I am reading this notes of Hitchin to understand about gerbes. He defines gerbe by giving a collection of $2$ cocycles $g_{\alpha\beta\gamma}:U_\alpha\cap U_\beta\cap U_\gamma\rightarrow S^1$ with some conditions as here.

He then

define a connection on a gerbe given by cocycles $g_{\alpha\beta\gamma}:U_\alpha\cap U_\beta\cap U_\gamma\rightarrow S^1$ by forms which satisfy $$G|_{U_\alpha}=dF_\alpha$$ $$F_\beta-F_\alpha=dA_{\alpha\beta}$$ $$i A_{\alpha\beta}+iA_{\beta\gamma}+iA_{\gamma\alpha}=g_{\alpha\beta\gamma}^{-1}dg_{\alpha\beta\gamma}.$$ We call the closed $3$ form $G$ the curvature of the gerbe connection.

We say that a connection on a gerbe is flat if its curvature $G$ vanishes. Suppose a gerbe is flat, we have $G=0$ i.e., $dF_{\alpha}=0$. As we are considering contractible $U_\alpha$ (good open cover) we see that $F_\alpha$ being closed implies $F_\alpha$ is exact (by Poincare lemma) i.e., $F_\alpha=dB_{\alpha}$ on $U_\alpha$.

On $U_\alpha\cap U_\beta$ we have $F_\beta-F_\alpha=d(B_\beta-B_\alpha)$.

As $F_\beta-F_\alpha=dA_{\alpha\beta}$ we have $d(B_\beta-B_\alpha)=dA_{\alpha\beta}$ i.e., $A_{\alpha\beta}-B_\beta+B_\alpha=df_{\alpha\beta}$ (again by Poincare lemma) on $U_\alpha\cap U_\beta$.

He then says,

as $iA_{\alpha\beta}+iA_{\beta\gamma}+iA_{\gamma\alpha}=g_{\alpha\beta\gamma}^{-1}dg_{\alpha\beta\gamma}$ , we have $d(if_{\alpha\beta}+if_{\beta\gamma}+if_{\gamma\alpha}-\log g_{\alpha\beta\gamma})=0$

I am not able to see how this is true. I did write down what $A_{\alpha\beta}$ and $f_{\alpha\beta}$ are but I got some relation that does not look anyway close to this.

Assuming $d(if_{\alpha\beta}+if_{\beta\gamma}+if_{\gamma\alpha}-\log g_{\alpha\beta\gamma})=0$, it says the following:

Of course $\log(g)$ is defined only modulo $2\pi i \mathbb{Z}$ so what we have here is a collection of constants $c_{\alpha\beta\gamma}\in 2\pi \mathbb{R}/\mathbb{Z}$. The cocycle $c_{\alpha\beta\gamma}/2\pi$ represents a Cech class in $H^2(X,\mathbb{R}/\mathbb{Z})$ which we call the holonomy of the connection.

I do not understand completely what this means.

As $if_{\alpha\beta}+if_{\beta\gamma}+if_{\gamma\alpha}-\log g_{\alpha\beta\gamma}:U_\alpha\cap U_\beta\cap U_\gamma\rightarrow S^1$, $d(if_{\alpha\beta}+if_{\beta\gamma}+if_{\gamma\alpha}-\log g_{\alpha\beta\gamma})=0$ implies that $if_{\alpha\beta}+if_{\beta\gamma}+if_{\gamma\alpha}-\log g_{\alpha\beta\gamma}$ is a constant and I guess this is what they are calling $c_{\alpha\beta\gamma}$. Identifying $S^1$ with $\mathbb{R}/\mathbb{Z}$ he says $c_{\alpha\beta\gamma}\in 2\pi \mathbb{R}/\mathbb{Z}$. Seeing constants $c_{\alpha\beta\gamma}$ as constant functions, this defines maps $c_{\alpha\beta\gamma}:U_\alpha\cap U_\beta\cap U_\gamma\rightarrow \mathbb{R}/\mathbb{Z}$. This defines $2$ cocycle, thus an element of $H^2(X,\mathbb{R}/\mathbb{Z})$, this they are calling it as a holonomy.

I am not very sure if this is what it means. Any comments are welcome.

Any reference for concept of holonomy on gerbes would be useful.

EDIT : I thank user Tsemo for proving the equality that I said I was not able to prove. As I have not clearly stated what my question is, I would like to say it now. Any thoughts on motivation behind calling this holonomy is welcome. Is this collection $\{c_{\alpha\beta\gamma}\}$ restircted to some subset is holonomy of some (line) bundle?