1D Landau-Ginzburg Superpotential of Big Quantum Cohomology of $\mathbb{CP}^2$

Abstract

Using the inverse period map of the Gauss-Manin connection associated with $QH^{*}\bigl(\mathbb{CP}^2\bigr)$ and the Dubrovin construction of Landau-Ginzburg superpotential for Dubrovin-Frobenius manifolds, we construct a one-dimensional Landau-Ginzburg superpotential for the quantum cohomology of $\mathbb{CP}^2$. In the case of small quantum cohomology, the Landau-Ginzburg superpotential is expressed in terms of the cubic root of the $j$-invariant function. For big quantum cohomology, the one-dimensional Landau-Ginzburg superpotential is given by Taylor series expansions whose coefficients are expressed in terms of quasi-modular forms. Furthermore, we express the Landau-Ginzburg superpotential for both small and big quantum cohomology of $QH^{*}\bigl(\mathbb{CP}^2\bigr)$ in closed form as the composition of the Weierstrass $\wp$-function and the universal coverings of $\mathbb{C} \setminus \bigl(\mathbb{Z} \oplus {\rm e}^{\frac{\pi {\rm i}}{3}}\mathbb{Z}\bigr)$ and $\mathbb{C} \setminus (\mathbb{Z} \oplus z\mathbb{Z})$, respectively.