A proof of N.Takahashi's conjecture for $(\mathbb{P}^2,E)$ and a refined sheaves/Gromov-Witten correspondence
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866911179639619584 |
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| author | Bousseau, Pierrick |
| author_facet | Bousseau, Pierrick |
| contents | 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 $y=e^{i \hbar}$, where $y$ is the refined/cohomological variable on the sheaf side, and $\hbar$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1909_02992 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | A proof of N.Takahashi's conjecture for $(\mathbb{P}^2,E)$ and a refined sheaves/Gromov-Witten correspondence Bousseau, Pierrick Algebraic Geometry High Energy Physics - Theory Symplectic Geometry 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 $y=e^{i \hbar}$, where $y$ is the refined/cohomological variable on the sheaf side, and $\hbar$ 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. |
| title | A proof of N.Takahashi's conjecture for $(\mathbb{P}^2,E)$ and a refined sheaves/Gromov-Witten correspondence |
| topic | Algebraic Geometry High Energy Physics - Theory Symplectic Geometry |
| url | https://arxiv.org/abs/1909.02992 |