Polynomial superpotential for Grassmannian $Gr(k,n)$ from a limit of vertex function

Abstract

In this note we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, $X=T^{\ast }Gr(k,n)$. This integral representation can be used to compute the $\hslash \to \infty$ limit of the vertex function, where $\hslash$ denotes the equivariant parameter of a torus acting on $X$ by dilating the cotangent fibers. We show that in this limit the integral turns into the standard mirror integral representation for the $A$-series of the Grassmannian $Gr(k,n)$ with the Laurent polynomial Landau-Ginzburg superpotential of Eguchi, Hori and Xiong. We also observe some Dwork type congruences for the coefficients of the $A$-series.

16 pages, 3 figures