{
  "abstract": "The blowup equations of Huang, Sun and Wang are bilinear relations satisfied by the refined topological string partition function of a local Calabi--Yau 3-fold. We prove the blowup equations for the local Hirzebruch surfaces \\(\\operatorname{Tot}_{\\mathbb F_\\ell}K_{\\mathbb F_\\ell}\\), \\(0\\leq\\ell\\leq2\\), as identities of torus-equivariant symmetrized \\(K\\)-theoretic stable pair invariants; for \\(\\ell=2\\) the invariants are localized indices. The proof identifies the stable pair vertex sum, after division by the fibre contribution, with the equivariant Euler characteristic of \\((\\det\\mathcal V)^\\ell\\) on the moduli space of framed rank \\(2\\) sheaves on \\(\\mathbb P^2\\). The blowup formulas of Nakajima--Yoshioka for framed sheaves then yield the unity and vanishing equations. For local \\(\\mathbb P^2\\) we obtain the blowup equations conditionally on two explicitly stated conjectures. We also state the Huang--Sun--Wang conjecture for general local Calabi--Yau 3-folds in the language of stable pairs, and formulate stable pair conjectures for the local rational elliptic surface involving the \\(E_8\\) lattice.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.25397",
  "author_total": 1,
  "authors": [
    {
      "name": "Lutian Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a67hrs80hxw0x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.25397",
  "fulltext_url": null,
  "id": "a67hrs80hxw0x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.25397",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204487112",
  "pdf_url": "https://arxiv.org/pdf/2608.25397",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a67hrs80hxw0x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:26Z",
      "native_id": "2608.25397",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.25397"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Topological String Blowup Equations via Stable Pairs",
  "title_text": "Topological String Blowup Equations via Stable Pairs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/a67hrs80hxw0x",
  "venue": "arXiv",
  "versions": []
}