Let $G$ be the gauge group whose Yang-Mill's theory one is looking at and $A$ be its connection and $C$ be a loop in the space-time and $R$ be a finite-dimensional representation of the gauge group $G$. Then the classical Wilson loop is defined as, $W_R(C)(A) = Tr_R[Hol(A,C)]$, the trace in the representation $R$ of the gauge field $A$ around the curve $C$.

- I want to know why the above can be written as, $W_R(C) = e^{i\int _D F}$ where $F$ is the curvature of the connection when $G$ is Abelian and $C$ I guess is the boundary of a contractible disk $D$.

{..the above claim reminds me of heuristic calculations (far from a proof!) that I know of where one shows that in the limit if infinitesimal loops, the eventual deviation of a vector on being parallel transported along it by the Riemann-Christoffel connection is proportional to the product of the corresponding Riemann curvature tensor and the area of the loop..}

In the above proof kindly indicate as to what is the subtlety with $G$ being non-Abelian? Isn't there a natural notion of "ordering" in some sense along the loop given by a parametrization or the trajectory of a particle?

In relation to discussions on confinement, what is the motivation for also looking at the cases where $R$ is a representation not of $G$ but of its simply connected cover? I mean - how does the definition for $W_R(C)$ even make sense if $R$ is not a representation of $G$?

I don't understand what is meant by statements like (from a lecture by Witten), *"..if $R$ is a representation of $G$, then there are physical processes contributing to $<W_R(C)>$ in which large portions of the Wilson line have zero-charge i.e carry trivial representations of $G$, because some particles in the theory have annihilated the charges on the Wilson line.."*

I would have thought that its only a representation of $G$ that can be fixed and I don't see this possible imagery of seeing a representation attached to every point on the loop.

- What is the subtlety about Wilson loops for those representations of $G$ which can come from a representation of its universal cover? If someone can precisely write down the criteria for when this will happen and then what happens...

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