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
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
\[ F^A = dA + \frac{1}{2} [A \wedge A] \]