Mirror-brane conjecture for Aganagic–Vafa branes

Let $X=X_\Delta^3$ be a Calabi–Yau 3-fold and let $L=L_{AV}$ be an Aganagic–Vafa A-brane. Let $X^\vee\subset(\mathbb{C}^\times)^2\times\mathbb{C}^2$ be its mirror, and let $W$ be the function appearing in the mirror construction. For $q_0\in\mathbb{C}^\times$, define $z_2(q_0)$ implicitly by

$$W(q_0,z_2(q_0))=0.$$

Let $\vec{Q}=(Q_0,Q_1,\ldots,Q_{m-3})$ be the open/closed Kähler parameters and $\vec{q}=(q_0,q_1,\ldots,q_{m-3})$ the open/closed complex-structure parameters. Define

$$F(Q_0,Q_1,\ldots,Q_{m-3})=\sum_{p=1}^{\infty}\sum_{b_0,\ldots,b_{m-3}\geq0}\frac{N_{b_0,\ldots,b_{m-3}}}{p^2}Q_0^{pb_0}\cdots Q_{m-3}^{pb_{m-3}},$$

where $N_{b_0,\ldots,b_{m-3}}$ is the instanton number of the effective relative class $(b_0,\ldots,b_{m-3})\in\mathbb{Z}^{m-2}\simeq H_2(X,L;\mathbb{Z})$, and define

$$\mathcal{W}(q_0,q_1,\ldots,q_{m-3})=\int_{\Gamma_{q_0,q_1,\ldots,q_{m-3}}}\Omega_{q_1,\ldots,q_{m-3}},$$

where $\Omega_{q_1,\ldots,q_{m-3}}$ is the volume form of $X^\vee$ and $\Gamma_{q_0,q_1,\ldots,q_{m-3}}$ is an open 3-chain in $X^\vee$. Mirror-brane conjecture. The corresponding mirror B-brane is

$$L^\vee=\left\{(u,v,z_1,z_2)\in X^\vee:\ uv=0,\ z_1=q_0,\ z_2=z_2(q_0)\right\},$$

and $L$ is mirror to $L^\vee$ in the sense that

$$F(\vec{Q})=\mathcal{W}(\vec{q})$$

up to the open/closed mirror map. This conjecture formulates the predicted correspondence between Aganagic–Vafa A-branes, mirror B-branes, open Gromov–Witten invariants, and Abel–Jacobi-type period integrals; the source gives no resolution status.