Quantum difference equation for the affine type $A$ quiver varieties I: General Construction

TL;DR

This paper constructs the quantum difference equation for affine type A quiver varieties using quantum toroidal algebra, defining wall sets via the universal R-matrix and illustrating with examples like instanton moduli spaces.

Contribution

It introduces a new construction of quantum difference equations for affine type A quiver varieties using the quantum toroidal algebra and universal R-matrix, linking to stable envelopes.

Findings

Explicit quantum difference operators for specific examples

Wall sets defined via universal R-matrix actions

Connection between wall sets and stable envelopes

Abstract

In this article we use the philosophy in [OS22] to construct the quantum difference equation of affine type $A$ quiver varieties in terms of the quantum toroidal algebra $U_{q,t}(\hat{\hat{\mathfrak{sl}}}_{r})$. In the construction, and we define the set of wall for each quiver varieties by the action of the universal $R$-matrix, which is shown to be almost equivalent to that of the $K$-theoretic stable envelope on each interval in $H^2(X,\mathbb{Q})$. We also give the examples of the instanton moduli space $M(n,r)$ and the equivariant Hilbert scheme $\text{Hilb}_{n}([\mathbb{C}^2/\mathbb{Z}_{r}])$ to show the explicit form of the quantum difference operator.