The all-genus log-open correspondence for maximal-boundary log Calabi–Yau surfaces

Let $Y(D)$ be a log Calabi–Yau surface with maximal boundary, let $d$ be a $D$-convex curve class, and use the all-genus generating functions $\mathsf{O}_{\iota^{-1}(d)}(Y^{\rm op}(D))$ and $\mathsf{N}^{\rm \log}_d(Y(D))$, together with the quantum numbers $[r]_q$ and $q=\exp(\mathrm{i}\hbar)$. The all-genus log-open correspondence.

$$\mathsf{O}_{\iota^{-1}(d)}(Y^{\rm op}(D))(-\mathrm{i}\log q)=[1]_q^{l-2}\frac{(-1)^{d\cdot D_l+1}}{[d\cdot D_l]_q}\prod_{k=1}^{l-1}\frac{(-1)^{d\cdot D_k+1}}{d\cdot D_k}\,\mathsf{N}^{\rm \log}_d(Y(D))(-\mathrm{i}\log q).$$

This is an all-genus $q$-analogue of the genus-zero log-open correspondence. The text motivates its simplicity through degeneration and multiple-cover contributions, but gives no resolution of the conjecture in general.