Symplectic Structure and Prequantum Bundles in Chern-Simons Theory — E8 Intelligence Research

FINDING: Chern-Simons theory quantization is governed by the Atiyah-Bott symplectic structure on the space of connections, with the integer level \(k\) as the symplectic form's cohomology class — the key insight is that the symplectic form \(\omega = \frac{k}{4\pi}\int_{\Sigma} \mathrm{Tr}(\delta A \wedge \delta A)\) is not merely geometric but defines a prequantum line bundle whose curvature is \(k\) times a generator of \(H^2(\mathcal{A}/\mathcal{G}, \mathbb{Z})\). | MATH: Atiyah-Bott symplectic form: \(\omega_{AB}(\alpha,\beta) = \frac{k}{2\pi}\int_{\Sigma} \mathrm{Tr}(\alpha \wedge \beta)\) for \(\alpha,\beta \in T_A\mathcal{A}\); quantization condition: \([\omega_{AB}/2\pi] \in H^2(\mathcal{A}/\mathcal{G}, \mathbb{Z})\) forces \(k \in \mathbb{Z}\). Combinatorial quantization (Aghaei) replaces the infinite-dimensional symplectic quotient with a finite-dimensional one using quantum groups, yielding the Reshetikhin-Turaev invariant with \(q = e^{2\pi i/(k+h^\vee)}\) where \(h^\vee\) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com