Groupoid Central Extension

\(S^1\)-gerbes are in 1-to-1 correspondence with Morita equiavalence classes of groupoid \(S^1\)-central extensions.

\(S^1\)-central extension of a Lie Groupoid#

Definition. An \(S^1\)-central extension of a Lie groupoid \(X_1 \rightrightarrows X_0\) is a Lie groupoid \(R=(R_1\rightrightarrows X_0)\) with a groupoid morphism \(\pi:R_1\to X_1\) such that \(\text{ker}\pi \cong X_0\times S^1\) lies in the center of \(R_1\).

• Naturally, \(\pi:R_1\to X_1\) is an \(S^1\)-principal bundle.
• If we take \(X\) to be the Cech groupoid \(\coprod_{ij}U_{ij}\rightrightarrows \prod_i U_i\) of a manifold \(M\), then we get essentially the picture of an \(S^1\)-gerbe over a manifold. The multiplication on \(\coprod_{ij} U_ij\times S^1\) is given by \((x_{ij},\lambda_1)\cdot(x_{jk},\lambda_2) = (x_{ij},\lambda_1\lambda_2 c_{ijk})\), and \(c_{ijk}:U_{ijk}\to S^1\) is a 2-cocycle which represents the Cech class in \(H^2(M,S^1) \cong H^3(M,\mathbb{Z})\).

In Poisson geometry, a certain prequantization of a symplectic groupoid naturally becomes an $S^1$-central extension of groupoids with a connection, see Extensions of symplectic groupoids and quantization.