Gromov-Witten Invariants of Local P^2 and Modular Forms

TL;DR

This paper constructs a sheaf of Fock spaces over the moduli space of elliptic curves to connect Gromov-Witten invariants of local P^2 with modular forms, confirming conjectures about their modularity and crepant resolution.

Contribution

It introduces a sheaf of Fock spaces and a global section that links Gromov-Witten potentials to modular forms, proving modularity and crepant resolution conjectures.

Findings

Gromov-Witten potentials are quasi-modular functions for Gamma_1(3)

Proves the Crepant Resolution Conjecture for [C^3/mu_3] in all genera

Establishes a geometric quantization framework for these invariants

Abstract

We construct a sheaf of Fock spaces over the moduli space of elliptic curves E_y with 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 P^2 and of the orbifold C^3/mu_3. This proves that the Gromov-Witten potentials of local P^2 are quasi-modular functions for the group Gamma_1(3), as predicted by Aganagic-Bouchard-Klemm, and proves the Crepant Resolution Conjecture for [C^3/mu_3] in all genera.