A Plücker coordinate mirror for type A flag varieties

Abstract

We introduce a superpotential for partial flag varieties of type A $A$ . This is a map W : Y ∘ → C $W: Y^\circ \rightarrow \mathbb {C}$ , where Y ∘ $Y^\circ$ is the complement of an anticanonical divisor on a product of Grassmannians. The map W $W$ 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 $Y$ , 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.