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The original papers by Gerard 't Hooft himself are quite readable. On the Phase Transition Towards Permanent Quark Confinement A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories Topology of the Gauge Condition and New Confinement Phases in Nonabelian Gauge Theories Whenever I open these papers, I'm always awestruck.

This is the definition of the gauge field. Suppose you have an SU(2) symmetry, for definiteness, consider isospin. So the notion of "proton" and "neutron" define two axes in isospin space, and you might want to say that it is arbitrary which two linear combinations of proton and neutron are the right basis vectors. So that someone defines one basis of ...

It is simple to describe mathematically. First I will recall what the equation for the Aharonov-Bohm phase means, and then I will describe (without proof) its relationship to parallel transport for $G$-bundles, which I define. The gauge potential $A$ is a connection on some principal $G$-bundle, where $G$ is the gauge group. Principal $G$-bundles over a ...

1) Let us write the Wilson-line of a simple open curve $\gamma: [s_i,s_f]\to \mathbb{R}^4$ as $$\tag{1} U(s_f,s_i) ~=~ \mathcal{P}\exp \left[ i\int_{\gamma} A_{\mu}~ dx^{\mu} \right]. $$ 2) The path-ordering $\mathcal{P}$ becomes important if the gauge potential $$\tag{2}A_{\mu}~=~A^a_{\mu} T_a$$ is non-abelian. Here $T_a$ are the generators of the ...

Possible answer to your first bullet: Stokes theorem. If $G=U(1)$, then $F=dA$ (not $dA+[A,A]$) and so $\int_{\partial D}A=\int_{D}dA=\int_D F.$ Further, if memory serves, $U(1)$ is the only Abelian Lie algebra (thinking of the Cartan classification) so $G$ is Abelian means $G=U(1)$ anyway. EDIT: Coordinate expression for the Holonomy. $Hol(A,C)$ is defined ...