Wilson loop in general representation and RG flow in 1D defect QFT

The generalized Wilson loop operator interpolating between the supersymmetric and the ordinary Wilson loop in N = 4 SYM theory provides an interesting example of renormalization group flow on a line defect: the scalar coupling parameter C has a non-trivial beta function and may be viewed as a running coupling constant in a 1D defect QFT. In this paper we continue the study of this operator, generalizing previous results for the beta function and Wilson loop expectation value to the case of an arbitrary representation of the gauge group and beyond the planar limit. Focusing on the scalar ladder limit where the generalized Wilson loop reduces to a purely scalar line operator in a free adjoint theory, and specializing to the case of the rank k symmetric representation of SU(N), we also consider a certain 'semiclassical' limit where k is taken to infinity with the product k zeta(2) fixed. This limit can be conveniently studied using a 1D defect QFT representation in terms of N commuting bosons. Using this representation, we compute the beta function and the circular loop expectation value in the large k limit, and use it to derive constraints on the structure of the beta function for general representation. We discuss the corresponding 1D RG flow and comment on the consistency of the results with the 1D defect version of the F-theorem.

Automated summary

The paper investigates the generalized Wilson loop operator in the N = 4 SYM theory, focusing on its beta function and Wilson loop expectation value. By generalizing previous results, the study considers arbitrary representations of the gauge group and goes beyond the planar limit. It is found that in a specific limit, the generalized Wilson loop reduces to a purely scalar line operator. The computation of the beta function and circular loop expectation value in this limit provides constraints on the structure of the beta function for general representations.