The SYZ mirror-map inversion conjecture for toric Calabi–Yau manifolds

Let $X$ be a toric Calabi–Yau manifold with mirror family $\check{X}_y$, and let

$$\phi:\mathcal{M}_K(X)\to\mathcal{M}_\mathbb{C}(\check{X})$$

be the SYZ map, defined by

$$y_a(q)=q_a\prod_{i=0}^{m-1}(1+\delta_i(q))^{Q_i^a},\qquad a=1,\ldots,l,$$

where $1+\delta_i(q)$ are generating functions of the genus $0$ open Gromov–Witten invariants $n_{\beta_i+\alpha}$. Let $\check{\Omega}_y$ be the holomorphic volume form on $\check{X}_y$. The SYZ mirror-map inversion conjecture. There exist integral cycles $\Gamma_1,\ldots,\Gamma_l$ forming part of an integral basis of the middle homology $H_n(\check{X}_y;\mathbb{Z})$ such that

$$q_a=\exp\left(-\int_{\Gamma_a}\check{\Omega}_{\phi(q)}\right),\qquad a=1,\ldots,l.$$

Equivalently, the SYZ map coincides with the inverse of a mirror map defined by period integrals. This conjecturally gives an enumerative interpretation of mirror maps in terms of genus $0$ open Gromov–Witten invariants; the supplied source does not state a resolution, so the conjecture is recorded as open.