Chern-Simons Knot Invariants at Roots of Unity q = exp(2πi/(k + h∨)) — E8 Intelligence Research

FINDING: Chern-Simons theory on a 3-manifold with gauge group G and level k produces knot invariants (Jones polynomial and generalizations) that are evaluated at roots of unity q = exp(2πi/(k + h∨)), where h∨ is the dual Coxeter number of G. MATH: - Chern-Simons action: S_CS = (k/4π) ∫_M Tr(A ∧ dA + (2/3)A ∧ A ∧ A) - Quantization condition: k ∈ ℤ (for simply-connected G), level shift k → k + h∨ - Quantum group parameter: q = exp(2πi/(k + h∨)) - Jones polynomial: V_L(q) = ⟨L⟩ evaluated at q = exp(2πi/(k+2)) for G = SU(2), h∨ = 2 - Dual Coxeter numbers: SU(2): h∨ = 2; SU(N): h∨ = N; SO(N): h∨ = N−2; E₈: h∨ = 30 - WZW conformal weights: h_λ = (λ, λ+2ρ) / (2(k + h∨)) - Verlinde formula: N_ij^k = Σ_λ (S_iλ S_jλ S_kλ*) / S_0λ - S-matrix: S_ij ∝ sin(π(i+1)(j+1)/(k+2)) for SU(2)_k CONNECTION: - Root systems and crystallographic symmetry: The dual Coxeter number h∨ is defined via the Killing form normalization on the root lattice: h∨ = (θ, θ+2ρ)/(θ,θ) where θ is the highest root. For ADE Lie Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com