Geometry of Uhlenbeck partial compactification of orthogonal instanton spaces and the K-theoretic Nekrasov partition functions

Abstract Let M n K be the moduli space of framed K -instantons over S 4 with instanton number n when K is a compact simple Lie group of classical type. Let U n K be the Uhlenbeck partial compactification of M n K . A scheme structure on U n K is endowed by Donaldson as an algebro-geometric Hamiltonian reduction of ADHM data. In this paper, for K = SO ( N , R ) , N ≥ 5 , we prove that U n K is an irreducible normal variety with smooth locus M n K . Hence, together with the author's previous results on USp ( N ) , the K-theoretic Nekrasov partition function for any simple classical group other than SO ( 3 , R ) , is interpreted as a generating function of Hilbert series of the instanton moduli spaces. Using this approach we also study the case K = SO ( 4 , R ) which is the unique semisimple but non-simple classical group.