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  "slug": "braid-group-representations-and-the-uniqueness-of-modular-data-the-ising-test-ca",
  "title": "Braid Group Representations and the Uniqueness of Modular Data: The Ising Test Case and General Limitations",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "We investigate whether the unitary braid group representation carried by a collection of non‑Abelian anyons uniquely determines the modular data—namely the S and T matrices—of the underlying modular tensor category (MTC). Using the Ising anyon theory as a concrete laboratory, we reconstruct the full modular data from the elementary braid eigenvalue R_σσ=e^-iπ/8 together with the fusion rule σ×σ=1+ψ. Explicit arithmetic yields the quantum dimension d_σ=√2≈1.4142, the total quantum dimension D=2, the S‑entry S_σ 1= √2/2≈0.7071, the vanishing diagonal S_σσ=0, and the T‑entry T_σσ=e^iπ/8≈0.9239+0.3827 i. We also compute the fusion‑space dimensions f(n,1)=2^n/2-1 for even n, and the braid generator’s ordinary order $16$ (projective order $4$). [...]",
  "abstract_markdown": "We investigate whether the unitary braid group representation carried by a collection of non‑Abelian anyons uniquely determines the modular data—namely the $S$ and $T$ matrices—of the underlying modular tensor category (MTC). Using the Ising anyon theory as a concrete laboratory, we reconstruct the full modular data from the elementary braid eigenvalue $R_{\\sigma\\sigma}=e^{-i\\pi/8}$ together with the fusion rule $\\sigma\\times\\sigma=1+\\psi$. Explicit arithmetic yields the quantum dimension $d_{\\sigma}=\\sqrt{2}\\approx1.4142$, the total quantum dimension $D=2$, the $S$‑entry $S_{\\sigma 1}= \\sqrt{2}/2\\approx0.7071$, the vanishing diagonal $S_{\\sigma\\sigma}=0$, and the $T$‑entry $T_{\\sigma\\sigma}=e^{i\\pi/8}\\approx0.9239+0.3827\\,i$. We also compute the fusion‑space dimensions $f(n,1)=2^{n/2-1}$ for even $n$, and the braid generator’s ordinary order $16$ (projective order $4$). [...]",
  "date": "2026-10-09",
  "first_published": "2026-10-09",
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  "url": "https://papers.qnfo.org/papers/braid-group-representations-and-the-uniqueness-of-modular-data-the-ising-test-ca",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Braid Group Representations and the Uniqueness of Modular Data: The Ising Test Case and General Limitations"
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        "description": "We investigate whether the unitary braid group representation carried by a collection of non‑Abelian anyons uniquely determines the modular data—namely the S and T matrices—of the underlying modular tensor category (MTC). Using the Ising anyon theory as a concrete laboratory, we reconstruct the full modular data from the elementary braid eigenvalue R_σσ=e^-iπ/8 together with the fusion rule σ×σ=1+ψ. Explicit arithmetic yields the quantum dimension d_σ=√2≈1.4142, the total quantum dimension D=2, the S‑entry S_σ 1= √2/2≈0.7071, the vanishing diagonal S_σσ=0, and the T‑entry T_σσ=e^iπ/8≈0.9239+0.3827 i. We also compute the fusion‑space dimensions f(n,1)=2^n/2-1 for even n, and the braid generator’s ordinary order $16$ (projective order $4$). [...]",
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