I’m tackling physics recreationally from a pure math perspective.
Right now I’m looking at just the outline of gauge theory. The Wikipedia article explains that gauge fields correspond to generators of the Lie algebra of the Lie group the Lagrangian is invariant under. And then gauge bosons are the quanta of these fields, so for example there are eight gluons since SU(3) is eight dimensional. Cool! But what I don’t get is the intermediate step from generators of the Lie algebra to gauge fields.
A gauge field is mathematically a connection on a principal bundle, which is a Lie algebra valued 1-form satisfying some conditions. How do these correspond to generators of the Lie algebra? Here are my thoughts:
Let $\pi: P \to M$ be a principal bundle. Let $\omega$ be a principal connection on $P$. Let $\phi: U \times G \to \pi^{-1}(U)$ be a local trivialization of $P$. Then $s(x)=\phi^{-1}(x,e)$ defines a section and $A=s^*\omega$ is a $\mathfrak{g}$-valued 1-form on $U$ Now we can write $A(x)=\sum c_i(x)T^i$ where $c_i$ is a 1-form on $U$ and the $T^i$ form a basis for $\mathfrak{g}$. Are the $c_i$ what we mean by the gauge fields corresponding to the generators of the Lie algebra? Or would it be the whole $c_iT^i$ terms? It seems more likely that it’s the latter.
Writing it out like this I suppose you could do the same thing on $P$. Just take the component 1-forms or project onto those subspaces. If this is the case then the projections would still have to satisfy the axioms of connections on $P$. And maybe that is really obvious to see but I’m too deep in speculation to verify it if it’s the case or not.