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  "slug": "braid-group-representations-and-modular-data-non-uniqueness-finite-images-and-th",
  "title": "Braid Group Representations and Modular Data: Non-Uniqueness, Finite Images, and the Ising Test Case",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "A central question in the theory of topological order asks whether the braid group representation B_N carried by N non-Abelian anyons uniquely determines the modular data — the S and T matrices — of the underlying topological quantum field theory. We reconcile three lines of analysis into a single assessment. Structurally, we formalize the question as a reduction map Φ from modular tensor category data to projective braid group representations, whose kernel is the locus of potential counterexamples. Two rigorous finiteness results constrain the problem: braid representations from twisted quantum doubles of finite groups factor through finite groups, and the SO(N)₂/O(N)₂ braid representations are Gaussian with finite image. We then perform an explicit computation on the Ising anyon model — the canonical non-Abelian order of Majorana zero modes — reconstructing the topological spins θ_σ = e^iπ/8, θ_ψ = -1, the total quantum dimension D=2, and the full S matrix from the braid eigenvalues, with all arithmetic shown. [...]",
  "abstract_markdown": "A central question in the theory of topological order asks whether the braid group representation $B_N$ carried by $N$ non-Abelian anyons uniquely determines the modular data — the $S$ and $T$ matrices — of the underlying topological quantum field theory. We reconcile three lines of analysis into a single assessment. Structurally, we formalize the question as a reduction map $\\Phi$ from modular tensor category data to projective braid group representations, whose kernel is the locus of potential counterexamples. Two rigorous finiteness results constrain the problem: braid representations from twisted quantum doubles of finite groups factor through finite groups, and the $SO(N)_2$/$O(N)_2$ braid representations are Gaussian with finite image. We then perform an explicit computation on the Ising anyon model — the canonical non-Abelian order of Majorana zero modes — reconstructing the topological spins $\\theta_\\sigma = e^{i\\pi/8}$, $\\theta_\\psi = -1$, the total quantum dimension $\\mathcal{D}=2$, and the full $S$ matrix from the braid eigenvalues, with all arithmetic shown. [...]",
  "date": "2026-10-01",
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  "url": "https://papers.qnfo.org/papers/braid-group-representations-and-modular-data-non-uniqueness-finite-images-and-th",
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    "titles": [
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        "title": "Braid Group Representations and Modular Data: Non-Uniqueness, Finite Images, and the Ising Test Case"
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        "description": "A central question in the theory of topological order asks whether the braid group representation B_N carried by N non-Abelian anyons uniquely determines the modular data — the S and T matrices — of the underlying topological quantum field theory. We reconcile three lines of analysis into a single assessment. Structurally, we formalize the question as a reduction map Φ from modular tensor category data to projective braid group representations, whose kernel is the locus of potential counterexamples. Two rigorous finiteness results constrain the problem: braid representations from twisted quantum doubles of finite groups factor through finite groups, and the SO(N)₂/O(N)₂ braid representations are Gaussian with finite image. We then perform an explicit computation on the Ising anyon model — the canonical non-Abelian order of Majorana zero modes — reconstructing the topological spins θ_σ = e^iπ/8, θ_ψ = -1, the total quantum dimension D=2, and the full S matrix from the braid eigenvalues, with all arithmetic shown. [...]",
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