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| Κύριοι συγγραφείς: | , , |
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| Μορφή: | Preprint |
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2020
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| Διαθέσιμο Online: | https://arxiv.org/abs/2004.09355 |
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| _version_ | 1866913781571911680 |
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| author | Cao, Yalong Kool, Martijn Monavari, Sergej |
| author_facet | Cao, Yalong Kool, Martijn Monavari, Sergej |
| contents | In 2008, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold $X$ using Gromov-Witten theory. Recently, Cao-Maulik-Toda proposed a conjectural description of these invariants in terms of stable pair theory. When $X$ is the total space of the sum of two line bundles over a surface $S$, and all stable pairs are scheme theoretically supported on the zero section, we express stable pair invariants in terms of intersection numbers on Hilbert schemes of points on $S$. As an application, we obtain new verifications of the Cao-Maulik-Toda conjectures for low degree curve classes and find connections to Carlsson-Okounkov numbers. Some of our verifications involve genus zero Gopakumar-Vafa type invariants recently determined in the context of the log-local principle by Bousseau-Brini-van Garrel. Finally, using the vertex formalism, we provide a few more verifications of the Cao-Maulik-Toda conjectures when thickened curves contribute and also for the case of local $\mathbb{P}^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_09355 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Stable pair invariants of local Calabi-Yau 4-folds Cao, Yalong Kool, Martijn Monavari, Sergej Algebraic Geometry High Energy Physics - Theory Mathematical Physics In 2008, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold $X$ using Gromov-Witten theory. Recently, Cao-Maulik-Toda proposed a conjectural description of these invariants in terms of stable pair theory. When $X$ is the total space of the sum of two line bundles over a surface $S$, and all stable pairs are scheme theoretically supported on the zero section, we express stable pair invariants in terms of intersection numbers on Hilbert schemes of points on $S$. As an application, we obtain new verifications of the Cao-Maulik-Toda conjectures for low degree curve classes and find connections to Carlsson-Okounkov numbers. Some of our verifications involve genus zero Gopakumar-Vafa type invariants recently determined in the context of the log-local principle by Bousseau-Brini-van Garrel. Finally, using the vertex formalism, we provide a few more verifications of the Cao-Maulik-Toda conjectures when thickened curves contribute and also for the case of local $\mathbb{P}^3$. |
| title | Stable pair invariants of local Calabi-Yau 4-folds |
| topic | Algebraic Geometry High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2004.09355 |