Moduli spaces of framed symplectic and orthogonal bundles on P2 and the K-theoretic Nekrasov partition functions

TL;DR

This paper investigates the geometric properties of moduli spaces of framed symplectic and orthogonal bundles on P2, linking them to the K-theoretic Nekrasov partition functions and providing new insights into their structure.

Contribution

It analyzes the geometric structure of the ADHM data and moduli spaces for symplectic and orthogonal instantons, establishing properties like irreducibility and normality, and connects these to Nekrasov's partition functions.

Findings

For USp(N/2), the ADHM data space is a normal irreducible variety.

For low N and n, similar geometric properties hold for SO(N,R) cases.

The work provides a mathematical interpretation of the K-theoretic Nekrasov partition function.

Abstract

Let $K$ be the compact Lie group $USp(N/2)$ or $SO(N, R)$. Let $M^K_n$ be the moduli space of framed K-instantons over $S^4$ with the instanton number $n$. By Donaldson (1984), $M^K_n$ is endowed with a natural scheme structure. It is a Zariski open subset of a GIT quotient of $\mu^{-1}(0)$, where $\mu$ is a holomorphic moment map such that $\mu^{-1}(0)$ consists of the ADHM data. The purpose of the paper is to study the geometric properties of $\mu^{-1}(0)$ and its GIT quotient, such as complete intersection, irreducibility, reducedness and normality. If $K=USp(N/2)$ then $\mu$ is flat and $\mu^{-1}(0)$ is an irreducible normal variety for any $n$ and even $N$. If $K = SO(N, R)$ the similar results are proven for low $n$ and $N$. As an application one can obtain a mathematical interpretation of the K-theoretic Nekrasov partition function of Nekrasov and Shadchin (2004).