This is along the lines of [this question](https://mathoverflow.net/questions/298437/confusion-in-definition-of-gerbes-in-hitchins-notes)



> 
> 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.



Connection on a Principal bundle and transition functions, as in Hitchin's notes
A principal $G$-bundle $P\to M$ can be described by an open cover $(U\_\alpha)$ of $M$ and a cocycle $g\_{\beta\alpha}: U\_\alpha\cap U\_\beta\to G$. The total space is the quotient of the disjoint union of the spaces $U\_\alpha\times G$ via the equivalence relation


$$ U\_\alpha\times g\ni (x, g')\sim (y, g'')\in U\_\beta\times G $$
iff
$$ x=y,\;\;g''=g\_{\beta\alpha}(x)\cdot g'. $$


A section of a $G$ bundle is then a collection of smooth maps 


$$\sigma\_\alpha: U\_\alpha\to G $$


satisfying the gluing conditions


$$ \sigma\_\beta(x)=g\_{\beta\alpha}(x)\cdot \sigma\_\alpha(x),\;\;\forall x\in U\_\alpha\cap U\_\beta. $$


