\mathrm{D}{\mathrm{T}}_3$ invariants on $Y$. Examples are computed for both compact and non-compact Fano 3-folds to support our conjecture.