Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology?

TL;DR

This paper explores the equivalence between Landau-Ginzburg link cohomology and Khovanov homology by analyzing ground state Hilbert spaces and employing asymptotic methods like WKB and spectral networks.

Contribution

It demonstrates the equivalence by leveraging invariance of ground state Hilbert spaces and introduces null-webs as a novel association of instantons with WKB line configurations.

Findings

Confirmed the invariance of ground state Hilbert spaces under homotopy.

Established a correspondence between instantons and null-webs.

Provided a new perspective connecting Landau-Ginzburg models with Khovanov homology.

Abstract

In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the ${\cal N}=(2,2)$ 2d Landau-Ginzburg theory in models describing link embeddings in ${\mathbb{R}}^3$ 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 Landau-Giznburg 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.