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A Pl\"ucker coordinate mirror for partial flag varieties and quantum Schubert calculus

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arxiv 2401.15640 v2 pith:M7Q2SN4E submitted 2024-01-28 math.AG math.CO

Changzheng Li , Konstanze Rietsch , Mingzhi Yang , Chi Zhang This is my paper · ORCID
classification math.AGmath.CO
keywords mirrorquantumbulletmathbbmathcalcalculuscohomologyconjecture
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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.

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