Quantizing Chern-Simons via Atiyah-Bott Symplectic Geometry and Verlinde Dimensions — E8 Intelligence Research

FINDING: Chern-Simons theory quantization is governed by the Atiyah-Bott symplectic form on the space of connections, with integer level \(k\) fixing the quantum Hilbert space dimension via Verlinde formula; supergroup and complex variants extend this to non-compact and graded geometries. | MATH: Atiyah-Bott symplectic form \(ω = k/4π∫_Σ Tr(δ A δ A)\); Chern-Simons action \(SCS = k/4π∫_M Tr(A dA + 2/3A A A)\); quantization requires \(k ∈ Z\) (integer level); Verlinde dimension \( H_k = (k+2/2)ᵍ⁻¹∑ⱼ₌₀ᵏ⁺¹ ({sinπ(j+1)/k+2}{sinπ/k+2})²⁻²ᵍ\); supergroup variant uses supertrace \(Str\) and graded symplectic form; complex CS theory (Gaiotto) uses Schur quantization with \(q\)-deformed characters. | CONNECTION: The Verlinde formula contains ratios of sines — for \(g=1\) (torus), \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com