Abstract
In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the mathcal{N}=left(2,2right) 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.
Highlights
In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the N = (2, 2) 2d Landau-Ginzburg theory in models describing link embeddings in R3 to Khovanov and Khovanov-Rozansky homologies
One associates to embedding of link L certain interface path ℘ˆ on parameter space P of the LG theory. The invariance of this construction for different link embeddings follows form the isomorphism of cohomologies: H∗(E(℘ˆ), Q) ∼= H∗(E(℘ˆ ), Q)
3.3 Sketchy review of Khovanov homology construction. As it was indicated in the introduction our final aim is to compare cohomological theory emerging in the Landau-Ginzburg model description of tangles to Khovanov homology
Summary
A quest for a TQFT describing Khovanov invariants [53], a categorification of Jones polynomials — Wilson loop averages in the 3d Chern-Simons theory [24, 69, 78], — or more desirably Khovanov-Rozansky invariants [54, 55], or even more generally superpolynomials [21], delivering categorification to HOMFLY-PT invariants, has led to a wide and profound development in the both physics and mathematics, see e.g. [3, 4, 6, 14,15,16,17, 41,42,43,44, 55, 60, 64, 73, 82,83,84].1 See a recent nice review [75]. One associates to embedding of link L certain interface path ℘ˆ on parameter space P of the LG theory (see figure 1) The invariance of this construction for different link embeddings follows form the isomorphism of cohomologies: H∗(E(℘ˆ), Q) ∼= H∗(E(℘ˆ ), Q). This analysis has been proven to be useful in estimates of eta-invariants of Dirac operators in the soliton background (see [39, appendix A]) and we use it to count h-critical. We argue that one can choose a distinguished path ℘ˆ0 on P such that the LG complex becomes isomorphic to Khovanov’s one, there is an intriguing question if one can distinguish path ℘ˆ0 from other representatives in its homotopy class giving different complexes a priori
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