Family Floer theory, non-abelianization, and Spectral Networks

2023/07/09 by Yoon Jae Nick Nho, Nho, Yoon Jae · 2 citations
Mathematics · #53D40 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2307.04213

openalex publication_date 2023/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ϕ defined on a closed Riemann surface C, let C be the complement of the poles of ϕ. In the case where the spectral curve Σ_ϕ is exact with respect to the canonical Liouville form on T^∗C, we show that an "almost flat" GL(1;ℂ)-local system L on Σ_ϕ defines a Floer cohomology local system HF_ε(Σ_ϕ,L;ℂ) on C for 0< ε≤ 1. Then we show that for small enough ε, the non-abelianization of L is isomorphic to the family Floer cohomology local system HF_ε(Σ_ϕ,L;ℂ)

Cited by

• Multi-valued Morse homotopy for the SYZ mirror of the complex projective plane
• Higher-dimensional Heegaard Floer homology and spectral networks

Related

• Spectral Floer theory and tangential structures
• Anchored Lagrangian submanifolds and their Floer theory
• Spectral Networks and Betti Lagrangians
• Higher rank local systems in Lagrangian Floer theory
• Higher-dimensional Heegaard Floer homology and spectral networks