{
  "abstract": "The $\\hat{Z}$ invariant of a three-manifold $M_{3}$, a half-index of the $3d$ $\\mathcal{N}=2$ theory $T[M_{3}]$, is well understood for negative-definite plumbed three-manifolds. For orientation-reversed manifolds, however, the known formulas fail to produce $q$-series, and making sense of the operation $q \\rightarrow q^{-1}$ remains a central open problem. Seifert manifolds, whose $\\hat{Z}$ invariants are built from false theta functions, offer a natural testing ground. We study ``dual false theta functions'', the objects that should replace false theta functions under orientation reversal. Two characterizations of these duals exist in the literature: a modular one, based on mock modularity with a prescribed shadow, and a resurgent one, based on Borel--Mordell integrals and transseries. We prove that the two characterizations are equivalent. We construct a new family of dual false theta functions for values of positive integer $p$ satisfying a Pell-equation condition. Its members are built from Zwegers' indefinite theta functions, have integer coefficients, and have effective central charge $c_{\\text{eff}} \\leq 1$. We also show that the mock theta functions of Li and Schwagenscheidt satisfy these characterizations and arise from a natural regularization of the divergent product $η^{3}(τ)ψ_{p, a}(- τ)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00516",
  "author_total": 1,
  "authors": [
    {
      "name": "Mrunmay Jagadale"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f8tvtsx1rjk3x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00516",
  "fulltext_url": null,
  "id": "f8tvtsx1rjk3x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00516",
  "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.GT"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7216218408",
  "pdf_url": "https://arxiv.org/pdf/2610.00516",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f8tvtsx1rjk3x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2610.00516",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00516"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Mock Modularity, Resurgence, and Dual False Theta Functions",
  "title_text": "Mock Modularity, Resurgence, and Dual False Theta Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/f8tvtsx1rjk3x",
  "venue": "arXiv",
  "versions": []
}