Family Floer theory, non-abelianization, and Spectral Networks

TL;DR

This paper explores the connection between spectral networks, non-abelianization, and Floer theory, establishing an isomorphism between non-abelianization of local systems and Floer cohomology local systems on Riemann surfaces.

Contribution

It demonstrates that under certain conditions, the non-abelianization map corresponds to a Floer cohomology local system, linking spectral network constructions with Floer theory.

Findings

Non-abelianization corresponds to Floer cohomology local systems.

Spectral curve exactness ensures the isomorphism.

Results connect spectral networks with Floer theory in symplectic geometry.

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 $\phi$ defined on a closed Riemann surface $C$, let $\tilde{C}$ be the 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" $GL(1;\mathbb{C})$-local system $\mathcal{L}$ on $\Sigma_{\phi}$ defines a Floer cohomology local system $HF_{\epsilon}(\Sigma_{\phi},\mathcal{L};\mathbb{C})$ on $\tilde{C}$ for $0< \epsilon\leq 1$. Then we show that for small enough $\epsilon$, the non-abelianization of $\mathcal{L}$ is isomorphic to the family Floer cohomology local system $HF_{\epsilon}(\Sigma_{\phi},\mathcal{L};\mathbb{C})$