Quantum K-theory of quiver varieties at roots of unity

Abstract Let $\Psi (\textbf{z},\textbf{a},q)$ be a fundamental solution matrix of the quantum difference equation of a Nakajima variety $X$. In this work, we prove that the operator $$ \begin{align*} & \Psi(\textbf{z},\textbf{a},q) \Psi(\textbf{z}^p,\textbf{a}^p,q^{p^2})^{-1} \end{align*} $$ has no poles at the primitive complex $p$-th roots of unity $q=\zeta _{p}$. As a byproduct, we show that the iterated product of the operators $\textbf{M}_{\mathcal{L}}(\textbf{z},\textbf{a},q )$ from the $q$-difference equation on $X$ $$ \begin{align*} & \textbf{M}_{\mathcal{L}} (\textbf{z} q^{(p-1)\mathcal{L}},\textbf{a},q) \cdots \textbf{M}_{\mathcal{L}} (\textbf{z} q^{\mathcal{L}},\textbf{a},q) \textbf{M}_{\mathcal{L}} (\textbf{z},\textbf{a},q) \end{align*} $$ evaluated at $q=\zeta _{p}$ has the same eigenvalues as $\textbf{M}_{\mathcal{L}} (\textbf{z}^{p},\textbf{a}^{p},q^{p})$. Upon a reduction of the quantum difference equation of $X$ to the quantum differential equation over the field of finite characteristic, the above iterated product transforms into a Grothendiek–Katz $p$-curvature of the corresponding quantum connection whereas $\textbf{M}_{\mathcal{L}} (\textbf{z}^{p},\textbf{a}^{p},q^{p})$ becomes a certain Frobenius twist of that connection. In this way, we give an explicit description of the spectrum of the $p$-curvature of quantum connection for Nakajima varieties.