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 $Q{H}^{\ast }\left({\mathbb{CP}}^2\right)$ 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 $Q{H}^{\ast }\left({\mathbb{CP}}^2\right)$ in closed form as the composition of the Weierstrass $\wp$-function and the universal coverings of $\mathbb{C}\setminus (\mathbb{Z}\oplus {\mathrm{e}}^{\frac{\pi \mathrm{i}}{3}}\mathbb{Z})$ and $\mathbb{C}\setminus (\mathbb{Z}\oplus z\mathbb{Z})$, respectively.