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

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 $U_{q,t}({\widehat{\widehat{\mathfrak{s}\mathfrak{l}}}}_r)$. 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 ${\text{Hilb}}_n([{\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_r])$ to show the explicit form of the quantum difference operator.