Floer theory and spectral networks

Abstract

This thesis consists of two papers. In the first 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 $HF_{\epsilon}(\Sigma_{\phi},\mathcal{L};\mathbb{C})$. In the second paper, we extend Groman and Solomon's reverse isoperimetric inequality to pseudoholomorphic curves with punctures at the boundary and whose boundary components lie in a collection of Lagrangian submanifolds with intersections locally modelled on $\RR^n\cap (\RR^{k}\times \sqrt{-1}\RR^{n-k})$ inside $\CC^n$. Our construction closely follows the methods used by Duval and Abouzaid and corrects an error appearing in the latter approach.