Non-archimedean cylinder counts are logarithmic Gromov-Witten invariants
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914105134153728 |
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| author | Hinault, Thorgal YU, Tony Yue |
| author_facet | Hinault, Thorgal YU, Tony Yue |
| contents | We establish a comparison result relating non-archimedean cylinder counts and logarithmic cylinder counts in a smooth affine log Calabi-Yau variety. Using the decomposition theorem and the gluing formula from log Gromov-Witten theory, we can express logarithmic cylinder counts in terms of wall type invariants. As a corollary, we show that in the surface case the non-archimedean scattering diagram from Keel-Yu and the logarithmic scattering diagram from Gross-Siebert coincide, and deduce that the two mirror constructions agree. Along the way, we prove the exponential formula, expressing the non-archimedean wall-crossing function as the exponential of a generating series of punctured log Gromov-Witten invariants. This provides the first explicit formula relating counts of non-archimedean curves with boundary to punctured log invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-archimedean cylinder counts are logarithmic Gromov-Witten invariants Hinault, Thorgal YU, Tony Yue Algebraic Geometry Symplectic Geometry Primary 14N35, Secondary 14J33, 14G22, 14A21 We establish a comparison result relating non-archimedean cylinder counts and logarithmic cylinder counts in a smooth affine log Calabi-Yau variety. Using the decomposition theorem and the gluing formula from log Gromov-Witten theory, we can express logarithmic cylinder counts in terms of wall type invariants. As a corollary, we show that in the surface case the non-archimedean scattering diagram from Keel-Yu and the logarithmic scattering diagram from Gross-Siebert coincide, and deduce that the two mirror constructions agree. Along the way, we prove the exponential formula, expressing the non-archimedean wall-crossing function as the exponential of a generating series of punctured log Gromov-Witten invariants. This provides the first explicit formula relating counts of non-archimedean curves with boundary to punctured log invariants. |
| title | Non-archimedean cylinder counts are logarithmic Gromov-Witten invariants |
| topic | Algebraic Geometry Symplectic Geometry Primary 14N35, Secondary 14J33, 14G22, 14A21 |
| url | https://arxiv.org/abs/2510.18319 |