Gopakumar–Vafa integrality conjecture for Gromov–Witten invariants

Let $X$ be the smooth projective variety under consideration, let $N^{\operatorname{GW}}_{g,\beta}(X)$ denote its Gromov–Witten invariants, let $u$ be a formal variable, and let $t^\beta$ record curve classes $\beta\in H_2(X,\mathbb Z)$. The symbols $\beta\geq 0$ and $\beta>0$ denote, respectively, the effective classes including the zero class and the nonzero effective classes. Gopakumar–Vafa integrality conjecture. There exist integers $n_g^\beta\in\mathbb Z$, for $g\geq 0$ and $\beta\in H_2(X,\mathbb Z)$, such that

$$\sum_{g\geq 0,\beta\geq 0}N^{\operatorname{GW}}_{g,\beta}(X)u^{2g-2}t^\beta=\sum_{\substack{g\geq 0,\beta> 0\\ k\in\mathbb Z_{\geq 1}}}\frac{n_g^\beta}{k}\left(2\sin\left(\frac{ku}{2}\right)^{2g-2}\right)t^{k\beta}.$$

The conjecture asserts the integrality of the Gopakumar–Vafa invariants and is motivated by string duality between type IIA string theory and M-theory; the displayed formula relates them to the generally rational Gromov–Witten invariants.