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 ${\mathfrak{gl}}_1$. 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.

arXiv comment77 page, 5 figures

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