Euler characteristics of SU(2) instanton moduli spaces on rational elliptic surfaces

Recently, Minahan, Nemeschansky, Vafa and Warner computed partition functions for N=4 topological Yang-Mills theory on rational elliptic surfaces. In particular they computed generating functions of Euler characteristics of SU(2)-instanton moduli spaces. In mathematics, they are expected to coincide with those of Gieseker compactifications. In this paper, we compute Euler characteristics of these spaces and show that our results coincide with theirs. We also check the modular property of Z_{SU(2)} and Z_{SO(3)} conjectured by Vafa and Witten.