Quantum difference equation for the affine type $A$ quiver varieties I: General Construction
Year: 2023
Type: preprint
Abstract: Following the philosophy of \cite{OS22}, we construct quantum difference equations for affine type $A$ quiver varieties in terms of the quantum toroidal algebra $U_{q,t}(\hat{\hat{\mathfrak{sl}}}_{n})$. As part of the construction, we define the wall set of each quiver variety through the action of the universal $R$-matrix and show that, on each interval in $H^2(X,\mathbb{Q})$, this wall set is essentially equivalent to the one determined by the $K$-theoretic stable envelope. We illustrate the construction by deriving explicit quantum difference operators for the instanton moduli space $M(n,r)$ and the equivariant Hilbert scheme $\text{Hilb}_{n}([\mathbb{C}^2/\mathbb{Z}_{r}])$.
Source: arXiv (Cornell University)
Author: Zhu, Tianqing
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4385901395