The Gopakumar–Vafa/Gromov–Witten correspondence conjecture

Let $X$ be a smooth projective Calabi–Yau $3$-fold. For a curve class $\beta \in H_2(X, \mathbb{Z})$ and a non-negative integer $g \in \mathbb{Z}_{\geq 0}$, let $\operatorname{GW}_{g, \beta} \in \mathbb{Q}$ and $n_{g, \beta} \in \mathbb{Z}$ denote the Gromov–Witten and Gopakumar–Vafa invariants of $X$. Gopakumar–Vafa/Gromov–Witten correspondence conjecture. The generating series satisfy

$$\sum_{g, \beta}\operatorname{GW}_{g, \beta}\lambda^{2g-2}t^\beta = \sum_{g, \beta, k \geq 0} \frac{n_{g, \beta}}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g-2}t^{k\beta}.$$

The conjecture predicts that Gromov–Witten curve-counting invariants are determined by the integral Gopakumar–Vafa invariants through the multiple-cover expansion. The paper places this correspondence in the context of local curves and studies degree-two Gopakumar–Vafa invariants; the supplied text does not state a resolution of the conjecture.