Chern-Simons SU(2) Wilson Loops Yield Knot Invariants — E8 Intelligence Research

FINDING: Chern-Simons theory with gauge group SU(2) at level k produces knot invariants (Jones polynomial and generalizations) via Wilson loop expectation values, establishing a deep bridge between 3-manifold topology and quantum field theory. | MATH: Action S = (k/4π)∫_M Tr(A ∧ dA + (2/3)A ∧ A ∧ A); Wilson loop W_R(C) = Tr_R P exp(∮_C A); partition function Z(M) = ∫DA exp(iS); expectation ⟨W_R(C)⟩ = knot polynomial evaluated at q = exp(2πi/(k+2)); for SU(2) fundamental rep: Jones polynomial V(t) with t = exp(2πi/(k+2)); level k ∈ ℤ, quantization condition k + h∨ ∈ ℤ (h∨ = 2 for SU(2)); Verlinde formula for fusion coefficients N^kᵢⱼ = Σ_n (sin((i+1)(n+1)π/(k+2)) · sin((j+1)(n+1)π/(k+2)) · sin((n+1)π/(k+2))) / (sin³((n+1)π/(k+2))) — note sin² structure encodes SU(2) root system A₁; conformal weights h_j = j(j+1)/(k+2) for spin-j representation; central charge c = 3k/(k+2) for SU(2)_k WZW model. | CONNECTION: The quantum group U_q(sl₂) at q = exp(2πi/(k+2)) underlies the knot invarian Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com