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 \(\omega = \frac{k}{4\pi}\int_\Sigma \mathrm{Tr}(\delta A \wedge \delta A)\); Chern-Simons action \(S_{CS} = \frac{k}{4\pi}\int_M \mathrm{Tr}(A\wedge dA + \frac{2}{3}A\wedge A\wedge A)\); quantization requires \(k \in \mathbb{Z}\) (integer level); Verlinde dimension \(\dim \mathcal{H}_k = \left(\frac{k+2}{2}\right)^{g-1}\sum_{j=0}^{k+1} \left(\frac{\sin\frac{\pi(j+1)}{k+2}}{\sin\frac{\pi}{k+2}}\right)^{2-2g}\); supergroup variant uses supertrace \(\mathrm{Str}\) 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), \(\dim \math Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com