nLab bundle 2-gerbe

Idea

A bundle 2-gerbe is a special presentation of the total space of a $\mathbf{B}^2 U(1)$-principal 3-bundle, where $\mathbf{B}^2 U(1)$ 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 $\mathcal{L} \to Y\times_X Y$;

• a morphims of bundle gerbes $\pi_1^* \mathcal{L} \otimes\pi_2^* \mathcal{L} \to \pi_1^* \mathcal{L}$;

• 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, Journal of Geometry and Physics

  Volume 21, Issue 2, January 1997, Pages 183-197 (arXiv:hep-th/9511169, doi:10.1016/S0393-0440(96)00014-9)

and further developed in

• Danny Stevenson, Bundle 2-gerbes, Proceedings of the London Mathematical Society, Volume 88, Issue 2, March 2004, Pages 405–435 (arXiv:math/0106018, doi:10.1112/S0024611503014357),

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:alg-geom/9601025, doi:10.1007/s002220050118)

A model for the supergravity C-field in terms of nonabelian bundle 2-gerbes:

• Paolo Aschieri, Branislav Jurčo, Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory, JHEP 0410:068 (2004) [arXiv:hep-th/0409200, doi:10.1088/1126-6708/2004/10/068]