A defect action for Wilson loops

An effective action is proposed to compute the expectation value of Wilson loops in $(S)U(N)$ gauge theories. The action consists of fermions localized on the loop and an Abelian gauge field that fixes the representation. The discussion is limited to weak coupling and Wilson loops in the fundamental representation extended along a smooth curve, but there are no restrictions on the matter content as long as the theory has a UV fixed point or it is conformal. For a circular Wilson loop it is found that the expectation value coincides at leading order with the exact result of the $1/2$ BPS Wilson loop of ${\cal N}=4$ super Yang-Mills, which is determined by a solvable Gaussian matrix model. This hints towards a universal connection to string theory duals and SYK models.