Imagine a 2D plane. Instead of the usual Cartesian or polar coordinates, we can use a different point of view: for each point, we can start from the x axis and move in the y direction. Also, we can start from (x,sin x) and move in the y direction. If we regard the first label of the point location as its x axis and the second label as the projected distance to starting point, then the two coordinates are related by (x,y)<->(x,y-sin x). We can see the y axis as a manifold growing out of a point in the base space x. After we construct R^2, the base space becomes arbitrary in a sense: we can use y=sin x instead of y=0 as the starting location.
Similar thing happens in S^2*R: the space is somewhat similar to R^3, but with part of H^3 compactified like the compactification ofthe base space R^2 into S^2. We can see that the base space S^2 can be “wobbled” around the origin, so that for the same point in the space, there are many ways of choosing the base space, hence the coordinates are different.
In a sense, the fixing of the base space is equivalent to declaring a gauge. After this is fixed, attaching a vector space is unique. However, trying to reconstruct our base space is, as we illustrated above, hopeless. In the textbooks, the vector space attached to each point of the base space is called “vertical”, and a base space is called a horizontal section. We see that projection into vertical components are generic but projection into horizontal components are not fixed. Fixing such component locally is called a local trivialisation. In this way you can view the local space as a product space, which is very convenient. When you patch everything together, non-trivial thing may happen as in the mobius strip. Space admits a global trivialisation is a product space, which is topologically unexciting. For spaces with nontrivial topology, we can’t choose a section that doesn’t vanish everywhere, hence patching them together is the best description.
One magic part of principal bundles is that the vertical subspace has a group structure, which means we can connect one point with another using a Lie group element g. This is good. We only need to know the collection of origins of the vertical components growing out of the base space to reconstruct everything. This is a trivialisation of a principal bundle: it is to choose a section!
What happens for composite Lie group? For example, the U(1) gauge theory in curved spacetime? The geometry of space time is diffeomorphism invariant, which means it admits a orthonormal SO(1,3) section. On top of this we have a principal U(1) section. In flat spacetime,
\(D_\mu \psi=(\partial_\mu+A_\mu)\psi\)
Promote the partial differential operator with covariant derivative in curved spacetime. Also, consider a vector V instead of the wavefunction scalar of T(M).
\(D_\mu V^\nu=\partial_\mu V^\nu+\Gamma^\nu_{\mu\rho}V^\rho+A_\mu V^\nu\)
What happens if A is a general Lie group adjoint representation and V is also a fundamental vector of A? Then the connection map(i.e. the linear transformation in the covariant derivative) becomes
\((\Gamma^\nu_{\mu\rho}\delta^\tau_\sigma+[A_\mu]^\tau_\sigma\delta_\rho^\nu ) [V^{\rho}]^\sigma\)
Therefore, the connection is actually L(G)*T(M) as expected, as the representation of V and A suggests.
An interesting question would be, can we reinterpret the principle bundle and absorb A into spacetime curvature? The answer is no. since they have different indices. But can we reinterpret L(G)*T(M) into another decomposition and say something meaningful? The easiest direction is to absorb L(G) as T(M’) where M’ is a compactified space, and we can view the base as M*M’. This is Kaluza-Klein compactification, where we write down the GUT naively and says the spacetime is higher-dimensional with some curved dimensions. Can we break T(M) and construct a different base space M’? This is an exercise to the readers.