Central extensions of Lie groups preserving a differential form

Let $M$ be a manifold with a closed, integral $(k+1)$-form $ω$, and let $G$ be a Fréchet-Lie group acting on $(M,ω)$. 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 $(S,β)$, where $β$ 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 $C^\infty(S, M)$.

No reviews yet.

No comments yet.