Why is Landau-Ginzburg link cohomology equivalent to Khovanov homology?

Abstract

Abstract In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the $$ \mathcal{N}=\left(2,2\right) $$ N = 2 2 2d Landau-Ginzburg theory in models describing link embeddings in ℝ3 to Khovanov and Khovanov-Rozansky homologies. To confirm the equivalence we exploit the invariance of Hilbert spaces of ground states for interfaces with respect to homotopy. In this attempt to study solitons and instantons in the LandauGiznburg theory we apply asymptotic analysis also known in the literature as exact WKB method, spectral networks method, or resurgence. In particular, we associate instantons in LG model to specific WKB line configurations we call null-webs.