Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 038, 65 pages      arXiv:2402.09574      https://doi.org/10.3842/SIGMA.2025.038

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

Guilherme F. Almeida ab
a) Mannheim University, Mannheim, Germany
b) Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany

Received February 16, 2024, in final form May 12, 2025; Published online May 30, 2025

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.

Key words: Dubrovin-Frobenius manifolds; big quantum cohomology; Landau-Ginzburg superpotential.

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