{
  "abstract": "We develop an operator-algebraic framework for boundary conditions and bulk-boundary correspondence in one-dimensional gapped phases with categorical symmetry. Working directly in the thermodynamic limit, we construct half-infinite fusion spin chains and commuting-projector boundary Hamiltonians from a unitary fusion category $\\mathcal{C}$, an indecomposable semisimple right $\\mathcal{C}$-module category $\\mathcal{M}$, a Q-system $Q\\in\\mathcal{C}$ specifying the bulk phase, and a right $Q$-module $K\\in\\mathcal{M}_{Q}$, regarded as an object of $\\mathcal{M}_{Q}^{\\mathrm{op}}$, specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor $\\mathcal{M}_{Q}^{\\mathrm{op}}\\to\\mathrm{BCond}$ is an equivalence, so simple boundary conditions are classified by simple objects of $\\mathcal{M}_{Q}$ and general boundary conditions by their finite direct sums. We also give a microscopic formulation of the boundary symmetry topological field theory using DHR bimodules of the boundary quasi-local algebra. For a half-infinite fusion spin chain, the boundary DHR category is monoidally equivalent to $(\\mathcal{C}_{\\mathcal{M}}^{\\vee})^{\\mathrm{rev}}$, and the canonical action of the bulk DHR category on it agrees with the categorical action of $Z_1(\\mathcal{C}^{\\mathrm{rev}})$. Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of $(\\mathcal{C}_{\\mathcal{M}}^{\\vee})^{\\mathrm{rev}}$ on $\\mathcal{M}_{Q}^{\\mathrm{op}}$. This yields a one-dimensional bulk-boundary correspondence: the enriched monoidal category describing the bulk is the enriched center of the enriched category describing the boundary.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.19137",
  "author_total": 3,
  "authors": [
    {
      "name": "Yizhou Ma"
    },
    {
      "name": "Gen Yue"
    },
    {
      "name": "Tian Lan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cdzh12jrny2vz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.19137",
  "fulltext_url": null,
  "id": "cdzh12jrny2vz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.19137",
  "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",
      "cond-mat.str-el"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7165136733",
  "pdf_url": "https://arxiv.org/pdf/2606.19137",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cdzh12jrny2vz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:31Z",
      "native_id": "2606.19137",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.19137"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.str-el",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Bulk-boundary correspondence of (1+1)D symmetric gapped phases",
  "title_text": "Bulk-boundary correspondence of (1+1)D symmetric gapped phases",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/cdzh12jrny2vz",
  "venue": "arXiv",
  "versions": []
}