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In this paper, we study the relationship between Gaiotto-Moore-Neitzke'snon-abelianization map and Floer theory. Given a complete GMN quadraticdifferential $\phi$ defined on a closed Riemann surface $C$, let $\tilde{C}$ bethe complement of the poles of $\phi$. In the case where the spectral curve$\Sigma_{\phi}$ is exact with respect to the canonical Liouville form on$T^{\ast}\tilde{C}$, we show that an "almost flat" $GL(1;\mathbb{C})$-localsystem $\mathcal{L}$ on $\Sigma_{\phi}$ defines a Floer cohomology local system$HF_t(\Sigma_{\phi},\mathcal{L};\mathbb{C})$ on $\tilde{C}$ for $0< t\leq 1$.Then we show that for small enough $t$, the non-abelianization of $\mathcal{L}$is isomorphic to the family Floer cohomology local system$HF_t(\Sigma_{\phi},\mathcal{L};\mathbb{C})$

Family Floer theory, non-abelianization, and Spectral Networks