{
  "abstract": "We develop an equivariant K-theoretic vertex formalism for a colored, abelianized higher-rank sector on smooth projective toric threefolds $X$. Nonzero first Chern classes are encoded as magnetic fluxes and absorbed into line-bundle twists, after which the virtual character is redistributed into finite perturbative, vertex, and edge contributions. We compute the resulting (unreduced and trace-free) virtual dimensions and analyze the interaction terms relevant to a possible reduction to rank one. We emphasize that, on a compact toric threefold, colored partitions do not classify all torus-fixed stable higher-rank sheaves: the resulting series is therefore a proposed abelianized/framed vertex series unless an additional comparison theorem with the full stable-sheaf moduli problem is supplied. When the threefold obstruction theory is not available in the desired form, we pass to the local Calabi--Yau fourfold $Y=\\operatorname{Tot}(K_X)$ and formulate the corresponding colored DT4 vertex. A $K_X$-twisted tautological insertion removes fixed points with nontrivial fiber layers, reducing solid partitions to plane partitions at the level of the local vertex. In rank one we give a scheme-theoretic explanation: the Hilbert functor of subschemes of $X$ is canonically isomorphic to the constrained Hilbert functor of subschemes of $Y$ lying scheme-theoretically in the fixed zero section. Under the hypotheses of the DT4 K-theoretic Lefschetz/virtual-pullback formalism, the associated support insertion reduces the rank-one fourfold theory to the threefold theory. We keep this global rank-one statement logically distinct from the higher-rank colored-vertex reduction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04965",
  "author_total": 3,
  "authors": [
    {
      "name": "Nicolo Piazzalunga"
    },
    {
      "name": "Artan Sheshmani"
    },
    {
      "name": "Shing-Tung Yau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/acq0dbnq8hysr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04965",
  "fulltext_url": null,
  "id": "acq0dbnq8hysr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04965",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7220703911",
  "pdf_url": "https://arxiv.org/pdf/2610.04965",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/acq0dbnq8hysr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2610.04965",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04965"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Counting higher-rank sheaves on toric threefolds",
  "title_text": "Counting higher-rank sheaves on toric threefolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/acq0dbnq8hysr",
  "venue": "arXiv",
  "versions": []
}