How would you explain gauge theory to your mom?

Well, since my mom is a mathematician, I'd say that, given a Lie group G, a principal G-bundle is a fiber bundle over a manifold that locally looks like U×G. The bundle is equipped with a right action of G that acts by right translation on the fibers. A gauge transformation is an automorphism of this bundle that preserves this right action, and locally it looks like left translation on the fibers. Gauge theory is then the study of objects defined on the bundle, such as connections, up to gauge equivalence, or, alternatively, objects that are invariant under gauge transformations.

I'd then illustrate with some examples of principal G-bundles, such as the frame bundle, and some examples of gauge-invariant objects, such as the trace of the curvature or the energy of a connection. I then might go on to explain how G-bundles provide a nice framework for generalizing the Gauss-Bonnet theorem to Chern-Weil theory.

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