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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.04213 |
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Table of 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 $HF_ε(Σ_ϕ,\mathcal{L};\mathbb{C})$