This is a joint work with A. Oblomkov exploring the relation between the HOMFLY-PT link homology and coherent sheaves over the Hilbert scheme of points on C^2.

We consider a special object in the 2-category related to the Hilbert scheme of n points on C^2. We define a homomorphism from the braid group on n strands to the monoidal category of endomorphisms of this object. We prove that the space of morphisms between the images of a braid and of the identity braid is the invariant of a link constructed by closing the braid. Conjecturally, this space is the triply-graded HOMFLY-PT homology.

From the TQFT point of view, we consider a B-twisted 3d N=4 SUSY YM with matter, whose Higgs branch is the Hilbert scheme.

Link homology appears as the Hilbert space of a 2-disk. Its boundary carries a flag variety-based sigma model, and the Kahler parameters of the flag variety braid as one goes around the disk.

From the IIA string theory point of view, the points on C^2 are BPS particles coming from a 2-disk shaped stack of n D2-branes located in one of the fibers at the North Pole of P^1, which forms the base of a resolved conifold. The D2-branes end on a stack of NS5 branes which form a closed braid in the other fiber.