The crepant transformation conjecture for the Landau–Ginzburg and Calabi–Yau models

Let $I\mathcal{X}$ and $I\mathcal{Y}$ be the inertia orbifolds of the local targets $\mathcal{X}$ and $\mathcal{Y}$, and let $\mathscr{L}^{\mathcal{X}}\subset\mathscr{V}^{\mathcal{X}}$ and $\mathscr{L}^{\mathcal{Y}}\subset\mathscr{V}^{\mathcal{Y}}$ be the Lagrangian cones of their equivariant genus-zero Gromov–Witten theories. Let $t=t^{\mathfrak{j}}$ be the coordinate dual to $\mathbb{1}_{\mathfrak{j}}$ on $H^*_{CR}(\mathcal{X})$, let $q$ be the exponential of the coordinate dual to the hyperplane class $H$ on $H^*(\mathcal{Y})$, and assume that a function $I^{\mathcal{Y}}(\mathbf{t},z)$ generating $\mathscr{L}^{\mathcal{Y}}$ is analytic near $q=0$. Via $q=t^{-d}$ and analytic continuation, regard $\mathscr{L}^{\mathcal{Y}}$ as continued from $q=0$ to $t=0$. The crepant transformation conjecture. The analytic continuation of $\mathscr{L}^{\mathcal{Y}}$ converges near $t=0$, and there exists a symplectic transformation

$$\mathbb{U}:\mathscr{V}^{\mathcal{X}}\to\mathscr{V}^{\mathcal{Y}}$$

which identifies $\mathscr{L}^{\mathcal{X}}$ with the analytic continuation of $\mathscr{L}^{\mathcal{Y}}$. This is the genus-zero crepant transformation prediction relating the Gromov–Witten theories of the two birational targets; the stated source does not provide evidence resolving it in general.