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

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