The Maurer-Cartan form of a principal bundle?

Question

I was reading this page of nlab: https://ncatlab.org/nlab/show/Maurer-Cartan+form I do here a small recall first of the Maure-Cartan form first: If you have a Lie group $G$, then you have the Maurer Cartan form which sends each tangent vector to the tangent space of $G$ at the identity. It is given by the following construction: For any $a$ and $b$ in $G$, there exists exactly one $\theta(a,b)$ such that $a\cdot \theta(a,b)=b$. It implies that for a smooth curve $b_t$ through $a$, $\theta(a,b_t)$ is a smooth curve through the identity. So the tangent vector defined by $b_t$ is sent to the tangent vector defined by $\theta(a,b_t)$. This is how the Maurer-Cartan form is defined on a Lie group.

But in this page of nlab, they say in the introduction that the concept of the Maurer-Cartan form can be generalized to a $G$-principal bundle. Since $G$ acts freely and transitively on the fibres, it follows that we can do the same definition than before with the $\theta$, and so we can send any vertical tangent vector to a tangent vector of $G$ at the identity.

My question is: how is this Maurer-Cartan form defined for non-vertical tangent vectors? The article of nlab does not give further informations, so I do not know how to extend the definition. I'm also asking myself if there is only one Maurer-Cartan form on principal bundle.

Thank you for your help!

Answer 1

You need to fix a connection in your principal bundle. That is, you introduce what an horizontal vector is, and then you can project any vector field to a vertical one. In this way, you can project and then go to the Lie algebra of the group.

Answer 2

For any right $G$-torsor $\mathcal{G}$ you have a division map $\operatorname{div}: \mathcal{G} \times \mathcal{G} \to G $ characterized by $$u.\operatorname{div}(u,v)=v.$$ Hence we often write $u^{-1}v := \operatorname{div}(u,v)$, even though technically there is no canonical multiplication in the torsor. If the elements $u$ and $v$ of your torsor are infinitesimally close to first order (i.e. first order neighbours in the sense of synthetic differential geometry) then they define a tangent vector. Hence, $u^{-1}v$ will be an infinitesimal neighbour of $e \in G$. This gives us a canonical Maurer-Cartan form $\omega: T\mathcal{G} \to \mathfrak{g}$ on any $G$-torsor. In particular, if $\mathcal{G}=G$ this reproduces the usual Maurer-Cartan form.

Now if $\pi: \mathcal{G} \to M$ is a principal $G$-bundle then it is by definition a fibrewise right $G$-torsor, so we have a fibrewise division map $\operatorname{div}: \mathcal{G} \times_M \mathcal{G} \to G $. Hence we have a canonical Maurer-Cartan form on each fibre, giving us a Maurer-Cartan form defined on the vertical bundle $\omega: V\mathcal{G} \to G $.

Now this Maurer-Cartan form on the vertical bundle can be extended to the whole tangent space of the principal bundle, but there is no canonical way to do this. The choice of an extension of $\omega$ to the entire tangent bundle $T\mathcal{G}$ is precisely a choice of principal connection on $\mathcal{G}$. i.e. we pick a complement $H\mathcal{G}$ such that $T\mathcal{G} \cong V\mathcal{G} \oplus H\mathcal{G}$, and we can then define $\omega$ to be zero on $H\mathcal{G}$.