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: \( S_{CS} = \frac{k}{4\pi} \int_M \text{Tr}(A \wedge dA + \frac{2}{3} A \wedge A \wedge A) \), with gauge group SU(2), level \( k \in \mathbb{Z} \). - WZW model central charge: \( c = \frac{3k}{k+2} \) for SU(2)_k. - Root lattice \( A_1^{(1)} \): affine Lie algebra with simple root \( \alpha \) satisfying \( |\alpha|^2 = 2 \), dual Coxeter number \( h^\vee = 2 \). - Level-rank duality: \( k \leftrightarrow 1/k \) in certain limits, linking to modular \( S \)-matrix entries \( S_{ab} = \sqrt{\frac{2}{k+2}} \sin\left(\frac{\pi (a+1)(b+1)}{k+2}\right) \). CONNECTION: - Golden ratio appears in limit \( k \to \infty \): \( c \to 3 \), but finite \( k \) yields rational conformal field theories with quantum dimensions \( d_j = \frac{\sin(\pi (2j+1)/(k+2))}{\sin(\pi/(k+2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com