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

Peter Koroteev, Andrey Smirnov

TL;DR

This work studies the quantum K-theory of Nakajima quiver varieties at roots of unity through the quantum difference equation (QDE) and its fundamental solution $\Psi$. It constructs a pole-cancellation intertwiner at primitive $p$-th roots of unity and proves an isospectrality between the $p$-th iterated quantum multiplication and its Frobenius-twisted counterpart, via a reduction to $p$-curvature in finite characteristic. Eigenvalues and eigenvectors of quantum multiplication are tied to Bethe Ansatz equations through Yang–Yang functions, with explicit asymptotics of vertex functions at roots of unity. The results yield a concrete description of the spectrum of the $p$-curvature of the quantum connection for Nakajima varieties and connect the $q$-difference framework to Grothendieck–Katz $p$-curvature, including an illustrative example $X=T^{*}\mathbb{P}^1$ and parallel developments in $p$-adic and modular settings.

Abstract

Let $Ψ(\textbf{z},\textbf{a},q)$ a the fundamental solution matrix of the quantum difference equation of a Nakajima variety $X$. In this work, we prove that the operator $$ Ψ(\textbf{z},\textbf{a},q) Ψ\left(\textbf{z}^p,\textbf{a}^p,q^{p^2}\right)^{-1} $$ has no poles at the primitive complex $p$-th roots of unity $q=ζ_p$. 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 $q=ζ_p$ 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.

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

TL;DR

This work studies the quantum K-theory of Nakajima quiver varieties at roots of unity through the quantum difference equation (QDE) and its fundamental solution . It constructs a pole-cancellation intertwiner at primitive -th roots of unity and proves an isospectrality between the -th iterated quantum multiplication and its Frobenius-twisted counterpart, via a reduction to -curvature in finite characteristic. Eigenvalues and eigenvectors of quantum multiplication are tied to Bethe Ansatz equations through Yang–Yang functions, with explicit asymptotics of vertex functions at roots of unity. The results yield a concrete description of the spectrum of the -curvature of the quantum connection for Nakajima varieties and connect the -difference framework to Grothendieck–Katz -curvature, including an illustrative example and parallel developments in -adic and modular settings.

Abstract

Let a the fundamental solution matrix of the quantum difference equation of a Nakajima variety . In this work, we prove that the operator has no poles at the primitive complex -th roots of unity . As a byproduct, we show that the iterated product of the operators from the -difference equation on : evaluated at has the same eigenvalues as . Upon a reduction of the quantum difference equation of to the quantum differential equation over the field of finite characteristic, the above iterated product transforms into a Grothendiek-Katz -curvature of the corresponding quantum connection whreas becomes a certain Frobenius twist of that connection. In this way, we give an explicit description of the spectrum of the -curvature of quantum connection for Nakajima varieties.
Paper Structure (25 sections, 16 theorems, 140 equations)

This paper contains 25 sections, 16 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. The Quantum Difference Equation
  3. Quantum K-theory
  4. Eigenvalues of $\mathcal{M}_{\mathcal{L}}(\mathbold{z},\mathbold{a})$ and Bethe Ansatz
  5. The Case $q^p=1$
  6. The Intertwiner
  7. $p$-Curvature and Frobenius
  8. Example $X=T^{*}\mathbb{P}^1$
  9. Acknowledgments
  10. Asymptotics of Vertex Functions
  11. Integral representations of the vertex functions
  12. Integrand of (\ref{['intrep']}) near roots of unity.
  13. Asymptotics of vertex functions via steepest descend method
  14. Bethe Equations
  15. Bethe Equations for Yang-Yang Functions
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\{\lambda_1(\mathbold{z},\mathbold{a}),\lambda_2(\mathbold{z},\mathbold{a}), \dots\}$ be the set of the eigenvalues of $\mathcal{M}_{\mathcal{L}}(\mathbold{z},\mathbold{a})$ then the eigenvalues of $\mathcal{M}_{\mathcal{L} \,\zeta_p}(\mathbold{z},\mathbold{a})$ are given by the set $\{\lambda_

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Corollary 2.4
  • proof
  • ...and 16 more