This is along the lines of this question
Gerbes are not just topological objects: we can do differential geometry with them too. We shall next describe what a connection on a gerbe is.
To begin with, let’s look at a connection on a line bundle which is given by transition functions $g_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow S^1\subseteq \mathbb{C}^*$. A connection on consists of $1$ forms $A_\alpha$ defined on $U_\alpha$ such that on a twofold intersection $U_\alpha\cap U_\beta$ we have $iA_\alpha-iA_\beta=g_{\alpha\beta}^{-1}dg_{\alpha\beta}$
I know what is a connection $1$ form (on principal bundle) but not as in above version. I am trying to relate what I know with what is given here.
Let $(P,M,\pi)$ be a principal $G$ bundle with $\mathfrak{g}=T_eG$ and transition functions $g_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow G$.
Definition : A connection form on $P$ is a $\mathfrak{g}$ valued $1$ form $\omega$ on $P$ such that
- $\omega(p)(A^*(p))=A$ for all $A\in \mathfrak{g}$ and $p\in P$
- $(\delta_g^*\omega)(p)(v)=Ad_{g^{-1}}(\omega(p)(v))$ for all $p\in P,g\in G$ and $v\in T_pP$.
Given a connection $1$ form $\omega$ on $P$ I am trying to associate a collection of $1$ forms $\{A_\alpha\}$ with some compatability conditions (here $A_\alpha$ is a $1$ form on $U_\alpha$).
Given local trivialization $\psi_\alpha$, we have a section of $\pi$ namely $\sigma_\alpha:U_\alpha\rightarrow P$ defined as $\sigma_\alpha(x)=\psi_\alpha^{-1}(x,e)$. Given local trivialization $\psi_\beta$, we have a section of $\pi$ namely $\sigma_\beta:U_\beta\rightarrow P$ defined as $\sigma_\beta(x)=\psi_\beta^{-1}(x,e)$.
Suppose $x\in U_\alpha\cap U_\beta$ then, we have $\sigma_\alpha(x)\in \pi^{-1}(x)$ and $\sigma_\beta(x)\in \pi^{-1}(x)$. Thus, there exists $g\in G$ (depending on $x$) such that $\sigma_\beta(x)=\sigma_\alpha(x)g$. Given $x\in U_\alpha\cap U_\beta$ there is an obvious choice for an element of $G$ namely $g_{\alpha\beta}(x)$. I see that we have $\sigma_\beta(x)=\sigma_\alpha(x)g_{\alpha\beta}(x)$ for all $x\in U_{\alpha}\cap U_\beta$ i.e., $\sigma_\beta=\sigma_\alpha g_{\alpha\beta}$.
Given a $1$ form $\omega$ on $P$ and we can pull back $\omega$ to $U_\alpha$ under $\sigma_\alpha$ to get $1$ form $\omega_\alpha=\sigma_\alpha^*\omega$ on $U_\alpha$ similarly we can pull back to $U_\beta$ to get $1$ form $\omega_\beta=\sigma_\beta^*\omega$ on $U_\beta$.
As $\sigma_\alpha$ and $\sigma_\beta$ are related by $\sigma_\alpha=\sigma_\beta g_{\alpha\beta}$, one can expect that $\omega_\alpha$ and $\omega_\beta$ are related some how. Given $g_{\alpha\beta}:U_{\alpha\beta}\rightarrow G$ we can produce a $1$ form on $U_\alpha\beta$ as pull back of $\theta$ on $G$ i.e., the canonical $1$ form on $G$ which is a left invariant $1$ form determined by $\theta(e)(A)=A$ for all $A\in \mathfrak{g}$. Let us denote pull back of $\theta$ to $U_\alpha\cap U_\beta$ by $\theta_{\alpha\beta}$. Then, I am expecting some compatibility relation between $1$ forms $\omega_\alpha,\omega_\beta$ and $\theta_{\alpha\beta}$ that should come from $\sigma_\alpha=\sigma_\beta g_{\alpha\beta}$.
EDIT : I could see in Kobayshi and Nomizu that these local connection forms $\sigma_\alpha$ are related by $$ \sigma_\beta(p)(X_p)=ad(g_{\alpha\beta}(p)^{-1})(\sigma_\alpha(p)(X_p))+\theta_{\alpha\beta}(p)(X_p) \text{ on } U_\alpha\cap U_\beta.$$ In short, we have $$\sigma_\beta=ad(g_{\alpha\beta}^{-1})\sigma_\alpha+\theta_{\alpha\beta} \text{ on } U_\alpha\cap U_\beta$$
I see that people use $g_{\alpha\beta}^{-1}dg_{\alpha\beta}$ (notation of Maurer Cartan differential of a map $g_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow G$) to denote what we have called as $\theta_{\alpha\beta}$. So, in this notation, we have
$$\sigma_\beta=ad(g_{\alpha\beta}^{-1})\sigma_\alpha+g_{\alpha\beta}^{-1}d g_{\alpha\beta}\text{ on } U_\alpha\cap U_\beta.$$
Question : I am trying to understand how $\sigma_\beta=ad(g_{\alpha\beta}^{-1})\sigma_\alpha+g_{\alpha\beta}^{-1}d g_{\alpha\beta}\text{ on } U_\alpha\cap U_\beta$ on principal bundles has its similar part as $iA_\alpha-iA_\beta=g_{\alpha\beta}^{-1}dg_{\alpha\beta}$ in case of line bundles.
Any suggestion/reference on how to see this is welcome.
