Two-Dimensional Torus Action on the Quadric Threefold and GW-DT-GV Correspondence

We compute Gromov–Witten (GW) and Donaldson-Thomas (DT) invariants (including descendant invariants) for local CY $4$-folds over a smooth quadric threefold $Q$, up to degree $4$. We use the virtual localization formula with respect to a two-dimensional torus action to compute DT invariants. Combining these results with the computation of the J-function of the variety $Q$, we confirm the correspondence between DT and Gopakumar–Vafa invariants suggested by Cao–Maulik–Toda in genus zero. Based on these results, we predict a genus one BPS invariant on $Q$.