A-type Quiver Varieties and ADHM Moduli Spaces

We study quantum geometry of Nakajima quiver varieties two different types - framed A-type quivers and ADHM quivers. While these spaces look completely we find a surprising connection between equivariant K-theories thereof with nontrivial match their parameters. In particular, demonstrate that K-theory $A_n$ in certain $n\to\infty$ limit reproduces the Hilbert scheme points on $\mathbb{C}^2$. analyze correspondence from point view enumerative geometry, representation theory integrable systems. also propose conjecture which relates spectra multiplication operators moduli solution elliptic Ruijsenaars-Schneider model.