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

Using the $3D$ mirror symmetry we construct a system of polynomials $T_s(z)$ with integral coefficients which solve the quantum differential equitation of $X=T^{\ast }Gr(k,n)$ 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 $s\to \mathrm{\infty }$. We prove that $T_s(z)$ satisfy Dwork-type congruences which lead to a new infinite product presentation of the vertex function modulo $p^s$.