Instanton counting, Macdonald function and the moduli space of D-branes

We argue the connection of Nekrasov's partition function in the Ω background and the moduli space of D-branes, suggested by the idea of geometric engineering and Gopakumar-Vafa invariants. In the instanton expansion of N = 2 SU(2) Yang-Mills theory the Nakrasov's partition function with equivariant parameters ε1,ε2 of toric action on C2 factorizes correctly as the character of SU(2)L x SU(2)R spin representation. We show that up to two instantons the spin contents are consistent with the Lefschetz action on the moduli space of D2-branes on (local) F0. We also present an attempt at constructing a refined topological vertex in terms of the Macdonald function. The refined topological vertex with two parameters of T2 action allows us to obtain the generating functions of equivariant χy and elliptic genera of the Hilbert scheme of n points on C2 by the method of topological vertex