{
  "id": "oai:papers.qnfo.org:a-two-index-framework-for-the-bulk-boundary-correspondence-of-anyon-condensation",
  "slug": "a-two-index-framework-for-the-bulk-boundary-correspondence-of-anyon-condensation",
  "title": "A Two-Index Framework for the Bulk-Boundary Correspondence of Anyon Condensation and Boundary Majorana Statistics",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "The bulk-boundary correspondence asserts that topological data of a (2+1)-dimensional phase determine the physics of its boundary, but in the setting of anyon condensation the correspondence is usually stated structurally rather than quantitatively. We propose a set of computable indices that quantify the correspondence between anyon condensation in a parent topological order and the emergence of Majorana statistics at a gapped boundary. The first index, κ = dim(A)², measures the \"size\" of the condensate, where A is the connected (étale) algebra of condensed anyons inside the parent modular tensor category C. For normal condensates — those in which every local A-module sector contributes once to the child, as in Cases I and II — κ coincides with D_C²/D_D², the ratio of squared total quantum dimensions of parent and child; for non-normal condensates (Case III) the two expressions differ, and we report both. The second index, μ = Δc/(1/2) = 2Δc, counts chiral Majorana edge modes forced by the mismatch Δc between the chiral central charges of parent and child phases. A third diagnostic, the non-Abelian excess fraction f_NA, measures the share of boundary quantum dimension carried by sectors with d > 1. We evaluate all indices explicitly for three canonical cases: condensing A = 1 ⊕ ψ in the Ising category (κ = 4, μ = 1, boundary σ defects with two fusion channels — the fingerprint of Majorana zero modes); condensing A = 1 ⊕ e in the toric code (κ = 4, μ = 0, single-channel fusion, no Majorana structure); and condensing A = 1 ⊕ (ψ,ψ) in the doubled Ising category (κ = dim(A)² = 4, child/parent dimension ratio D_D/D_C = √(3/7), f_NA = 1/3, non-Abelian boundary carrier (σ,σ)). The contrast between the first two cases shows that κ alone does not detect Majorana statistics; μ and f_NA do. All numerical results are derived by explicit arithmetic from standard category data, divergences between independent drafts are documented in Appendix A, and limitations of the framework",
  "abstract_markdown": "The bulk-boundary correspondence asserts that topological data of a (2+1)-dimensional phase determine the physics of its boundary, but in the setting of anyon condensation the correspondence is usually stated structurally rather than quantitatively. We propose a set of computable indices that quantify the correspondence between anyon condensation in a parent topological order and the emergence of Majorana statistics at a gapped boundary. The first index, κ = dim(A)², measures the \"size\" of the condensate, where A is the connected (étale) algebra of condensed anyons inside the parent modular tensor category C. For normal condensates — those in which every local A-module sector contributes once to the child, as in Cases I and II — κ coincides with D_C²/D_D², the ratio of squared total quantum dimensions of parent and child; for non-normal condensates (Case III) the two expressions differ, and we report both. The second index, μ = Δc/(1/2) = 2Δc, counts chiral Majorana edge modes forced by the mismatch Δc between the chiral central charges of parent and child phases. A third diagnostic, the non-Abelian excess fraction f_NA, measures the share of boundary quantum dimension carried by sectors with d > 1. We evaluate all indices explicitly for three canonical cases: condensing A = 1 ⊕ ψ in the Ising category (κ = 4, μ = 1, boundary σ defects with two fusion channels — the fingerprint of Majorana zero modes); condensing A = 1 ⊕ e in the toric code (κ = 4, μ = 0, single-channel fusion, no Majorana structure); and condensing A = 1 ⊕ (ψ,ψ) in the doubled Ising category (κ = dim(A)² = 4, child/parent dimension ratio D_D/D_C = √(3/7), f_NA = 1/3, non-Abelian boundary carrier (σ,σ)). The contrast between the first two cases shows that κ alone does not detect Majorana statistics; μ and f_NA do. All numerical results are derived by explicit arithmetic from standard category data, divergences between independent drafts are documented in Appendix A, and limitations of the framework",
  "date": "2026-10-02",
  "first_published": "2026-10-02",
  "last_revised": "2026-10-02T00:00:00Z",
  "modified": "2026-10-02T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/a-two-index-framework-for-the-bulk-boundary-correspondence-of-anyon-condensation",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "A Two-Index Framework for the Bulk-Boundary Correspondence of Anyon Condensation and Boundary Majorana Statistics"
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        "description": "The bulk-boundary correspondence asserts that topological data of a (2+1)-dimensional phase determine the physics of its boundary, but in the setting of anyon condensation the correspondence is usually stated structurally rather than quantitatively. We propose a set of computable indices that quantify the correspondence between anyon condensation in a parent topological order and the emergence of Majorana statistics at a gapped boundary. The first index, κ = dim(A)², measures the \"size\" of the condensate, where A is the connected (étale) algebra of condensed anyons inside the parent modular tensor category C. For normal condensates — those in which every local A-module sector contributes once to the child, as in Cases I and II — κ coincides with D_C²/D_D², the ratio of squared total quantum dimensions of parent and child; for non-normal condensates (Case III) the two expressions differ, and we report both. The second index, μ = Δc/(1/2) = 2Δc, counts chiral Majorana edge modes forced by the mismatch Δc between the chiral central charges of parent and child phases. A third diagnostic, the non-Abelian excess fraction f_NA, measures the share of boundary quantum dimension carried by sectors with d > 1. We evaluate all indices explicitly for three canonical cases: condensing A = 1 ⊕ ψ in the Ising category (κ = 4, μ = 1, boundary σ defects with two fusion channels — the fingerprint of Majorana zero modes); condensing A = 1 ⊕ e in the toric code (κ = 4, μ = 0, single-channel fusion, no Majorana structure); and condensing A = 1 ⊕ (ψ,ψ) in the doubled Ising category (κ = dim(A)² = 4, child/parent dimension ratio D_D/D_C = √(3/7), f_NA = 1/3, non-Abelian boundary carrier (σ,σ)). The contrast between the first two cases shows that κ alone does not detect Majorana statistics; μ and f_NA do. All numerical results are derived by explicit arithmetic from standard category data, divergences between independent drafts are documented in Appendix A, and limitations of the framework",
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