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 $H^{k-1}(M,\mathbb{R}{)}^{\ast }$. We show that the image of $H_{k-1}(M,\mathbb{Z})$ in $H^{k-1}(M,\mathbb{R}{)}^{\ast }$ 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 $C^{\infty }(S,M)$.

Published version