Hilbert schemes of points on surfaces and conjectural link homology groups

Gukov et al. suggested triply graded link homology groups via refined BPS counting on the deformed conifold. Through large N duality they identify their Poincar\'e polynomials for the Hopf link as refined topological vertices. There is also a recent study by Awata-Kanno. I further apply the geometric engineering to interpret them as equivariant holomorphic Euler characteristics of natural vector bundles over Hilbert schemes of points on the affine plane. Then they perfectly make sense mathematically. As in the relation between Donaldson invariants wall-crossing and Nekrasov partition function, studied by G\"ottsche, Yoshioka and myself, they also gave similar holomorphic Euler characteristic for arbitrary projective surfaces.

nakajima@math.kyoto-u.ac.jp