nLab connection on a 3-bundle

Definition

For $G$ a Lie 3-group, a connection on a $G$-principal 3-bundle coming from a cocycle $g : X \to \mathbf{B}G$ is a lift of the cocycle to the 3-groupoid of Lie 3-algebra valued forms $\mathbf{B}G_{conn}$

$$\array{ && \mathbf{B}G_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G }$$

Examples

• 11-dimensional supergravity can be understood as the theory of 3-connections with coefficients in the super L-infinity algebra called the supergravity Lie 3-algebra. See D'Auria-Fre formulation of supergravity.

Related concepts

• connection on a bundle

• connection on a 2-bundle / connection on a gerbe / connection on a bundle gerbe

• connection on a 3-bundle / connection on a bundle 2-gerbe

  • chiral anomaly, baryogenesis

  • supergravity C-field

• connection on an ∞-bundle

References

The higher parallel transport of local 3-connections is considered in

• Joao Faria Martins, Roger Picken, The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module (arXiv:abs/0907.2566)

Discussion in terms of Gray-categories is in

• Wei Wang, On 3-gauge transformations, 3-curvature and Gray-categories (arXiv:1311.3796)

Examples of 3-connections obtained from fibrations of Courant algebroids are discussed in

• Olivier Brahic, On the infinitesimal Gauge Symmetries of closed forms (arXiv)

A discussion of fully general local 3-connections is in

• Hisham Sati, Urs Schreiber, Jim Stasheff, $L_\infty$-connections (arXiv:0801.3480)

and the globalization is in

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech cocycles for differential characteristic classes

For a discussion of all this in a more comprehensive context see section xy of

• Urs Schreiber, differential cohomology in a cohesive topos

See also connection on an infinity-bundle for the general theory.