{
  "abstract": "In quantum mechanics of genus-1 potentials, perturbation theory around the topologically trivial saddle and non-perturbative instanton saddle are constructively related via the P/NP relation. Whether such a relation persists for higher-genus potentials has remained an open problem. In this work, we resolve it using tools from complex algebraic geometry combined with exact WKB. The classical energy conservation relation $p^2 = 2(E-V(q))$ defines a genus-g Riemann surface $X$, whose periods are naturally organized by its Jacobian variety $J(X) = \\mathbb{C}^{\\text{g}}/Λ$ via the Abel-Jacobi map, and independently by the solution basis of the associated Picard-Fuchs equations. We show that the physically relevant, WKB-active cycles are generally related to the Picard-Fuchs basis by a linear transformation that lies in $\\text{SL}(2\\text{g},\\mathbb{Z})$ but not in the symplectic group $\\text{Sp}(2\\text{g},\\mathbb{Z})$ once $\\text{g}\\geq2$, in contrast to the genus-1 case, where the two groups coincide. This mismatch is the obstruction that has made an explicit higher-genus P/NP relation elusive. We show that it can be resolved by passing to a modified Riemann bilinear identity, built from a suitably transformed intersection matrix, which the WKB-active periods do satisfy exactly. The result is an all-orders quantum P/NP relation valid for potentials of arbitrary genus, which we derive explicitly for genus-2 (quintic and sextic) and genus-3 (septic and octic) anharmonic oscillators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10940",
  "author_total": 2,
  "authors": [
    {
      "name": "Mustafa Türe"
    },
    {
      "name": "Mithat Ünsal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/33vw82dqy292j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10940",
  "fulltext_url": null,
  "id": "33vw82dqy292j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10940",
  "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-ph"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221091476",
  "pdf_url": "https://arxiv.org/pdf/2610.10940",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/33vw82dqy292j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:54Z",
      "native_id": "2610.10940",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10940"
    }
  ],
  "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"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Quantum geometry at arbitrary genus - I: Anharmonic potentials",
  "title_text": "Quantum geometry at arbitrary genus - I: Anharmonic potentials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/33vw82dqy292j",
  "venue": "arXiv",
  "versions": []
}