From quantum difference equation to Dubrovin connection of affine type A quiver varieties

TL;DR

This paper analyzes the quantum difference equations of affine type A quiver varieties in equivariant K-theory, showing their degeneration to the Dubrovin connection in quantum cohomology and describing the monodromy operators explicitly.

Contribution

It provides a detailed representation of quantum difference operators and their monodromy, connecting quantum difference equations to Dubrovin connections in affine type A quiver varieties.

Findings

Representation of quantum difference operators in $U_q(sl_2)$ and $U_q( ilde{gl}_1)$ forms

Analysis of the connection matrix in the nodal limit $p o 0$

Degeneration of quantum difference equations to Dubrovin connection

Abstract

This is the continuation of the article \cite{Z23}. In this article we will give a detailed analysis of the quantum difference equation of the equivariant $K$-theory of the affine type $A$ quiver varieties. We will give a good representation of the quantum difference operator $\mathbf{M}_{\mathcal{L}}(z)$ such that the monodromy operator $\mathbf{B}_{\mathbf{m}}(z)$in the formula can be written in the $U_{q}(\mathfrak{sl}_2)$-form or in the $U_{q}(\hat{\mathfrak{gl}}_1)$-form. We also give the detailed analysis of the connection matrix for the quantum difference equation in the nodal limit $p\rightarrow0$. Using these two results, we prove that the degeneration limit of the quantum difference equation is the Dubrovin connection for the quantum cohomology of the affine type $A$ quiver varieties, and the monodromy representation for the Dubrovin connection is generated by the monodromy operators $\mathbf{B}_{\mathbf{m}}$.