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

Abstract

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

25 pages, expanded sections 4 and 5, other minor changes