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

TL;DR

This paper studies the quantum difference equations related to the Hilbert scheme of points in a2^2, providing explicit descriptions of their monodromy through representation theory and algebraic geometry, and connecting them to K-theoretic R-matrices.

Contribution

It offers two explicit descriptions of the monodromy of quantum difference equations for a2^2, linking representation theory with algebraic geometry and identifying monodromy matrices with K-theoretic R-matrices.

Findings

Monodromy acts via explicit elements in quantum toroidal algebra a1.

Transition matrices between stable envelope bases describe monodromy.

Monodromy matrices correspond to K-theoretic R-matrices of cyclic quiver varieties.

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.