Principal connection

It is a particular case of connection on a fiber bundle.

Definition

A connection on a principal $G$-bundle $P\stackrel{\pi }{\to }M$, with $dim(M)=n$, is a $n$-dimensional distribution (named horizontal distribution)

on $P$ such that the following conditions hold for every $u\in P$:

• $T_{u}P=V_{u}P\oplus H_{u}P$, where $V_{u}P=Ker((d\pi {)}_u)$.
• $H_{{\rho }_g(u)}P=(d{\rho }_g{)}_u(H_{u}P)$, for every $g\in G$, and being ${\rho }_g$ the right action of $g$ in $P$
  Here $VP$ is the vertical bundle.

The simplest example of a connection is the trivial connection on the trivial bundle $P=M\times G$, that is, $$H_{(m,g)} P=T_m M \subseteq T_{(m,g)} P.$$

Connection 1-form

It can be shown (@vakar2021), without much difficulty, that a connection can be alternatively defined as a 1-form $\omega \in {\mathrm{\Omega }}^1(P)\otimes \mathfrak{g}$, being $\mathfrak{g}$ the Lie algebra of $G$, that satisfies:

• $\omega (\tilde{A})=A$, for every $A\in \mathfrak{g}$ and being $\tilde{A}$ the fundamental vector field of $A.$
• For every $g\in G$ the following diagram commutes:
  
  being $G\stackrel{\mathrm{Ad}}{⟶}GL(\mathfrak{g})$ the adjoint representation of $G$. This can be written

which is the same expression that we find for the Maurer-Cartan form (see Maurer-Cartan form#MC form and left and right actions). The connection is then an extension of the Maurer-Cartan form of the bundle to the whole $T_{u}P$ but the values are still in $\mathfrak{g}$. A Cartan connection is similar, but the values are taken in a bigger Lie algebra.

Observe that given the horizontal distribution $H$ we can construct the associated 1-form $\omega$ by projecting every vector $X\in T_{u}P$ to $V_{u}P$ and identifying the latter with $\mathfrak{g}$ by using the Maurer-Cartan form and any trivialization (see [Belgun 2020] for more details). Conversely, given a connection by the 1-form $\omega$, we recover the horizontal distribution by means of

Idea

See the idea in the particular case of affine connection.

Another characterization

But connections can be characterized in one more way. Given an atlas $\{(U_{\alpha },{\psi }_{\alpha }){\}}_{\mathrm{\Lambda }}$ for the $G$-bundle $P\stackrel{\pi }{\to }M$ and the connection defined by $\omega$, we can use the collection of distinguished local $G$-frames (local sections) $p_{\alpha }:U_{\alpha }\to P$ (see principal bundle#Alternative approach) to define the family of local 1-forms

satisfying

where ${\mathrm{\Theta }}_{\alpha \beta }$ denote $g_{\alpha \beta }^{\ast }(\mathrm{\Theta })$, being $\mathrm{\Theta }$ the Maurer-Cartan form. Every ${\omega }_{\alpha }$ is called a Yang-Mills field (see [Schuller 2013] page 185).

Reciprocally, any collection $\{(U_{\alpha },{\psi }_{\alpha },{\omega }_{\alpha })\}$, being $\{(U_{\alpha },{\psi }_{\alpha })\}$ an atlas for the bundle $P\stackrel{\pi }{\to }M$, defines a connection on it if $\{{\omega }_{\alpha }\}$ satisfies condition (!) (see [Marathe 1992] for a proof).

The idea behind Yang-Mills fields is what follows. The local section $p_{\alpha }$ is a moving frame, that is, a choice of a $G$-frame for every point $m\in U$. The 1-form $p_{\alpha }^{\ast }(\omega )$ evaluated on a vector $v\in T_{m}M$ tells us how the frame varies when we move in the direction $v$ (assuming the notion of derivative provided by the connection $\omega$). But this "variation" is measured in terms of elements of $\mathfrak{g}$.

Exterior covariant derivative

An important feature is exterior covariant derivative.