Castelnuovo bound conjecture for Gopakumar–Vafa invariants

Let $X$ be a projective Calabi–Yau 3-fold of Picard number one and degree $n$. For integers $g$ and $d$, let $\mathsf{GV}_{g,d}$ denote the genus $g$, degree $d$ Gopakumar–Vafa invariant of $X$.

Castelnuovo bound conjecture. The invariant vanishes when

$$g > \frac{1}{2n}d^2+\frac{1}{2}d+1.$$

This conjecture is the A-model conjecture used in the physical derivation of Gromov–Witten invariants of projective Calabi–Yau 3-folds. It predicts an effective vanishing range for the Gopakumar–Vafa invariants, supplying many of the initial conditions needed to determine higher-genus Gromov–Witten generating series; the source does not specify whether the conjecture has been resolved.