Genus-zero DT$_4$–Gromov–Witten correspondence for local Fano threefolds

Let $Y$ be one of the Fano threefolds considered in the paper, let $|K_Y|$ denote its local Calabi–Yau $4$-fold, let $\beta$ be a curve class with degree $d=\beta\mathbin{\cdot}[\mathrm{line}]$, and let $\gamma\in H^4(Y,\mathbb Z)$. Denote by $n_{0,\beta}(\gamma)$ the genus-zero Gopakumar–Vafa invariant, by $\mathrm{DT}_4(|K_Y|)(d|\gamma)$ the corresponding DT$_4$ invariant, and by $\mathrm{GW}_{0,\beta}(Y)^{\mathrm{twist}}(\gamma)$ the twisted genus-zero Gromov–Witten invariant.

Local Fano threefold genus-zero correspondence. One has

$$n_{0,\beta}(\gamma)=\mathrm{DT}_4(|K_Y|)(d|\gamma)$$

and

$$\mathrm{GW}_{0,\beta}(Y)^{\mathrm{twist}}(\gamma)=\sum_{k|\beta}\frac{1}{k^2}\mathrm{DT}_4(|K_Y|)(d/k|\gamma).$$

The paper checks this correspondence computationally for the local Fano threefolds $V_5$ and $V_{22}$ in degrees $1\leq d\leq3$; the displayed assertion is presented as a rewriting of the preceding general conjecture rather than as a new independent conjecture.