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

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 $U_q({\widehat{\mathfrak{gl}}}_1)$-form. We also give the detailed analysis of the connection matrix for the quantum difference equation in the nodal limit $p\to 0$. 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 ${\mathbf{B}}_{\mathbf{m}}$.

56 pages. Revised version. Comments are welcome!