The trilogy of the momentum map
Let \(\omega\) be a closed 2-form on a convex differentiable manifold \(X\) and denote by \(P_\omega\) the group of periods of the form \(\omega\). The factor space \(T_\omega= \mathbb{R}/P_\omega\) is called the torus of periods of the form \(\omega\). Every principal bundle of base \(X\) with the structural group \(T_\omega\) that admits a connection of curvature \(\omega\) is called an integration bundle of the form \(\omega\). In the first section of the paper, the author proves that the set of integration bundles of the form \(\omega\) is determined, up to an equivalence of principal bundles, by the first group of extensions of \(H_1(X, \mathbb{Z})\) with coefficients in \(P_\omega\), i.e. \(\text{Ext}(H_1(X, \mathbb{Z}), P_\omega)\) (Theorem 1.5). In the following three sections, the author generalizes the known construction from the case where the form \(\omega\) is integer (i.e. the group of periods \(P_\omega\) is either null or isomorphic to \(\mathbb{Z}\)) of the trilogy of the Moment Map to the case where \(\omega\) is an arbitrary closed 2-form. In section 5 the obtained construction is illustrated by some examples. Finally, the author gives an appendix which presents the category of differentiable spaces used in this paper. This category of differentiable spaces is necessary because the torus of periods \(T_\omega\) is not a Lie group if the form \(\omega\) is not integer.
scientific article; zbMATH DE number 773330
scientific article