A Pl\"ucker coordinate mirror for type A flag varieties

TL;DR

This paper constructs a superpotential for type A partial flag varieties using Plücker coordinates, generalizing previous Grassmannian mirrors and connecting with quantum Schubert calculus.

Contribution

It introduces a new superpotential for partial flag varieties that unifies and extends existing mirror constructions using Plücker coordinates.

Findings

Superpotential expressed in Plücker coordinates for flag varieties.

Agreement with previous mirror constructions in specific cluster charts.

Application of quantum Schubert calculus as a key tool.

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 Pl\"ucker coordinates of the Grassmannian factors. This construction generalizes the Marsh--Rietsch Pl\"ucker 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.