The notion of connection on a 2-bundle generalizes the notion of connection on a bundle from principal bundles to principal 2-bundles / gerbes.
It comes with a notion of 2-dimensional parallel transport.
For an exposition of the concepts here see also at infinity-Chern-Weil theory introduction the section Connections on principal 2-bundles .
For $G$ a Lie 2-group, a connection on a $G$-principal 2-bundle coming from a cocycle $g : X \to \mathbf{B}G$ is a lift of the cocycle to the 2-groupoid of Lie 2-algebra valued forms $\mathbf{B}G_{conn}$
When the underlying principal 2-bundle over a smooth manifold $X$ is topologically trivial, then the connections on it are identified with Lie 2-algebra valued differential forms on $X$.
Recall from the discussion there what such form data looks like.
Let $\mathfrak{g}$ be some Lie 2-algebra. For instance for discussion of connections on $G$-gerbes ($G$ a Lie group) this would be the derivation Lie 2-algebra of the Lie algebra of $G$.
Let $\mathfrak{g}_0$ and $\mathfrak{g}_1$ be the two vector spaces involved and let
be a dual basis, respectively. The structure of a Lie 2-algebra is conveniently determined by writing out the most general Chevalley-Eilenberg algebra
with these generators.
We thus have
for collections of structure constants $\{C^a{}_{b c}\}$ (the bracket on $\mathfrak{g}_0$) and $\{r^i_a\}$ (the differential $\mathfrak{g}_1 \to \mathfrak{g}_0$) and $\{\alpha^i{}_{a j}\}$ (the action of $\mathfrak{g}_0$ on $\mathfrak{g}_1$) and $\{r_{a b c}\}$ (the “Jacobiator” for the bracket on $\mathfrak{g}_0$).
These constants are subject to constraints (the weak Jacobi identity and its higher coherence laws) which are precisely equivalent to the condition
Over a test space $U$ a $\mathfrak{g}$-valued form datum is a morphism
from the Weil algebra $W(\mathfrak{g})$.
This is given by a 1-form
and a 2-form
The curvature of this is $(\beta, H)$, where the 2-form component (“fake curvature”) is
and whose 3-form component is
We spell out the data of a connection on a 2-bundle over a smooth manifold $X$ with respect to a given open cover $\{U_i \to X\}$, following (FSS, SchreiberCohesive)
(…)
A differential string structure (untwisted) is a 2-connection with coefficients in the string 2-group / string Lie 2-algebra.
The worldvolume theory of the fivebrane is expected to be a 6d (2,0)-supersymmetric QFT containing a self-dual higher gauge theory whose fields are 2-connections (see Self-dual 2-connections there).
A connection on a twisted vector bundle is naturally a 2-connection.
connection on a 2-bundle / connection on a gerbe / connection on a bundle gerbe
Connections on 2-bundles with vanishing 2-form curvature and arbitrary 3-form curvature are defined in terms of their higher parallel transport are discussed in
Urs Schreiber, Konrad Waldorf,
Smooth Functors and Differential Forms, Homology, Homotopy Appl., 13(1), 143-203 (2011) (arXiv:0802.0663)
Connections on non-abelian gerbes and their holonomy, Theory and Applications of Categories, Vol. 28, 2013, No. 17, pp 476-540. (TAC, arXiv:0808.1923, web)
expanding on
Much further discussion and illustration and relation to tensor networks is in
Examples of 2-connections with vanishing 2-form curvature obtained from geometric quantization are discusssed in
The cocycle data for 2-connections with coeffcients in automorphism 2-groups but without restrictions on the 2-form curvature have been proposed in
Lawrence Breen, William Messing, Differential Geometry of Gerbes Advances in Mathematics, Volume 198, Issue 2, 20 (2005) (arXiv:math/0106083)
Lawrence Breen, Differential Geometry of Gerbes and Differential Forms (arXiv:0802.1833)
and
A discussion of fully general local 2-connections is in
and the globalization is in
For a discussion of all this in a more comprehensive context see section xy of
See also connection on an infinity-bundle for the general theory.
Nonabelian 2-connections appear for instance as orientifold B-fields in type II string theory, as differential string structure in heterotic string theory, and as fields in non-abelian 7-dimensional Chern-Simons theory. See at these pages for references.
An appearance in 4-dimensional Yang-Mills theory and 4d TQFT is reported in