FINDING: Fibonacci anyon braid group representations are classified by modular tensor categories (MTCs) with fusion rules derived from the Fibonacci (golden ratio) quantum group $\text{SU}(2)_3$ , yielding non-Abelian statistics and universal topological quantum computation. MATH: - Fibonacci anyon fusion rule: $\tau \times \tau = 1 + \tau$ (where $\tau$ is the non-Abelian anyon, 1 is the vacuum). - Quantum dimension of $\tau$ : $d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618$ . - Braid matrix eigenvalues: $e^{\pm i\pi/5}$ and $e^{\pm i3\pi/5}$ , derived from the $R$ -matrix of the quantum group $U_q(\mathfrak{sl}_2)$ at $q = e^{i\pi/5}$ . - Modular $S$ -matrix for Fibonacci MTC: $S = \frac{1}{\sqrt{\phi+2}} \begin{pmatrix} 1 & \phi \\ \phi & -1 \end{pmatrix}$ with $\phi = 1.618$ . - Central charge $c = 4/5$ , related to the minimal model $M(5,4)$ of conformal field theory. CONNECTION: - The golden ratio $ \phi = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Fibonacci anyon braid group representations are classified by modular tensor categories (MTCs) with fusion rules derived from the Fibonacci (golden ratio) quantum group \( \text{SU}(2)_3 $, yielding non-Abelian statistics and universal topological quantum computation. MATH: - Fibonacci anyon fusion rule: $ \tau \times \tau = 1 + \tau $ (where $ \tau $ is the non-Abelian anyon, 1 is the vacuum). - Quantum dimension of $ \tau $: $ d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 $. - Braid matrix eigenvalues: $ e^{\pm i\pi/5} $ and $ e^{\pm i3\pi/5} $, derived from the $ R $-matrix of the quantum group $ U_q(\mathfrak{sl}_2) $ at $ q = e^{i\pi/5} $. - Modular $ S $-matrix for Fibonacci MTC: \[ S = \frac{1}{\sqrt{\phi+2}} \begin{pmatrix} 1 & \phi \\ \phi & -1 \end{pmatrix} \] with $ \phi = 1.618 $. - Central charge $ c = 4/5 $, related to the minimal model $ M(5,4) $ of conformal field theory. CONNECTION: - The golden ratio \( \phi = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 23, 20260 citations

Quantum Representations of Modular Groups and Maslov Indices

Vladimir Georgievich Turaev

De Gruyter eBooksDec 31, 19921 citation

Modular S and T Matrices: Linking CFT, Quantum Groups, and Topological Phases — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Modular S and T matrices from CFT encode braiding and fusion in quantum group representations at roots of unity, linking topological phases to geometric invariants. | MATH: Modular group SL(2,Z) generators: S (modular transformation τ → -1/τ), T (τ → τ+1). S and T satisfy S² = (ST)³ = C (charge conjugation), S⁴ = I. Quantum group U_q(g) at q = e^{π i / k} yields finite-dimensional representations with fusion rules N_{ij}^k. S-matrix elements: S_{ab} = (1/√N) Σ_c d_c e^{2π i (h_a + h_b - h_c)}. T-matrix: T_{ab} = δ_{ab} e^{2π i h_a - c/24}. | CONNECTION: S-matrix eigenvalues relate to golden ratio φ = 1.618 in SU(2)_k at k=1: S = (1/√2)[1 1; 1 -1]. For k=3, S entries involve φ: S_{11} = (1/√5) sin(π/5) = φ/√5 ≈ 0.7236. Fusion rules for Fibonacci anyons: τ × τ = 1 + τ, with quantum dimension d_τ = φ = 1.618. Braiding eigenvalues: R^{ττ}_1 = e^{-4π i/5}, R^{ττ}_τ = e^{3π i/5}, giving phases related to 0.618 (1/φ). | DEPTH: 8 — Directly connects modular tensor categories to topolo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Modular S and T matrices from CFT encode braiding and fusion in quantum group representations at roots of unity, linking topological phases to geometric invariants. | MATH: Modular group SL(2,Z) generators: S (modular transformation τ → -1/τ), T (τ → τ+1). S and T satisfy S² = (ST)³ = C (charge conjugation), S⁴ = I. Quantum group U_q(g) at q = e^{π i / k} yields finite-dimensional representations with fusion rules N_{ij}^k. S-matrix elements: S_{ab} = (1/√N) Σ_c d_c e^{2π i (h_a + h_b - h_c)}. T-matrix: T_{ab} = δ_{ab} e^{2π i h_a - c/24}. | CONNECTION: S-matrix eigenvalues relate to golden ratio φ = 1.618 in SU(2)_k at k=1: S = (1/√2)[1 1; 1 -1]. For k=3, S entries involve φ: S_{11} = (1/√5) sin(π/5) = φ/√5 ≈ 0.7236. Fusion rules for Fibonacci anyons: τ × τ = 1 + τ, with quantum dimension d_τ = φ = 1.618. Braiding eigenvalues: R^{ττ}_1 = e^{-4π i/5}, R^{ττ}_τ = e^{3π i/5}, giving phases related to 0.618 (1/φ). | DEPTH: 8 — Directly connects modular tensor categories to topolo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 27, 20260 citations

Modular variables in quantum theory : final report

Yakir Aharonov, Aage Petersen

Jan 1, 19670 citations

FINDING: Fibonacci anyon braid group representations are classified by modular tensor categories (MTCs) with fusion rules derived from the Fibonacci (golden ratio) quantum group

\text{SU}(2)_3

, yielding non-Abelian statistics and universal topological quantum computation. MATH: - Fibonacci anyon fusion rule:

\tau \times \tau = 1 + \tau

(where

\tau

is the non-Abelian anyon, 1 is the vacuum). - Quantum dimension of

\tau

d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618

. - Braid matrix eigenvalues:

e^{\pm i\pi/5}

and

e^{\pm i3\pi/5}

, derived from the

R

-matrix of the quantum group

U_q(\mathfrak{sl}_2)

q = e^{i\pi/5}

. - Modular

S

-matrix for Fibonacci MTC:

S = \frac{1}{\sqrt{\phi+2}} \begin{pmatrix} 1 & \phi \\ \phi & -1 \end{pmatrix}

with

\phi = 1.618

. - Central charge

c = 4/5

, related to the minimal model

M(5,4)

of conformal field theory. CONNECTION: - The golden ratio \( \phi = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com