Contents

Idea

A bundle 2-gerbe is a special presentation of the total space of a $B^{2}U(1)$-principal 3-bundle, where $B^{2}U(1)$ is the circle Lie 3-group.

A connection on a bundle 2-gerbe is a special cocycle representative for circle n-bundles with connection, hence for degree 4 Deligne cohomology, hence for degree 4 Cheeger-Simons differential characters.

The definition is built by iteration on the definition of bundle gerbe:

a bundle 2-gerbe over a manifold $X$ is

• a surjective submersion $Y\to X$;

• on the fiber product $Y{\times }_{X}Y$ a bundle gerbe $ℒ\to Y{\times }_{X}Y$;

• a morphims of bundle gerbes ${\pi }_1^{*}ℒ\otimes {\pi }_2^{*}ℒ\to {\pi }_1^{*}ℒ$;

• which is associative up to a choice of coherent 2-morphisms.

Examples

• Chern-Simons 2-gerbe

Related concepts

• principal bundle / torsor / associated bundle

• principal 2-bundle / gerbe / bundle gerbe

• principal 3-bundle / 2-gerbe / bundle 2-gerbe

• principal ∞-bundle / associated ∞-bundle / ∞-gerbe

References

Bundle 2-gerbes were briefly introduced in

• Alan Carey, Michael Murray, B.L. Wang, Higher bundle gerbes and cohomology classes in gauge theories (arXiv:hep-th/9511169)

and further developed in

• Danny Stevenson, Bundle 2-gerbes (arXiv:math/0106018),

drawing on ideas from Stevenson’s PhD thesis (arXiv:math/0004117).

A general picture of bundle $n$-gerbes (with connection) as circle (n+1)-bundles with connection classified by Deligne cohomology is in

• Pawel Gajer, Geometry of Deligne cohomology Invent. Math., 127(1):155–207 (1997) (arXiv)