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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2307.04213 |
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| _version_ | 1866913476050419712 |
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| author | Nho, Yoon Jae |
| author_facet | Nho, Yoon Jae |
| contents | In this paper, we study the relationship between Gaiotto-Moore-Neitzke's non-abelianization map and Floer theory. Given a complete GMN quadratic differential $ϕ$ defined on a closed Riemann surface $C$, let $\tilde{C}$ be the complement of the poles of $ϕ$. In the case where the spectral curve $Σ_ϕ$ is exact with respect to the canonical Liouville form on $T^{\ast}\tilde{C}$, we show that an "almost flat" $GL(1;\mathbb{C})$-local system $\mathcal{L}$ on $Σ_ϕ$ defines a Floer cohomology local system $HF_ε(Σ_ϕ,\mathcal{L};\mathbb{C})$ on $\tilde{C}$ for $0< ε\leq 1$. Then we show that for small enough $ε$, the non-abelianization of $\mathcal{L}$ is isomorphic to the family Floer cohomology local system $HF_ε(Σ_ϕ,\mathcal{L};\mathbb{C})$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_04213 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Family Floer theory, non-abelianization, and Spectral Networks Nho, Yoon Jae Symplectic Geometry 53D40 In this paper, we study the relationship between Gaiotto-Moore-Neitzke's non-abelianization map and Floer theory. Given a complete GMN quadratic differential $ϕ$ defined on a closed Riemann surface $C$, let $\tilde{C}$ be the complement of the poles of $ϕ$. In the case where the spectral curve $Σ_ϕ$ is exact with respect to the canonical Liouville form on $T^{\ast}\tilde{C}$, we show that an "almost flat" $GL(1;\mathbb{C})$-local system $\mathcal{L}$ on $Σ_ϕ$ defines a Floer cohomology local system $HF_ε(Σ_ϕ,\mathcal{L};\mathbb{C})$ on $\tilde{C}$ for $0< ε\leq 1$. Then we show that for small enough $ε$, the non-abelianization of $\mathcal{L}$ is isomorphic to the family Floer cohomology local system $HF_ε(Σ_ϕ,\mathcal{L};\mathbb{C})$ |
| title | Family Floer theory, non-abelianization, and Spectral Networks |
| topic | Symplectic Geometry 53D40 |
| url | https://arxiv.org/abs/2307.04213 |