nLab nonabelian bundle gerbe

Idea

A nonabelian bundle gerbe (as studied by Aschieri-Cantini-Jurco) is a model for the Lie groupoid which is the total space of a smooth $AUT(H)$-principal 2-bundle for $AUT(H)$ the Lie 2-group that is the automorphism 2-group of a Lie group $H$.

Specifically, a nonabelian bundle gerbe on a smooth manifold $X$ is given by a surjective submersion $Y \to X$ and an $H$-bibundle $P \to Y\times_X Y$ together with a morphism of $H$-bibundles

$$\mu : \pi_0^* P \otimes \pi_2^* P \to \pi_1^* P$$

that is associative in the evident sense. This construction serves to model pullbacks of Lie 2-groupoids of the form

$$\array{ \tilde P &\to& \mathbf{E}AUT(H) \\ \downarrow && \downarrow \\ C(Y) &\stackrel{g}{\to}& \mathbf{B}AUT(H) \\ \downarrow^{\mathrlap{\simeq}} \\ X } \,,$$

where on the right we have the universal principal 2-bundle.

The resulting Lie groupoid $\tilde P$ is an extension of the Cech groupoid $C(Y)$ by $AUT(H)$. This generalizes the case of ordinary bundle gerbes, which are models for $\mathbf{B}U(1)$-principal 2-bundles, for $\mathbf{B}U(1)$ the circle 2-group.

This can all be extended to topological groupoids, and to structure 2-groups given by more general crossed modules than $H\to Aut(H)$.

References

• Paolo Aschieri, Luigi Cantini, Branislav Jurco, Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory (arXiv:hep-th/0312154)