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

Abstract

Following the philosophy of \cite{OS22}, we construct quantum difference equations for affine type $A$ quiver varieties in terms of the quantum toroidal algebra $U_{q,t}({\widehat{\hat{\mathfrak{sl}}}}_n)$. As part of the construction, we define the wall set of each quiver variety through the action of the universal $R$-matrix and show that, on each interval in $H^2(X,\mathbb{Q})$, this wall set is essentially equivalent to the one determined by the $K$-theoretic stable envelope. We illustrate the construction by deriving explicit quantum difference operators for the instanton moduli space $M(n,r)$ and the equivariant Hilbert scheme ${\text{Hilb}}_n([{\mathbb{C}}^2/{\mathbb{Z}}_r])$.

arXiv comment53 pages. Revised version. Comments are welcome!