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

Abstract

We construct a Plücker 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.

41 pages