Bousseau's refined sheaves/Gromov–Witten correspondence

Let $S$ be a del Pezzo surface, let $E$ be a suitable anticanonical divisor, and let $M_{\beta}$ be the moduli space of stable one-dimensional sheaves of class $\beta$ and Euler characteristic one. Define the shifted Poincaré polynomial

$$\Omega_{\beta}\left(y^{1/2}\right)=y^{-\frac12\operatorname{dim}_{\mathbb C}M_{\beta}}\sum_{j=0}^{\operatorname{dim}_{\mathbb C}M_{\beta}}b_{2j}(M_{\beta})y^j.$$

Let $\bar F^{NS}$ be the refined sheaf generating series and let $\bar F^{S/E}$ be the generating series of maximal-contact Gromov–Witten invariants of $(S,E)$. Bousseau's refined correspondence. Under the change of variables $y=e^{i\hbar}$,

$$\bar F^{NS}=\bar F^{S/E}.$$

This correspondence implies the Choi–van Garrel–Katz–Takahashi divisibility conjecture; it is known for $S=\mathbb P^2$, but its general status is conjectural.