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

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 = eπ i / k yields finite-dimensional representations with fusion rules Nᵢⱼ^k. S-matrix elements: Sab = (1/√N) Σ_c d_c e2π i (h_a + h_b - h_c). T-matrix: Tab = δab e2π 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: S₁₁ = (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: Rττ_1 = e-4π i/5, Rττ_τ = e3π 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