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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.

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  • $\begingroup$ 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? $\endgroup$ – KCd Feb 16 at 14:46
  • $\begingroup$ @KCd I updated my answer. Please let me know if it's clear now. Thanks for your time. $\endgroup$ – stressed out Feb 16 at 14:57
  • $\begingroup$ I'm pretty sure this has been asked and answered before, at several different levels, on the physics site or here. $\endgroup$ – Lee David Chung Lin Feb 16 at 14:59
  • $\begingroup$ @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. $\endgroup$ – stressed out Feb 16 at 15:00
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    $\begingroup$ @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. $\endgroup$ – anakhro Feb 27 at 17:14
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Sorry to answer a question about one book by mentioning others, but... have you looked at Naber's "Topology, Geometry, and Gauge Fields: Foundations" (he wrote a follow-up book "Topology, Geometry, and Gauge Fields: Interactions") or Frenkel's "The Geometry of Physics"?

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  • $\begingroup$ No, I had never heard about them before. I googled them. They seem to be very good books. Thanks. Although, the last one is available at my university's library and it's 655 pages which seems suitable for a two-semester course! Plus, I'm eager to understand Atiyah's book. :( $\endgroup$ – stressed out Feb 16 at 15:41

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