{
  "abstract": "We construct indefinite theta representations for a family of signed generalized Nahm sums of arbitrary rank \\(r\\geq2\\). After explicit normalization and boundary correction, one signed branch gives the even components of a \\((2r-1)\\)-component holomorphic theta vector associated with a lattice of signature \\((1,1)\\). Euler-type multiple sums provide the odd components, yielding formulas uniform in both the rank and the component index. Using Zwegers' theory, we determine the weight-one modular transformation laws of the real-analytic completion, with an explicit finite sine matrix for the \\(S\\)-transformation. We decompose the nonholomorphic correction into a finite sum of products of unary theta functions and one-dimensional error-function series, and compute the antiholomorphic derivative of the completion. Together with the completed transformation laws, these formulas give the inhomogeneous transformation law of the holomorphic vector. We also evaluate a linear combination of two signed branches as a difference of Jacobi product quotients. This identity and further evaluations for two families of symmetrizable matrices and their duals, combined with known results, give multiple-sum representations for every component of four associated product vectors. These representations are finite combinations of generalized Nahm sums with the prescribed matrices and symmetrizers; in the dual families, half-period translates are absorbed into signs and constant phase factors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00967",
  "author_total": 2,
  "authors": [
    {
      "name": "Dandan Chen"
    },
    {
      "name": "Siyu Yin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1tca0szswr4wq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00967",
  "fulltext_url": null,
  "id": "1tca0szswr4wq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00967",
  "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",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7215897272",
  "pdf_url": "https://arxiv.org/pdf/2610.00967",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1tca0szswr4wq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.00967",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00967"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Indefinite theta vectors and generalized Nahm sums of arbitrary rank",
  "title_text": "Indefinite theta vectors and generalized Nahm sums of arbitrary rank",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/1tca0szswr4wq",
  "venue": "arXiv",
  "versions": []
}