Central Extensions of Lie Groups Preserving a Differential Form

Abstract

Let |$M$| be a manifold with a closed, integral |$(k+1)$|-form |$\omega $|⁠, and let |$G$| be a Fréchet–Lie group acting on |$(M,\omega )$|⁠. As a generalization of the Kostant–Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of |${\mathfrak{g}}$| by |${\mathbb{R}}$|⁠, indexed by |$H^{k-1}(M,{\mathbb{R}})^*$|⁠. We show that the image of |$H_{k-1}(M,{\mathbb{Z}})$| in |$H^{k-1}(M,{\mathbb{R}})^*$| corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of |$G$| by the circle group |${\mathbb{T}}$|⁠. The idea is to represent a class in |$H_{k-1}(M,{\mathbb{Z}})$| by a weighted submanifold |$(S,\beta )$|⁠, where |$\beta $| is a closed, integral form on |$S$|⁠. We use transgression of differential characters from |$ S$| and |$ M $| to the mapping space |$ C^\infty (S, M) $| and apply the Kostant–Souriau construction on |$ C^\infty (S, M) $|⁠.