{
  "id": "2312.13188",
  "version": "v1",
  "published": "2023-12-20T16:59:51.000Z",
  "updated": "2023-12-20T16:59:51.000Z",
  "title": "Quantum cohomology of the Hilbert scheme of points on an elliptic surface",
  "authors": [
    "Georg Oberdieck",
    "Aaron Pixton"
  ],
  "comment": "46 pages, comments welcome",
  "categories": [
    "math.AG",
    "math.RT"
  ],
  "abstract": "We determine the quantum multiplication with divisor classes on the Hilbert scheme of points on an elliptic surface $S \\to \\Sigma$ for all curve classes which are contracted by the induced fibration $S^{[n]} \\to \\Sigma^{[n]}$. The formula is expressed in terms of explicit operators on Fock space. The structure constants are meromorphic quasi-Jacobi forms of index $0$. Combining with work of Hu-Li-Qin, this determines the quantum multiplication with divisors on the Hilbert scheme of elliptic surfaces with $p_g(S)>0$. We also determine the equivariant quantum multiplication with divisor classes for the Hilbert scheme of points on the product $E \\times \\mathbb{C}$. The proof of our formula is based on Nesterov's Hilb/PT wall-crossing, a newly established GW/PT correspondence for the product of an elliptic surface times a curve, and new computations in the Gromov-Witten theory of an elliptic curve.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-20T16:59:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hilbert scheme",
      "quantum cohomology",
      "divisor classes",
      "elliptic surface times",
      "equivariant quantum multiplication"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}