# Does gauge invariance of scalars/fermions in the adjoint representation induce the existence of Wilson loop and (then) covariant derivative?

First, for the known case of $$U(N)$$ gauge invariance we have scalars (it works for fermions too) transforming as (fundamental representation) $$\phi(x)\to V(x)\phi(x), \ \ V(x)\in U(N)$$ So then we see that the (usual) derivative can not be used to build the kinetic term, so we then conclude that a comparator $$U(x,y)$$ is needed. You can see that discussion, for instance, in Peskin-Schroeder, page 482-490.

Then expanding it as $$U(x+\varepsilon n,x)=1+ig \varepsilon n^\mu A_\mu^a(x) t^a +\mathcal{O}(\epsilon^2)$$ we can find the covariant derivative $$D_\mu\phi(x)=\partial_\mu \phi(x) -ig A_\mu^a t^a \phi(x)$$ Taking the covariant derivative along the direction of the curve of the comparator leads to a sort of parallel transport equation for the comparator. Solving that equation gives something like $$U_\gamma=\mathcal{P} e^{ig\int_\gamma A^a_\mu t^adx^\mu}$$ If we take the trace we will have the expression of the Wilson loop.

Now, my question is if this well known idea will work for scalars/fermions transforming as (Adjoint representation) $$\phi(x)\to V(x)\phi(x)V^\dagger(x), \ \ V(x)\in U(N)$$

I dont know how to expand $$U$$ in order to get $$D_\mu\phi(x)=\partial_\mu \phi(x) -ig [A_\mu ,\phi(x)]$$ Also I should find a parallel transport equation and then show that the comparator is basicaly the Wilson loop. Is there any reference for that?

Could this idea work also for $$\mathcal{N}=4$$ Super Yang-Mills?

If there is something unclear just comment.