Nekrasov Partition Function Research Articles

Instanton counting, Macdonald function and the moduli space ofD-branes

We argue the connection of Nekrasov's partition function in the \Omega 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 \epsilon_1, \epsilon_2 of toric action on C^2 factorizes correctly as the character of SU(2)_L \times 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) F_0. 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 T^2 action allows us to obtain the generating functions of equivariant \chi_y and elliptic genera of the Hilbert scheme of n points on C^2 by the method of topological vertex.

Asymptotic form of Gopakumar–Vafa invariants from instanton counting

We study the asymptotic form of the Gopakumar–Vafa invariants at all genera for Calabi–Yau toric threefolds which have the structure of fibration of the A n singularity over P 1. We claim that the asymptotic form is the inverse Laplace transform of the corresponding instanton amplitude in the prepotential of N=2 SU( n+1) gauge theory coupled to external graviphoton fields, which is given by the logarithm of the Nekrasov's partition function.