Instantons and Bows for the Classical Groups

TL;DR

This paper classifies holomorphic instantons on multi-Taub-NUT spaces, linking them to bow solutions and Coulomb branches, and provides a monad construction for instanton bundles with classical Lie groups.

Contribution

It extends the classification of instantons to ALF multi-Taub-NUT spaces and connects them to bow solutions and mirror gauge theories, offering a monad construction for instanton bundles.

Findings

Classification of instantons on multi-Taub-NUT spaces established.

Connection between bow solutions, Coulomb branches, and instantons demonstrated.

Monad construction for holomorphic instanton bundles provided for classical Lie groups.

Abstract

The construction of Atiyah, Drinfeld, Hitchin, and Manin [ADHM78] provided complete description of all instantons on Euclidean four-space. It was extended by Kronheimer and Nakajima to instantons on ALE spaces, resolutions of orbifolds $\mathbb{R}^4/\Gamma$ by a finite subgroup $\Gamma\subset SU(2).$ We consider a similar classification, in the holomorphic context, of instantons on some of the next spaces in the hierarchy, the ALF multi-Taub-NUT manifolds, showing how they tie in to the bow solutions to Nahm's equations [Che09] via the Nahm correspondence. Recently in [Nak18a] and [NT17], based on [Nak03], Nakajima and Takayama constructed the Coulomb branch of the moduli space of vacua of a quiver gauge theory, tying them to the same space of bow solutions. One can view our construction as describing the same manifold as the Higgs branch of the mirror gauge theory [COS11]. Our construction also yields the monad construction of holomorphic instanton bundles on the multi-Taub-NUT space for any classical compact Lie structure group.