The remodeling conjecture for open Gromov–Witten generating functions

Consider the topological A-model on a local toric Calabi–Yau 3-fold $Y_{\bm{l}}$ with a special Lagrangian submanifold $L$. Let $F_n^{(g)}(\mathsf{x}_1,\ldots,\mathsf{x}_n)$ be the generating functions of open Gromov–Witten invariants for genus-$g$ Riemann surfaces with $n$ boundaries mapped to $Y_{\bm{l}}$, with boundaries mapped to $L$. Let $\Sigma_{Y_{\bm{l}}^{\vee}}^{K}$ be the mirror curve, let $B(z_1,z_2)$ be its Bergman kernel, and let $\omega_n^{(g)}$ be the multilinear meromorphic differentials produced by topological recursion. The points $z_i$ lie on the mirror curve in a local coordinate, and $z_i^*$ are reference points at which the relevant integrals vanish. Via the closed and open mirror maps, the open Gromov–Witten generating functions are expressed, up to framing ambiguity, by the topological-recursion integrals

$$F_1^{(0)}(\mathsf{x}_1)=\int_{z_1^*}^{z_1}\omega(x(z_1')),\qquad \omega(x(z))=\log y(x(z))\,\frac{dx(z)}{x(z)},$$ $$F_2^{(0)}(\mathsf{x}_1,\mathsf{x}_2)=\int_{z_1^*}^{z_1}\int_{z_2^*}^{z_2}\left(B(z_1',z_2')-\frac{dx(z_1')dx(z_2')}{(x(z_1')-x(z_2'))^2}\right),$$

and

$$F_n^{(g)}(\mathsf{x}_1,\ldots,\mathsf{x}_n)=\int_{z_1^*}^{z_1}\cdots\int_{z_n^*}^{z_n}\omega_n^{(g)}(z_1',\ldots,z_n')$$

for $(g,n)\ne(0,1),(0,2)$.