{
  "abstract": "FINDING: Quantum geometric Langlands duality is fundamentally tied to symplectic duality and level-shift by the dual Coxeter number, with affine Weyl group structure governing the self-duality condition. | MATH: The key structure is the quantum geometric Langlands correspondence: for a reductive group \\(G\\) and its Langlands dual \\({}^L G\\), the duality relates \\(G\\)-modules at level \\(k\\) to \\({}^L G\\)-modules at level \\(k'\\) where \\(k' = \\frac{1}{k}\\) (up to shift). The critical shift is \\(k \\to k + h^\\vee\\) where \\(h^\\vee\\) is the dual Coxeter number. Self-duality occurs when \\(k = \\pm \\sqrt{h^\\vee \\, {}^L h^\\vee}\\) or more precisely when the level satisfies \\(k^2 = h^\\vee \\, {}^L h^\\vee\\) (for simply-laced, \\(h^\\vee = {}^L h^\\vee\\), giving \\(k = h^\\vee\\)). The affine Weyl group \\(\\widetilde{W}\\) acts on the level space, and the duality is an involution generated by the affine Weyl group element \\(s_0\\) (the affine simple reflection). The \"quantum\" parameter \\(q = e^{2\\pi i/(k+h^\\ve\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
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  ],
  "bibtex": "https://research.draconitic.com/paper/e1pwekra6vyk3.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23635303",
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  "id": "e1pwekra6vyk3",
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  "landing_url": "https://zenodo.org/records/23635303",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e1pwekra6vyk3.ris",
  "sources": [
    {
      "first_seen": "2026-11-08T12:21:56Z",
      "native_id": "23635303",
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      "url": "https://zenodo.org/records/23635303"
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  "subjects": [],
  "title": "Quantum Geometric Langlands Duality Linked to Symplectic Duality and Affine Weyl Structure — E8 Intelligence Research",
  "title_text": "Quantum Geometric Langlands Duality Linked to Symplectic Duality and Affine Weyl Structure — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-08T12:21:56Z",
  "url": "https://research.draconitic.com/paper/e1pwekra6vyk3",
  "venue": "Zenodo",
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}