{
  "abstract": "We propose a uniform construction of affine schemes encoding the Higgs branches and Macdonald indices of the Argyres--Douglas theories $\\mathcal{D}_p(ADE)$, with $\\gcd(p,h^\\vee)=1$. The construction captures conjectural OPE decoupling relations via the Jacobian ring of a single invariant polynomial potential, obtained from a polynomial twisted period of the corresponding Coxeter orbit space. We show that these potentials are ``syzygy-maximizing'' in the first jet schemes and recover the Higgs branches as their reduced critical loci. We obtain closed determinant expressions for the classical types and explicit potentials for $E_6$, $E_7$, and $E_8$. Powers of the nilradical define an intrinsic filtration on the resulting Jacobian rings, from which we compute candidate Macdonald indices with the correct Schur and Hall--Littlewood limits, in the examples studied. Restricting the potentials to Slodowy slices extends the prescription to nilpotently Higgsed theories. We further conjecture that these Jacobian rings are isomorphic to Zhu's $C_2$-algebras of simple affine VOAs at boundary admissible levels and their quantum Drinfeld--Sokolov reductions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33151",
  "author_total": 3,
  "authors": [
    {
      "name": "Monica Jinwoo Kang"
    },
    {
      "name": "Craig Lawrie"
    },
    {
      "name": "Jaewon Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3fntzbrs78r8h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33151",
  "fulltext_url": null,
  "id": "3fntzbrs78r8h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33151",
  "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.AG"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7223936455",
  "pdf_url": "https://arxiv.org/pdf/2610.33151",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3fntzbrs78r8h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2610.33151",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33151"
    }
  ],
  "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"
    }
  ],
  "title": "One ring to rule them all: syzygy maximization for Argyres-Douglas",
  "title_text": "One ring to rule them all: syzygy maximization for Argyres-Douglas",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/3fntzbrs78r8h",
  "venue": "arXiv",
  "versions": []
}