Higher-dimensional Heegaard Floer homology and spectral networks

TL;DR

This paper constructs a new algebraic homomorphism linking braid skein algebras of surfaces and spectral curves using advanced Floer homology techniques, revealing deep connections between topology, geometry, and algebra.

Contribution

It introduces a novel homomorphism from the braid skein algebra of a surface to a matrix-valued algebra of a spectral curve via higher-dimensional Heegaard Floer homology, combining Floer and Morse theories.

Findings

Established a homomorphism between braid skein algebras using HDHF.

Connected holomorphic curve counts with Morse gradient graphs.

Provided a sketch of equivalence with a hybrid Floer-Morse approach.

Abstract

Given a closed surface $C$ and a real exact Lagrangian $\Sigma \subset T^*C$ associated to a spectral curve, we construct a homomorphism $\operatorname{BSk}_\kappa(C)\to\operatorname{Mat}(N^{\kappa},\operatorname{BSk}_\kappa(\Sigma))$ from the braid skein algebra of $C$ to the matrix-valued braid skein algebra of $\Sigma$ 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.