Higher-dimensional Heegaard Floer homology and spectral networks

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