Higher-dimensional Heegaard Floer homology and spectral networks

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.