Hosono's Gamma Conjecture for Calabi–Yau manifolds

Let $X$ be a Calabi–Yau manifold equipped with a symplectic form $\omega$, and let $\{Z_t\}_{t\in \Delta^*}$ be a family of Calabi–Yau manifolds parametrized by $t$ in a small punctured disc $\Delta^*$ corresponding to $(X,\omega)$ under mirror symmetry. For a suitable choice of a holomorphic volume form $\Omega_t$ on $Z_t$ and of a coordinate $t$, suppose that a Lagrangian cycle $C_t\subset Z_t$ is mirror to a coherent sheaf $E$ on $X$. Gamma Conjecture in the Calabi–Yau case. Then

$$\int_{C_t\subset Z_t} \Omega_t = \int_X t^{-\omega} \cdot \widehat{\Gamma}_X \cdot (2\pi \mathtt{i})^{\deg/2} \operatorname{ch}(E) + O\left(t^\epsilon\right)$$

as $t\to 0$ in a fixed angular sector, for some $\epsilon>0$, where $\mathtt{i}=\sqrt{-1}$ is the imaginary unit. The conjecture is an asymptotic mirror-symmetry statement relating periods of the mirror family to the Gamma class and Chern character on the original Calabi–Yau manifold; as stated, it depends on the choice of mirror pair and is therefore not mathematically precise without further SYZ-type data.