Chern-Simons Quantization via SU(2) Volume Normalization — E8 Intelligence Research

Abstract

FINDING: Non-Abelian Chern-Simons theory quantizes the second Chern number via the integral of the trace of (A∧dA + (2/3)A∧A∧A), yielding the level quantization constant 1/(8π²) — the volume of SU(2) as a group manifold. | MATH: The key object is the Chern-Simons 3-form: CS(A) = Tr(A∧dA + (2/3)A∧A∧A) The action is S = 2πi · k · (1/8π²) ∫ CS(A) The normalization 1/(8π²) is exactly the volume of SU(2) (the 3-sphere S³) under the Killing form. For SU(2), the second Chern number (instanton number) is: c₂ = (1/8π²) ∫ Tr(F∧F) ∈ ℤ where F = dA + A∧A. The ratio 1/8π² = 1/(2·(2π)²) — note the factor 2 from the trace normalization (Tr(TᵃTᵇ) = ½δᵃᵇ). | CONNECTION: The volume of S³ = 2π². The normalization 1/8π² = 1/(4·2π²) — the factor 4 arises from the SU(2) structure constants f_abc = ε_abc (totally antisymmetric, with ε₁₂₃ = 1). The ratio 1/8π² ≈ 0.012665 — not a golden ratio, but the *structure* of SU(2) as S³ (Hopf fibration S¹→S³→S²) encodes the quaternionic unit sphere. The secon Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com