Higher-dimensional Heegaard Floer homology and spectral networks

2026/01/22 by Ko Honda, Yin Tian, Tianyu Yuan · 1 voice · 1 citation
Mathematics · #math.SG

paper · pdf

Abstract

Given a closed surface C and a real exact Lagrangian Σ⊂ T^*C associated to a spectral curve, we construct a homomorphism BSk_κ(C)\toMat(N^κ,BSk_κ(Σ)) from the braid skein algebra of C to the matrix-valued braid skein algebra of Σ using Floer theory and in particular higher-dimensional Heegaard Floer homology (HDHF). We sketch a proof that this map coincides with a hybrid Floer-Morse approach which counts HDHF-type holomorphic curves coupled with certain Morse gradient graphs -- called fold\-ed Morse trees -- using a variant of the adiabatic limit theorems of Fukaya-Oh and Ekholm, which compares holomorphic curves and Morse flow trees.

Citations

• Spectral Networks and Betti Lagrangians
• Family Floer theory, non-abelianization, and Spectral Networks
• Adiabatic compactness for holomorphic curves with boundary on nearby Lagrangians
• Higher-dimensional Heegaard Floer homology and Hecke algebras
• Applications of higher-dimensional Heegaard Floer homology to contact topology
• q-nonabelianization for line defects
• Counting embedded curves in symplectic 6-manifolds
• DAHAs and skein theory
• Skeins on Branes
• Spectral networks
• On the wrapped Fukaya category and based loops
• Rigidity and uniruling for Lagrangian submanifolds
• Morse flow trees and Legendrian contact homology in 1–jet spaces
• Morse theory for Lagrangian intersections

Cited by

• From Morse Trees to J-Holomorphic Discs -- Rigid Y-Graphs

Discussions

• Ko Honda, Yin Tian, Tianyu Yuan: Higher-dimensional Heegaard Floer homology and spectral networks https://arxiv.org/abs/2601.15923 https://arxiv.org/pdf/2601.15923 https://arxiv.org/html/2601.15923 [bsky, 0 points, 0 comments]

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