{
  "id": "oai:papers.qnfo.org:root-of-unity-quantization-in-su-2-k-chern-simons-fusion-categories",
  "slug": "root-of-unity-quantization-in-su-2-k-chern-simons-fusion-categories",
  "title": "Root‑of‑Unity Quantization in SU(2)_k Chern‑Simons Fusion Categories",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "The fusion categories underlying three‑dimensional SU(2)_k Chern‑Simons theory are known to require the deformation parameter q to be a complex root of unity, q=e^iπ/(k+2), yet a transparent derivation of this constraint from first principles is rarely presented. Existing literature treats the root‑of‑unity condition as a post‑ulate of quantum group theory without linking it to the physical data of the topological field theory. We close this gap by analysing the Dirac quantisation of monopole charge, the one‑loop shift of the Chern‑Simons level, and the consistency of the modular S‑matrix. Using the explicit formula for quantum dimensions d_j=sin big((2j+1)π/(k+2)big)/sin big(π/(k+2)big) we compute the total quantum dimension D for several low levels and verify that q must satisfy q^k+2= -1 to render all d_j real and positive. For k=2 we obtain q=e^iπ/4= (√2)/(2)+i(√2)/(2) and D=2≈2.00000000, while for k=1 we find q=e^iπ/3= 1/2+i(√3)/(2) and D=√2≈1.41421356. The result explains why the fusion category is a specialization of the quantum group U_q(su(2)) at a root of unity and predicts that any deformation away from q=e^iπ/(k+2) would break modularity. Consequently, topological quantum computing platforms that rely on SU(2)_k anyons must enforce this quantisation to preserve fault‑tolerance.",
  "abstract_markdown": "The fusion categories underlying three‑dimensional SU(2)$_{k}$ Chern‑Simons theory are known to require the deformation parameter $q$ to be a complex root of unity, $q=e^{i\\pi/(k+2)}$, yet a transparent derivation of this constraint from first principles is rarely presented. Existing literature treats the root‑of‑unity condition as a post‑ulate of quantum group theory without linking it to the physical data of the topological field theory. We close this gap by analysing the Dirac quantisation of monopole charge, the one‑loop shift of the Chern‑Simons level, and the consistency of the modular $S$‑matrix. Using the explicit formula for quantum dimensions $d_j=\\sin\\!\\big((2j+1)\\pi/(k+2)\\big)/\\sin\\!\\big(\\pi/(k+2)\\big)$ we compute the total quantum dimension $D$ for several low levels and verify that $q$ must satisfy $q^{k+2}= -1$ to render all $d_j$ real and positive. For $k=2$ we obtain $q=e^{i\\pi/4}= \\frac{\\sqrt2}{2}+i\\frac{\\sqrt2}{2}$ and $D=2\\approx2.00000000$, while for $k=1$ we find $q=e^{i\\pi/3}= \\frac12+i\\frac{\\sqrt3}{2}$ and $D=\\sqrt2\\approx1.41421356$. The result explains why the fusion category is a specialization of the quantum group $U_q(\\mathfrak{su}(2))$ at a root of unity and predicts that any deformation away from $q=e^{i\\pi/(k+2)}$ would break modularity. Consequently, topological quantum computing platforms that rely on SU(2)$_{k}$ anyons must enforce this quantisation to preserve fault‑tolerance.",
  "date": "2026-11-11",
  "first_published": "2026-11-11",
  "last_revised": "2026-11-11T00:00:00Z",
  "modified": "2026-11-11T00:00:00Z",
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  "doi": null,
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  "url": "https://papers.qnfo.org/papers/root-of-unity-quantization-in-su-2-k-chern-simons-fusion-categories",
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  "version_url": "https://papers.qnfo.org/papers/root-of-unity-quantization-in-su-2-k-chern-simons-fusion-categories/v/1.0.0",
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    "creators": [
      {
        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Root‑of‑Unity Quantization in SU(2)_k Chern‑Simons Fusion Categories"
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    "publicationYear": "2026",
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        "description": "The fusion categories underlying three‑dimensional SU(2)_k Chern‑Simons theory are known to require the deformation parameter q to be a complex root of unity, q=e^iπ/(k+2), yet a transparent derivation of this constraint from first principles is rarely presented. Existing literature treats the root‑of‑unity condition as a post‑ulate of quantum group theory without linking it to the physical data of the topological field theory. We close this gap by analysing the Dirac quantisation of monopole charge, the one‑loop shift of the Chern‑Simons level, and the consistency of the modular S‑matrix. Using the explicit formula for quantum dimensions d_j=sin big((2j+1)π/(k+2)big)/sin big(π/(k+2)big) we compute the total quantum dimension D for several low levels and verify that q must satisfy q^k+2= -1 to render all d_j real and positive. For k=2 we obtain q=e^iπ/4= (√2)/(2)+i(√2)/(2) and D=2≈2.00000000, while for k=1 we find q=e^iπ/3= 1/2+i(√3)/(2) and D=√2≈1.41421356. The result explains why the fusion category is a specialization of the quantum group U_q(su(2)) at a root of unity and predicts that any deformation away from q=e^iπ/(k+2) would break modularity. Consequently, topological quantum computing platforms that rely on SU(2)_k anyons must enforce this quantisation to preserve fault‑tolerance.",
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