Categorification of Quantum Invariants via Matrix Factorizations in Non-Perturbative Chern-Simons Gauge Theory

Mathematical Applications of Science Fiction We construct a rigorous isomorphism between the triply-graded Khovanov-Rozansky homology of a link L ⊂ S3and the hypercohomology of a specific perverse sheaf sup-ported on the affine Grassmannian of the gauge groupG = SL(N, C). By analyzing the Landau-Ginzburg modelsassociated with the decomposition of the potential W in theChern-Simons path integral, we derive a spectral sequenceconverging to the motivic cohomology of the Hilbert schemeof points on the singular surface defined by the spectralcurve of the link. We demonstrate that the Euler charac-teristic of this homology theory recovers the quantum slNinvariant, while the higher-order differentials encode non-perturbative instanton corrections. Furthermore, we provethat the category of matrix factorizations associated withthe link diagram is equivalent to the derived category ofcoherent sheaves on a specific derived stack, providing a ge-ometric interpretation of the so-called "anomalous" gradingshifts in terms of Etale homotopy types.