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Mirror symmetry for \(\mathbb P^2\) and tropical geometry. (English) Zbl 1190.14038
The author explores the connection between mirror symmetry for \(\mathbb{P}^{2}\) at the level of big quantum cohomology, and tropical geometry. The mirror of \(\mathbb{P}^{2}\) is taken to be \(((\mathbb{C}^{\times})^{2},W)\) where \(W\) is a Landau-Ginzburg potential. The author shows that \(W\) can be deformed by counting Maslov index two tropical disks, and the natural parameters appearing in the deformation are the flat coordinates of the complex moduli space of the mirror. Moreover, the author proves that the mirror symmetry for \(\mathbb{P}^{2}\) is equivalent to the tropical curve counting formulas including those for gravitational descendent invariants.

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14T05 Tropical geometry (MSC2010)
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