{
  "abstract": "Line operators of 4d $\\mathcal{N}=2$ theories of class $\\mathcal{S}$ of type $A_{N-1}$ in a half Omega background provide a physical model for $SL_N$ skein algebras. In this paper we extend this correspondence to $GL_N$ skein algebras and prove that the $GL_2$ skein algebra of the punctured torus is isomorphic to the $GL_2$ spherical double affine Hecke algebra ${{\\mathbf{S}\\ddot{\\mathbf{H}}}}^{q,t}_2$. The construction is based on $\\mathfrak{q}$-nonabelianization for the $SU(2)$ $\\mathcal{N}=2^*$ gauge theory, which realizes the skein algebra as a quantum torus algebra associated with the Seiberg-Witten curve. Different regions of the Coulomb branch yield distinct presentations of ${{\\mathbf{S}\\ddot{\\mathbf{H}}}}^{q,t}_2$. These range from the Macdonald $q$-difference module in Fenchel-Nielsen (weak coupling) charts to cluster-type realizations in Fock-Goncharov (strong coupling) charts. Transitions between these descriptions are governed by the vanilla BPS spectrum of the $\\mathcal{N}=2^*$ theory via framed wall-crossing, which provides a unified physical framework for several representations of spherical DAHA.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07911",
  "author_total": 2,
  "authors": [
    {
      "name": "Kunal Gupta"
    },
    {
      "name": "Pietro Longhi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2g77j4eq48262.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07911",
  "fulltext_url": null,
  "id": "2g77j4eq48262",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07911",
  "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.MP"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "math.RT"
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    [
      "arxiv",
      "math.SG"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7212055159",
  "pdf_url": "https://arxiv.org/pdf/2609.07911",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2g77j4eq48262.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:48Z",
      "native_id": "2609.07911",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07911"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.SG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Spherical DAHA as an algebra of framed BPS states",
  "title_text": "Spherical DAHA as an algebra of framed BPS states",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/2g77j4eq48262",
  "venue": "arXiv",
  "versions": []
}