How is the Lie algebra connection form a potential energy field? I'm pretty familiar with the math of principle bundles etc, but I have no intuition for physics at all, so I'm curious how some facts fit together.
When a potential is undetermined up to a gauge, we can choose gauges from a symmetry group on a fiber over the point by a section. For example, elements of $U(1)$ in electromagnetic potential get added to the EM potential. This makes sense, except I have no idea of the following two ideas:

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*Why is the gauge symmetry acting on a potential field, but the connection, here the Lie algebra of the gauge group, is being called the potential?

For example, the electromagnetic four potential is the connection on the $U(1)$ fibers, but it is elements of the group that go into the potential. So I don't see how the Lie algebra is a potential in this case. I'm just confused about the physical significance of these things and how they work together.

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*Where is the symmetry? Obviously the lagrangian changes when we choose a gauge because it is in terms of the potential, so how is this symmetric? Maybe because when we minimize the action the gauge goes away no matter what we choose?


Maybe the first is just a language confusion, and the lie algebra isn't a potential. I have also seen it called a gauge field. But this seems to be the way it is described, as a potential field, despite the mismatch between group elements acting on the potential and the lie algebra being the potential.
I would think of it as saying as we move along fibers, how does the gauge change, making the connection a lie algebra as the tangent space of the group. This makes sense, but then, this doesn't fix the above questions for me.
If you can help, thank you! Maybe there is a good paper or book I can use to see what the math to physics dictionary is, and how these things are used in physical theory.
 A: I don't know exactly the etiquette of answering my own questions, but I think I have cleared it up for me and want to share.
I used this paper a lot: https://arxiv.org/abs/2009.02162 it turns out, my problem is, I didn't know what gauge theory was at all!
The physical ingredients seem to be this:
As @ChiralAnomaly and @duality have discussed with me, a gauge field is not actually a potential, but a field of connection forms that keeps track of the choice of gauge. This is a connection between fibers when the section gives us a choice of gauge from gauge group elements in the fibers. Of course, when this gauge doesn't change from fiber to fiber, the connection is flat, and that's why a dynamical connection has a force from the two form $F=dA$. With this connection we can go into the associated matter field and transform its lagrangian using a covariant derivative to keep the symmetry.
For $\rm U(1)$, I wonder if this is just a special case, because the gauge is also the actual phase of the particle, so the connection is also the actual potential field of the electromagnetic field. This is like saying as we choose a gauge we choose a rescaling of the actual waveform, so the connection tells us how the actual wave behaves. And of course, one component of $\rm U(1)$ symmetry corresponds to an electron, with a charge from $\rm U(1)$ representation. Maybe its the same in other theories that I don't know, like Yang Mills theory, but this clears up for me why the potential coincides with the connection, and its relation to gauge symmetry.
I won't accept my own answer in case a better one is offered, as a matter of courtesy I think.
