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We prove N.Takahashi's conjecture determining the contribution of eachcontact 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 aquestion in Gromov-Witten theory which can be fully solved despite the presenceof contracted components and multiple covers. The proof relies on a tropicalcomputation of the Gromov-Witten invariants and on the interpretation of thetropical picture as describing wall-crossing in the derived category ofcoherent sheaves on $\mathbb{P}^2$, giving a translation of the originalGromov-Witten question into a known statement about Euler characteristics ofmoduli spaces of one-dimensional Gieseker semistable sheaves on $\mathbb{P}^2$.  The same techniques allow us to prove a new sheaves/Gromov-Wittencorrespondence, relating Betti numbers of moduli spaces of one-dimensionalGieseker semistable sheaves on $\mathbb{P}^2$, or equivalently refinedgenus-$0$ Gopakumar-Vafa invariants of local $\mathbb{P}^2$, with higher-genusmaximal contact Gromov-Witten theory of $(\mathbb{P}^2,E)$. The correspondenceinvolves the non-trivial change of variables $y=e^{i \hbar}$, where $y$ is therefined/cohomological variable on the sheaf side, and $\hbar$ is the genusvariable on the Gromov-Witten side. We explain how this correspondence can beheuristically motivated by a combination of mirror symmetry and hyperk\"ahlerrotation.

A proof of N.Takahashi's conjecture for $(\mathbb{P}^2,E)$ and a refined sheaves/Gromov-Witten correspondence