Central $U(1)$-Extensions

Central $U(1)$-extensions from prequantization

The result is from NV03, Thm 3.4. See also Integrability of central extensions of the Poisson Lie algebra via prequantization

Theorem(NV03, Thm 3.4) . Let $(M,\omega)$ be a prequantizable symplectic manifold with a Hamiltonian action of a connected Lie group $G$. The pullback of the prequantization central extension by the action $G\to \operatorname{Ham}(M,\omega)$ provides a central Lie group extension $$ 1 \to U(1)\to \hat{G}\to G \to 1 $$

The setting is a connected prequantizable symplectic manifold $(M,\omega)$ with a prequantum $U(1)$-bundle $L\to M$ with connection 1-form $\theta$, a connected Lie group $G$ and a Hamiltonian $G$-action on $M$. Let $\operatorname{Ham}(M,\omega)$ be the Hamiltonian symplectomorphism group of $M$ and let $\operatorname{Quant}(L,\theta)$ be the quantomorphism group of connection-preserving automorphisms of $L$. A well-known result is the Kostant-Souriau extension:

$$ 1\to U(1) \to \operatorname{Quant}(L,\theta)_0 \to \operatorname{Ham}(M,\omega) \to 1 $$

Here the subscript $0$ in $\operatorname{Quant}(L,\theta)$ denotes the identity component. This is a central $U(1)$-extension of the group $\operatorname{Ham}(M,\omega)$. The $G$-action is Hamiltonian, so we have a homomorphism $G\to \operatorname{Ham}(M,\omega)$. Pull the quantomorphism group $\operatorname{Quant}(L,\theta)_0$ back by $G\to \operatorname{Ham}(M,\omega)$:

Created by potrace 1.16, written by Peter Selinger 2001-2019

yields a central $U(1)$-extension $\hat{G}$ of $G$. This construction remains valid when $M$ and $G$ are infinite dimensional, even if $\operatorname{Ham}(M,\omega)$ and $\operatorname{Quant}(L,\theta)$ might fail to be Lie groups.