We prove N. Takahashi’s conjecture determining the contribution of each contact point in genus-0 maximal contact Gromov–Witten theory of ${\mathbb{P}}^2$ relative to a smooth cubic E. This is a new example of a question in Gromov–Witten theory that can be fully solved despite the presence of contracted components and multiple covers. The proof relies on a tropical computation of the Gromov–Witten invariants and on the interpretation of the tropical picture as describing wall-crossing in the derived category of coherent sheaves on ${\mathbb{P}}^2$.

The same techniques allow us to prove a new sheaves/Gromov–Witten correspondence, relating Betti numbers of moduli spaces of one-dimensional Gieseker semistable sheaves on ${\mathbb{P}}^2$, or equivalently, refined genus-0 Gopakumar–Vafa invariants of local ${\mathbb{P}}^2$, with higher-genus maximal contact Gromov–Witten theory of $({\mathbb{P}}^2,E)$. The correspondence involves the nontrivial change of variables $y=e^{i\hslash }$, where y is the refined/cohomological variable on the sheaf side, and ℏ is the genus variable on the Gromov–Witten side. We explain how this correspondence can be heuristically motivated by a combination of mirror symmetry and hyper-Kähler rotation.