Gromov–Witten invariants of local P2 and modular forms

Abstract

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.