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

TL;DR

This paper investigates the geometric structure of Uhlenbeck partial compactifications of orthogonal instanton moduli spaces, establishing their irreducibility and normality for certain groups, and connects these findings to K-theoretic Nekrasov partition functions.

Contribution

It proves irreducibility and normality of Uhlenbeck compactifications for $SO(N,R)$ with $N extgreater 4$, and relates these geometric results to Nekrasov partition functions.

Findings

Uhlenbeck compactifications are irreducible normal varieties for $SO(N,R)$, $N extgreater 4$.

The K-theoretic Nekrasov partition function is a generating function of Hilbert series of instanton moduli spaces.

The case $SO(4,R)$ is also studied, revealing unique properties.

Abstract

Let $M^K_n$ 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 $U^{K}_{n}$ be the Uhlenbeck partial compactification of $M^{K}_{n}$. A scheme structure on $U^{K}_{n}$ is endowed by Donaldson as an algebro-geometric Hamiltonian reduction of ADHM data. In this paper, for $K=SO(N,R)$, $N\ge5$, we prove that $U^{K}_{n}$ is an irreducible normal variety with smooth locus $M^{K}_{n}$. 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.