Quantum differential and difference equations for $\mathrm{Hilb}^{n}(\mathbb{C}^2)$

Abstract: We consider the quantum difference equation of the Hilbert scheme of points in $\mathbb{C}^2$. This equation is the K-theoretic generalization of the quantum differential equation discovered by A. Okounkov and R. Pandharipande. We obtain two explicit descriptions for the monodromy of these equations - representation-theoretic and algebro-geometric. In the representation-theoretic description, the monodromy acts via certain explicit elements in the quantum toroidal algebra $\frak{gl}_1$. In the algebro-geometric description, the monodromy features as transition matrices between the stable envelope bases in the equivariant K-theory and elliptic cohomology. Using the second approach we identify the monodromy matrices for the differential equation with the K-theoretic $R$-matrices of cyclic quiver varieties, which appear as subvarieties in the $3D$-mirror Hilbert scheme. Most of the results in the paper are illustrated by explicit examples for cases $n=2$ and $n=3$ in the Appendix.