BKMP conjecture for toric Calabi–Yau threefolds

Let ${\mathfrak X}$ be a toric Calabi–Yau 3-fold, with a Lagrangian brane $L$ and framing parameters $\mathbf t$. Let $\mathcal W_{g,n}({\mathfrak X},L,\mathbf t;x_1,\dots,x_n)$ denote the open and closed Gromov–Witten amplitudes, and let $\widehat{\mathfrak X}$ be the framed mirror curve of ${\mathfrak X}$. Let $W_{g,n}(\widehat{\mathfrak X};x_1,\dots,x_n)$ be the invariants obtained from the topological recursion on this mirror spectral curve. BKMP conjecture. For every $(g,n)\in\mathbb N^2$,

$$\mathcal W_{g,n}({\mathfrak X},L,\mathbf t;x_1,\dots,x_n)\,dx_1\otimes\dots\otimes dx_n=W_{g,n}(\widehat{\mathfrak X};x_1,\dots,x_n).$$

This conjecture identifies the A-model Gromov–Witten invariants of a toric Calabi–Yau 3-fold with the topological-recursion invariants of its framed mirror curve, including open and closed sectors. The source presents it as the BKMP conjecture; no resolution status is supplied here.