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

Abstract

In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the $\mathcal{N}=(2,2)$ 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.

arXiv comment61 page, 14 figures

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Cited by in corpus (7)

• Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?
• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians
• Tunnels Under Geometries (or Instantons Know Their Algebras)
• Towards Catastrophe theory for Khovanov-Rozansky homology
• Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs
• Khovanov--Rozansky matrix factorization reduction for bipartite links