Skein traces from curve counting
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914107584675840 |
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| author | Ekholm, Tobias Longhi, Pietro Park, Sunghyuk Shende, Vivek |
| author_facet | Ekholm, Tobias Longhi, Pietro Park, Sunghyuk Shende, Vivek |
| contents | Given a 3-manifold $M$, and a branched cover arising from the projection of a Lagrangian 3-manifold $L$ in the cotangent bundle of $M$ to the zero-section, we define a map from the skein of $M$ to the skein of $L$, via the skein-valued counting of holomorphic curves. When $M$ and $L$ are products of surfaces and intervals, we show that wall crossings in the space of the branched covers obey a skein-valued lift of the Kontsevich-Soibelman wall-crossing formula.
Holomorphic curves in cotangent bundles correspond to Morse flow graphs; in the case of branched double covers, this allows us to give an explicit formula for the the skein trace. After specializing to the case where $M$ is a surface times an interval, and additionally specializing the HOMFLYPT skein to the $\mathfrak{gl}(2)$ skein on $M$ and the $\mathfrak{gl}(1)$ skein on $L$, we recover an existing prescription of Neitzke and Yan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19041 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Skein traces from curve counting Ekholm, Tobias Longhi, Pietro Park, Sunghyuk Shende, Vivek Symplectic Geometry High Energy Physics - Theory Geometric Topology Quantum Algebra Given a 3-manifold $M$, and a branched cover arising from the projection of a Lagrangian 3-manifold $L$ in the cotangent bundle of $M$ to the zero-section, we define a map from the skein of $M$ to the skein of $L$, via the skein-valued counting of holomorphic curves. When $M$ and $L$ are products of surfaces and intervals, we show that wall crossings in the space of the branched covers obey a skein-valued lift of the Kontsevich-Soibelman wall-crossing formula. Holomorphic curves in cotangent bundles correspond to Morse flow graphs; in the case of branched double covers, this allows us to give an explicit formula for the the skein trace. After specializing to the case where $M$ is a surface times an interval, and additionally specializing the HOMFLYPT skein to the $\mathfrak{gl}(2)$ skein on $M$ and the $\mathfrak{gl}(1)$ skein on $L$, we recover an existing prescription of Neitzke and Yan. |
| title | Skein traces from curve counting |
| topic | Symplectic Geometry High Energy Physics - Theory Geometric Topology Quantum Algebra |
| url | https://arxiv.org/abs/2510.19041 |