$M$ are classified by the second hypercohomology $H^2(M;\mathscr{D}_2)$ with coefficients in the second Deligne complex. For each bundle gerbe with connection one can define a field strength, which is a 3-form on $M$; and the bundles gerbes with connection having vanishing field strength, the so-called *flat gerbes*, are classified by $H^2(M;{\operatorname{U}}(1))$, i.e. by ordinary (singular) cohomology with coefficients in the constant sheaf ${\operatorname{U}}(1)$. Hence, flat gerbes can be described in purely topological terms.

More generally, we will say that for any topological space $T$ the cohomology $H^n(T;{\operatorname{U}}(1))$ classifies flat $n-1$-gerbes on $T$, i.e. a flat gerbe on $T$ can be described by a singular cocycle $\theta \in Z^n(T;{\operatorname{U}}(1))$.

The Wess-Zumino terms for the sigma model of a flat gerbe $\theta \in Z^n(T;{\operatorname{U}}(1))$ on a space $T$ can now be understood as follows: For a given map $\psi : {\Sigmait}{\longrightarrow}T$ from a closed oriented $n$-dimensional manifold ${\Sigmait}$ to $T$ (${\Sigmait}$ can be understood as a (higher dimensional) worldsheet), the Wess-Zumino term is given by $$\begin{aligned}
\langle\psi^* \theta,\mu_{\Sigmait}\rangle, \label{wzwamplitude}\end{aligned}$$ i.e. by the evaluation of the pullback of $\theta$ along $\psi$ on the fundamental class $\mu_{\Sigmait}$ of ${\Sigmait}$. It is also referred to as the *(higher) holonomy of $\theta$ with respect to $\psi$*. 