Integer Chern-Simons Level and Rational Symplectic Volume via Verlinde Formula — E8 Intelligence Research

FINDING: Chern-Simons level quantization is integer due to topological invariance of the path integral; symplectic volume of moduli space is computed via Verlinde formula and yields rational numbers related to level \(k\) and rank \(g\). MATH: - Chern-Simons action: \( SCS = k/4π ∫_M Tr(A dA + 2/3 A A A) \), with \(k ∈ Z\) (level quantization). - Symplectic volume of moduli space \(M(G, Σ)\) of flat connections on Riemann surface \(Σ\) of genus \(g\): \[ Vol(M) = {1}{(k+h^)ᵍ⁻¹} ∑λ ( ∏α>0 2sinπ(α,λ)/k+h^ )²⁻²ᵍ \] where \(h^\) is dual Coxeter number, sum over integrable highest weights \(λ\). - Verlinde formula: dimension of conformal blocks = \(( k+h^/2 )ᵍ⁻¹ ∑λ ∏α>0 ( 2sinπ(α,λ)/k+h^ )²⁻²ᵍ\). - Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com