{
  "abstract": "We study surface defects in four-dimensional $\\mathcal{N}=2$ theories of class $\\mathcal{S}$ and $\\mathcal{N}=1$ theories of class $\\mathcal{S}_k$ placed on $L(r,1)\\times S^1$. The lens space index depends on the fugacities $p$ and $q$, and the effect of the defects on it is captured by difference operators acting on the fugacities and holonomies of flavor symmetries. For theories of type $A_1$ we determine in closed form the difference operators for the basic defects, labeled by $p^r$, $q^r$ and $pq$, and show that for odd $r$ there exist additional elementary defects, whose operators shift the holonomy by one half modulo $r$. All the operators commute, and the elementary operators generate the others. For even $r$ the same holds once the holonomy is extended to a twisted sector. We show that the elementary operators are traces of $L$-operators whose entries generate a Sklyanin algebra, with nome $q^r$ and shift $pq$. The lens space enters only through the parameters of the algebra and through its representation, which is realized on a doubled holonomy label. The $p$- and $q$-type algebras form a lens space version of the elliptic modular double. For class $\\mathcal{S}_k$ theories we propose, and test in several ways, that the defect operators are transfer matrices of inhomogeneous spin chains built from these $L$-operators. Throughout, we use the TQFT structure of the lens space index, which follows from S-duality, both to fix and to check the operators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28165",
  "author_total": 3,
  "authors": [
    {
      "name": "Kazunobu Maruyoshi"
    },
    {
      "name": "Toshihiro Ota"
    },
    {
      "name": "Masahito Yamazaki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/466tt01bgqsn2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28165",
  "fulltext_url": null,
  "id": "466tt01bgqsn2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28165",
  "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",
      "nlin.SI"
    ],
    [
      "openalex",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7223088729",
  "pdf_url": "https://arxiv.org/pdf/2610.28165",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/466tt01bgqsn2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:04Z",
      "native_id": "2610.28165",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28165"
    }
  ],
  "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 on lens space and transfer matrices",
  "title_text": "Surface defects on lens space and transfer matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/466tt01bgqsn2",
  "venue": "arXiv",
  "versions": []
}