A Plücker coordinate mirror for type A flag varieties

Abstract

We introduce a superpotential for partial flag varieties of type $A$. This is a map $W:Y^{\circ }\to \mathbb{C}$, where $Y^{\circ }$ is the complement of an anticanonical divisor on a product of Grassmannians. The map $W$ is expressed in terms of Plücker coordinates of the Grassmannian factors. This construction generalizes the Marsh--Rietsch Plücker coordinate mirror for Grassmannians. We show that in a distinguished cluster chart for $Y$, our superpotential agrees with earlier mirrors constructed by Eguchi--Hori--Xiong and Batyrev--Ciocan-Fontanine--Kim--van Straten. Our main tool is quantum Schubert calculus on the flag variety.