What is gauge field? It is a Lie Algebra Valued 1-form on space time.

This 1-form eats tangent vector (direction of motion) and returns something linear, but this linear thing is not a number but an infinitesimal symmetery generator!

Let’s illustrate by a concrete example: SU(2) gauge field acting on a spacetime vector.

Then we can study curvature: Curvature measures the failure of the horizontal distribution to close under Lie brackets.

Lastly, we need to differentiate the word “field” in gauge field and quantum field:

at the most basic level, field is a function on spacetime.

They live in different mathematical worlds.

GEOMETRY QUANTUM THEORY

Connection A → Operator field Â
Curvature F → Field-strength operator
Holonomy → Wilson loop operator

A quantized gauge field is a quantum field,
but not every quantum field is a gauge field.