Quantum Differential and Difference Equations for \(\textrm{Hilb}^{n}(\mathbb {C}^2)\)

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 in [27]. 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 . In the algebro-geometric description, the monodromy features as transition matrices between the stable envelope bases in 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.