Modular Invariance of Chern-Simons Theory Quantized via A1 Root Lattice — E8 Intelligence Research

FINDING: SL(2,Z) modular invariance of Chern-Simons partition function on torus is quantized via root lattice A1, linking topological quantum field theory to lattice geometry. | MATH: Chern-Simons action \( S = k/4π ∫_M Tr(A dA + 2/3 A A A) \); partition function \( Z(τ) \) transforms under SL(2,Z) as \( Z(τ+1) = Z(τ) \), \( Z(-1/τ) = τc/2 Z(τ) \); for gauge group SU(2) (A1 root lattice), quantization condition: \( k ∈ Z \), level \( k \) related to central charge \( c = 3k/(k+2) \); modular parameter \( τ \) of torus; A1 root lattice vectors \( α = ± √2 \) in normalized units. | CONNECTION: A1 root lattice (SU(2) Lie algebra) has root length ratio \( √2 \), linking to golden ratio via \( √2 ≈ 1.414 \) (not 1.618, but related by \( φ^2 = φ + 1 \)); modular invariance imposes quantization of coupling \( k \) in steps of 1, mirroring discrete harmonic ratios; torus mod Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com