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  "slug": "braid-group-representations-modular-data-and-the-classification-of-majorana-zero",
  "title": "Braid Group Representations, Modular Data, and the Classification of Majorana Zero Mode Fusion Rules in 2D Topological Superconductors",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "We investigate whether the braid group representation on N anyons uniquely determines the modular data (S and T matrices) of a non-Abelian topological order, and whether this framework classifies all fusion rules for Majorana zero modes (MZMs) in 2D topological superconductors. We show that the local N-anyon braid group representation constrains the modular data up to finite gauge ambiguity but does not uniquely fix the global S-matrix without specifying the underlying spin structure. For MZMs, the braid generators satisfy the Ising anyon R-matrix with the asymptotic Hecke relation σ_i² = 1 (up to phase), which rigidly fixes the local F and R symbols of the Ising category. Consequently, the fusion rules are strictly of Ising type (σ × σ = 1 + ψ), precluding alternative non-Abelian fusion rules. The T-matrix is determined by topological spins, but the full S-matrix depends on the spin structure of the underlying manifold, reflecting the super-modular structure of the theory.",
  "abstract_markdown": "We investigate whether the braid group representation on $N$ anyons uniquely determines the modular data ($S$ and $T$ matrices) of a non-Abelian topological order, and whether this framework classifies all fusion rules for Majorana zero modes (MZMs) in 2D topological superconductors. We show that the local $N$-anyon braid group representation constrains the modular data up to finite gauge ambiguity but does not uniquely fix the global $S$-matrix without specifying the underlying spin structure. For MZMs, the braid generators satisfy the Ising anyon $R$-matrix with the asymptotic Hecke relation $\\sigma_i^2 = 1$ (up to phase), which rigidly fixes the local $F$ and $R$ symbols of the Ising category. Consequently, the fusion rules are strictly of Ising type ($\\sigma \\times \\sigma = 1 + \\psi$), precluding alternative non-Abelian fusion rules. The $T$-matrix is determined by topological spins, but the full $S$-matrix depends on the spin structure of the underlying manifold, reflecting the super-modular structure of the theory.",
  "date": "2026-09-07",
  "first_published": "2026-09-07",
  "last_revised": "2026-09-14T00:41:50Z",
  "modified": "2026-09-14T00:41:50Z",
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  "url": "https://papers.qnfo.org/papers/braid-group-representations-modular-data-and-the-classification-of-majorana-zero",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Braid Group Representations, Modular Data, and the Classification of Majorana Zero Mode Fusion Rules in 2D Topological Superconductors"
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        "description": "We investigate whether the braid group representation on N anyons uniquely determines the modular data (S and T matrices) of a non-Abelian topological order, and whether this framework classifies all fusion rules for Majorana zero modes (MZMs) in 2D topological superconductors. We show that the local N-anyon braid group representation constrains the modular data up to finite gauge ambiguity but does not uniquely fix the global S-matrix without specifying the underlying spin structure. For MZMs, the braid generators satisfy the Ising anyon R-matrix with the asymptotic Hecke relation σ_i² = 1 (up to phase), which rigidly fixes the local F and R symbols of the Ising category. Consequently, the fusion rules are strictly of Ising type (σ × σ = 1 + ψ), precluding alternative non-Abelian fusion rules. The T-matrix is determined by topological spins, but the full S-matrix depends on the spin structure of the underlying manifold, reflecting the super-modular structure of the theory.",
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