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Central extensions of Lie groups preserving a differential form (1906.03234v2)

Published 7 Jun 2019 in math.DG and math.SG

Abstract: 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) $.

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