On the Quantum K-theory of Quiver Varieties at Roots of Unity

TL;DR

This paper investigates the quantum K-theory of Nakajima quiver varieties at roots of unity, revealing spectral properties of associated quantum difference equations and their relation to p-curvature and Frobenius twists.

Contribution

It proves the pole-freeness of a specific operator at roots of unity and relates the spectrum of p-curvature to Frobenius twists in quantum connections of Nakajima varieties.

Findings

Operator has no poles at primitive p-th roots of unity.

Eigenvalues of iterated operators match p-th powers of original operators.

Explicit description of p-curvature spectrum for quantum connections.

Abstract

Let $\Psi(\textbf{z},\textbf{a},q)$ a the fundamental solution matrix of the quantum difference equation of a Nakajima variety $X$. In this work, we prove that the operator $$ \Psi(\textbf{z},\textbf{a},q) \Psi\left(\textbf{z}^p,\textbf{a}^p,q^{p^2}\right)^{-1} $$ 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 ${\bf M}_{\mathcal{L}}(\textbf{z},\textbf{a},q )$ from the $q$-difference equation on $X$: $$ {\bf M}_{\mathcal{L}} (\textbf{z} q^{(p-1)\mathcal{L}},\textbf{a},q) \cdots {\bf M}_{\mathcal{L}} (\textbf{z} q^{\mathcal{L}},\textbf{a},q) {\bf M}_{\mathcal{L}} (\textbf{z} ,\textbf{a},q) $$ evaluated at $q=\zeta_p$ has the same eigenvalues as ${\bf M}_{\mathcal{L}} (\textbf{z}^p,\textbf{a}^p,q^p)$. 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 whreas ${\bf M}_{\mathcal{L}} (\textbf{z}^p,\textbf{a}^p,q^p)$ 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.