{
  "abstract": "In this work, we propose a mathematical description of the critical point of boundary phase transitions in 2+1D topological orders. Given two boundary phases, $\\mathcal{C}_A$ and $\\mathcal{C}_B$, with corresponding Lagrangian algebras $A$ and $B$ in a 2+1D topological order $\\mathcal{C}$, respectively, we propose that the critical point of the phase transition between these two phases, if it exists, corresponds to $D = A \\times_{M} B$, which is the fiber product of $A$ and $B$ over $A$-$B$-algebra $M$. We prove that $D$ is also a condensable algebra. Additionally, we present an alternative classification of condensable algebras in $Z(Vec_G^ω)$ and reveal splitting phenomenon that arises during the condensation of certain algebras. We also perform explicit computations for the condensable algebras within several examples of $Z(Vec_G^ω)$, covering both Abelian and non-Abelian groups $G$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12079",
  "author_total": 2,
  "authors": [
    {
      "name": "Hanlin Lin"
    },
    {
      "name": "Hao Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/80hasg1gcrn28.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12079",
  "fulltext_url": null,
  "id": "80hasg1gcrn28",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12079",
  "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",
      "math-ph"
    ],
    [
      "arxiv",
      "math.CT"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7221085032",
  "pdf_url": "https://arxiv.org/pdf/2610.12079",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/80hasg1gcrn28.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:50Z",
      "native_id": "2610.12079",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12079"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Fiber product of condensable algebras and critical points of boundary phase transitions",
  "title_text": "Fiber product of condensable algebras and critical points of boundary phase transitions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/80hasg1gcrn28",
  "venue": "arXiv",
  "versions": []
}