Flux homomorphisms and principal bundles over infinite dimensional manifolds
In the article under review, the concept of flux homomorphism for a closed differential form (see, e.g., [\textit{A. Banyaga}, The structure of classical diffeomorphism groups, Kluwer (1997; Zbl 0874.58005)]) is generalized to the case of a closed \(p\)-form \(\omega\) on a smooth manifold \(M\) modelled on a locally convex space, taking its values in a sequentially complete, locally convex space \({\mathfrak z}\). For \(p=2\) and connected \(M\), the flux homomorphism is used to analyze which automorphisms of \((M,\omega)\) can be lifted to automorphisms of a given principal \(Z\)-bundle \(P\to M\) with connection \(1\)-form \(\theta\) and curvature \(\omega\), where \(Z:={\mathfrak z}/\Gamma\) for a discrete subgroup of \({\mathfrak z}\). Generalizing a result of \textit{B. Konstant} in the finite-dimensional case [Lect.\ Notes Math. 170, 87-208 (1970; Zbl 0223.53028)], the latter are characterized as those automorphisms of \((M,\omega)\) which preserve the holonomy of piecewise smooth closed curves in \(M\). The results are then used to construct central extensions of infinite-dimensional Lie groups.
scientific article; zbMATH DE number 1984997