From quantum difference equation to Dubrovin connection of affine type A quiver varieties (2405.02473v3)
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 $\mathbf{B}{\mathbf{m}}$.
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