Local coordinates of one form on a principal bundle

Question

I am reading "Natural and Gauge Natural Formalism for Classical Field Theory" by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates.

Let's say we have a principal bundle $\mathcal{P}=(P, M, \pi ; G)$ and the isomorphism $T_{e} L_{p}: \mathfrak{g} \longrightarrow V_{p}(\pi): T_{A} \mapsto \lambda_{A}(p)$. He fixes a point $p=[x, g]_{(\alpha)}$. $\theta_{(L)}^{A}=\bar{L}_{a}^{A}(g) \mathrm{d} g^{a}$ is a local basis of left invariant 1-forms dual to $\lambda_{A}=L_{A}^{a}(g) \partial_{a}$ where the two matrices are inverses of each other. Then the connection 1-form can be written as \begin{equation} \bar{\omega}_{p}=\left[\theta_{(L)}^{A}(p)+\mathrm{Ad}_{B}^{A}\left(g^{-1}\right) \omega_{\mu}^{B}(x) \mathrm{d} x^{\mu}\right] \otimes T_{A}. \end{equation} Pulling back with a section, he gets \begin{equation}\sigma^{*} \bar{\omega}=\left[\bar{L}_{a}^{A}(g) \partial_{\mu} g^{a}(x)+\mathrm{Ad}_{B}^{A}\left(g^{-1}\right) \omega_{\mu}^{B}(x)\right] \mathrm{d} x^{\mu} \otimes T_{A}. \end{equation} However, then he goes on saying "We remark that the local gauge $\sigma$ also induces a local trivialization of $\mathcal{P}$. In the induced local trivialization, the section $\sigma$ has the expression $\sigma: x^{\mu} \mapsto\left(x^{\mu}, e\right)$ and the vector potential is of the form" \begin{equation} \sigma^{*} \bar{\omega}=\omega_{\mu}^{A}(x) \mathrm{d} x^{\mu} \otimes T_{A} \end{equation} He also states that the induces connection is of the form \begin{equation} \omega=\mathrm{d} x^{\mu} \otimes\left(\partial_{\mu}-\omega_{\mu}^{A}(x) \rho_{A}\right) \end{equation}

1. My first question is how one can derive the expression for $\bar{\omega}_{p}$?
2. My second question is why every other book on the subject I know uses $\sigma^{*} \bar{\omega}=\omega_{\mu}^{A}(x) \mathrm{d} x^{\mu} \otimes T_{A}$ as the definition of a form in local coordinates; even this seems not to be true for a general section.
3. I am also not sure how one can derive the form of the induced connection.

Cross posted: https://math.stackexchange.com/questions/4128191/local-coordinates-of-one-form-on-a-principal-bundle

Answer 1

Since you cross-posted, I'll cross-answer my reply from MSE. I'M rather new to the MO forum so I could use some reputation...

• I assume by $T_A$ you mean a set of basis vectors of $\mathfrak{g}$.
• I assume by $\omega$ (without bar) you mean the connection of an associated vector bundle, associated via some representation of $G$.

For (1.):

By definition a connection form is a differential form $\mathcal{A}\in\Omega^1(P,\mathfrak{g})$, which is both of type $\text{Ad}$, i.e. $(R_g)^*\mathcal{A}=\text{Ad}\circ\mathcal{A}$, and in a point $p\in P$ the map $\mathcal{A}\big|_p:T_pP\to\mathfrak{g}$ has a certain behavior on vertical tangent vectors. That is, by being of type $\text{Ad}$, a connection form is determined already on a whole fiber by only one single value in this fiber. On the other hand, the difference between two connection forms is also a $1$-form of type $\text{Ad}$ and vanishes on vertical tangent vectors (since both have the same prescribed values on these). Such $1$-forms with the latter property are called horizontal. Conversely, you can show that for any connection form $\mathcal{A}$ and any horizontal $1$-form $\omega$ of type $\text{Ad}$, also $\mathcal{A}+\omega$ is a connection form. In other words, the set of connection forms is an affine space with spatial structure of the space of horizontal $1$-forms of type $\text{Ad}$.

Now, on a trivial bundle $U\times G$ you have a canonical flat connection form, which essentially does only what it should on vertical tangent vectors, and nothing else. Up to some issues of identification, I think this should be more or less what you have labeled $\theta_{(L)}$, or $\theta_{(L)}^A$ in some basis of $\mathfrak{g}$. Moreover, if you have a local trivialization $\phi:\pi^{-1}(U)\to U\times G$, then you can pull back the distinguished connection form on $U\times G$ to $\pi^{-1}(U)$.

Hence, your local trivialization did distinguish a connection form on $\pi^{-1}(U)$. As we have seen above, any other connection form, restricted to $\pi^{-1}(U)$, is then our distinguished one, plus some horizontal $1$-form of type $\text{Ad}$ on $\pi^{-1}(U)$, call it $\eta$. The local trivialization of $\pi^{-1}(U)$, moreover, distinguishes a section $s$ in $\pi^{-1}(U)$ (the one you have mentioned), and a $1$-form of type $\text{Ad}$ is fully determined by its values on the range of this section (which contains one point in each fiber). Then, if you pull back this $1$-form through a section $s:U\to\pi^{-1}(U)$, you end up with a $1$-form $s^*\eta$ on $U$ with values in $\mathfrak{g}$ and your original $\eta$ may be fully recovered from only the values of $s^*\eta$, namely, more or less, by the second term of your formula for $\bar{\omega}_p$ (where you additionally expanded in the basis $T_A$).

Taken together, locally on some $U$ on which both the principal bundle admits a trivialization and the basis manifold admits a chart, your formula is just expanding an arbitrary connection form into the basis $\text{d}x^\mu$, the basis $T_A$ and the part (first term) with the distinguished connection form on $\pi^{-1}(U)$.

For (2.):

I'm not sure how to answer this. To me it seems that these "every other books" concentrate on the non-trivial part of a connection form which I have discussed above, and just call this the connection form. This is quite sloppy as not always possible, but suitable for many applications. From your first sentence I conclude that you head up to physics applications. Note that, e.g. Minkowski space is contractible (just $\mathbb{R}^4$), and thus every vector bundle (and the associated frame bundles) over it is trivializable. Therefore, such a bundle admits a distinguished (global) connection form on the frame bundle and it suffices to regard any other connection form only by its deviation from the distinguished one. Also if you head to GR applications, a metric also distinguishes a connection form on the frame bundle and we have the same situation.

For (3.):

As I said, I assume you mean by $\omega$ the connection on an associated vector bundle (probably on the adjoint bundle, as you expand $\omega$ with in indices $A$. Is that right?). Also on a vector bundle you can show that the set of connections is an affine space, but now with the spatial structure of $1$-forms on the basis manifold, which take values in the endomorphism bundle of the given vector bundle. Moreover, you find a one-on-one association between connections on a vector bundle, and connection forms on its frame bundle. Given a local trivialisation of the former, you also have a local trivialization of the latter. Moreover, given a local trivialization of the vector bundle you can distinguish a certain connection/covariant derivative, which is essentially just the Cartan derivative $\text{d}$ (concretely acting in your formula by $\partial_\mu$), pulled back through the local trivialization. This connection is, under the above one-on-one association, mapped to the trivial flat connection form on the respective part of the frame bundle, if you do everything correctly (I know, that's a stupid thing to say...). Moreover, this one-on-one association maps the deviation from the respective distinguished connection/connection forms to each other (where the latter is first pulled back through the right section, above called $s^*\eta$ to reproduce your forms $\omega_\mu$ or $\omega_\mu^A$, resp.). That's why you find your $\omega_\mu^A$ again in this connection, as deviation of your given connection form the distinguished trivial connection acting by $\partial_\mu$.