Current browse context:
math.RT
Change to browse by:
References & Citations
Mathematics > Representation Theory
Title: Quantum difference equation for the affine type $A$ quiver varieties I: General Construction
(Submitted on 1 Aug 2023 (v1), last revised 7 May 2024 (this version, v4))
Abstract: In this article we use the philosophy in [OS22] to construct the quantum difference equation of affine type $A$ quiver varieties in terms of the quantum toroidal algebra $U_{q,t}(\hat{\hat{\mathfrak{sl}}}_{r})$. In the construction, and we define the set of wall for each quiver varieties by the action of the universal $R$-matrix, which is shown to be almost equivalent to that of the $K$-theoretic stable envelope on each interval in $H^2(X,\mathbb{Q})$. We also give the examples of the instanton moduli space $M(n,r)$ and the equivariant Hilbert scheme $\text{Hilb}_{n}([\mathbb{C}^2/\mathbb{Z}_{r}])$ to show the explicit form of the quantum difference operator.
Submission history
From: Tianqing Zhu [view email][v1] Tue, 1 Aug 2023 13:47:02 GMT (618kb)
[v2] Thu, 3 Aug 2023 05:39:30 GMT (600kb)
[v3] Wed, 6 Dec 2023 14:00:57 GMT (44kb)
[v4] Tue, 7 May 2024 03:48:23 GMT (47kb)
Link back to: arXiv, form interface, contact.