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:
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?
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?
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:
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.
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.
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.