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

## TL;DR

This paper explicitly inverts the period map associated with the quantum cohomology of , expressing the inverse via Eisenstein series and quasi-modular forms, revealing deep connections between quantum cohomology and modular forms.

## Contribution

It provides an explicit inversion of the period map for quantum cohomology of , linking it to Eisenstein series and quasi-modular forms, and extends to the big quantum cohomology case.

## Key findings

- Explicit inverse for small quantum cohomology using Eisenstein series.
- Perturbative inverse for big quantum cohomology as a Taylor series.
- Connection established between quantum cohomology and modular forms.

## 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.