Connections on principal fiber bundles ed
useful:
There are several equivalent definitions of connections on a principal bundle \[(P, \pi, M, G)\]...
As horizontal subspaces ed
Definition ed
- vertical subspaces
- connections = horizontal spaces
- \[ R^\star_g \, Th_p = Th_{gp} \]
Parallel transport ed
A curve \[\gamma\] on the base manifold \[M\] can be horizontally lifted onto \[P\] along the connection. For a given starting point \[p\in P\] of the lift, this is unique.
If \[\gamma\] is a closed loop, start point \[p\] and end point \[p'\] might be different but have to lie in the same fiber. Therefore there is a \[g \in G\] with \[p' = g \cdot p\].
A connection is called flat if always \[p=p'\] or \[g=e\]. Then the connection is integrable to a surface.
Connection forms ed
Alternatively, one can define
Definition ed
A connection form \[A\] is a \[\mathfrak g\]-valued 1-form on \[P\] with
- \[A(\tilde X) = X\] for fundamental fields \[\tilde X\] for \[X \in \mathfrak g\]
- \[R^\star_g \, A = \operatorname{Ad}(g^{-1}) \circ A\]
The first condition completely defines the vertical behavior (\[A_v = \mu_G\]). The remaining degrees of freedom are equivalent to choosing a horizontal subspace via \[Th_p = \operatorname{ker} A_p\].
The second condition sets the equivariant transformation behavior along the fibers. Informally, moving \[A\] transforms its vector-input and \[\mathfrak g\]-output. To mimic the horizontal spaces, the input transforms trivially, while the output transforms like the fundamental vector fields.
Curvature ed
\[ F^A = dA + \frac{1}{2} [A \wedge A] \]