On the Quantum K-theory of Quiver Varieties at Roots of Unity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908296939569152 |
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| author | Koroteev, Peter Smirnov, Andrey |
| author_facet | Koroteev, Peter Smirnov, Andrey |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_19383 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Quantum K-theory of Quiver Varieties at Roots of Unity Koroteev, Peter Smirnov, Andrey Algebraic Geometry High Energy Physics - Theory Mathematical Physics Number Theory Representation Theory 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. |
| title | On the Quantum K-theory of Quiver Varieties at Roots of Unity |
| topic | Algebraic Geometry High Energy Physics - Theory Mathematical Physics Number Theory Representation Theory |
| url | https://arxiv.org/abs/2412.19383 |