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

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.

131 pages, 9 figures; fully commented source code included as ancillary file; for video of talk, see: https://www.youtube.com/watch?v=raqkmHxCJYI and https://www.youtube.com/watch?v=sRMESF1TSOA v2: final version, to appear in Kyoto Journal of Mathematics

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Cited by in corpus (3)

• Crepant resolution and the holomorphic anomaly equation for C^3/Z_3
• Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface
• Quasi-modularity and holomorphic anomaly equation for the twisted Gromov-Witten theory: $\mathcal{O}(3)$ over ${\mathbb{P}}^2$