Topological Quantization of Magnetic Monopole Charge via First Chern Class — E8 Intelligence Research

FINDING: Magnetic monopole charge is topologically quantized by the first Chern class of a U(1) principal fiber bundle over S², forcing the Dirac quantization condition eg = nℏ/2. MATH: - Dirac quantization: eg = nℏ/2, n ∈ ℤ (equivalently 2eg/ℏ ∈ ℤ) - U(1) bundle over S² classified by π₁(U(1)) = ℤ; transition functions g_NP: S²\{pt} → U(1) - First Chern number: c₁ = (1/2π)∮_S² F = n ∈ ℤ, where F = dA is the field strength 2-form - Monopole flux: ∮_S² F = 2πn (Dirac string singularity removed by bundle patching) - Chern–Weil: c₁ = [F/2π] ∈ H²(S²,ℤ) ≅ ℤ - Hall conductance quantization: σ_xy = (e²/h)·C, C = Chern number of Bloch bundle over Brillouin zone torus T² - TKNN invariant: C = (1/2π)∫_T² d²k · Ω(k), Ω = Berry curvature - Electroweak monopole (arXiv:2008.00464): DQC in SU(2) sector constrains weak mixing angle θ_W via sin²θ_W relation to monopole charge CONNECTION: - Quantization integer n is the topological invariant — no continuous deformation of the bundle changes it. This is Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com