# 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.

• See eg equation (3.2) in arxiv.org/abs/1305.6471 Apr 10, 2018 at 2:22
• @DavidRoberts Thanks, that definitely helps :) Apr 10, 2018 at 4:23

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 $\pi:P\rightarrow M$ Apr 12, 2018 at 8:42
• I should have said that the collection $(U_\alpha)$ is an open cover of $M$. Apr 14, 2018 at 15:33