Floer theory and spectral networks

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_ε(\Sigma_\phi,\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(\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 \RRⁿ\cap (\RR^k\times \sqrt{-1}\RR^n-k) inside \CCⁿ. Our construction closely follows the methods used by Duval and Abouzaid and corrects an error appearing in the latter approach.

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2024

Author dc:creator

• Nho, Yoon Jae

Advisor dc:contributor.advisor

• Keating, ailsa

• symplectic
• spectral curves
• quadratic differentials

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9bd05024-5745-4459-a316-a591acfbd5b1/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.110686

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/371527