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arXiv · 2302.03092

The p-adic approximations of vertex functions via 3D-mirror symmetry

Andrey Smirnov·Alexander Varchenko
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Abstract

Using the $3D$ mirror symmetry we construct a system of polynomials $T_s(z)$ with integral coefficients which solve the quantum differential equitation of $X=T^{*} Gr(k,n)$ modulo $p^s$, where $p$ is a prime number. We show that the sequence $T_s(z)$ converges in the $p$-adic norm to the Okounkov's vertex function of $X$ as $s\to \infty$. We prove that $T_s(z)$ satisfy Dwork-type congruences which lead to a new infinite product presentation of the vertex function modulo $p^s$.

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Andrey Smirnov, Alexander Varchenko. 2023-02-06. The p-adic approximations of vertex functions via 3D-mirror symmetry. https://arxiv.org/abs/2302.03092

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