Cao–Maulik–Toda genus-zero DT$_4$–Gopakumar–Vafa correspondence

Let $X$ be a Calabi–Yau $4$-fold, let $\beta e 0$ be a curve class, and let $\tau_0(\gamma)$ be the primary insertion for $\gamma\break\relax gin H^4(X,\mathbb Z)$. Denote by $n_{0,\beta}(\relax gamma)$ the genus-zero Gopakumar–Vafa invariant and by $\mathrm{GW}_{0,\beta}^{X}(\relax gamma)$ the corresponding Gromov–Witten invariant.

Cao–Maulik–Toda's genus-zero correspondence conjecture. Via some suitable choice of orientation on the moduli space, one has

$$n_{0,\beta}(\gamma)=\mathrm{DT}_4(X)(\beta|\gamma),\qquad \mathrm{GW}_{0,\beta}^{X}(\gamma)=\sum_{k|\beta}\frac{1}{k^2}\mathrm{DT}_4(X)(\beta/k|\gamma).$$

The two equations are equivalent via the definition of $n_{0,\beta}(\gamma)$. The conjecture is known for Calabi–Yau $4$-folds with an elliptic structure and for local curves and surfaces, while the general case remains open.