A PROOF OF N. TAKAHASHI'S CONJECTURE FOR (P2, E) AND A REFINED SHEAVES/GROMOV-WITTEN CORRESPONDENCE

We prove N. Takahashi's conjecture determining the contribution of each contact point in genus-0 maximal contact Gromov-Witten theory of P2 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 P2.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 P2, or equivalently, refined genus-0 Gopakumar-Vafa invariants of local P2, with higher-genus maximal contact Gromov-Witten theory of (P2, E). The correspondence involves the nontrivial change of variables y D ein, where y is the refined/cohomological variable on the sheaf side, and h 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-Kahler rotation.