{
  "abstract": "We determine the surface-defect difference operators of the 4d $\\mathcal{N}=2$ lens space index for class $\\mathcal{S}$ theories of type $A_{N-1}$ on $(\\mathbb{S}^3/\\mathbb{Z}_r)\\times\\mathbb{S}^1$. For generic fugacities $(p,q,t)$ and arbitrary rank $N$, orbifold order $r$ and discrete holonomies, their coefficients are finite products of theta functions with nomes $p^r$ and $q^r$. After an explicit gauge transformation, Gaussian in the holonomies, each operator factorizes into two elliptic Ruijsenaars--Schneider-type operators with step $pq$, one with nome $p^r$ and one with nome $q^r$, acting on variables glued along a lattice. Using this factorization and a Wronski-type relation, we prove that all the operators commute. We also establish a sufficient criterion for self-adjointness with respect to the bilinear vector multiplet pairing. It applies to the basic mixed operator and, in suitable parameter chambers, to the pure $p$- and $q$-type operators; for general mixed operators the contour prescription remains open. These results support the expectations from S-duality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30849",
  "author_total": 1,
  "authors": [
    {
      "name": "Masahito Yamazaki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1jk6vw4wjc9bk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30849",
  "fulltext_url": null,
  "id": "1jk6vw4wjc9bk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30849",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "nlin.SI"
    ],
    [
      "openalex",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7223341802",
  "pdf_url": "https://arxiv.org/pdf/2610.30849",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1jk6vw4wjc9bk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:34Z",
      "native_id": "2610.30849",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30849"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "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:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Surface Defects in the $\\mathcal{N}=2$ Lens Space Index and Elliptic Integrability",
  "title_text": "Surface Defects in the 𝒩=2 Lens Space Index and Elliptic Integrability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/1jk6vw4wjc9bk",
  "venue": "arXiv",
  "versions": []
}