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.
Categories: math.RT hep-th math-ph math.MP math.QA