We construct a sheaf of Fock spaces over the moduli space of elliptic curves $E_y$ with ${\mathrm{\Gamma }}_1(3)$-level structure, arising from geometric quantization of $H^1(E_y)$, and a global section of this Fock sheaf. The global section coincides, near appropriate limit points, with the Gromov–Witten potentials of local ${\mathbb{P}}^2$ and of the orbifold $[{\mathbb{C}}^3∕{\mathrm{\mu }}_3]$. This proves that the Gromov–Witten potentials of local ${\mathbb{P}}^2$ are quasimodular functions for the group ${\mathrm{\Gamma }}_1(3)$, as predicted by Aganagic, Bouchard, and Klemm, and it proves the crepant resolution conjecture for $[{\mathbb{C}}^3∕{\mathrm{\mu }}_3]$ in all genera.

We give a detailed proof of a result of Gabber (unpublished) on the lifting problem of quasi-excellent rings, extending the previous work of Nishimura and Nishimura. As a corollary, we establish that an ideal-adic completion of an excellent (resp., quasi-excellent) ring is excellent (resp., quasi-excellent).

We prove an equivalence of two $A_{\mathrm{\infty }}$-functors, via Orlov’s Landau–Ginzburg/ Calabi–Yau (LG/CY) correspondence. One is the Polishchuk–Zaslow mirror symmetry functor of elliptic curves, and the other is a localized mirror functor from the Fukaya category of $T^2$ to a category of noncommutative matrix factorizations. As a corollary, we prove that the noncommutative mirror functor $\mathcal{L}{\mathcal{M}}_{gr}^{{\mathbb{L}}_t}$ realizes homological mirror symmetry for any t.