  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 $p^s$.
