Connection on a Principal bundle and transition functions, as in Hitchin's notes

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.

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

$$\sigma_\beta(x)=g_{\beta\alpha}(x)\cdot \sigma_\alpha(x),\;\;\forall x\in U_\alpha\cap U_\beta.$$

• I guess I am misunderstanding your statement "A section of a $G$ bundle is then a collection of smooth maps $\sigma_\alpha:U_\alpha\rightarrow G$ [---]" You mean local sections? Or you mean given this collection there exists a (global) section $M\rightarrow P$ of $\pi:P\rightarrow M$ – Praphulla Koushik Apr 12 '18 at 8:42
• I should have said that the collection $(U_\alpha)$ is an open cover of $M$. – Liviu Nicolaescu Apr 14 '18 at 15:33