Symplectic Reduction and Moment Maps in Gauge Theories — E8 Intelligence Research

Abstract

FINDING: Atiyah-Bott symplectic reduction identifies Yang-Mills functional on a Riemann surface as the norm-square of a moment map; modern extensions include homotopy moment maps for Chern-Simons and geometric quantization on b-symplectic manifolds. | MATH: Let \( \mathcal{A} \) be affine space of connections on a principal \(G\)-bundle over a Riemann surface \(\Sigma\). The gauge group \(\mathcal{G}\) acts symplectically on \(\mathcal{A}\) with moment map \(\mu: \mathcal{A} \to \mathfrak{g}^*\) given by curvature \(F_A\) (up to sign). Yang-Mills action \(S_{YM}(A) = \frac{1}{2}\|F_A\|^2 = \frac{1}{2}\|\mu(A)\|^2\). Symplectic quotient \(\mu^{-1}(0)/\mathcal{G} = \mathcal{M}_{\text{flat}}\) (moduli of flat connections). For Chern-Simons on 3-manifold, homotopy moment map \(\tilde{\mu}: \mathcal{A} \to \mathfrak{g}^*[1]\) satisfies \(d\tilde{\mu} = \iota_{\xi}\omega_{CS}\) up to homotopy, with level \(k \in \mathbb{Z}\) entering via \(S_{CS} = \frac{k}{4\pi}\int \text{Tr}(A\wedge dA + \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com