  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.
