Role of connection in Lagrangian of a field theory

I understand some of the basics of differential geometry. I have two questions about gauge theory from the physical perspective:

• I understand the connection is literally a "connection" between fibers, a horizontal tangent space, to provide the parallel transport of a lifted point to its endpoint based on the curve it gets lifted from. I know this is also a 1 form on the fibres themselves, and takes the value of a Lie algebra when these fibres are principle fibers. In what way is this connection a Lagrangian? What physical features do they share?

• Following this, the parallel transport is the integral of a connection. I think I am clear on why this is, because of covariant derivatives and whatnot, but what physical intuition do we have for this being the action?

Obviously this is in a classical field, else we would have action $$\to$$ path integral - I think - and I would accept either a general answer or a specific case study. For example, in an electromagnetic field.

• The Lagrangian takes in a connection and outputs a number. If you integrate the Lagrangian over all of space, you get the action. Connections which extremize the action are solutions to the classical equations of motion. Mar 11 '21 at 18:25
• Ok, I follow you. Is this what is called a gauge transformation on the G fibres? Putting a connection into L and giving back a number? Mar 11 '21 at 19:07
• No. Look up the "principle of least action" in physics, which is a very general thing. If you have some system, maybe a ball moving near earth's surface, if you fix the start point and end point of the ball, and the time interval it takes to travel from one point to the other, the path the ball takes will be the one which minimizes a functional of the paths called the "action." In other words, the "action" for system gives the dynamics, or the "equations of motion," saying how the system will evolve in one moment of time to the next. Mar 11 '21 at 19:11
• In a standard yang mills gauge theory, the lagrangian is $L = -\tfrac{1}{4} Tr( F^{\mu \nu} F_{\mu \nu})$ and the action is $S = \int d^4 x L$. Mar 11 '21 at 19:12
• I know about least action, sorry for explaining myself poorly. I mean why is the connection what goes into the lagrangian? I thought it was related to the symmetry of the group, because usually a term like this shows up in L. What is the physical meaning of the connection, beyond a connection between different vertical fibres? Mar 11 '21 at 19:15