Mirror symmetry for P2 and tropical geometry

This paper explores the relationship between mirror symmetry for P2, at the level of big quantum cohomology, and tropical geometry. The mirror of P2 is typically taken to be (2(C×),W), where W is a Landau–Ginzburg potential of the form x+y+1/xy. The complex moduli space of the mirror is the universal unfolding of W, and oscillatory integrals produce a Frobenius manifold structure on this universal unfolding. We show that W can be deformed by counting Maslov index two tropical disks, and the natural parameters appearing in this deformation are then the flat coordinates on the moduli space. Furthermore, the oscillatory integrals are shown to read off directly tropical curve counts from the potential. Thus we show in fact that mirror symmetry for P2 is equivalent in a strong sense to tropical curve counting formulas, including tropical formulas for gravitational descendent invariants.