A Plücker coordinate mirror for partial flag varieties and quantum Schubert calculus

Abstract: We construct a Pl\"ucker coordinate superpotential $\mathcal{F}-$ that is mirror to a partial flag variety $\mathbb{ F}\ell(n\bullet)$. Its Jacobi ring recovers the small quantum cohomology of $\mathbb{ F}\ell(n_\bullet)$ and we prove a folklore conjecture in mirror symmetry. Namely, we show that the eigenvalues for the action of the first Chern class $c_1(\mathbb{ F}\ell(n_\bullet))$ on quantum cohomology are equal to the critical values of $\mathcal{F}-$. We achieve this by proving new identities in quantum Schubert calculus that are inspired by our formula for $\mathcal{F}-$ and the mirror symmetry conjecture.