The $SO(3)$-instanton moduli space and tensor products of ADHM data

TL;DR

This paper proves flatness of the moment map for the $SO(3)$-instanton moduli space, completing the geometric interpretation of Nekrasov partition functions for classical groups and providing explicit ADHM descriptions for tensor products.

Contribution

It establishes the flatness of the moment map for $SO(3)$-instantons, enabling a complete geometric interpretation of Nekrasov partition functions for classical groups.

Findings

Proves the flatness of the moment map for $SO(3)$-instantons.

Provides explicit ADHM data for tensor products of instanton bundles.

Completes the geometric interpretation of Nekrasov partition functions for classical groups.

Abstract

Let $M^K_n$ be the moduli space of framed $K$-instantons with instanton number $n$ when $K$ is a compact simple Lie group of classical type. Due to Donaldson's theorem, its scheme structure is given by the regular locus of a GIT quotient of $\mu^{-1}(0)$ where $\mu$ is the moment map on the associated symplectic vector space of ADHM data. A main theorem of this paper asserts that $\mu$ is flat for $K=SO(3,\mathbb{R})$ and any $n\ge0$. Hence we complete the interpretation of the K-theoretic Nekrasov partition function for the classical groups in Nekrasov-Shadchin's work in term of Hilbert series of the instanton moduli spaces together with the author's previous results. We also write ADHM data for the second symmetric and exterior products of the associated vector bundle of an instanton. This gives an explicit quiver-theoretic description of the isomorphism $M^{K}_{n}\cong M^{K'}_{n}$ for all the pairs $K,K'$ with isomorphic Lie algebras.