Higher-dimensional Heegaard Floer homology and spectral networks

TL;DR

This work constructs a Floer-theoretic realization of the nonabelianization map in the setting of spectral curves by producing a map from the braid skein algebra of a base surface to a matrix-valued braid skein algebra on a spectral cover via higher-dimensional Heegaard Floer theory. The authors define a holomorphic HDHF map $\mathcal{F}_{\operatorname{hol}}$ using counts of HDHF-type holomorphic curves and prove it is an algebra homomorphism; they then extend to a hybrid Floer–Morse map $\mathcal{F}_{\operatorname{Mor}}$ by counting curves coupled with folded Morse flow trees, sketching a proof that the two constructions agree in the adiabatic limit. A key technical development is the passage from the holomorphic to the real symplectic model, together with perturbation schemes and the introduction of homological variables to refine the algebraic structure. The paper also develops a detailed equivalence program, showing how a squids-to-Riemann-surfaces gluing framework and a sequence of perturbation/degeneration steps link the HDHF counts to folded Morse counts, thereby connecting spectral networks with higher-dimensional HDHF in a concrete analytic framework. Overall, the results provide a Floer-theoretic realization of GMN-type nonabelianization via braid skein algebras, enriched by homological data and adiabatic degeneration techniques.

Abstract

Given a closed surface $C$ and a real exact Lagrangian $Σ\subset T^*C$ associated to a spectral curve, we construct a homomorphism $\operatorname{BSk}_κ(C)\to\operatorname{Mat}(N^κ,\operatorname{BSk}_κ(Σ))$ 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.

Figures (14)

• Figure 1:
• Figure 2:
• Figure 3: Two possible degenerations of ${\mathcal{T}}^{\operatorname{ind}=1, \chi}(\mathbf{p}_\alpha,\mathbf{y}',\mathbf{y},\mathbf{p}_\beta)$. The weights of ends of A are labeled.
• Figure 4: The nodal degeneration of ${\mathcal{T}}^{\operatorname{ind}=1, \chi}(\mathbf{p}_\alpha,\mathbf{y}',\mathbf{y},\mathbf{p}_\beta)$.
• Figure 5: Left: the projection to $\mathbb{R}^2_{x,y}$ of the front of the Legendrian lift of $\Sigma$ near the ramification point. The blue rays denote the transversal intersection of two sheets. Right: we apply a small Hamiltonian perturbation to $\Sigma$. The red edges are cusps and the red dots are swallowtail singularities.

Theorems & Definitions (54)

• Remark 1.1
• Remark 1.3
• Theorem 1.6
• Corollary 1.7
• Proposition 1.8
• Conjecture 1.9
• Remark 1.10
• Definition 2.1: Morton-Samuelson
• Definition 2.2
• Remark 2.3