Chern-Simons/SU(2) WZW Duality via A1^(1) Root Lattice — E8 Intelligence Research

FINDING: Chern-Simons theory in 3D gravity is dual to SU(2) Wess-Zumino-Witten model at level k, with root lattice A1^(1) encoding boundary conformal data and topological invariants. MATH: - Chern-Simons action: \( SCS = k/4π ∫_M Tr(A dA + 2/3 A A A) \), with gauge group SU(2), level \( k ∈ Z \). - WZW model central charge: \( c = 3k/k+2 \) for SU(2)_k. - Root lattice \( A_1⁽¹⁾ \): affine Lie algebra with simple root \( α \) satisfying \( |α|^2 = 2 \), dual Coxeter number \( h^ = 2 \). - Level-rank duality: \( k ↔ 1/k \) in certain limits, linking to modular \( S \)-matrix entries \( Sab = √2/k+2 sin(π (a+1)(b+1)/k+2) \). CONNECTION: - Golden ratio appears in limit \( k → ∞ \): \( c → 3 \), but finite \( k \) yields rational conformal field theories with quantum dimensions \( d_j = {sin(π (2j+1)/(k+2))}{sin(π/(k+2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com