{
  "id": "2401.15640",
  "version": "v1",
  "published": "2024-01-28T12:28:40.000Z",
  "updated": "2024-01-28T12:28:40.000Z",
  "title": "A Plücker coordinate mirror for partial flag varieties and quantum Schubert calculus",
  "authors": [
    "Changzheng Li",
    "Konstanze Rietsch",
    "Mingzhi Yang",
    "Chi Zhang"
  ],
  "comment": "41 pages",
  "categories": [
    "math.AG",
    "math.CO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-28T12:28:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J33",
      "14M15",
      "14N15",
      "14N35"
    ],
    "keywords": [
      "partial flag variety",
      "quantum schubert calculus",
      "plücker coordinate mirror",
      "mirror symmetry conjecture",
      "small quantum cohomology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}