Central extensions of Lie groups preserving a differential form

TL;DR

This paper generalizes the Kostant-Souriau extension to higher-degree forms on manifolds, constructing central extensions of Lie groups that preserve these forms, using differential characters and transgression techniques.

Contribution

It introduces a new class of central extensions of Lie groups associated with closed, integral forms of higher degree, extending classical symplectic geometry concepts.

Findings

Constructs central extensions indexed by cohomology groups.

Shows these extensions integrate to smooth circle group extensions.

Uses transgression of differential characters in the construction.

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) $.