A Plücker coordinate mirror for partial flag varieties and quantum Schubert calculus
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914673328128000 |
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| author | Li, Changzheng Rietsch, Konstanze Yang, Mingzhi Zhang, Chi |
| author_facet | Li, Changzheng Rietsch, Konstanze Yang, Mingzhi Zhang, Chi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_15640 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Plücker coordinate mirror for partial flag varieties and quantum Schubert calculus Li, Changzheng Rietsch, Konstanze Yang, Mingzhi Zhang, Chi Algebraic Geometry Combinatorics 14J33 (Primary), 14M15, 14N15, 14N35 (Secondary) 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. |
| title | A Plücker coordinate mirror for partial flag varieties and quantum Schubert calculus |
| topic | Algebraic Geometry Combinatorics 14J33 (Primary), 14M15, 14N15, 14N35 (Secondary) |
| url | https://arxiv.org/abs/2401.15640 |