Instanton counting on blowup. I. 4-dimensional pure gauge theory

We give a mathematically rigorous proof of Nekrasov's conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on R-4 gives a deformation of the Seiberg-Witten prepotential for N = 2 SUSY Yang-Mills theory. Through a study of moduli spaces on the blowup of R-4, we derive a differential equation for the Nekrasov's partition function. It is a deformation of the equation for the Seiberg-Witten prepotential, found by Losev et al., and further studied by Gorsky et al.