Group actions, gauge transformations, and the calculus of variations
Let \(M\) be a Banach manifold with a smooth Riemannian metric and let \(G\) be a Banach Lie group acting isometrically on \(M\). Also, it is assumed that \(\varphi\) is a \(G\)-invariant smooth map from \(M\) into a Banach space. The present paper is devoted to the study of the geometry of \(M^ 0_ g = \varphi^{-1}(0)/G\). In particular, the authors study the Riemannian metric and curvature of \(M^ 0_ g\) induced by the metric of \(M\), using a variational problem for gauge transformations. Applications are dealing with the space of metrics or connections, especially the moduli space of self-dual connections on bundles over 4-manifolds, the Weil-Petersson metric on Teichmüller space and Hamiltonian group actions (\(\varphi\) being now a moment map).
scientific article; zbMATH DE number 126350
scientific article