The precise crepant resolution conjecture for analytic big A-model VSHS

Let $\mathcal{X}$ be an orbifold with projective coarse moduli space $X$, and let $\pi:Y\to X$ be a crepant resolution. Suppose that the big quantum products $\bullet_\tau$ for $\mathcal{X}$ and $Y$ are convergent as functions of $\tau$ and $Q$, so that their analytic big A-model VSHS with Novikov variables specialized to $1$ are defined on open subsets of $H^\bullet_{\rm orb}(\mathcal{X};\mathbb{C})$ and $H^\bullet(Y;\mathbb{C})$. Let $\mathbb{E}^{\mathcal{X}}_\tau\subset\mathcal{H}_{\mathcal{X}}|_{Q_i=1}$ and $\mathbb{E}^Y_\tau\subset\mathcal{H}_Y|_{Q_i=1}$ be the moving-subspace realizations of these VSHS. Define

$$\mathbb{E}^{\mathcal{X}}_{{\rm \lim},\tau}=\lim_{\sigma\to\mathrm{l.r.l.}}e^{-\sigma/z}\mathbb{E}^{\mathcal{X}}_{\tau+\sigma},$$

where $\sigma\in H^2(\mathcal{X};\mathbb{C})$ and the large-radius limit means $\Re(\int_d\sigma)\to-\infty$ for every $d\in\operatorname{Eff}(\mathcal{X})$. The precise crepant resolution conjecture. There is a symplectic transformation $\overline{\mathbb{U}}:\mathcal{H}_{\mathcal{X}}|_{Q_i=1}\to\mathcal{H}_Y|_{Q_i=1}$ and a map $\Upsilon$ between open subsets of the corresponding cohomology spaces such that, after analytic continuation if necessary,

$$\overline{\mathbb{U}}\bigl(\mathbb{E}^{\mathcal{X}}_\tau\bigr)=\mathbb{E}^Y_{\Upsilon(\tau)}.$$

Moreover: (a) $\overline{\mathbb{U}}$ is degree-preserving and $\mathbb{C}\{z,z^{-1}\}$-linear; (b) $\overline{\mathbb{U}}(\rho\cup)=(\pi^*(\rho)\cup)\overline{\mathbb{U}}$ for every non-twisted $\rho\in H^2(\mathcal{X};\mathbb{C})$, using the Chen–Ruan orbifold cup product on the left and the usual cup product on the right; and (c) there is $\tau_0\in H^2_{\rm orb}(\mathcal{X};\mathbb{C})$ such that the standard opposite subspaces are opposite to $\mathbb{E}^{\mathcal{X}}_{{\rm \lim},\tau_0}$ and to $\overline{\mathbb{U}}(\mathbb{E}^{\mathcal{X}}_{{\rm \lim},\tau_0})$, respectively. This precise formulation refines the introductory crepant resolution conjecture and is intended to capture the symplectic transformation and its compatibility with grading, divisor operators, and limiting Hodge structures.