{
  "abstract": "We determine the equal-mass two-dimensional three- and four-loop sunset Feynman integrals $I^{(3)}_\\circleddash$ and $I^{(4)}_\\circleddash$ at large Euclidean momentum in terms of the genus-zero Gromov--Witten theory of the local Calabi--Yau fourfold and fivefold $X_L=\\operatorname{Tot}(K_{F_L})$, where $F_L\\subset(\\mathbb P^1)^{L+1}$, $L=3,4$, is a smooth hypersurface of multidegree $(1,\\ldots,1)$. At two loops the sunset integral is governed by the prepotential of a local Calabi--Yau threefold. For $L\\geq3$ the moduli spaces of unmarked stable maps have positive virtual dimension and there is no prepotential, therefore the Feynman integral is governed instead by the one-point potential $\\mathcal F_{γ_L}$ with insertion $γ_L=H^{L-1}/(L-1)!$, by its gravitational descendants, and by $ζ(3)$ boundary terms coming from the Gamma class. At both loop orders the coefficient of $R^{L-1}$ in the Frobenius expansion in the mirror coordinate $R$ is $-θ_Q\\mathcal F_{γ_L}$ with the Novikov variable $Q=\\exp(R)$. At three loops the whole Feynman integral is an explicit expression in $\\mathcal F_{γ_3}$ alone, because genus-zero recursion determines all descendant series from it; at four loops two diagonal series suffice, and the $ζ(3)$ term is itself corrected by $θ_Q\\mathcal F_{γ_4}$. At three loops we also derive the analytic normalization independently, from the Picard--Fuchs equation and the polar part of a Mellin transform, prove integrality of the multiple-cover transforms for arbitrary integral insertions together with an explicit quantum-$K$ formula, and compute the exact period connection polynomials. Finally we give endpoint values along the Euclidean axis, bounds on the Taylor radii in the Novikov variable $Q$, a computer-assisted proof of the four-loop endpoint nonanalyticity, and a comparison of the normalizations at two, three and four loops.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10821",
  "author_total": 2,
  "authors": [
    {
      "name": "Hartmut Maennel"
    },
    {
      "name": "Pierre Vanhove"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/86n3fg5znb08g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10821",
  "fulltext_url": null,
  "id": "86n3fg5znb08g",
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  "landing_url": "https://hal.science/hal-05788327v1",
  "language": "en",
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  "openalex": "W7221130157",
  "pdf_url": "https://arxiv.org/pdf/2610.10821",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/86n3fg5znb08g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:52Z",
      "native_id": "hal-05788327",
      "source": "hal",
      "url": "https://hal.science/hal-05788327v1"
    },
    {
      "first_seen": "2026-10-30T12:29:50Z",
      "native_id": "2610.10821",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10821"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "The three- and four-loop equal-mass sunset integrals and local Calabi--Yau fourfolds and fivefolds",
  "title_text": "The three- and four-loop equal-mass sunset integrals and local Calabi--Yau fourfolds and fivefolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/86n3fg5znb08g",
  "venue": "HAL",
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}