Katz genus-zero GV/DT3 correspondence for Calabi–Yau threefolds

Let $Y$ be a smooth projective Calabi–Yau $3$-fold, let $\beta\in H_2(Y,\mathbb Z)$, and let $n^{\mathrm{GW}}_{0,\beta}$ be the genus-zero Gopakumar–Vafa invariant determined by the Gromov–Witten multiple-cover expansion. Let $M_\beta$ be the moduli space of one-dimensional stable sheaves $E$ on $Y$ with $[E]=\beta$ and $\chi(E)=1$, and define

$$\mathrm{DT}_3(\beta)=\int_{[M_\beta]^{\mathrm{vir}}}1.$$

Katz's genus-zero GV/DT conjecture.

$$n^{\mathrm{GW}}_{0,\beta}=\mathrm{DT}_3(\beta).$$

This identifies genus-zero Gopakumar–Vafa invariants with sheaf-counting invariants on Calabi–Yau threefolds. It is used in the paper as a standard conjectural correspondence.