Abelian gerbes can arise from obstructions to lifting a principal $C$-bundle to a principal $B$-bundle given some central extension $0\to A \to B \to C \to 0$ or as a representative of a cohomology cocycle.

**Are there explicit examples of non-trivial (see below), non-abelian gerbes with non-flat connection?**

Especially useful would be a go-to example for physicists; akin to the Dirac Monopole example for abelian gerbes.

Note: I'm aware that there is no definition for "explicit " or "example" but I think my question is useful in spite of this.

For clarity, here is my definition of a non-abelian gerbe with connection:

Definition: A non-abelian gerbe, $\mathcal{G}$, with connection on a manifold, $M$, with structure (2-)group given by the crossed module $(H \xrightarrow{{t}} G, \alpha)$, consists of the data:

- An good (in the Cech sense) open cover $\{ U_{i} \}$ of $M$.
- For each open set $U_i$, differential forms $A_i \in \Omega^1(U_i, \mathfrak{g})$ and $B_i \in \Omega^2(U_i, \mathfrak{h})$
- For each intersection, $U_{ij}$, functions and forms $g_{ij} \in \Omega^0(U_{ij}, G)$ and $a_{ij} \in \Omega^1(U_{ij}, \mathfrak{h})$
- For each triple intersection, $U_{ijk}$, functions $f_{ijk} \in \Omega^0(U_{ijk}, H)$.

satsifying some compatibility conditions, where $\mathfrak{g}$ and $\mathfrak{h}$ are the Lie algebras of the Lie groups $G$ and $H$, respectively.

If you're unfamiliar with studying these non-abelian gerbes, it is easy to mention that a principal $G$ bundle with connection is given by the above data where $f_{ijk} =1$, $a_{ij}=0$, and $B_i = 0$.

Suggested conditions for a non-abelian gerbe with connection to be non-trivial include:

- The gerbe $\mathcal{G}$ is not equivalent to a gerbe $\mathcal{G}'$ where the crossed module $(G' \to H')$ has $H’$ is an abelian group.
- The 3-curvature $H_i = dB_i + [A_i \wedge B_i]$ is non-zero.
- The $\mathcal{G}$ is not equivalent to a gerbe $\mathcal{G}'$ where $f'_{ijk}=1$.