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

Abstract

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 which 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 ${\mathbb{P}}^2$, giving a translation of the original Gromov-Witten question into a known statement about Euler characteristics of moduli spaces of one-dimensional Gieseker semistable 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 non-trivial change of variables $y=e^{i\hslash }$, where $y$ is the refined/cohomological variable on the sheaf side, and $\hslash$ 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 hyperkähler rotation.

54 pages. Final version published in Duke Mathematical Journal