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 $K=SO(N,R)$, $N\ge 5$, we prove that $U_n^K$ is an irreducible normal variety with smooth locus $M_n^K$. Hence, together with the author's previous result, 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.

40 pages, v.2: typos corrected, v.3: dedicatory and acknowledgement messages added, references updated, to appear in Adv. Math