Family Floer theory, non-abelianization, and Spectral Networks

Abstract

In this paper, we study the relationship between Gaiotto-Moore-Neitzke's non-abelianization map and Floer theory. Given a complete GMN quadratic differential $\varphi$ defined on a closed Riemann surface $C$, let $\tilde{C}$ be the complement of the poles of $\varphi$. In the case where the spectral curve ${\Sigma }_{\varphi }$ 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 ${\Sigma }_{\varphi }$ defines a Floer cohomology local system $H{F}_{\epsilon }({\Sigma }_{\varphi },\mathcal{L};\mathbb{C})$ on $\tilde{C}$ for $0<\epsilon \le 1$. Then we show that for small enough $\epsilon$, the non-abelianization of $\mathcal{L}$ is isomorphic to the family Floer cohomology local system $H{F}_{\epsilon }({\Sigma }_{\varphi },\mathcal{L};\mathbb{C})$

108 pages, 17 figures. Comments welcome! 29/11/2023 minor fix v2 89 pages, 11 figures, significant changes in the exposition. 18/03/2024 typo fixes v 21/08 2024 Thesis accepted version

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