A-type Quiver Varieties and ADHM Moduli Spaces

TL;DR

This paper explores the quantum geometry of A-type quiver varieties and ADHM moduli spaces, revealing surprising connections in their equivariant K-theories and linking them to integrable systems and enumerative geometry.

Contribution

It uncovers a novel relationship between A-type quiver varieties and ADHM moduli spaces through equivariant K-theory and proposes a conjecture connecting quantum multiplication spectra to the elliptic Ruijsenaars-Schneider model.

Findings

Quantum K-theory of A_n quiver varieties approaches that of Hilbert schemes as n→∞.

A conjecture relates spectra of quantum multiplication operators to elliptic Ruijsenaars-Schneider solutions.

The study bridges enumerative geometry, representation theory, and integrable systems.

Abstract

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