Moduli spaces of framed symplectic and orthogonal bundles on P2P2 and the KK-theoretic Nekrasov partition functions

Let KK be the compact Lie group USp(N/2) or SO(N,R). Let MnK be the moduli space of framed KK-instantons over S4S4 with the instanton number nn. By Donaldson (1984), MnK is endowed with a natural scheme structure. It is a Zariski open subset of a GIT quotient of μ−1(0)μ−1(0), where μμ is a holomorphic moment map such that μ−1(0)μ−1(0) consists of the ADHM data.The purpose of the paper is to study the geometric properties of μ−1(0)μ−1(0) and its GIT quotient, such as complete intersection, irreducibility, reducedness and normality. If K=USp(N/2) then μμ is flat and μ−1(0)μ−1(0) is an irreducible normal variety for any nn and even NN. If K=SO(N,R) the similar results are proven for low nn and NN.As an application one can obtain a mathematical interpretation of the KK-theoretic Nekrasov partition function of Nekrasov and Shadchin (2004).