Higher-Dimensional Heegaard Floer Homology and Spectral Networks

Let C be a closed surface and Σ⊂T*C a real exact Lagrangian surface associated to a spectral curve. In this talk we will first try to explain the context of this work (e.g., Higgs bundles and spectral curves). We then construct a homomorphism from the κ-strand braid skein algebra of C to the κ-strand matrix-valued braid skein algebra of Σ via higher-dimensional Heegaard Floer homology (HDHF). Finally we explore the adiabatic limit of this homomorphism, which yields a hybrid count of HDHF-type holomorphic curves coupled with certain Morse gradient graphs, called folded Morse trees. This is joint work with Tianyu Yuan and Yin Tian.