Weak Riemannian structures on gauge-group orbits
Let \(\pi\) : \(P\to M\) be a principal bundle. The Sobolev extended group of gauge transformations \({\mathcal G}^{k+1}\) is a Hilbert Lie group and acts smoothly on the Sobolev extended space of connections \({\mathcal C}^ k\) (for sufficiently large k). The topological space \({\mathcal C}^ k/{\mathcal G}^{k+1}\) (which is not a smooth manifold unless one restricts oneself to irreducible connections or replaces \({\mathcal G}^{k+1}\) by \({\mathcal G}^{k+1}/center)\) admits a stratification into Hilbert manifolds; i.e. one has a countable family (\({\mathcal C}^ k_{\alpha})\) of \({\mathcal G}^{k+1}\)-invariant submanifolds and for each \(\alpha\) the orbit manifolds \({\mathcal C}^ k_{\alpha}/{\mathcal G}^{k+1}\) exist. The authors show that \({\mathcal C}^ k_{\alpha}/{\mathcal G}^{k+1}\) are (weak) Riemannian manifolds, the weak metric corresponding to the ''kinematic'' part of the Lagrangian considered in Yang-Mills theories.
scientific article; zbMATH DE number 3994586