Let $M$ be a manifold with a closed, integral $(k+1)$-form $\omega$, and let $G$ be a Fr\'echet-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) $.
Categories: math.DG math.SG