Deformed Khovanov-Rozansky homology and the isospectral Hilbert scheme
A family of conjectures by Gorsky-Negut-Rasmussen asserts a close relationship between Khovanov-Rozansky link homology and Hilbert schemes of points in C^2, which categorifies a known relationship between the HOMFLY-PT link polynomial and the ring of symmetric functions.
In this talk I will discuss a deformed (in the spirit of Batson-Seed) version of Khovanov-Rozansky homology, as constructed and studied in joint work with Eugene Gorsky. Our main result explicitly computes the "full-twist algebra" formed by taking the direct sum of homologies of the torus links T(n,nk) for fixed n and k\geq 0. Proj of this algebra is the "isospectral Hilbert scheme", by work of Haiman. By setting the deformation parameters equal to zero, we obtain a concrete connection between the usual Khovanov-Rozansky homology and Hilbert schemes. This can be viewed as a step in the direction of the GNR conjectures.