The Period map for quantum cohomology of $\mathbb{P}^2$

Abstract

We invert the period map defined by the second structure connection of quantum cohomology of $\mathbb{P}^2$. For small quantum cohomology the inverse is given explicitly in terms of the Eisenstein series $E_4$ and $E_6$, while for big quantum cohomology the inverse is determined perturbatively as a Taylor series expansion whose coefficients are quasi-modular forms.Comment: 58 pages. Typos fixed. The previous version was extended in the following way: we proved the Laplace transform version of the $\Gamma$-conjecture for $\mathbb{P}^2$, worked out the structure of a Schwarz triangular map, and added an appendix explaining the relation to vertex algebras and W-constraints. Version accepted in Adv. in Mat