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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.