Polynomial Superpotential for Grassmannian \({\text {Gr}}(k,n)\) from a Limit of Vertex Function

In this note, we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, \(X=T^{*}{\text {Gr}}(k,n)\) $X=T^{\ast }\text{Gr}(k,n)$ . This integral representation can be used to compute the \(\hbar \rightarrow \infty \) $ħ\to \infty$ limit of the vertex function, where \(\hbar \) $ħ$ 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 of the A-series of the Grassmannian \({\text {Gr}}(k,n)\) $\text{Gr}(k,n)$ with the Laurent polynomial Landau–Ginzburg superpotential of Eguchi, Hori and Xiong.