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 \( S = \frac{k}{4\pi} \int_M \text{Tr}(A \wedge dA + \frac{2}{3} A \wedge A \wedge A) \); partition function \( Z(\tau) \) transforms under SL(2,Z) as \( Z(\tau+1) = Z(\tau) \), \( Z(-1/\tau) = \tau^{c/2} Z(\tau) \); for gauge group SU(2) (A1 root lattice), quantization condition: \( k \in \mathbb{Z} \), level \( k \) related to central charge \( c = 3k/(k+2) \); modular parameter \( \tau \) of torus; A1 root lattice vectors \( \alpha = \pm \sqrt{2} \) in normalized units. | CONNECTION: A1 root lattice (SU(2) Lie algebra) has root length ratio \( \sqrt{2} \), linking to golden ratio via \( \sqrt{2} \approx 1.414 \) (not 1.618, but related by \( \phi^2 = \phi + 1 \)); 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