Refined sheaf-counting conjecture for del Pezzo surfaces

Let $S$ be a del Pezzo surface with a smooth anticanonical divisor, and let $\beta$ be a curve class. The higher genus relative Gromov–Witten invariants of $S$ relative to the divisor, with maximal tangency and insertion of the top lambda class, are compared with refined counts of dimension-one stable sheaves on the local Calabi–Yau threefold

$$\operatorname{Tot} K_S,$$

the total space of the canonical line bundle of $S$. Refined sheaf-counting conjecture. These higher genus relative Gromov–Witten invariants are related to the refined counts of dimension-one stable sheaves on $\operatorname{Tot} K_S$. This proposes a connection between higher genus relative Gromov–Witten theory and refined BPS or stable-sheaf invariants of the associated local Calabi–Yau threefold; the precise meaning and validity of the proposed relation remain open in the statement given here.