BKMP conjecture for open Gromov–Witten potentials of toric Calabi–Yau threefolds

Let $\mathcal{S}=(\Gamma_X,C,\Delta,\log U,\log V)$ be the mirror spectral curve of a toric Calabi–Yau threefold $X$, with a polarization whose cycles $A_i$ have the stated large-complex-structure normalization. Let $\mathcal{S}_f$ be obtained by $U\to UV^f$, $V\to V$ for $f\in\mathbb{Z}$, and let $W_h^{(g)}$ be the correlators produced by Eynard–Orantin recursion. For $2g+h>1$, define the integrated correlators by

$$\mathcal{F}^{(\widehat{X},\widehat{L})}_{g,h}=\int W_h^{(g)}(p_1,\ldots,p_h)\frac{dp_1}{p_1}\dots\frac{dp_h}{p_h}.$$

BKMP conjecture. These integrated correlators are equal to the A-model framed open Gromov–Witten potentials of $(X,L)$, where $L$ is the mirror brane to $\Gamma_X\subset\widehat X$, after substituting the closed and open mirror maps.

This conjecture identifies the Eynard–Orantin topological recursion on the mirror curve with open Gromov–Witten theory of toric Calabi–Yau threefolds. The source gives no resolution status.