# What constitutes a gauge theory? Help me understand electromagnetism as the prototype of all gauge theories

In The Geometry of Physics and Knots, Sir Michael Atiyah says that electromagnetism is the prototype of all gauge theories:

The prototype of all gauge theories is electromagnetism. From the geometrical point of view the electromagnetic potential $$a_{\mu}$$ $$(\mu=1,\cdots,4)$$ defines a connection for a $$U(1)$$ bundle over Minkowski space $$M$$. The field is the corresponding curvature $$f_{\mu,\nu}=\partial_{\mu}a_{\nu}-\partial_\nu a_{\mu} \hspace{10px}(\partial_{\mu}=\partial/\partial x_{\mu}).$$ Maxwell's equations in vacuo take the form $$df=0, \hspace{10px}d^*f=0$$ where $$f$$ is now viewed as a $$2$$-form, $$d$$ is the exterior derivative and $$d^{*}$$ is its formal adjoint (relative to the Minkowski metric).

He then goes on to say how non-Abelian gauge theories are obtained by modifications to the example above.

So, my goal is to understand the part that I have quoted it from the book, page $$2$$. I know what electric and gravitational potentials are. I understand how having some potential function defined on a space leads to forces acting on each particle in the field. My first question arises naturally now:

1. How does a potential field (electric potential) give a curvature on space? In what sense $$f_{\mu,\nu}$$ is a curvature? And what is the intuition behind associating a connection to a potential field?

2. How does Maxwell's equations take that form? I know what exterior derivative is. How is $$f$$ viewed as a $$2$$-form? And how is $$d^{*}$$ defined?

3. What is a $$U(1)$$ bundle? Does this mean that each fiber is diffeomorphic to $$U(1)$$? What's special about $$U(1)$$ here?

I know that reading this book is a huge leap for me considering my mathematical background, but my goal is to read and analyze the first chapter step by step with your help in a month. Maybe if I understood the first chapter well, I would attempt the second chapter too. So, I earnestly request you to bear with me and try to write your answers with minimal mathematical background please. Thanks in advance.

Edit: my mathematical background:

1. I have passed two courses in fundamentals of physics covering classical mechanics, thermodynamics and the classical theories of electromagnetism and gravitation. I have self-studied special relativity and I know that general relativity attempts to generalize SR to curved space-time by Einstein's field equations.

2. I haven't passed a differential geometry course yet, but I have read Barret O'Neill's Elementary Differential Geometry up to the chapter where the shape operator is defined and Do Carmo's Differential Forms and Applications, excluding Gauss-Bonnet's proof. I have also looked at some chapters from Lie's Introduction to Smooth Manifolds and I know what covariant differentiation, geodesics, Lie derivative, Lie algebra and Lie groups are.

3. I know what a vector bundle is (it's the abstraction of tangle bundle. Right?). I don't know much about principal bundles, except the explanations and definitions on Wikipedia.

• You ask for answers assuming a "minimal mathematical background". Can you say something about your actual background in math and physics? I see you said you know what the exterior derivative is (in what settings?), but what about other topics?
– KCd
Feb 16, 2019 at 14:46
• @KCd I updated my answer. Please let me know if it's clear now. Thanks for your time. Feb 16, 2019 at 14:57
• I'm pretty sure this has been asked and answered before, at several different levels, on the physics site or here. Feb 16, 2019 at 14:59
• @LeeDavidChungLin I prefer to have someone with a math background answer this question to be honest. Due to my background, I understand better when a mathematician explains a concept like this rather than a physicist. And I looked for this on MSE, but didn't find any. If you found any similar question, feel free to post it as related. Feb 16, 2019 at 15:00
• @stressedout A very nice book that you might want to check out is "Gauge fields, knots and gravity" by John Baez and Javier P. Munian. It develops (or reviews) a lot of the necessary concepts in differential geometry in the first chapter (with application to electromagnetism) before turning attention to gauge fields. Highly recommended.
– anak
Feb 27, 2019 at 17:14